![source on instruments and machinery is detrimental as it might affect the performance. An example is
optical recording and experiments, where every minute vibration of the optical table would cause op-
tical distortions. Similarly, vibration due to transportation of equipment may cause damage. Therefore,
study of externally induced vibration on objects is also important.
Another concern for the vibration of machinery and radiated noise is the environmental problem.
Probably, the noise pollution created by vehicles is the most serious problem in urban life. With rapid
urbanisation and random use of automobiles on streets, rail and air traffic, loud public address systems,
construction machinery, etc., the environment is subjected to a high level of noise pollution. The
excessive sound in the atmosphere can cause permanent damage to our eardrum, or partial loss of hear-
ing [1]. Noise can cause ischemic heart disease, hypertension, cardiovascular problems, etc. [2].
Increased noise levels can create stress, increased aggression and workplace accident rates, and pos-
sibly enhance antisocial behaviours [3]. Loud noise may affect the nervous system of the brain and
create psychological disorders [1–3]. Birth defects such as harelip, cleft palate, and defects in the spine
are also possible from high-intensity sounds such as airport environment. Typical noise levels of sound
originated from normal human conversation can be 60 dB, while noise of ground vehicles can be 80–
90 dB and jet engine at take-off can be 120 dB [4]. However, continuous noise level 80 dB is dangerous
for human health. In India, the noise intensity limit in the day time is 75 dB (A) in industrial areas,
whereas 55 dB (A) in residential areas as per the noise pollution rules [5].
Earthquakes are the most important example of the ill effect of vibration. It is a seismic wave on the
earth surface and its devastating effect is well experienced by mankind. It results from a sudden release
of huge energy in the earth’s crust. These waves can be violent causing loss of life and can even destroy
an entire civilisation around. The intensities of such seismic waves are measured in a comparative scale
called ‘Richter’, named after the seismologist Charles Francis Richter. It is a logarithmic scale, where,
a magnitude of 3 on the Richter scale is not dangerous, while 7 on the scale can cause large-scale
devastation. Richter scale is developed as an empirical formula using the logarithm of amplitudes
at the recording seismometer, arbitrary value, and the epicentral distance of the recording station as
parameters [6].
Sound and vibration are also important subjects of study for underwater objects. Acoustic signals
are used for bathymetric seabed mapping studies, to detect and identify underwater objects like rocks,
submarines, mines, and even to detect marine animals such as a cluster of fishes, sharks, whales, etc.,
using a device called sound navigation and ranging (SONAR). There are two methods for the detection
of an underwater object, namely active and passive detections. In active detection system, the SONAR
sends a sound pulse to get a reflection from any object having a difference in specific acoustic imped-
ance compared to water and analyse the reflected signal to determine the type of object and its
coordinates (location), while passive SONAR only receives the underwater noise radiated from any
object nearby. For the defence forces, naval ships and submarines need to be as silent as possible
to avoid detection by the passive SONAR. Vibrations of machinery inside the vessels are transmitted
to the hull of the vessel through internal structures and in turn transmitted to the seawater as radiated
noise, which is picked up by the passive SONAR. Since there is a number of such machinery in a
submarine/ship, there would be a particular underwater acoustic signature of the ship/submarine.
The vibration spectrum would depend on the type of machinery, their design, manufacturing process,
installation methodologies, and also on ageing effects. In general, vibrations of heavy machinery such
as diesel alternators and turbines in a ship or submarine produce low-frequency radiated noise in the
sea. The cavitation due to rotation of propellers of ships/submarines also produces sound like noise in
3
1.2 Importance of the study of vibration](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-13-320.jpg)
![the seawater. The cavitation noise spectrum would depend on the number of blades, rpm of the
propeller, cavitation volume, skin friction of the blades, the thrust, and propeller loading [7]. Both
the radiated noise due to machinery and the cavitation can be detected by passive SONARs.
The study of vibration and sound propagation in different media is very important for prediction and
enhancing the fatigue life of an equipment, prediction, and mitigation of catastrophic failure of machin-
eries, reduction of environmental acoustic pollution, studies on acoustic noise-related health hazard of
the society, and for design of acoustically stealthy strategic objects of defence forces.
1.3 Simple harmonic motion (SHM)
A simple and periodic oscillation of a point object is called harmonic motion if it passes through a
reference point after a definite interval and follows the same locus periodically. The number of such
motion per unit time is termed as ‘frequency’ and the interval is the ‘time period’, while the length
between two consecutive points having the same phase is called the ‘wavelength’. It is explained in
Fig. 1.1A and B. Let us imagine a point rotating in a circular path repeatedly at a constant angular speed.
Therefore, the point would cross any reference point on the circle at a regular interval. The locus of the
point is the circle as shown in Fig. 1.1A, and when the movement of the point is plotted against time,
the locus describes a perfect sinusoidal curve as in Fig. 1.1B. From Fig. 1.1B, the wavelength is the
distance AB, the time taken to cover this distance is ‘time period’, and number of such distance
travelled in one second is the ‘frequency’ of the wave. It means, one wave is equivalent to one rotation
of the point on the circle and correspondingly the distance traveled by the wave is AB, which is
the wavelength. The speed of the wave propagation is thus the product of the frequency and the
wavelength. The convention of symbols and units for wave properties are given in Table 1.1. Simple
vibration, in the basic analysis, is assumed to be harmonic motion, that is, a sinusoidal wave, though the
actual vibration signature may contain various periodicities (frequency). A nonharmonic vibration is a
random vibration with no regularity in periodicity.
FIG. 1.1
SHM: (A) locus of a point in SHM, (B) time-dependent displacement of the point.
4 Chapter 1 Fundamentals of vibration damping](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-14-320.jpg)
![as 9.81 m/s2
. Secondly, all vibration intensities are expressed as a ratio of acceleration normalised by
the force [a(t)/F0].
1.3.2 Free vibration and natural frequency
If a body is allowed to vibrate by an initial impact force, the vibration is called free vibration and the
frequency of the vibration is termed as the natural frequency of the body. Every matter or object has a
natural frequency of vibration depending on its physical characteristics such as elastic modulus, mass,
size, and shape [8, 9]. As an example, if a string with a fixed length in a guitar is plucked, there will be a
sound of a particular frequency and every time the same tune will be heard. There can be subsequent
higher harmonics which are multiples of the first natural frequency of the object. In the case of a thin
beam, modes in the axial direction is enough to study vibration, for a flat plate of negligible thickness,
two directions are to be considered, and for a three-dimensional body, there will be natural frequencies
and modals in three directions, for example, a cylinder will have axial, circumferential, and radial nat-
ural frequencies and higher modes of each too.
1.3.3 Forced vibration and resonance
When an external dynamic force is applied to an object for a period greater than its own time period of
oscillation, the vibration is termed as forced vibration. The object vibrates at the same frequency as
is imposed by the external force. Running machinery (such as motor, engine, pump, centrifuge,
etc.) is subjected to forced vibration.
FIG. 1.2
Displacement, velocity, and acceleration of a simple harmonic motion.
6 Chapter 1 Fundamentals of vibration damping](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-17-320.jpg)
![When the frequency of an external dynamic force coincides with the natural frequency of the body,
there will be a very large amplitude of displacement and the phenomenon is termed as resonance. It is
also the frequency, at which the potential energy of the object is totally converted to kinetic energy.
Therefore, at resonance, the vibration amplitude attains a maximum value. There can be several modes
of resonances, which are higher harmonics of the first natural frequency. In each mode, the intensity
peaks are observed. System resonance for machinery with very large amplitude may cause severe
damage or catastrophic failure. Thus, the study of the vibration response of a structure with respect
to time and frequency is very important to take measures to avoid such damages or failures.
1.4 Random vibration
If the magnitude of external dynamic force at a given time is known, then the vibration is deterministic
vibration. When the magnitudes cannot be determined at a given time, then it is random vibration. In a
random vibration scenario, a cluster of vibration intensities and frequencies would exist. Therefore,
whatever be the complexity of a random vibration signature, it can be assumed as a sum of many pure
sine waves of different amplitudes with corresponding harmonic frequencies such as
f t
ð Þ ¼ u0 + u1 sin ωtφ1
ð Þ + u2 sin 2ωtφ2
ð Þ + u3 sin 3ωtφ3
ð Þ + ⋯ + un sin nωtφn
ð Þ
Generally, the recording of a random vibration spectrum is done with respect to time (time domain).
Fourier Transform is applied to find the individual intensities in the frequency scale after selection of a
time domain window of the random signal. The example of random vibration is wind velocity, earth-
quakes, etc. If large data is available, a statistical analysis may be done to determine the different mag-
nitudes and frequencies of the random vibration [10, 11]. Typical Fourier transform of random
vibration in time domain and frequency domain taking the limits of ∞ to +∞ for frequency and time
are expressed mathematically as
Ψ f
ð Þ ¼
ð
+∞
∞
φ t
ð Þexp iωt
ð Þdt
φ t
ð Þ ¼
ð
+∞
∞
Ψ f
ð Þexp iωt
ð Þdω
The infinite limits, however, is only theoretical, and the limits can be decided where the intensity is below
1% of the highest intensity in the frequency scale. Present-day Fast Fourier Transform (FFT) analysis of
vibration and shock spectra is computerised as standard software available with all vibration analysers.
1.5 Undamped and damped vibration
A free or forced vibration may or may not experience resistances like internal friction or inelastic de-
formation of the object. An undamped vibration is a case where there is no resistance to the vibration
and the vibration is not damped or attenuated with time, which implies that the dynamic deformation is
perfectly elastic in nature. Therefore, in an undamped system, there is no loss of mechanical energy in
the system.
7
1.5 Undamped and damped vibration](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-18-320.jpg)
![tan φ ¼
v0
u0ωn
(1.10)
The phase angle (φ) can be very small in case of a system with a high natural frequency, or in another
way, high stiffness and low mass.
1.5.2 Expressions for free damped vibration
Continuing with the undamped system as a spring-mass combination, a viscous dashpot can be intro-
duced to represent a damped vibration. A viscous pot consists of a cylinder filled with a viscous fluid
such as silicone oil and a piston in the cylinder. The viscous dissipation takes place for a force applied to
the piston due to shearing of the viscous fluid. The force is proportional to the velocity of the piston:
F ¼ λ
du
dt
(1.11)
where λ is called Damping coefficient of the viscous dashpot.
The damped vibrating system is assumed to be a combination of spring, mass, and a dashpot. A
typical sketch of the spring-mass-dashpot system is shown in Fig. 1.5. The force balance of such a sys-
tem with an SDOF, under free vibration, would be [12]
m
d2
u
dt2
+ λ
du
dt
+ ku ¼ 0 (1.12)
The solution to this equation depends on the value of the damping coefficient ‘λ’. If the damping is
small, as for metallic beams, the system will continue to oscillate but with diminishing amplitude with
time. The system is said to be Underdamped. If the damping is just enough to stop the oscillation, then it
is termed as Critically Damped. When the damping exceeds the critical value, the system is called
Overdamped. The value of critical damping is
λC ¼ 2
ffiffiffiffiffiffi
km
p
(1.13)
FIG. 1.5
A schematic representation of a free vibration with damping system for SDOF.
10 Chapter 1 Fundamentals of vibration damping](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-21-320.jpg)
![turn failed; but it was to be on St. Peter's Day, and was not that the
fittest day of all?
The Archbishop of Westminster, in the name of the committee,
spoke, on May 25, for nearly two hours. Indeed, morning by
morning the committee availed itself of the right of reply granted to
its members exclusively, by setting up one of them to refute the
objections advanced in the previous sitting. Kenrick says that he
knew not which to admire most—Manning's diction, his delivery, his
power, of command and frankness, or his ardour in urging and
almost commanding the new definition.[413]
I thought, says Kenrick, of what used to be said of
Englishmen living in Ireland, that they were more Irish than the
Irish themselves. The Archbishop is certainly more Catholic than
all the Catholics I have known hitherto. He himself feels no
doubt as to pontifical infallibility, personal, separate, and
absolute; and he will not permit others to feel any. He asserts
that the doctrine is of faith, and as such he hardly asks the
Council to define it, but rather predicts that it will do so—
perhaps after the manner of those prophets who strive to bring
events to pass by foretelling them. So far as concerns myself—
as one whom sixty years that have passed over me since I
began to learn the rudiments of the faith, have perhaps left as
well instructed on the point in question as one who joined the
Church about twenty years ago—I dare to assert that the
opinion, as it is found in the proposed Decree, is not a doctrine
of faith, and that it cannot become such by any definition
whatsoever, even that of a Council. We are custodians of the
deposit of faith, not lords of it. We are teachers indeed of the
faithful committed to our care, in so far as we are witnesses.
Manning resented, graviter illud tulit, the attempt which had been
made to raise a case of conscience in the mind of the bishops by
asserting that any bishop would incur the guilt of a mortal sin who
gave a vote in favour of infallibility without having duly investigated](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-28-320.jpg)
![the question for himself; because his act would contribute to impose
a new yoke on the faithful. This Manning held to be injurious to the
dignity and the honour of the bishops; as if, says Kenrick, he denied
that bishops could sin, or denied that they would be guilty of mortal
sin if through negligence or idleness they failed rightly to inform
their judgments.
Manning contended that infallibility was a supernatural grace—
charisma—and, therefore, that it properly attached to a person. He
would not hear of conditions being connected with the exercise of
infallibility. He asserted that he who had bestowed this supernatural
grace would also give the means for its due exercise.[414] Moreover,
he took the ground that the Council had already, in the conclusion of
the Decree which had been passed, committed itself to the doctrine
of infallibility, and that it could not now recede. Kenrick replied that
the assertion of Manning was one of several things which he had
heard with stupefaction. They had been assured, he stated, as we
have already seen, in the clearest terms by the reporter of the
committee, that the clause referred to contained no doctrine, and
that it was only a fitting conclusion to the four chapters of the
Decree. Then follows the statement that the reporter had either
himself been deceived or had knowingly deceived the minority.
In the sitting of May 25, MacEvilly, Bishop of Galway, also referred to
Kenrick's argument, drawn from the fact that the Catholics of
England and Ireland had been admitted to equal civil rights on the
faith of repeated declarations, and even of oaths, to the effect that
the doctrine of Papal infallibility was not binding on Catholics, and
that consequently such edicts of Pontiffs as the Bull Unam Sanctam
had not doctrinal authority. To this MacEvilly replied that the
Catholics in England had been admitted to equal civil rights, not
because of their declarations, but because the English government
feared a civil war. The reply of Kenrick to this straightforward
utterance is worthy of being given word for word—
The doctrine of Papal infallibility was always odious to the
English government, and had it been really a doctrine of the](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-29-320.jpg)
![faith, Protestants would have understood Papal doctrine better
than English and Irish Catholics; for they knew that Roman
Pontiffs had claimed the highest power in temporal things for
themselves, and had attempted to drive several English kings
from the throne by absolving their subjects from the oath of
allegiance.
Catholics, by public oath repeatedly made, denied that such
power belonged to the Roman Pontiff in the realm of England,
and had they not done so, they never would have been or ought
to have been admitted to equal civil rights.[415] How the faith
thus pledged to the British government is to be reconciled with
the definition of Papal infallibility may be looked to by those of
the Irish prelate who have taken that oath as I myself did I
cannot solve the difficulty as yet. I am Davus, not Ædipus.
Nevertheless those civil rights were conceded to Catholics by
men who through a long life had strongly opposed that course.
They did indeed apprehend civil war; but they did not dread it in
this sense, that a war of that kind could not be otherwise hurtful
to the power of the government than by causing a disturbance
of the peace for a certain time.
They feared the occurrence of a war, not the result of it, as to
which no sensible man could have been uncertain. Those great
men preferred to yield rather than to conquer by the slaughter
of a brilliant nation, and of a people worthy of a better fate,
even in what seemed to them its errors. Oh that here the same
spirit of moderation which they exhibited may be displayed by
the majority of the bishops who are listening to these words,
and that by a prevision of the calamities which may arise to us
from this hapless controversy, they may, in circumstances calling
for consummate moderation, ward off from us, who are fewer,
but who represent a greater number of Catholics than those
who are opposed to us, evils which it is not possible to
anticipate without horror, and which it would be impossible to
repair by a late repentance.](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-30-320.jpg)
![On the one hand, we cannot but regret that these words, fitly
written, were not actually spoken in the deaf ears of the resolved
majority. On the other hand, we remember that had they been
spoken, they would have sunk into the Vatican archives, and would
never have been heard of more till those graves give up their dead.
They now belong to history, and furnish a living link in a chain of
memorable professions and performances. The denationalizing
influence of the Papacy had still left something of the citizen alive in
the soul of Kenrick. During his stay in Rome, when witnessing the
paltry tyrannies that flounced about under the dependent banner of
the Pope, all of the citizen that was left in him must have turned
with fresh respect to the two flags of the free under which he had
spent his days—the flags of England and America. And yet there
were those sitting there, each with all the rights of a free man in his
hands, planning to reconstruct the society of England and America
on the degraded and fettered model of the States of the Roman
Bishop. There is a crime which no code has defined—the crime, not
of breaking one specific law of one's country, but of contriving, with
a foreign pretender, how to overturn everything vital in a venerable
and generous legislation.
It was not merely by a pupil of Maynooth that the eager ex-Anglican
was considered extreme in his views. Clifford, Bishop of Clifton,
spoke on the same day, refuting the notions of Manning about the
favourable effects to be produced by his beloved dogma in England,
and appealing to him as a witness that an eminent statesman had
represented the influence of the recent course of the Curia upon
public opinion in England as being much to the disadvantage of their
own cause, and greatly to the encouragement of extreme
Protestants.[416]
In the next Congregation, on the 28th, it was Senestrey who took
the post occupied on the last morning by Manning, that of official
respondent against attacks. On that day, a scene was raised by
Verot, of Florida. He declared that they were making innovations in
the Church, and that such an innovation as the personal infallibility](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-31-320.jpg)
![of the Pope was sacrilege. That horrid word applied in the sacred
place to an object so dear to the Pope, touched indeed the apple of
the eye. Sacrilege! The Cardinals de Angelis and Capalti, says
Vitelleschi, quite lost their temper; and a scene ensued which for
anger and excitement is said to have fallen but little short of
Strossmayer's scene in March.[417] The odious, and to well-tuned
Curialistic ears the inconceivable, task of hearing the infallibility of
the Pope denied, and of seeing his pleasure daily thwarted under the
roof of St. Peter's, was not to be endured any longer. The word
passed that the power given by the new Rules to close the debate
must be called into requisition.
A trusty American was set up in the next meeting, by the committee,
to repair the mischief done by Verot—Spalding, of Baltimore. Here,
again, we are indebted for light to Kenrick's unspoken speech.
Referring to the moral question which had been raised by Kenrick, to
which we have already seen allusions, Spalding said that it called for
as much investigation to justify one in giving a negative as in giving
an affirmative vote on the question of Papal infallibility, and that in
withholding an affirmative vote one would confirm the celebrated
Gallican articles.
On May 31, Valerga, the Patriarch of Jerusalem, made a vigorous
attack on the minority, speaking cleverly, and hitting hard. Spirited,
piquant, and insolent, is the description of Quirinus. Soon
afterwards, another American was in the desk, Purcell, of Cincinnati.
Quirinus says that he affirmed that the Americans abhorred every
doctrine opposed to civil and spiritual freedom; and that the
American sons of the Church loved her, because she was the freest
society in the world. He also took the position that, as kings existed
for the good of the people, so the Pope existed for the good of the
Church. On the same day spoke Conolly, Archbishop of Halifax. He
seems to be the only one in the Council who really related a
theological experience, declaring that he had formerly believed in
the personal infallibility of the Pope, and had come to Rome
believing that the Augsburg Gazette had circulated a calumny in](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-32-320.jpg)
![representing the dogmatizing of this opinion as the real object of the
Council. He went on to say that, on finding what was expected of
him, he determined to sift the arguments of the Roman theologians
and the proofs by which they supported them. He now bore witness
to the result upon his own views. All antiquity, he declared,
explained the passages harped upon by those theologians, in a
sense different from theirs. All antiquity bore witness against the
notion that the Pope alone, and separate from the bishops, was
infallible. He further took the ground that to found a dogma on the
rejection of the traditional interpretation of Scripture was pure
Protestantism. I will have nothing, he said, turned into dogma but
the indubitable Word of God. Ten thousand theologians do not
suffice for me, and on the present subject no theologian should be
quoted who lived subsequent to the Isidorean forgeries. To define
the dogma would be to bring the Vatican Council into contradiction
with the three General Councils which had condemned Pope
Honorius as a heretic, to narrow the gates of heaven, to repel the
East, and to proclaim, not peace, but war. In reply to Manning, he
protested that no one was justified in calling an opinion proximate
heresy when it had not been condemned as such by the Church.[418]
On June 3, Gilooly, Bishop of Elphin, replying to some observation of
Purcell as to the oaths and declarations, said[419] that Catholics had
not denied that they held the infallibility of the Pope as a doctrine of
the faith, but as a dogma of the faith; that is as a dogma defined by
a General Council. To this, Kenrick's unspoken speech replies, If
that is what was meant, which I do not believe, we might be
reproached, and that rightfully and deservedly, with not shrinking, in
a very grave matter, from the concealment of our meaning by
scholastic distinctions.[420] According to Quirinus (p. 661), Cardinal
Bonnechose prevailed upon Cardinal de Angelis to ask the Pope,
directly, if he would not consent to a prorogation of the Council on
account of the heat, now intolerable to all but Romans, or men from
the southward of Rome. The reply was stern and, according to
many, savage. Whatever were the terms of it, the substance was](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-33-320.jpg)
![indubitable—no adjournment was to be allowed till the Decree of
Infallibility was passed. It is said that when Bishop Domenec, of
Pittsburg, in America, began his discourse, he was greeted with
laughter by the majority, and when he made the very plain and
simple statement—one which he might have picked up from any
intelligent or travelled Italian any day in the year—that American
Catholics were not merely nominal ones, as the Italians were,
Cardinal Capalti imperiously commanded silence.[421] Strossmayer
had spoken at length on June 2, and with such moderation as to
escape even a call to order, yet, it is said, with very great force. On
the 3rd, Moriarty, of Kerry, took the side of Purcell, Kenrick, and
MacHale, but we have no particulars of his speech.[422] That day
Maret was in the desk speaking in the loud and labouring tone of a
deaf man, arguing, not only against the convictions and feelings of
the majority, but against their personal detestation of himself. He
made a point that either the Council was to give infallibility to the
Pontiff, in which case the Council must be a higher authority than
he, or else the Pontiff was to give to himself an infallibility which he
had not previously possessed, in which case he would change the
constitution of the Church by his own power alone. Then Cardinal
Bilio interrupted, and cried, The Council does not give anything, nor
can it give anything. It gives its suffrage, and the Holy Father
decides what he pleases.[423] The representative of all that was left
of the once courageous Gallican liberties asked if he might be
allowed to proceed, and did so. The minority had a long list of
speakers still inscribed. Kenrick was waiting for his turn, and so were
Haynald, Dupanloup, and many others; but a fresh surprise was at
this point sprung upon them. The Presidents produced a requisition
for the close of the general debate, signed by above one hundred
and fifty bishops.[424] De Angelis at once called on those who were
for the closing of the debate to stand up. He then declared, A large
majority have stood up, and by the power conferred upon us by Our
Most Holy Lord (the capitals are official), we close the debate on the
general question. The Acta Sanctæ Sedis say that about fifty
remained sitting. No wonder that, after hearing sixty-five speakers,](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-34-320.jpg)
![FOOTNOTES:
[402] Vitelleschi, p. 158.
[403] Quirinus, p. 532.
[404] Quirinus, p. 533.
[405] Frond, vol. ii.
[406] Documenta ad Illustrandum, ii. 209.
[407] Meum honorem graviter læserunt.—Documenta ad
Illustrandum, i. 189.
[408] Documenta ad Illustrandum, pp. 187-224.
[409] Aliquid humani passum esse.
[410] He showed that Tischendorf read προβἁτια in both cases,
and that other editors had read πρὁβατα in both. Of course, in
the fifteenth verse, the word lambs—ἁρνἱα—is the proper
translation.
[411] Acta Sanctæ Sedis. As to MacHale, Kenrick omits what
Frond states, that he was of a very ancient family.
[412] Documenta, ii. pp. 415-24.
[413] Documenta, i. 209.
[414] Documenta, i. 223.
[415] Quod si non fecissent nunquam ad libertatis civilis
consortium admissi fuissent aut debuissent (p. 219).
[416] Quirinus, 584.
[417] Vitelleschi, p. 168.
[418] Quirinus, p. 597.
[419] Documenta, i. 215.
[420] Ibid., 215.
[421] Quirinus, p. 661.
[422] His name does not occur in the Acta Sanctæ Sedis for the
third.
[423] Quirinus, p. 608.](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-37-320.jpg)
![[424] Acta Sanctæ Sedis. Friedberg, p. 47, says there were two
hundred and fifty signatures, but this is evidently a mistake.
CHAPTER VI
To the Close of the Special Debate on Infallibility, July 4—
Proposal of the Minority to resist—They yield once more—
Another Protest—Efforts to procure Unanimity—Hope of the
Minority in Delay—Pope disregards the Heat—Disgrace of
Theiner—Decree giving to Pope ordinary Jurisdiction
everywhere—His Superiority to Law—Debate on Infallibility
—Speech of Guidi—Great Emotion—Scene with the Pope—
Close of the Debate—Present view of the Civiltá as to
Politics—Specimens of the Official Histories—Exultation.
ANY one who had observed the course of the minority in
emergencies would have probably foretold that, under the new trial,
they would feel indignant, would speak of doing something, and
would end with a protest. So it proved. The very day of the forcible
conclusion of the general debate, the French bishops met, and were
favourable to some determined action.[425] But the next day, eighty
congregated in the rooms of Cardinal Rauscher. The Hungarians,
French, and Americans, with Strossmayer, Clifford, and Conolly, are
named by Quirinus as recommending that the Fathers of the
Opposition should cease to take any part in the Council, reserving
themselves for the final vote, and should then give their Non placet.
The Germans, however, always marplots, urged that the better
course would be to adopt a protest, and continue to take part in the
proceedings. This counsel prevailed. Rauscher drew up a form of
protest, which was signed by some eighty prelates, and many of the
bishops took a trip to Naples or elsewhere.](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-38-320.jpg)
![Among the things represented by Quirinus as having been said on
this occasion, one was to the effect that in a Parliament speeches
were of some use, for if they did not influence votes, they did
enlighten public opinion; but in this Council, most of the hearers
were, from their degree of culture, quite incapable of apprehending
theological arguments, not to add that, in a moral point of view,
many of them stood so low that even if convinced they would not
act on their convictions. The ground taken in the protest is clear,
namely, that the right of supporting their votes by a statement of
reasons, is one which, by the very nature of a Council, belongs not
only to some of its members, but to them all, and that such a right
could not be taken away by any vote of a majority.[426]
The Hungarians now declared that they would take no further part in
the debates. On the other hand, the Unitá Cattolica foretold how
those who had written or spoken as Gallicans would be converted by
a miracle of the Holy Ghost, even in the Council Hall; and as the
Galileans had been constrained to speak in other tongues, so would
the Gallicans be constrained to proclaim in that Hall before the
astonished multitudes the doctrine they had gainsaid.
The absorbing care of the Curia and its instruments was now
directed to the one end of constraining all to vote placet. The victory
was no longer doubtful, but to procure unanimity was of great
practical moment. The Pope himself was indefatigable. His admirers
resented such epithets as unscrupulous when applied to his
conduct. But they took good care not to grapple with the details of
alleged facts which, if they could be credibly told about the conduct
of one of our sovereigns in respect to his nobles or to Parliament,
would be described in much stronger epithets than unscrupulous.
His tongue was evermore scattering rebukes or blandishments, and
enlivening the city with crackling sparks of gossip. There were but
few bishops of note among the minority whose portraits, etched by
the infallible acid, were not handed round the salons, lay and
clerical. His letters were bitter and undignified. Quirinus quotes the
words of a French bishop (p. 627): There is no longer any scruple](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-39-320.jpg)
![as to what is done to gain votes. It is a horror. There has never been
anything like it in the Church. These words recall to us a scene in
Rome. A remarkable head—one of those heads which bear on the
brow a diploma of gifts and letters—was stooping in the light of a
lamp by which pages had been penned that had been heard of
beyond Italy. The stoop was pensive, and the thinker said, I saw so
much of what was done during that Council, that it has destroyed all
my faith in anything that ever was done in the Church before.
It would seem as if, at the last, argument and appeal had begun to
tell on some of those who were of a milder mood among the
Curialists. It is said that even of the chosen three champions,
Manning, Deschamps, and Pie, the last wished to find some formula
less offensive than the one projected. Martin of Paderborn even
proposed a note which contained a recognition of the teaching
authority of bishops, though in an indirect way. On the other hand,
the members of the Opposition tried to discover some turn of
expression which would save the Church from the shame of being
publicly disavowed by her wilful lord. Conolly spoke of proposing, as
a formula which would still give her a recognised voice, words
declaring the Pope infallible when he spoke, as head of the Church
teaching with him. Others again wished to reinstate the formula of
St. Antoninus, of Florence, declaring the Pope infallible when he acts
with the counsel of the universal Church.[427]
Men now began to realize the full effect of the proposed dogma,
both in its executive and in its retrospective aspects. Many must
have remembered how happy they had been in argument, or in
diplomacy, when the ambiguous state of the case, as it had hitherto
existed, enabled them to evade the charge that such and such were
the principles of the Church. It was so convenient to be able to say
No, they have never been sanctioned by a Council; they are only the
words of a Papal Decree. Now, however, all these words were to
have fresh life breathed into them, and whatever they contained
affecting a general principle of belief, or practice, was to be taken for
divine,—was, in fact, to rank as the word of God.](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-40-320.jpg)
![Delay now became the forlorn hope of the minority, and expedition
the watchword of the majority. The minority were sure that the Pope
would not be so cruel as to force them to continue in Rome during
the summer heats. Hence, they thought that by delay they were
certain of a prorogation before the fatal deed was done. They forgot
the history of the Pope's prisons and executions. Perhaps they had
never read it, or had used their fatal facility of calling an unpleasant
statement a lie. Antonelli had generally carried away the chief part
of the blame for the blood of the political victims. However, he
seems completely to have escaped reproach for the broiling of the
bishops. Whether the fierce language ascribed to the Pope was
correct or not, nobody doubted its aptness.[428] When even the
faithful M. Veuillot said, Since they have put the Council upon the
gridiron, they shall broil (ii. p. 352), everyone treated him as only
echoing the language of his idol. When once the heats had begun to
tell, the feelings of majority and minority, as Vitelleschi points out,
changed. Men from the north, accustomed to the bracing air and
pure streams of Germany, could ill bear up against the miasma from
the Roman marshes and the torrid heats that were withering the city
and making even natives look pale. They therefore began to long for
an escape, and not a few of them took their way homewards. They
received not only ready but glad permission. Thus every day was
diminishing the strength of the Opposition. The majority, on the
other hand, consisting of Italians, South Americans, and Spaniards,
were inured to the heats, if not to the malaria, and felt that the sun
and the marshes were conspiring with them. Apollo had come to
camp shooting over the heads of the natives, but laying low the men
from beyond the sea.
There was now only one consideration that would make the Pope
anxious for despatch, and that was the daily pressure upon his
finances caused by supporting his three hundred boarders. This
certainly had proved a useful ground of appeal for funds. The sums
collected everywhere had been great. The Civiltá reproaches the
Liberal Catholics with not sending money any more than they had
sent men to fight for the Holy Father, and sets in contrast with their](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-41-320.jpg)
![stinginess and want of military spirit the fact that the Univers alone
had sent in more than nine thousand pounds (234,410 francs).[429]
The Holy Father said, They fear making the Pope infallible, but they
do not fear making him fail.[430] But M. Veuillot, on the contrary,
did not fear making him infallible, and did everything possible to
prevent him from failing. Hence it was no wonder that he should
have briefs to publish which would perform a service for the
exchequer of the Univers similar to what the Univers performed for
the exchequer of the author of the briefs. The words of the Pope
spoken to the deputation of scientific men were representative
words, Here I am to receive your offerings.
Theiner, the celebrated Prefect of the Vatican archives, now fell
publicly under displeasure. He had allowed Hefele and Strossmayer,
and perhaps others, to see the order of procedure of the Council of
Trent, and probably had in other ways shown leanings not
acceptable to the Jesuits. He was ordered to give up his keys to
Cardoni, who had been the first chosen secretly to prepare Drafts of
Decrees on Infallibility before intentions were disclosed, and had
kept his counsel well. The archives were actually closed against
Theiner. It is said that the passage into them from his own rooms
was walled up. The disgrace of Theiner, and the honour of Cardoni,
sharply symbolized the favourite saying that the dogma must
conquer history. Here again Antonelli escaped all reproach of a share
in the blundering injustice. Cardoni was one singled out by name in
a celebrated letter of Döllinger as having largely employed falsified
authorities. But that charge, to us so revolting, is a familiar sound
wherever the shadow of the Curia extends.[431] We ourselves once
heard a member of the Congregation of the Index claim, unmindful
of the presence of a Protestant, You must never trust any edition of
any work whatever that has passed through the hands of the
Jesuits.
The exciting matters now remaining to be treated in the Council
were the all-important particulars of those Drafts which had already
been under a general review. The two chapters teaching the](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-42-320.jpg)
![institution of the primacy in the person of Peter, and the
transmission of that primacy through the Roman Pontiffs as his
successors, were speedily disposed of. Had all the fathers attempted
to answer the arguments of Desanctis on these points, arguments
familiar to many Italians, they would not have found it light work.
But the third chapter was one of immense importance. It defined the
scope and nature of primacy, distending that term till it was made to
cover absolute, immediate, and ordinary control in the whole domain
of the Church—control over bishops and people, control over not
only all matters ordinarily included under the expression faith and
morals, but over all things held to be necessary for the government
or discipline of the Church. This last expression, as any one
acquainted with the views of those in authority, even so far as they
are recorded in our preceding pages, must know, covers almost
every possible question that can arise. The words of Vitelleschi (p.
174) are well considered. He speaks of the supreme jurisdiction,
ordinary and universal, of the Pope over all Churches, singly and
collectively, over pastors as well as flocks; from which doctrine it
follows that bishops in exercising any jurisdiction or authority, only
do so as official delegates of the Pope. Dr. Langen puts it thus:
Seeing that there can be only one bishop in a diocese, as soon as
the Pope is declared to have ordinary jurisdiction in that diocese, he
becomes its Ordinary, and the other person called a bishop is
nothing more than his delegate and representative.[432] Men who
cover a dominion of this sort under the pretext of primacy, and who
advance a claim of primacy in order to deduce from it an absolute
dictatorship, never do anything more sensible than when they decry
reason and relegate Scripture to the tradition-heap; when they call
for pictures instead of books, and processions and fireworks instead
of a free press and free discussion. There was political philosophy in
M. Veuillot's exclamation on witnessing the Easter rejoicings in
Rome, especially the fireworks representing the heavenly
Jerusalem, that it was impossible not to respect a people for whom
such entertainments were provided.](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-43-320.jpg)
![never minimises. She always interprets her own documents as a
legatee interprets a will, that is, in her own favour.
On June 15 the Council disposed of all the matters that stood in the
way of the great question. Seventy-five speakers had entered their
names. Two speeches were actually made on that day by Cardinals
Mathieu and Rauscher.[433] The latter said that he could never
assent to the doctrine of the Draft without mortal sin. We knew all
that from your pamphlet, cried Deschamps, interrupting. But you
have never refuted it, replied the Austrian.[434] The following day
was the grand procession of the Corpus Christi. If the good press
was parsimonious in information regarding debates and decrees, it
was profuse in description of the spectacles. On the 17th, Pius IX
entered on the twenty-fifth year of his pontificate. This year,
according to Roman tradition, is fatal to the Pontiffs, it being held
that Peter reigned twenty-five years, and that none of his successors
was to reign longer. Vitelleschi declares that the twenty-fifth year
proved fatal to Pius IX, as well as to the rest, because in the course
of it he ceased to be a mere mortal. This phrase from a Liberal
Catholic will seem natural when set beside one of M. Veuillot, on the
day on which Pius IX completed the twenty-fifth year of his
pontificate: We are reminded of the radiance of Jordan and of
Tabor, of the thunders of the Temple, 'This is my beloved Son, hear
ye him' (vol. ii. p. 468). On the next page he says, God has left us
His priest, His angel, the sacred interpreter of His law, the anointed
intercessor between Him and the world ... a second Peter, a second
Moses on the threshold of a new world. It remains to be seen
whether the twenty-fifth year of Pius IX was or was not that of the
final fall of the temporal power. If the speeches on the doctrine and
polity of the Church were concealed, the Pope's speech this day, in
reply to the Sacred College, was blazed abroad. He divided the
bishops into three classes—the ignorant, the time-serving, and the
good. So flowed abroad fresh streams from that fountain which, all
the time, was sending forth both sweet waters and bitter.](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-47-320.jpg)
![to be heard, and that no one had given the right of the Presidents to
the bishops; but he added, You will have the opportunity of saying
Placet or Non placet. Hereupon, from all ranks of the Opposition
burst out a cry of Optime! optime!—excellent! excellent! Do you
agree with us? asked a bishop of Manning. The Cardinal's head is
bewildered, was the reply. On this, says Quirinus, a bishop could
not refrain from saying to the powerful Archbishop of Westminster,
It is your own head, Monsignor, that is bewildered, and more than
half Protestant. If this language was really used, we must doubt
whether it was infallible.
Guidi went on to advocate a change in the wording of the Decree, to
the effect that the Pope acted with the concurrence of the bishops,
and that after having, at their request, occasioned by prevalent
errors, made inquiry in other Churches, he acted with the consent of
his brethren, or with that of a collective Council. He contended that
this was the doctrine of St. Thomas; that the word final implied
something to precede, and that supreme teacher and judge
presupposed other teachers and tribunals. He concluded by
proposing two Canons, the first of which declared Papal Decrees or
Constitutions to be entitled to cordial faith and reverence, and not to
be reformable; but the second said, If any one shall say that, in
issuing such Decrees, the Pope can act arbitrarily without the
counsel of the bishops as testifying to the tradition of the Church, let
him be anathema.[435] On finishing his discourse, he at once handed
his manuscript to the secretaries.
Quirinus relates that Valerga audibly said, in reply to some question,
Guidi is misguided. But his neighbour replied that Guidi's speech
contained nothing but the truth. Yes, rejoined the Patriarch of
Jerusalem, but it is not always expedient to speak the truth. The
excitement was great. Groups of prelates who had left the Hall might
be seen standing about everywhere in earnest conversation, while
within doors Bonnechose and Cullen were discoursing to a thin
audience with absent minds. It was related that Guidi did not speak
as a solitary individual, but represented fifteen bishops belonging to](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-49-320.jpg)
![the Order of Dominicans. He had gathered them together in the
central convent of the Minerva, where he himself resided. They had
considered the question, and accepted the views which he had now
presented to the Council. This was much against the feeling of
Father Jandel, their general, who was perfectly free from any taint of
the episcopal system, a thoroughly right-minded Papist. Guidi asked
how the Cardinals had taken his speech, and Cardinal Mathieu
replied, With serious and silent approval.
Rumours were soon afloat in Rome as to what followed between
Guidi and his royal master. What we now give is traced by Quirinus
to the authority of the Pope himself, who is notoriously fond of
telling the people with whom he chats how he has lectured this or
that dignitary.[436]
The creature was summoned to the presence of his master soon
after the sitting, and was greeted with the words, You are my
enemy. You are the coryphæus of my opponents. Ungrateful towards
my person, you have propounded heretical doctrine. My speech is
in the hands of your Presidents, if your Holiness will read it and
detect what is supposed to be heretical in it. I gave it at once to the
Under-Secretary, that people might not be able to say that anything
had been interpolated into it.[437] You have given great offence to
the majority of the Council. All five Presidents are against you, and
are displeased. Some material error may have escaped me, but
certainly not a formal one. I have simply stated the doctrine of
tradition, and of St. Thomas. I am tradition. I will require you to
make the profession of faith anew. La tradizione son' io, vi faro far
nuovamente la professione di fide. I am and remain subject to the
authority of the Holy See, but I venture to discuss a question not yet
made an article of faith. If your Holiness decides to be such in a
Constitution, I certainly shall not dare to oppose it. The value of
your speech may be measured by those whom it has pleased. Who
has been eager to testify to you his joy? That Bishop Strossmayer,
who is my personal enemy, has embraced you. You are in collusion
with him. I do not know him, and have never before spoken to](https://image.slidesharecdn.com/33805-250304015404-1796880f/85/Polymers-for-vibration-damping-applications-B-C-Chakraborty-50-320.jpg)
Polymers for vibration damping applications B. C. Chakraborty
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- 6. Polymers for Vibration Damping Applications
- 7. Polymers for Vibration Damping Applications B.C. Chakraborty Debdatta Ratna
- 8. Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-12-819252-8 For information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisition Editor: Edward Payne Editorial Project Manager: John Leonard Production Project Manager: Anitha Sivaraj Cover Designer: Victoria Pearson Typeset by SPi Global, India
- 9. Preface This book is dedicated to polymer scientists associated with vibration damping. Vibrations in structures and machines cause component fatigue and human discomfort if they are not properly controlled. For warships and submarines, controlling vibration of hull structure is a very important aspect of un- derwater acoustic stealth technology. One of the most effective methods of vibration damping is passive method, wherein, a suitable polymer system with appropriate viscoelastic property is used according to the application envelope. This requires understanding the basics of shock and vibration, development of suitable materials through synthesis, modification, formulation, and finally, engineer- ing design of damping treatment of the machinery or structure. Therefore, a combined effort of physicists, mechanical engineers, materials scientists, and chemists is absolutely necessary for the suc- cessful development of vibration damping systems. We have done extensive work in the field of materials and developed several products suitable for civil and defence applications. While working in this area for a long time, we felt the necessity of a self-sufficient book in this field. Therefore, we decided to compile our fundamental understanding and long research experience in the development of polymers and composites for vibration damping and present in the form of a book. This book is divided into six chapters. It starts with a general introduction to vibration in Chapter 1. In this chapter, basic mathematical expressions on free and forced vibration of objects, both undamped and with viscoelastic damping in single degree of freedom, are discussed. Examples of numerical calculation of undamped and damped free and forced vibration are provided with graphical repre- sentations. An appendix is added to demonstrate the selection of an appropriate elastomer from several compositions for machinery mount. Chapter 2 includes fundamental mathematical expressions of shock pulse and shock response spectra for a single degree of freedom system. Graphical representations of various response spectra (calculated by simple equations of response) are included. Viscoelastic properties are covered in Chapter 3. Only linear viscoelasticity is discussed in this chapter. Various physical and mathematical models for polymers are discussed with examples. Dy- namic viscoelasticity is dealt with some details including phenomenological relaxation theory with nu- merical examples of frequency and temperature dependence and time–temperature superpositions as well. A brief discussion on the comparison of static and dynamic properties is included with numerical examples. Chapter 4 deals with the design of polymer system for vibration damping applications and review of recent advances in the field. This chapter includes the various classification of polymers, basic prin- ciple for the selection of polymers and design of polymer systems for vibration damping applications. Various strategies, which are to be adopted to achieve vibration damping in a broad frequency range, have been elaborated. The mode of damping and the corresponding design rules required to be adopted for efficient vi- bration damping with some examples have been presented in Chapter 5. Various mathematical expres- sions for constrained and unconstrained layer damping are described with examples. Chapter 6 describes equipment, instruments, and methods of characterisation and testing, which are relevant to the development and selection of polymers for vibration and shock damping. xi
- 10. With such broad technical contents covering the basic concepts, practical examples, and recent advances, we are sure that this book will serve as a useful textbook-cum-handbook for the students, researchers, engineers, R&D scientists from academia, research laboratories, and industries related to the application of polymers for vibration and shock damping. We would like to dedicate this book to our late parents. We are indebted to the members of our families for their patience and for always being the source of inspiration, without which this book would not have been a reality. Dr. Chakraborty thanks his wife (Mitali), son (Abhishek), and daughter (Anwesha) for their encouragement and support. Dr. Ratna would like to place on record his sincere thanks to his wife (Sujata) and sons (Saptarshi and Debarshi). Although it has been our endeavour to make the book comprehensive and of a high standard, errors that may creep in are sincerely regretted. We are thankful to the publication team of M/s Elsevier, United Kingdom, for their cooperation and encouragement. We are also thankful to Dr. M. Patri, Director, and other colleagues of NMRL, especially Sri Praveen Srinivasan, for their cooperation and suggestion and Sri Ramakant Khushwaha for his help in preparing the book. Debdatta Ratna Bikash Chandra Chakraborty xii Preface
- 11. CHAPTER Fundamentals of vibration damping 1 Abbreviations DSA dynamic signal analyser FFT Fast Fourier Transform SHM simple harmonic motion SDOF single degree of freedom SONAR sound navigation and ranging VEM viscoelastic material Symbols with units A amplitude of vibration (displacement) (m) a acceleration (m/s2 ) C1, C2 constants of integration E modulus of elasticity (N/m2 (Pa)) F force (N) f frequency (Hz) g acceleration due to gravity (9.81 m/s2 ) h height or thickness (m) i imaginary quantity (¼ ffiffiffiffiffiffiffi 1 p ) k stiffness (spring constant) (N/m) m mass (kg) n mode number (dimensionless) P power (W or J/s) Q quality factor (dimensionless) R ratio of vibration intensities (dimensionless) T time period of oscillation (s) t time (s) u displacement (m) v velocity (m/s) W weight (N) δ deflection (m) ε transmissibility (dimensionless) η viscoelastic loss factor (dimensionless) Δ logarithmic decrement (dimensionless) Polymers for Vibration Damping Applications. https://doi.org/10.1016/B978-0-12-819252-8.00001-X # 2020 Elsevier Inc. All rights reserved. 1
- 12. φ phase angle (radian) λ damping coefficient (N s/m) ω angular frequency (rad/s) ξ damping factor (dimensionless) 1.1 Vibration Vibration is a very common phenomenon. In fact, everything vibrates—it is only the question of in- tensity and its effect on man, machine, and environment that can be a concern to the mankind. Vibration may be defined as a time-dependent movement of a particle around its equilibrium position. The dynamic (time-dependent) displacement is either uniform in timescale (harmonic) or nonuniform (non- harmonic). However, the difference in oscillation and vibration is that in oscillation, the matter moves periodically around an equilibrium position without any deformation of the body, but in case of vibra- tion, periodic deformation of a structure would also be involved. Vibrations also result in a wave, which is purely a mechanical pressure wave that can propagate through a medium such as gas, liquid, and solid and is termed ‘Sound’. The human ear can listen airborne sound within the frequency limits of 16–20,000 Hz. Typical examples of sound produced by vibration are the vehicle horns, vocal cord, and all musical instruments. For a three-dimensional object, vibrations can be in all three coordinates and hence the degree of freedom will be 3 or even more. However, for simplicity, only single-degree of freedom (SDOF) in vibration is being addressed in this book. The vibration reduction mechanism in any degree of freedom will be the same by passive damping materials such as polymeric viscoelastic materials (VEM). 1.2 Importance of the study of vibration The studies of sound and vibration are closely related. Sound waves, in turn, can also induce the vi- bration of objects, for example, tuning fork. In many occasions, sound and vibration are required. As an example, sound is necessary for the form of music, speeches, verbal communication, signalling, etc. Vibrations are required for material handling size separation, sieving, mechanical operations such as pneumatic drilling, medical treatments and health care, physiotherapy, etc. In many other cases, vibration and sound are undesirable. Undesirable sound is termed as ‘noise’ and the quality of an acous- tic signal is decided by the ‘signal to noise ratio’. Since sound is a result of material vibration, primarily, attempt to reduce noise is related to controlling the vibration at the source. Vibrations are closely as- sociated with the man and machinery. The vibrations of the machinery are a result of the imbalance of rotational movements (eccentricity) such as for motors, engines, turbines, etc. Machinery vibrations can cause serious damages to the machine due to excessive dynamic displacements apart from undesirable radiated noise in the atmosphere. Heavy machinery under vibration can be damaged permanently due to fatigue resulting from the cyclic variation of induced stresses. Furthermore, the vibration causes more rapid wear of machine parts such as bearings and gears and also creates excessive noise. In machines, vibration can loosen fasteners such as nuts, etc. In metal cutting processes, vibration can cause chatter, which leads to a poor surface finish. Vibration induced by the external 2 Chapter 1 Fundamentals of vibration damping
- 13. source on instruments and machinery is detrimental as it might affect the performance. An example is optical recording and experiments, where every minute vibration of the optical table would cause op- tical distortions. Similarly, vibration due to transportation of equipment may cause damage. Therefore, study of externally induced vibration on objects is also important. Another concern for the vibration of machinery and radiated noise is the environmental problem. Probably, the noise pollution created by vehicles is the most serious problem in urban life. With rapid urbanisation and random use of automobiles on streets, rail and air traffic, loud public address systems, construction machinery, etc., the environment is subjected to a high level of noise pollution. The excessive sound in the atmosphere can cause permanent damage to our eardrum, or partial loss of hear- ing [1]. Noise can cause ischemic heart disease, hypertension, cardiovascular problems, etc. [2]. Increased noise levels can create stress, increased aggression and workplace accident rates, and pos- sibly enhance antisocial behaviours [3]. Loud noise may affect the nervous system of the brain and create psychological disorders [1–3]. Birth defects such as harelip, cleft palate, and defects in the spine are also possible from high-intensity sounds such as airport environment. Typical noise levels of sound originated from normal human conversation can be 60 dB, while noise of ground vehicles can be 80– 90 dB and jet engine at take-off can be 120 dB [4]. However, continuous noise level 80 dB is dangerous for human health. In India, the noise intensity limit in the day time is 75 dB (A) in industrial areas, whereas 55 dB (A) in residential areas as per the noise pollution rules [5]. Earthquakes are the most important example of the ill effect of vibration. It is a seismic wave on the earth surface and its devastating effect is well experienced by mankind. It results from a sudden release of huge energy in the earth’s crust. These waves can be violent causing loss of life and can even destroy an entire civilisation around. The intensities of such seismic waves are measured in a comparative scale called ‘Richter’, named after the seismologist Charles Francis Richter. It is a logarithmic scale, where, a magnitude of 3 on the Richter scale is not dangerous, while 7 on the scale can cause large-scale devastation. Richter scale is developed as an empirical formula using the logarithm of amplitudes at the recording seismometer, arbitrary value, and the epicentral distance of the recording station as parameters [6]. Sound and vibration are also important subjects of study for underwater objects. Acoustic signals are used for bathymetric seabed mapping studies, to detect and identify underwater objects like rocks, submarines, mines, and even to detect marine animals such as a cluster of fishes, sharks, whales, etc., using a device called sound navigation and ranging (SONAR). There are two methods for the detection of an underwater object, namely active and passive detections. In active detection system, the SONAR sends a sound pulse to get a reflection from any object having a difference in specific acoustic imped- ance compared to water and analyse the reflected signal to determine the type of object and its coordinates (location), while passive SONAR only receives the underwater noise radiated from any object nearby. For the defence forces, naval ships and submarines need to be as silent as possible to avoid detection by the passive SONAR. Vibrations of machinery inside the vessels are transmitted to the hull of the vessel through internal structures and in turn transmitted to the seawater as radiated noise, which is picked up by the passive SONAR. Since there is a number of such machinery in a submarine/ship, there would be a particular underwater acoustic signature of the ship/submarine. The vibration spectrum would depend on the type of machinery, their design, manufacturing process, installation methodologies, and also on ageing effects. In general, vibrations of heavy machinery such as diesel alternators and turbines in a ship or submarine produce low-frequency radiated noise in the sea. The cavitation due to rotation of propellers of ships/submarines also produces sound like noise in 3 1.2 Importance of the study of vibration
- 14. the seawater. The cavitation noise spectrum would depend on the number of blades, rpm of the propeller, cavitation volume, skin friction of the blades, the thrust, and propeller loading [7]. Both the radiated noise due to machinery and the cavitation can be detected by passive SONARs. The study of vibration and sound propagation in different media is very important for prediction and enhancing the fatigue life of an equipment, prediction, and mitigation of catastrophic failure of machin- eries, reduction of environmental acoustic pollution, studies on acoustic noise-related health hazard of the society, and for design of acoustically stealthy strategic objects of defence forces. 1.3 Simple harmonic motion (SHM) A simple and periodic oscillation of a point object is called harmonic motion if it passes through a reference point after a definite interval and follows the same locus periodically. The number of such motion per unit time is termed as ‘frequency’ and the interval is the ‘time period’, while the length between two consecutive points having the same phase is called the ‘wavelength’. It is explained in Fig. 1.1A and B. Let us imagine a point rotating in a circular path repeatedly at a constant angular speed. Therefore, the point would cross any reference point on the circle at a regular interval. The locus of the point is the circle as shown in Fig. 1.1A, and when the movement of the point is plotted against time, the locus describes a perfect sinusoidal curve as in Fig. 1.1B. From Fig. 1.1B, the wavelength is the distance AB, the time taken to cover this distance is ‘time period’, and number of such distance travelled in one second is the ‘frequency’ of the wave. It means, one wave is equivalent to one rotation of the point on the circle and correspondingly the distance traveled by the wave is AB, which is the wavelength. The speed of the wave propagation is thus the product of the frequency and the wavelength. The convention of symbols and units for wave properties are given in Table 1.1. Simple vibration, in the basic analysis, is assumed to be harmonic motion, that is, a sinusoidal wave, though the actual vibration signature may contain various periodicities (frequency). A nonharmonic vibration is a random vibration with no regularity in periodicity. FIG. 1.1 SHM: (A) locus of a point in SHM, (B) time-dependent displacement of the point. 4 Chapter 1 Fundamentals of vibration damping
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- 16. 1.3.1 Displacement, velocity, and acceleration A SHM can be represented by a sinusoidal wave which also represents the expression for time- dependent displacement: u t ð Þ ¼ u0 sin ωt (1.1) where the angular frequency is given by: ω ¼ 2πf and f is the frequency in cycles per second or Hz of the dynamic displacement. One common term in a circular motion is revolution per minute (rpm), which is used for rotating machine like a motor. The corresponding frequency is then f ¼ (rpm/60) Hz. The velocity and acceleration can be derived from Eq. (1.1) as v t ð Þ ¼ ωu0 cos ωt ¼ ωu0 sin π 2 ωt (1.2) a t ð Þ ¼ ω2 u0 sin ωt ¼ ω2 u t ð Þ (1.3) Graphical representation of displacement, velocity, and acceleration is shown in Fig. 1.2. It can be seen from Eqs (1.2), (1.3), and Fig. 1.2: (i) The ratios of velocities or accelerations are the same as the displacement ratio. This is used in the calculation of vibration sensitivity of objects in a comparative evaluation. (ii) The velocity changes with the first power of frequency and the phase difference between the displacement and the velocity is 90°, which means that when the displacement is at either positive or negative maximum, the velocity is zero and vice-versa. (iii) The acceleration increases with the second power of frequency and the phase difference between the displacement and acceleration is 180°, which means that the acceleration is positive maximum when the displacement is at negative maximum and vice-versa. Also, at the neutral point, both acceleration and displacement are zero. (iv) The displacement, the velocity, and the dynamic force can be calculated from the above expressions at any given time by measuring the amplitude of the acceleration corresponding to a frequency when the mass of the object is known. It can be seen from Fig. 1.2 that the numerical value of acceleration amplitude is quite high compared to displacement. At the same dynamic force, a vibration amplitude reduces with increasing frequency, but the acceleration increases with the square of the frequency. This is used in the measurement of vibration studies where test protocols define acceleration both as input and measuring parameters. In addition, for the study of structural vibration and transients (shock), the acceleration is conven- tionally represented as multiple of acceleration due to gravity (g) and in SI unit, the value of ‘g’ is taken Table 1.1 Properties of a wave: symbols and units Property Symbol SI unit Wavelength λ m Frequency f Hz Time period T s Speed c m/s 5 1.3 Simple harmonic motion (SHM)
- 17. as 9.81 m/s2 . Secondly, all vibration intensities are expressed as a ratio of acceleration normalised by the force [a(t)/F0]. 1.3.2 Free vibration and natural frequency If a body is allowed to vibrate by an initial impact force, the vibration is called free vibration and the frequency of the vibration is termed as the natural frequency of the body. Every matter or object has a natural frequency of vibration depending on its physical characteristics such as elastic modulus, mass, size, and shape [8, 9]. As an example, if a string with a fixed length in a guitar is plucked, there will be a sound of a particular frequency and every time the same tune will be heard. There can be subsequent higher harmonics which are multiples of the first natural frequency of the object. In the case of a thin beam, modes in the axial direction is enough to study vibration, for a flat plate of negligible thickness, two directions are to be considered, and for a three-dimensional body, there will be natural frequencies and modals in three directions, for example, a cylinder will have axial, circumferential, and radial nat- ural frequencies and higher modes of each too. 1.3.3 Forced vibration and resonance When an external dynamic force is applied to an object for a period greater than its own time period of oscillation, the vibration is termed as forced vibration. The object vibrates at the same frequency as is imposed by the external force. Running machinery (such as motor, engine, pump, centrifuge, etc.) is subjected to forced vibration. FIG. 1.2 Displacement, velocity, and acceleration of a simple harmonic motion. 6 Chapter 1 Fundamentals of vibration damping
- 18. When the frequency of an external dynamic force coincides with the natural frequency of the body, there will be a very large amplitude of displacement and the phenomenon is termed as resonance. It is also the frequency, at which the potential energy of the object is totally converted to kinetic energy. Therefore, at resonance, the vibration amplitude attains a maximum value. There can be several modes of resonances, which are higher harmonics of the first natural frequency. In each mode, the intensity peaks are observed. System resonance for machinery with very large amplitude may cause severe damage or catastrophic failure. Thus, the study of the vibration response of a structure with respect to time and frequency is very important to take measures to avoid such damages or failures. 1.4 Random vibration If the magnitude of external dynamic force at a given time is known, then the vibration is deterministic vibration. When the magnitudes cannot be determined at a given time, then it is random vibration. In a random vibration scenario, a cluster of vibration intensities and frequencies would exist. Therefore, whatever be the complexity of a random vibration signature, it can be assumed as a sum of many pure sine waves of different amplitudes with corresponding harmonic frequencies such as f t ð Þ ¼ u0 + u1 sin ωtφ1 ð Þ + u2 sin 2ωtφ2 ð Þ + u3 sin 3ωtφ3 ð Þ + ⋯ + un sin nωtφn ð Þ Generally, the recording of a random vibration spectrum is done with respect to time (time domain). Fourier Transform is applied to find the individual intensities in the frequency scale after selection of a time domain window of the random signal. The example of random vibration is wind velocity, earth- quakes, etc. If large data is available, a statistical analysis may be done to determine the different mag- nitudes and frequencies of the random vibration [10, 11]. Typical Fourier transform of random vibration in time domain and frequency domain taking the limits of ∞ to +∞ for frequency and time are expressed mathematically as Ψ f ð Þ ¼ ð +∞ ∞ φ t ð Þexp iωt ð Þdt φ t ð Þ ¼ ð +∞ ∞ Ψ f ð Þexp iωt ð Þdω The infinite limits, however, is only theoretical, and the limits can be decided where the intensity is below 1% of the highest intensity in the frequency scale. Present-day Fast Fourier Transform (FFT) analysis of vibration and shock spectra is computerised as standard software available with all vibration analysers. 1.5 Undamped and damped vibration A free or forced vibration may or may not experience resistances like internal friction or inelastic de- formation of the object. An undamped vibration is a case where there is no resistance to the vibration and the vibration is not damped or attenuated with time, which implies that the dynamic deformation is perfectly elastic in nature. Therefore, in an undamped system, there is no loss of mechanical energy in the system. 7 1.5 Undamped and damped vibration
- 19. If the vibrating body experiences frictional or other types of loss of mechanical energy, the vibration intensity is attenuated or damped with time. The time dependence of response causes a phase shift for the state variable such as strain and is mathematically expressed as a complex quantity. The phase shift represents the extent of damping or loss of energy in a damped oscillatory system. The loss mechanism could be inelastic (dashpot) or viscoelastic damping, magneto-rheological or electro-rheological damp- ing, or active control or shear thinning-type damping. 1.5.1 Expressions for free, undamped vibration An SDOF system of a mass attached to a spring is shown in Fig. 1.3. The rotating or oscillatory ma- chines and many such real-life systems can be modelled as a mass-spring system as shown in the figure. Most metallic objects, such as machines and equipment, have very low inherent losses and the vibra- tions are approximately undamped. The key assumptions for the above system are that the spring has very low mass, and can be neglected and its behaviour is Hookean, that is, the force is linearly proportional to the deflection of the spring. The proportionality constant is termed as Spring Constant, denoted by ‘k’. The response of the spring is instantaneous, which means there is no time lag between the force and deformation and also the spring comes to its undeformed state instantly on withdrawal of the force. Therefore, there is no loss of energy due to deformation and retraction cycle. Considering the force balance, the force exerted by the stretching of the spring is balanced by the force due to the acceleration of the mass: m d2 u dt2 ¼ ku (1.4) One solution to the above second-order differential equation can be u ¼ u0 cos ωnt + v0 ωn sin ωnt (1.5) where u0 is the initial displacement, v0 is the initial velocity, and ωn is the Natural Angular Frequency of the spring-mass system and is given by ωn ¼ ffiffiffiffi k m r (1.6) FIG. 1.3 Spring-mass arrangement in an SDOF system: undamped free vibration. 8 Chapter 1 Fundamentals of vibration damping
- 20. Unit of natural angular frequency (ωn) is radian/s. The expression of natural frequency is significant for the design of machinery mount since the stiffness of the mount and the mass of the machine would decide the system resonance. A system of high stiffness and lower mass would have a high natural frequency, and the damping system should be tuned to cater for high damping at such frequency to reduce the high vibration intensity at resonance. In an elastic vibration system, when the mass is placed over the spring, and there is a static deflec- tion δs as a result of the weight of the mass, the resonance frequency of such a vibrating system can be expressed as ωn ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffi mg=δs m r ¼ ffiffiffiffi g δs r (1.7) where ‘g’ is acceleration due to gravity. The above expression for natural frequency is only valid for a system which is both linear and elas- tic in behaviour since static and dynamic stiffness of an elastic material are same and does not depend on the frequency of vibration. The expression is not valid for high damping materials since they are neither linear nor elastic in behaviour. Another solution to Eq. (1.4) can be of general expression as u t ð Þ ¼ A sin ωntφ ð Þ (1.8) The wave represents a sinusoidal curve with a phase offset of ‘φ’ and the displacement varies from A to +A as shown in Fig. 1.4. The amplitude A and phase φ are related to the initial conditions u0 and v0 as A ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 0 + v0 ωn 2 s (1.9) FIG. 1.4 Undamped vibration compared to a pure sinusoidal wave. 9 1.5 Undamped and damped vibration
- 21. tan φ ¼ v0 u0ωn (1.10) The phase angle (φ) can be very small in case of a system with a high natural frequency, or in another way, high stiffness and low mass. 1.5.2 Expressions for free damped vibration Continuing with the undamped system as a spring-mass combination, a viscous dashpot can be intro- duced to represent a damped vibration. A viscous pot consists of a cylinder filled with a viscous fluid such as silicone oil and a piston in the cylinder. The viscous dissipation takes place for a force applied to the piston due to shearing of the viscous fluid. The force is proportional to the velocity of the piston: F ¼ λ du dt (1.11) where λ is called Damping coefficient of the viscous dashpot. The damped vibrating system is assumed to be a combination of spring, mass, and a dashpot. A typical sketch of the spring-mass-dashpot system is shown in Fig. 1.5. The force balance of such a sys- tem with an SDOF, under free vibration, would be [12] m d2 u dt2 + λ du dt + ku ¼ 0 (1.12) The solution to this equation depends on the value of the damping coefficient ‘λ’. If the damping is small, as for metallic beams, the system will continue to oscillate but with diminishing amplitude with time. The system is said to be Underdamped. If the damping is just enough to stop the oscillation, then it is termed as Critically Damped. When the damping exceeds the critical value, the system is called Overdamped. The value of critical damping is λC ¼ 2 ffiffiffiffiffiffi km p (1.13) FIG. 1.5 A schematic representation of a free vibration with damping system for SDOF. 10 Chapter 1 Fundamentals of vibration damping
- 22. Damping is often expressed by a dimensionless quantity called Damping Factor ‘ξ’ defined as the ratio of the damping coefficient to the critical damping: ξ ¼ λ λC ¼ λ 2 ffiffiffiffiffiffi km p (1.14) The solutions for the three cases of damped vibration are described below with examples. 1.5.2.1 Case (1)—Underdamped system u t ð Þ ¼ u0eξωnt sin ffiffiffiffiffiffiffiffiffiffiffiffi 1ξ2 q ωnt + φ (1.15) In the above expression, we can identify a modified damped natural frequency ωd defined as ωd ¼ ωn ffiffiffiffiffiffiffiffiffiffiffiffi 1ξ2 q (1.16) The nature of the curve is sinusoidal with an exponential decay with a phase shift of φ compared to an undamped system as shown in Fig. 1.6. Eq. (1.15) of the underdamped system suggests that at high frequency (beyond 10 Hz), the damping is very fast, in a fraction of a second at a damping factor of even 0.1. It will be seen in consecutive chapters that the damping factor of most mounts and vibration damping materials are low at low frequencies except for active vibration damping or smart damping, where damping can be enhanced by a smart or active system at even very low frequencies. FIG. 1.6 Underdamped and undamped system in free vibration. 11 1.5 Undamped and damped vibration
- 23. The time period of damped oscillation, Td is modified since the damped resonance frequency is expressed by Eq. (1.16). Td is defined as Td ¼ 2π ωd ¼ 2π ffiffiffiffiffiffiffiffiffiffiffiffi 1ξ2 p (1.17) The decay of the wave can be expressed by taking the ratio of two consecutive amplitudes as An An + 1 ¼ exp 2πξ ffiffiffiffiffiffiffiffiffiffiffiffi 1ξ2 p ! (1.18) The natural logarithm of the amplitude ratio also termed as logarithmic decrement (Δ) expressed as Δ ¼ ln An An + 1 ¼ 2πξ ffiffiffiffiffiffiffiffiffiffiffiffi 1ξ2 p ! (1.19) This gives us an important conclusion that in an underdamped free vibration with an SDOF, the log- arithmic decrement depends only on the damping factor. Fig. 1.7 shows the variation in logarithmic decrement Δ, on damping factor ξ. 1.5.2.2 Case (2)—Critically damped system In this case, λ ¼ λC ¼ 2 ffiffiffiffiffiffi km p and ξ ¼ 1. The solution to the Eq. (1.12) could be u t ð Þ ¼ C1 + C2t ð Þeωnt Solving for C1 and C2, using initial conditions as: u(t) ¼ u0 and v(t) ¼ v0 u t ð Þ ¼ u0 + v0 + u0ωn ð Þt ½ exp ωnt ð Þ (1.20) FIG. 1.7 Dependence of logarithmic decrement on damping factor. 12 Chapter 1 Fundamentals of vibration damping
- 24. It can be seen that the displacement can have one overshoot compared to the undamped initial displace- ment. Fig. 1.8 depicts the nature of the decay, which is an exponential curve having no sinusoidal component. 1.5.2.3 Case (3)—Overdamped system The vibration response is overdamped when the damping factor is 1, that is, λ/λC or ξ 1. The solution to Eq. (1.12) can be u t ð Þ ¼ C1 exp ξ + ffiffiffiffiffiffiffiffiffiffiffiffi 1ξ2 q ωnt + C2 ξ ffiffiffiffiffiffiffiffiffiffiffiffi 1ξ2 q ωnt (1.21) where C1 and C2 are the coefficients of integration and can be found out by solving the equation using initial conditions u0 and v0. The solution becomes u t ð Þ ¼ u0ωn ξ + ffiffiffiffiffiffiffiffiffiffiffiffi ξ2 1 p + v0 2ωn ffiffiffiffiffiffiffiffiffiffiffiffi ξ2 1 p exp ξ + ffiffiffiffiffiffiffiffiffiffiffiffi ξ2 1 q ωnt + u0ωn ξ ffiffiffiffiffiffiffiffiffiffiffiffi ξ2 1 p v0 2ωn ffiffiffiffiffiffiffiffiffiffiffiffi ξ2 1 p exp ξ ffiffiffiffiffiffiffiffiffiffiffiffi ξ2 1 q ωnt (1.22) This is also a uniform exponential decay curve without any vibration characteristics. A typical response curve is given in Fig. 1.9. Example of an underdamped system is common, including cured rubber bushes and mounts, while a typical example of an overdamped system is the unvulcanised butyl rubber latex. FIG. 1.8 Free vibration of an SDOF system, undamped and critically damped. 13 1.5 Undamped and damped vibration
- 25. 1.5.3 Forced vibration When an external oscillatory excitation force is applied onto a body, the resulting vibration is termed as forced vibration. In the event of a frequency sweep by an external exciter, when the sweeping frequency coincides with that of the natural frequency of the system, the phenomenon is called resonance and at that point, the amplitude of vibration would be maximum. In a random vibration, such as a running turbine, there can be many subsequent higher modes. 1.5.3.1 Frequency response in undamped forced vibration Let F0sin(ωt) be the external oscillating force on a body without damping (mass-spring system). The force balance would be F0 sinωt ¼ m d2 u dt2 + ku (1.23) Solution to the above equation at steady state is u t ð Þ ¼ F=k 1 ω=ωn ð Þ2 sin ωt (1.24) where ωn is the natural frequency of the system, defined by Eq. (1.6). In a real-world application of forced vibration, it is important to study the force transmitted to the foundation due to a vibrating machine or vice versa. The transmissibility of output force compared to input force can be calculated and can be measured for a vibrating system. The force transmissibility (ε) in such a spring-mass system is defined by the ratio of exciter force amplitude to the resulting force amplitude. Following Eq. (1.24), we get, transmissibility, ε: ε ¼ 1 1 ω=ωn ð Þ2 (1.25) FIG. 1.9 Overdamped SDOF system in free vibration, compared with undamped and critically damped systems. 14 Chapter 1 Fundamentals of vibration damping
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- 28. turn failed; but it was to be on St. Peter's Day, and was not that the fittest day of all? The Archbishop of Westminster, in the name of the committee, spoke, on May 25, for nearly two hours. Indeed, morning by morning the committee availed itself of the right of reply granted to its members exclusively, by setting up one of them to refute the objections advanced in the previous sitting. Kenrick says that he knew not which to admire most—Manning's diction, his delivery, his power, of command and frankness, or his ardour in urging and almost commanding the new definition.[413] I thought, says Kenrick, of what used to be said of Englishmen living in Ireland, that they were more Irish than the Irish themselves. The Archbishop is certainly more Catholic than all the Catholics I have known hitherto. He himself feels no doubt as to pontifical infallibility, personal, separate, and absolute; and he will not permit others to feel any. He asserts that the doctrine is of faith, and as such he hardly asks the Council to define it, but rather predicts that it will do so— perhaps after the manner of those prophets who strive to bring events to pass by foretelling them. So far as concerns myself— as one whom sixty years that have passed over me since I began to learn the rudiments of the faith, have perhaps left as well instructed on the point in question as one who joined the Church about twenty years ago—I dare to assert that the opinion, as it is found in the proposed Decree, is not a doctrine of faith, and that it cannot become such by any definition whatsoever, even that of a Council. We are custodians of the deposit of faith, not lords of it. We are teachers indeed of the faithful committed to our care, in so far as we are witnesses. Manning resented, graviter illud tulit, the attempt which had been made to raise a case of conscience in the mind of the bishops by asserting that any bishop would incur the guilt of a mortal sin who gave a vote in favour of infallibility without having duly investigated
- 29. the question for himself; because his act would contribute to impose a new yoke on the faithful. This Manning held to be injurious to the dignity and the honour of the bishops; as if, says Kenrick, he denied that bishops could sin, or denied that they would be guilty of mortal sin if through negligence or idleness they failed rightly to inform their judgments. Manning contended that infallibility was a supernatural grace— charisma—and, therefore, that it properly attached to a person. He would not hear of conditions being connected with the exercise of infallibility. He asserted that he who had bestowed this supernatural grace would also give the means for its due exercise.[414] Moreover, he took the ground that the Council had already, in the conclusion of the Decree which had been passed, committed itself to the doctrine of infallibility, and that it could not now recede. Kenrick replied that the assertion of Manning was one of several things which he had heard with stupefaction. They had been assured, he stated, as we have already seen, in the clearest terms by the reporter of the committee, that the clause referred to contained no doctrine, and that it was only a fitting conclusion to the four chapters of the Decree. Then follows the statement that the reporter had either himself been deceived or had knowingly deceived the minority. In the sitting of May 25, MacEvilly, Bishop of Galway, also referred to Kenrick's argument, drawn from the fact that the Catholics of England and Ireland had been admitted to equal civil rights on the faith of repeated declarations, and even of oaths, to the effect that the doctrine of Papal infallibility was not binding on Catholics, and that consequently such edicts of Pontiffs as the Bull Unam Sanctam had not doctrinal authority. To this MacEvilly replied that the Catholics in England had been admitted to equal civil rights, not because of their declarations, but because the English government feared a civil war. The reply of Kenrick to this straightforward utterance is worthy of being given word for word— The doctrine of Papal infallibility was always odious to the English government, and had it been really a doctrine of the
- 30. faith, Protestants would have understood Papal doctrine better than English and Irish Catholics; for they knew that Roman Pontiffs had claimed the highest power in temporal things for themselves, and had attempted to drive several English kings from the throne by absolving their subjects from the oath of allegiance. Catholics, by public oath repeatedly made, denied that such power belonged to the Roman Pontiff in the realm of England, and had they not done so, they never would have been or ought to have been admitted to equal civil rights.[415] How the faith thus pledged to the British government is to be reconciled with the definition of Papal infallibility may be looked to by those of the Irish prelate who have taken that oath as I myself did I cannot solve the difficulty as yet. I am Davus, not Ædipus. Nevertheless those civil rights were conceded to Catholics by men who through a long life had strongly opposed that course. They did indeed apprehend civil war; but they did not dread it in this sense, that a war of that kind could not be otherwise hurtful to the power of the government than by causing a disturbance of the peace for a certain time. They feared the occurrence of a war, not the result of it, as to which no sensible man could have been uncertain. Those great men preferred to yield rather than to conquer by the slaughter of a brilliant nation, and of a people worthy of a better fate, even in what seemed to them its errors. Oh that here the same spirit of moderation which they exhibited may be displayed by the majority of the bishops who are listening to these words, and that by a prevision of the calamities which may arise to us from this hapless controversy, they may, in circumstances calling for consummate moderation, ward off from us, who are fewer, but who represent a greater number of Catholics than those who are opposed to us, evils which it is not possible to anticipate without horror, and which it would be impossible to repair by a late repentance.
- 31. On the one hand, we cannot but regret that these words, fitly written, were not actually spoken in the deaf ears of the resolved majority. On the other hand, we remember that had they been spoken, they would have sunk into the Vatican archives, and would never have been heard of more till those graves give up their dead. They now belong to history, and furnish a living link in a chain of memorable professions and performances. The denationalizing influence of the Papacy had still left something of the citizen alive in the soul of Kenrick. During his stay in Rome, when witnessing the paltry tyrannies that flounced about under the dependent banner of the Pope, all of the citizen that was left in him must have turned with fresh respect to the two flags of the free under which he had spent his days—the flags of England and America. And yet there were those sitting there, each with all the rights of a free man in his hands, planning to reconstruct the society of England and America on the degraded and fettered model of the States of the Roman Bishop. There is a crime which no code has defined—the crime, not of breaking one specific law of one's country, but of contriving, with a foreign pretender, how to overturn everything vital in a venerable and generous legislation. It was not merely by a pupil of Maynooth that the eager ex-Anglican was considered extreme in his views. Clifford, Bishop of Clifton, spoke on the same day, refuting the notions of Manning about the favourable effects to be produced by his beloved dogma in England, and appealing to him as a witness that an eminent statesman had represented the influence of the recent course of the Curia upon public opinion in England as being much to the disadvantage of their own cause, and greatly to the encouragement of extreme Protestants.[416] In the next Congregation, on the 28th, it was Senestrey who took the post occupied on the last morning by Manning, that of official respondent against attacks. On that day, a scene was raised by Verot, of Florida. He declared that they were making innovations in the Church, and that such an innovation as the personal infallibility
- 32. of the Pope was sacrilege. That horrid word applied in the sacred place to an object so dear to the Pope, touched indeed the apple of the eye. Sacrilege! The Cardinals de Angelis and Capalti, says Vitelleschi, quite lost their temper; and a scene ensued which for anger and excitement is said to have fallen but little short of Strossmayer's scene in March.[417] The odious, and to well-tuned Curialistic ears the inconceivable, task of hearing the infallibility of the Pope denied, and of seeing his pleasure daily thwarted under the roof of St. Peter's, was not to be endured any longer. The word passed that the power given by the new Rules to close the debate must be called into requisition. A trusty American was set up in the next meeting, by the committee, to repair the mischief done by Verot—Spalding, of Baltimore. Here, again, we are indebted for light to Kenrick's unspoken speech. Referring to the moral question which had been raised by Kenrick, to which we have already seen allusions, Spalding said that it called for as much investigation to justify one in giving a negative as in giving an affirmative vote on the question of Papal infallibility, and that in withholding an affirmative vote one would confirm the celebrated Gallican articles. On May 31, Valerga, the Patriarch of Jerusalem, made a vigorous attack on the minority, speaking cleverly, and hitting hard. Spirited, piquant, and insolent, is the description of Quirinus. Soon afterwards, another American was in the desk, Purcell, of Cincinnati. Quirinus says that he affirmed that the Americans abhorred every doctrine opposed to civil and spiritual freedom; and that the American sons of the Church loved her, because she was the freest society in the world. He also took the position that, as kings existed for the good of the people, so the Pope existed for the good of the Church. On the same day spoke Conolly, Archbishop of Halifax. He seems to be the only one in the Council who really related a theological experience, declaring that he had formerly believed in the personal infallibility of the Pope, and had come to Rome believing that the Augsburg Gazette had circulated a calumny in
- 33. representing the dogmatizing of this opinion as the real object of the Council. He went on to say that, on finding what was expected of him, he determined to sift the arguments of the Roman theologians and the proofs by which they supported them. He now bore witness to the result upon his own views. All antiquity, he declared, explained the passages harped upon by those theologians, in a sense different from theirs. All antiquity bore witness against the notion that the Pope alone, and separate from the bishops, was infallible. He further took the ground that to found a dogma on the rejection of the traditional interpretation of Scripture was pure Protestantism. I will have nothing, he said, turned into dogma but the indubitable Word of God. Ten thousand theologians do not suffice for me, and on the present subject no theologian should be quoted who lived subsequent to the Isidorean forgeries. To define the dogma would be to bring the Vatican Council into contradiction with the three General Councils which had condemned Pope Honorius as a heretic, to narrow the gates of heaven, to repel the East, and to proclaim, not peace, but war. In reply to Manning, he protested that no one was justified in calling an opinion proximate heresy when it had not been condemned as such by the Church.[418] On June 3, Gilooly, Bishop of Elphin, replying to some observation of Purcell as to the oaths and declarations, said[419] that Catholics had not denied that they held the infallibility of the Pope as a doctrine of the faith, but as a dogma of the faith; that is as a dogma defined by a General Council. To this, Kenrick's unspoken speech replies, If that is what was meant, which I do not believe, we might be reproached, and that rightfully and deservedly, with not shrinking, in a very grave matter, from the concealment of our meaning by scholastic distinctions.[420] According to Quirinus (p. 661), Cardinal Bonnechose prevailed upon Cardinal de Angelis to ask the Pope, directly, if he would not consent to a prorogation of the Council on account of the heat, now intolerable to all but Romans, or men from the southward of Rome. The reply was stern and, according to many, savage. Whatever were the terms of it, the substance was
- 34. indubitable—no adjournment was to be allowed till the Decree of Infallibility was passed. It is said that when Bishop Domenec, of Pittsburg, in America, began his discourse, he was greeted with laughter by the majority, and when he made the very plain and simple statement—one which he might have picked up from any intelligent or travelled Italian any day in the year—that American Catholics were not merely nominal ones, as the Italians were, Cardinal Capalti imperiously commanded silence.[421] Strossmayer had spoken at length on June 2, and with such moderation as to escape even a call to order, yet, it is said, with very great force. On the 3rd, Moriarty, of Kerry, took the side of Purcell, Kenrick, and MacHale, but we have no particulars of his speech.[422] That day Maret was in the desk speaking in the loud and labouring tone of a deaf man, arguing, not only against the convictions and feelings of the majority, but against their personal detestation of himself. He made a point that either the Council was to give infallibility to the Pontiff, in which case the Council must be a higher authority than he, or else the Pontiff was to give to himself an infallibility which he had not previously possessed, in which case he would change the constitution of the Church by his own power alone. Then Cardinal Bilio interrupted, and cried, The Council does not give anything, nor can it give anything. It gives its suffrage, and the Holy Father decides what he pleases.[423] The representative of all that was left of the once courageous Gallican liberties asked if he might be allowed to proceed, and did so. The minority had a long list of speakers still inscribed. Kenrick was waiting for his turn, and so were Haynald, Dupanloup, and many others; but a fresh surprise was at this point sprung upon them. The Presidents produced a requisition for the close of the general debate, signed by above one hundred and fifty bishops.[424] De Angelis at once called on those who were for the closing of the debate to stand up. He then declared, A large majority have stood up, and by the power conferred upon us by Our Most Holy Lord (the capitals are official), we close the debate on the general question. The Acta Sanctæ Sedis say that about fifty remained sitting. No wonder that, after hearing sixty-five speakers,
- 35. the Fathers were weary. Yet, no wonder, on the other hand, that the minority should allege that, while it was perfectly reasonable to close a debate in this manner when the object was that of making temporal laws liable to be unmade, or re-made, a year later, it was neither reasonable nor fair, and above all, it was not agreeable to any precedent, to past professions, or to any ecclesiastical principle, to close a debate upon a dogma while yet there were prelates wanting to bear witness to the tradition of their respective Churches. According to all their theologians, dogma was not to be made by mere opinion, but by evidence of the fact that the opinion in question had been believed from the beginning. Protestants would naturally say that it was time to bury this pretence under any heap; but men whose life had been spent under the illusion of the pretence naturally felt otherwise. They had not seen that when the Church adopted the principle of tradition instead of that of Scripture, the Spouse, while professing only to supplement the word of her Lord, really entered on a course which must lead to setting it aside in favour of her own word, and that when she had adopted the principle of general consent, instead of that of clear apostolical tradition, she had set aside the principle of antiquity for that of a majority amounting to a moral whole, and that now she was only proceeding a step further in substituting the principle of a numerical majority for that of moral unanimity. But one step more remained, and that was not far off. The Spouse who had put aside the authority of her Lord to exalt her own, was to find, not only her authority, but even her consent, formally repudiated before all men by the master whom she had, in the house of her Lord, set up in His place. In that house the talk was evermore of her authority, her wisdom, her infallibility, her glory, her stores of merit and her streams of blessing, and but rarely was her Lord heard of, except as having conferred the regency on her. Now drew nigh the day when the self-asserting Spouse was, before all men whom her loud vauntings had aroused, to receive on her brow such a stigma from her self-chosen Master as has seldom in set terms been affixed to a society by its head. Meantime the blow which had just been dealt seemed fatal to all the hopes of the minority. So once more they
- 36. dragged their robes down the marble way of St. Peter's with defeat behind them, but this time with annihilation close before, though not till after further strange experiences.
- 37. FOOTNOTES: [402] Vitelleschi, p. 158. [403] Quirinus, p. 532. [404] Quirinus, p. 533. [405] Frond, vol. ii. [406] Documenta ad Illustrandum, ii. 209. [407] Meum honorem graviter læserunt.—Documenta ad Illustrandum, i. 189. [408] Documenta ad Illustrandum, pp. 187-224. [409] Aliquid humani passum esse. [410] He showed that Tischendorf read προβἁτια in both cases, and that other editors had read πρὁβατα in both. Of course, in the fifteenth verse, the word lambs—ἁρνἱα—is the proper translation. [411] Acta Sanctæ Sedis. As to MacHale, Kenrick omits what Frond states, that he was of a very ancient family. [412] Documenta, ii. pp. 415-24. [413] Documenta, i. 209. [414] Documenta, i. 223. [415] Quod si non fecissent nunquam ad libertatis civilis consortium admissi fuissent aut debuissent (p. 219). [416] Quirinus, 584. [417] Vitelleschi, p. 168. [418] Quirinus, p. 597. [419] Documenta, i. 215. [420] Ibid., 215. [421] Quirinus, p. 661. [422] His name does not occur in the Acta Sanctæ Sedis for the third. [423] Quirinus, p. 608.
- 38. [424] Acta Sanctæ Sedis. Friedberg, p. 47, says there were two hundred and fifty signatures, but this is evidently a mistake. CHAPTER VI To the Close of the Special Debate on Infallibility, July 4— Proposal of the Minority to resist—They yield once more— Another Protest—Efforts to procure Unanimity—Hope of the Minority in Delay—Pope disregards the Heat—Disgrace of Theiner—Decree giving to Pope ordinary Jurisdiction everywhere—His Superiority to Law—Debate on Infallibility —Speech of Guidi—Great Emotion—Scene with the Pope— Close of the Debate—Present view of the Civiltá as to Politics—Specimens of the Official Histories—Exultation. ANY one who had observed the course of the minority in emergencies would have probably foretold that, under the new trial, they would feel indignant, would speak of doing something, and would end with a protest. So it proved. The very day of the forcible conclusion of the general debate, the French bishops met, and were favourable to some determined action.[425] But the next day, eighty congregated in the rooms of Cardinal Rauscher. The Hungarians, French, and Americans, with Strossmayer, Clifford, and Conolly, are named by Quirinus as recommending that the Fathers of the Opposition should cease to take any part in the Council, reserving themselves for the final vote, and should then give their Non placet. The Germans, however, always marplots, urged that the better course would be to adopt a protest, and continue to take part in the proceedings. This counsel prevailed. Rauscher drew up a form of protest, which was signed by some eighty prelates, and many of the bishops took a trip to Naples or elsewhere.
- 39. Among the things represented by Quirinus as having been said on this occasion, one was to the effect that in a Parliament speeches were of some use, for if they did not influence votes, they did enlighten public opinion; but in this Council, most of the hearers were, from their degree of culture, quite incapable of apprehending theological arguments, not to add that, in a moral point of view, many of them stood so low that even if convinced they would not act on their convictions. The ground taken in the protest is clear, namely, that the right of supporting their votes by a statement of reasons, is one which, by the very nature of a Council, belongs not only to some of its members, but to them all, and that such a right could not be taken away by any vote of a majority.[426] The Hungarians now declared that they would take no further part in the debates. On the other hand, the Unitá Cattolica foretold how those who had written or spoken as Gallicans would be converted by a miracle of the Holy Ghost, even in the Council Hall; and as the Galileans had been constrained to speak in other tongues, so would the Gallicans be constrained to proclaim in that Hall before the astonished multitudes the doctrine they had gainsaid. The absorbing care of the Curia and its instruments was now directed to the one end of constraining all to vote placet. The victory was no longer doubtful, but to procure unanimity was of great practical moment. The Pope himself was indefatigable. His admirers resented such epithets as unscrupulous when applied to his conduct. But they took good care not to grapple with the details of alleged facts which, if they could be credibly told about the conduct of one of our sovereigns in respect to his nobles or to Parliament, would be described in much stronger epithets than unscrupulous. His tongue was evermore scattering rebukes or blandishments, and enlivening the city with crackling sparks of gossip. There were but few bishops of note among the minority whose portraits, etched by the infallible acid, were not handed round the salons, lay and clerical. His letters were bitter and undignified. Quirinus quotes the words of a French bishop (p. 627): There is no longer any scruple
- 40. as to what is done to gain votes. It is a horror. There has never been anything like it in the Church. These words recall to us a scene in Rome. A remarkable head—one of those heads which bear on the brow a diploma of gifts and letters—was stooping in the light of a lamp by which pages had been penned that had been heard of beyond Italy. The stoop was pensive, and the thinker said, I saw so much of what was done during that Council, that it has destroyed all my faith in anything that ever was done in the Church before. It would seem as if, at the last, argument and appeal had begun to tell on some of those who were of a milder mood among the Curialists. It is said that even of the chosen three champions, Manning, Deschamps, and Pie, the last wished to find some formula less offensive than the one projected. Martin of Paderborn even proposed a note which contained a recognition of the teaching authority of bishops, though in an indirect way. On the other hand, the members of the Opposition tried to discover some turn of expression which would save the Church from the shame of being publicly disavowed by her wilful lord. Conolly spoke of proposing, as a formula which would still give her a recognised voice, words declaring the Pope infallible when he spoke, as head of the Church teaching with him. Others again wished to reinstate the formula of St. Antoninus, of Florence, declaring the Pope infallible when he acts with the counsel of the universal Church.[427] Men now began to realize the full effect of the proposed dogma, both in its executive and in its retrospective aspects. Many must have remembered how happy they had been in argument, or in diplomacy, when the ambiguous state of the case, as it had hitherto existed, enabled them to evade the charge that such and such were the principles of the Church. It was so convenient to be able to say No, they have never been sanctioned by a Council; they are only the words of a Papal Decree. Now, however, all these words were to have fresh life breathed into them, and whatever they contained affecting a general principle of belief, or practice, was to be taken for divine,—was, in fact, to rank as the word of God.
- 41. Delay now became the forlorn hope of the minority, and expedition the watchword of the majority. The minority were sure that the Pope would not be so cruel as to force them to continue in Rome during the summer heats. Hence, they thought that by delay they were certain of a prorogation before the fatal deed was done. They forgot the history of the Pope's prisons and executions. Perhaps they had never read it, or had used their fatal facility of calling an unpleasant statement a lie. Antonelli had generally carried away the chief part of the blame for the blood of the political victims. However, he seems completely to have escaped reproach for the broiling of the bishops. Whether the fierce language ascribed to the Pope was correct or not, nobody doubted its aptness.[428] When even the faithful M. Veuillot said, Since they have put the Council upon the gridiron, they shall broil (ii. p. 352), everyone treated him as only echoing the language of his idol. When once the heats had begun to tell, the feelings of majority and minority, as Vitelleschi points out, changed. Men from the north, accustomed to the bracing air and pure streams of Germany, could ill bear up against the miasma from the Roman marshes and the torrid heats that were withering the city and making even natives look pale. They therefore began to long for an escape, and not a few of them took their way homewards. They received not only ready but glad permission. Thus every day was diminishing the strength of the Opposition. The majority, on the other hand, consisting of Italians, South Americans, and Spaniards, were inured to the heats, if not to the malaria, and felt that the sun and the marshes were conspiring with them. Apollo had come to camp shooting over the heads of the natives, but laying low the men from beyond the sea. There was now only one consideration that would make the Pope anxious for despatch, and that was the daily pressure upon his finances caused by supporting his three hundred boarders. This certainly had proved a useful ground of appeal for funds. The sums collected everywhere had been great. The Civiltá reproaches the Liberal Catholics with not sending money any more than they had sent men to fight for the Holy Father, and sets in contrast with their
- 42. stinginess and want of military spirit the fact that the Univers alone had sent in more than nine thousand pounds (234,410 francs).[429] The Holy Father said, They fear making the Pope infallible, but they do not fear making him fail.[430] But M. Veuillot, on the contrary, did not fear making him infallible, and did everything possible to prevent him from failing. Hence it was no wonder that he should have briefs to publish which would perform a service for the exchequer of the Univers similar to what the Univers performed for the exchequer of the author of the briefs. The words of the Pope spoken to the deputation of scientific men were representative words, Here I am to receive your offerings. Theiner, the celebrated Prefect of the Vatican archives, now fell publicly under displeasure. He had allowed Hefele and Strossmayer, and perhaps others, to see the order of procedure of the Council of Trent, and probably had in other ways shown leanings not acceptable to the Jesuits. He was ordered to give up his keys to Cardoni, who had been the first chosen secretly to prepare Drafts of Decrees on Infallibility before intentions were disclosed, and had kept his counsel well. The archives were actually closed against Theiner. It is said that the passage into them from his own rooms was walled up. The disgrace of Theiner, and the honour of Cardoni, sharply symbolized the favourite saying that the dogma must conquer history. Here again Antonelli escaped all reproach of a share in the blundering injustice. Cardoni was one singled out by name in a celebrated letter of Döllinger as having largely employed falsified authorities. But that charge, to us so revolting, is a familiar sound wherever the shadow of the Curia extends.[431] We ourselves once heard a member of the Congregation of the Index claim, unmindful of the presence of a Protestant, You must never trust any edition of any work whatever that has passed through the hands of the Jesuits. The exciting matters now remaining to be treated in the Council were the all-important particulars of those Drafts which had already been under a general review. The two chapters teaching the
- 43. institution of the primacy in the person of Peter, and the transmission of that primacy through the Roman Pontiffs as his successors, were speedily disposed of. Had all the fathers attempted to answer the arguments of Desanctis on these points, arguments familiar to many Italians, they would not have found it light work. But the third chapter was one of immense importance. It defined the scope and nature of primacy, distending that term till it was made to cover absolute, immediate, and ordinary control in the whole domain of the Church—control over bishops and people, control over not only all matters ordinarily included under the expression faith and morals, but over all things held to be necessary for the government or discipline of the Church. This last expression, as any one acquainted with the views of those in authority, even so far as they are recorded in our preceding pages, must know, covers almost every possible question that can arise. The words of Vitelleschi (p. 174) are well considered. He speaks of the supreme jurisdiction, ordinary and universal, of the Pope over all Churches, singly and collectively, over pastors as well as flocks; from which doctrine it follows that bishops in exercising any jurisdiction or authority, only do so as official delegates of the Pope. Dr. Langen puts it thus: Seeing that there can be only one bishop in a diocese, as soon as the Pope is declared to have ordinary jurisdiction in that diocese, he becomes its Ordinary, and the other person called a bishop is nothing more than his delegate and representative.[432] Men who cover a dominion of this sort under the pretext of primacy, and who advance a claim of primacy in order to deduce from it an absolute dictatorship, never do anything more sensible than when they decry reason and relegate Scripture to the tradition-heap; when they call for pictures instead of books, and processions and fireworks instead of a free press and free discussion. There was political philosophy in M. Veuillot's exclamation on witnessing the Easter rejoicings in Rome, especially the fireworks representing the heavenly Jerusalem, that it was impossible not to respect a people for whom such entertainments were provided.
- 44. The first assertion in the Decree of ordinary and immediate jurisdiction over all Churches, oddly does not describe that jurisdiction as belonging to the Pope, but as belonging to the Roman Church (par. 2). No sooner, however, has principality been ascribed to the Roman Church than it is instantly transferred to the Pontiff, and is again instantly affirmed to be a truly episcopal power. This confusion, in such a document, would be amusing if the matter were not so serious. That a Church should be a bishop is certainly new; and that a truly episcopal power should reside in a Church which is not a bishop, is one of the many mysteries created by the Vatican Council. But that the source of the Pontiff's authority should in this very Decree be sought in the Church, is a proof how hard a task is theirs who determine to make dogma conquer history. In the very language of the Decree, history conquers the dogma. If the document contains this one taint of dualism as between Church and Pope, it is clear of all reproach of dualism as between the Pope and Princes. The latter are legislated out of all rights that could possibly conflict with those of their Lord Paramount. Notwithstanding the slight dualism as between Pope and Church, the latter is also legislated out of all her ancient claims; but incidentally she appears in clauses which, if she was only infallible without the consent of the Pope, as he is infallible without her consent, might in time prove very awkward. He has only as much infallibility as she has: that is a clumsy admission just before the assertion that he is infallible without her consent. However, wherever the power resides, or springs from, it is a power over all pastors and all believers, and extends, as we have said, not only to faith and morals, but to all things which affect the government of the Church. Thus it includes every mixed question whatsoever, and all things of any kind which in the estimation of the Pope of Rome may relate to the interests of that kingdom of which he is the king. This power, moreover, is immediate, and as such can act without being legally restricted to any processes, any agencies, or any forms. Being ordinary, it can never be obliged to wait until the ordinary jurisdiction has been tried and failed. Being immediate, it can never be told that it must take
- 45. this, that, or the other line of procedure. This language for ever settles the point which had been contested in the famous passage of letters with Darboy. How it could be necessary to add another word after these affirmations we can hardly see. Even Councils, or the pastors collectively, had but one office assigned to them—the office of obeying. After this the abstract proclamation of Infallibility, or Irreformability, or Inerrancy, could add nothing to a power that was universal, ordinary, and immediate, and towards which the people or bishops, singly or collectively, stood in one relation only—that of subjects in presence of an authority which they were bound absolutely to obey. It naturally follows that it is in this obedience that Rome finds unity. That is, in fact, her ideal of unity. Christians are Churchmen, not by being Christians, but by obeying the Roman Pontiff. Under the Papacy a Christian is outside the family of God if he does not obey the Cæsar of the Church. Absolute authority over bishops and people having been asserted, next comes the assertion of authority over princes. This is done in a paragraph in which only students would see anything of the kind. The fourth paragraph of the third chapter begins by speaking of the Pope's right to free communication with the pastors and flocks of the whole Church. What could appear more natural, or less dangerous? Had we not seen how much the communications of the Pope amount to, we should have taken that as a meek and harmless claim. But the close of the paragraph shows that what the Pope means is the right of giving to his own edicts the binding force of a higher law in every country, whether the government consents or does not consent. As primacy means dictatorship, so communication means promulging laws in regard to which no human being has the right of reply, inquiry, complaint, or appeal; has, we repeat, no office whatever except that of obedience. We have seen that teach in our Lord's commission to the apostles means so to give law to the nations that they can never be justified in resisting. No prince can have any title to exercise an exequatur, placet, or any other form of check upon an edict of the Pope. Every man who denies the validity
- 46. of a Papal law, because it is prohibited by the government of the country, is solemnly condemned; he interrupts the communication between the authority of the Pontiff and the conscience of his subjects. Indeed, the condemnation extends to all who even say that his decrees may be lawfully impeded in their execution. The reason of this appears in the next paragraph. The Pope is there formally declared the Supreme Judge of the faithful. Therefore all may justly resort to his judgment in all matters subject to ecclesiastical inquiry, and none may appeal from his judgment, for there is no authority greater than his. Matters subject to ecclesiastical inquiry must always include all those wherein the interests of the Papacy are in anywise involved. Next, even the old appeal to a General Council is formally condemned. Yet even that condemnation is bungled. None may appeal from the judgment of the Pope to a General Council as an authority superior to the Roman Pontiff. Then, will lawyers say, we can only appeal to a General Council as an authority equal to the Roman Pontiff. If these fourth and fifth paragraphs of the third chapter of the Decree on Primacy were read by a dozen educated Englishmen unused to Roman Catholic interpretations of Papal laws, nearly all of them would put aside clause after clause as not being of importance. They would take the damnamus and reprobamus as so much sulphur, and let it pass. Far otherwise Vitelleschi. From a practical point of view, he says, the declarations of infallibility could add nothing to the weight of this paragraph (p. 177). Vitelleschi looks upon the express declaration of infallibility, in the next chapter, as no more than indulgence in the luxury of self-assertion, to which absolute principles are prone. Yet when Mr. Gladstone pointed out the true range of the authority here set up, many of our politicians treated him as a statesman who had strayed out of his domain into theology. Since then, specimens of minimizing interpretation have been put into our own tongue, as curious as any furnished by the history of finesse. If there be one Canon expressing a rule absolute that needs no exception to prove it, we have it in the words, Rome
- 47. never minimises. She always interprets her own documents as a legatee interprets a will, that is, in her own favour. On June 15 the Council disposed of all the matters that stood in the way of the great question. Seventy-five speakers had entered their names. Two speeches were actually made on that day by Cardinals Mathieu and Rauscher.[433] The latter said that he could never assent to the doctrine of the Draft without mortal sin. We knew all that from your pamphlet, cried Deschamps, interrupting. But you have never refuted it, replied the Austrian.[434] The following day was the grand procession of the Corpus Christi. If the good press was parsimonious in information regarding debates and decrees, it was profuse in description of the spectacles. On the 17th, Pius IX entered on the twenty-fifth year of his pontificate. This year, according to Roman tradition, is fatal to the Pontiffs, it being held that Peter reigned twenty-five years, and that none of his successors was to reign longer. Vitelleschi declares that the twenty-fifth year proved fatal to Pius IX, as well as to the rest, because in the course of it he ceased to be a mere mortal. This phrase from a Liberal Catholic will seem natural when set beside one of M. Veuillot, on the day on which Pius IX completed the twenty-fifth year of his pontificate: We are reminded of the radiance of Jordan and of Tabor, of the thunders of the Temple, 'This is my beloved Son, hear ye him' (vol. ii. p. 468). On the next page he says, God has left us His priest, His angel, the sacred interpreter of His law, the anointed intercessor between Him and the world ... a second Peter, a second Moses on the threshold of a new world. It remains to be seen whether the twenty-fifth year of Pius IX was or was not that of the final fall of the temporal power. If the speeches on the doctrine and polity of the Church were concealed, the Pope's speech this day, in reply to the Sacred College, was blazed abroad. He divided the bishops into three classes—the ignorant, the time-serving, and the good. So flowed abroad fresh streams from that fountain which, all the time, was sending forth both sweet waters and bitter.
- 48. On June 18, the debate on the fourth chapter, that is, on infallibility, really began. It was a day of Cardinals. Pitra, Guidi, Bonnechose, and Cullen were the sole orators. Hitherto, what with the heat and what with the feeling that all was over, no interest had attached to the renewed debates after the violent close of the general discussion. But the torpor was suddenly shaken. A speech by a Roman, a Dominican and a Cardinal (Guidi), came upon the city, says Vitelleschi, like a sudden thunderclap in a cloudless sky. The Cardinal, like nearly all the members of the Sacred College, was a creature of Pius IX. According to Vitelleschi, he began his speech as a Cardinal should, but, according to Quirinus, he offended at the very first. Unhappily, in a matter of difference of this kind, the writers who enjoyed the radiance of infallibility give us no light. So we are left at the mercy of those whose assertions were all lies in general, but somehow, when attacked in detail, generally proved to be truths in particular. In the present case, we do not remember that even M. Veuillot attempts to impugn any of the facts stated. However Guidi may have begun, he affirmed that the doctrine of Papal infallibility, as contained in the proposed Decree, was unknown to the Church up to the close of the fourteenth century. Proofs of this doctrine were to be sought in vain in either Scripture or tradition. As a practical question, when had the Pope ever defined one dogma alone, and without the Church? An act, he continued, might be infallible, but a person never. Hitherto infallible acts had proceeded from the Church, either by counsel of the Church dispersed, or by a Council. Inquiry was indispensable to ascertain what was believed everywhere, and whether all Churches were in agreement with the Roman Church. After such inquiry, the Pope sanctioned finally, as St. Thomas says; and thus only could it be said that all taught through the Pope. Quoting Bellarmine, and even the modern Jesuit Perrone, he showed that the Popes had never acted by themselves alone in defining doctrine, or by themselves alone in condemning heresies. At these words, Spaccapietra, an Italian, but Bishop of Smyrna, led in a disturbance. One bishop cried Scoundrel! another cried Brigand! Vitelleschi even speaks of violent gestures (p. 189). Guidi said he had the right
- 49. to be heard, and that no one had given the right of the Presidents to the bishops; but he added, You will have the opportunity of saying Placet or Non placet. Hereupon, from all ranks of the Opposition burst out a cry of Optime! optime!—excellent! excellent! Do you agree with us? asked a bishop of Manning. The Cardinal's head is bewildered, was the reply. On this, says Quirinus, a bishop could not refrain from saying to the powerful Archbishop of Westminster, It is your own head, Monsignor, that is bewildered, and more than half Protestant. If this language was really used, we must doubt whether it was infallible. Guidi went on to advocate a change in the wording of the Decree, to the effect that the Pope acted with the concurrence of the bishops, and that after having, at their request, occasioned by prevalent errors, made inquiry in other Churches, he acted with the consent of his brethren, or with that of a collective Council. He contended that this was the doctrine of St. Thomas; that the word final implied something to precede, and that supreme teacher and judge presupposed other teachers and tribunals. He concluded by proposing two Canons, the first of which declared Papal Decrees or Constitutions to be entitled to cordial faith and reverence, and not to be reformable; but the second said, If any one shall say that, in issuing such Decrees, the Pope can act arbitrarily without the counsel of the bishops as testifying to the tradition of the Church, let him be anathema.[435] On finishing his discourse, he at once handed his manuscript to the secretaries. Quirinus relates that Valerga audibly said, in reply to some question, Guidi is misguided. But his neighbour replied that Guidi's speech contained nothing but the truth. Yes, rejoined the Patriarch of Jerusalem, but it is not always expedient to speak the truth. The excitement was great. Groups of prelates who had left the Hall might be seen standing about everywhere in earnest conversation, while within doors Bonnechose and Cullen were discoursing to a thin audience with absent minds. It was related that Guidi did not speak as a solitary individual, but represented fifteen bishops belonging to
- 50. the Order of Dominicans. He had gathered them together in the central convent of the Minerva, where he himself resided. They had considered the question, and accepted the views which he had now presented to the Council. This was much against the feeling of Father Jandel, their general, who was perfectly free from any taint of the episcopal system, a thoroughly right-minded Papist. Guidi asked how the Cardinals had taken his speech, and Cardinal Mathieu replied, With serious and silent approval. Rumours were soon afloat in Rome as to what followed between Guidi and his royal master. What we now give is traced by Quirinus to the authority of the Pope himself, who is notoriously fond of telling the people with whom he chats how he has lectured this or that dignitary.[436] The creature was summoned to the presence of his master soon after the sitting, and was greeted with the words, You are my enemy. You are the coryphæus of my opponents. Ungrateful towards my person, you have propounded heretical doctrine. My speech is in the hands of your Presidents, if your Holiness will read it and detect what is supposed to be heretical in it. I gave it at once to the Under-Secretary, that people might not be able to say that anything had been interpolated into it.[437] You have given great offence to the majority of the Council. All five Presidents are against you, and are displeased. Some material error may have escaped me, but certainly not a formal one. I have simply stated the doctrine of tradition, and of St. Thomas. I am tradition. I will require you to make the profession of faith anew. La tradizione son' io, vi faro far nuovamente la professione di fide. I am and remain subject to the authority of the Holy See, but I venture to discuss a question not yet made an article of faith. If your Holiness decides to be such in a Constitution, I certainly shall not dare to oppose it. The value of your speech may be measured by those whom it has pleased. Who has been eager to testify to you his joy? That Bishop Strossmayer, who is my personal enemy, has embraced you. You are in collusion with him. I do not know him, and have never before spoken to
- 51. him. It is clear you have spoken so as to please the world, the Liberals, the Revolution, and the government of Florence. Holy Father, have the goodness to have my speech given to you. It was said that the Pope stated afterwards that he had not sent for Guidi as a Cardinal, but as Brother Guidi, whom he had himself lifted out of the dust. The saying, I am tradition, made an impression in Rome much like the celebrated one of the French monarch, I am the State. It simply packed up and labelled the thought that had been more or less confusedly before the minds of all. Quirinus speaks of having often had the words I am the Church in his thoughts—l'Eglise c'est moi. We do not see that the Pope could have said anything more sensible or more exactly representing the theology and history which the favourite champions had put before the world. Quirinus very properly thinks that this formula fits well with the pregnant saying of Boniface VIII, The Pope holds all rights locked up in his breast. Truths and rights go together. Tradition consists of truths, and the Pope is all truth. Rights are based upon the truths, and the Pope holds them all in his own breast. And if the poor old man himself at last uttered these sad words, it was only after the incense had smoked around him thousands and thousands of times, hiding the realities of heaven from him by clouds that were only fumes. For this others were responsible, at least in part. Under the influence of it, what wonder if his senses had become confused? Mankind will have reason to be thankful that one Pope lived long enough to be thoroughly overcome by the smoke of the sacrifices. The ordinary reason assigned in Rome for Popes being short-lived is, that it is necessary to prevent the effects of their power upon themselves. The gravamen of Guidi's offence could not be removed by any subsequent submission. Seeing that the Canon he proposed had emerged into the light, the record could not be got out of the book of history that a Dominican, a divine of repute, a Cardinal in high credit, did up to that last hour of liberty hold that it was a heresy worthy of anathema to affirm the very doctrine which was soon to be part of the faith. The record could not be prevented from going
- 52. down to future ages that what was, on June 18, and under the dome of St. Peter's, liable to be called a heresy, was on July 18 under the same dome, promulged by the voice of the Pope as truth, and as binding on every human being who would be saved. Nor can craft ever blot out from the history of the eccentricities of intellect the instance offered by the fact that after this had been done, grave and learned men, even of advanced age and high office, went throughout the civilized world soberly affirming that the only reason why the dogma was then proclaimed, was that it had been clearly revealed by our Lord and His apostles, and had in every age been held as revealed truth by all Catholics, in all places. Vitelleschi is not quite clear as to whether all the incidents reported of the interview between the Pope and the Cardinal were correct. To him that is of no importance; Roman-like, he did not want anything to illustrate the relation of the Pope to his courtiers or to the Church. A few such scenes, more or less, would to him make no difference whatever. As if to prepare for the deeds directly tending to the restoration of facts when the Council should have completed the restoration of ideas, the tales of the Crusaders of St. Peter continued to appear side by side with the notices of the legislative proceedings in the successive numbers of the Civiltá. To us one episode comes near home. It was on an April day that a company leaving Rome bore across the Campagna, with all the solemnity of a relic of the saints, the heart of one whose body, in the Agro Verano, the cemetery of St. Lorenzo, slept close by the tombs of the ancient martyrs, and amid those of the martyrs of Mentana. As the party reached a point on the hill within a few steps of the village,—a point from which St. Peter's appeared in the distance,—they saw a block of white marble, surrounded by four little columns, hung round by an iron chain. Here, cried some zouaves who were of the party,—Here is the spot to which Julian pushed on, chasing the enemies of God with fire and sword, passing through a thousand bullets, of which one carried away his cap; and here he fell shot down at point blank. Above the marble block rose the cross of Mentana, and on it was cut the
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