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Machine foundations require a special consideration because they transmit dynamic
loads to soil in addition to static loads due to weight of foundation, machine and
accessories.
The dynamic load due to operation of the machine is generally small compared to the
static weight of machine and the supporting foundation.
In a machine foundation the dynamic load is applied repetitively over a very long
period of time but its magnitude is small and therefore the soil behaviour is essentially
elastic, or else deformation will increase with each cycle of loading and may become
unacceptable.
The amplitude of vibration of a machine at its operating frequency is the most
important
parameter to be determined in designing a machine foundation, in addition to the
natural frequency of a machine foundation soil system.

There are many types of machines that generate different periodic forces. The most
important categories are:
1. Reciprocating machines: The machines that produce periodic unbalanced forces
(such as steam engines) belong to this category. The operating speeds of such machines
are usually less than 600r/min. Rotary motion of crank is converted into the translatory
motion. For analysis of their foundations, the unbalanced forces can be considered to
vary sinusoidally.
2. Impact machines: These machines produce impact loads, for instance, forging
hammers. Their speeds of operation usually vary from 60 to 150 blows per minute.
Their dynamic loads attain a peak in a very short interval and then practically die out.
3. Rotary machines: High-speed machines like turbogenerators or rotary compressors
may have speeds of more than 3,000r/min and up to 12,000r/min.

One way to classify machines is based on their speed of operation:
Low to medium frequencies 0-500 rpm (large reciprocating engines,
compressors and blowers)
Medium to high frequencies 300-1000 rpm (diesel and gas engines)
Very high frequencies usually greater than 1000 rpm (turbo-generators
and high speed internal combustion engines)

For compressors and reciprocating machines, a block foundation is generally provided.
Block foundation consists of a massive block of concrete resting directly on soil or
supported on piles or a pedestal resting on a footing.
If two or more machines of similar type are to be installed in a shop, these can
profitably be mounted on one continuous mat.
A block foundation has a large mass and, therefore, a smaller natural frequency.
The block has large bending and torsional stiffness and easy to construct. To modify the
block foundation at a later time is extremely difficult.

However, if a relatively lighter foundation is desired, a box or a caisson type
foundation may be provided.
The mass of the foundation is reduced and its natural frequency increases.
Box or Caisson foundation consists of a hollow concrete block (can be used as
operational space) that supports the machine on its top. Hammers may also be mounted
on block foundations, but their details would be quite different than those for
reciprocating machines.
It has high static stiffness just like a plate foundation and is not easily amenable to
alterations at a later date.

Steam turbines have complex foundations that may consist of a system of walls
columns, beams and slabs.
This type is usually adopted for very high-speed machines requiring large operational
space below for connecting pipes and additional equipment.
It can be made or either RCC or steel frames. Although the frame made of steel is easy
to alter at a later date, its behaviour under dynamic loading is not as good as that of an
RCC frame.
Each element of such a foundation is relatively flexible as compared to a rigid block
and box or a caisson-type foundation.

Plate foundation
Plate foundation consists of a continuous plate made of concrete resting directly on soil
or supported on piles. The machine is placed on the plate.
The plate has high static stiffness and is easy to construct. Subsequent alterations to this
type of foundation are difficult to execute.
Spring foundation
Spring foundation consists of a concrete slab or a steel frame on which the machine
rests. The slab/frame is supported on a set of springs.
This type is used, if the subsoil is weak or we need to isolate the subsoil from the
transmitted vibrations.

Design Criteria for Machine Foundations
i. It should be safe from a bearing capacity failure under static and dynamic loads,
ii. The settlement must be less than the prescribed ones,
iii. The dynamic amplitudes of the machine-foundation-soil system must be within the
prescribed limits under service conditions.
iv. There should be no resonance, i.e. the natural frequency of the machine-
foundation-soil system should not coincide with the operating frequency of the
machine,
v. Preferably, the centre of gravity of the machine should lie in the same vertical line
as the centre of gravity of the foundation system.
When design criteria (iii) to (v) are satisfied then

the machine itself is not damaged by the vibrations generated,
the structure in which the machine is housed and adjacent structures do not suffer any
vibration induced damage,
performance of machines located in its vicinity is not impaired, and
employees working around the machine are not bothered by the vibrations.

One of the design criteria for machine foundations is that there should be no resonance.
To achieve this it is necessary to tune the machine foundation. Tuning is done by
adjusting the natural frequency of the machine-foundation-soil system so that it is far
away from the operating frequency of the machine.
Shifting the natural frequency of the machine-foundation-soil system well above the
operating frequency is called high tuning, or in the jargon of Mechanical Engineers
sub-critical run.
On the other hand shifting the natural frequency below the operating frequency is
called low tuning or supercritical run.
There are thus two choices at your disposal to avoid resonance.
High tuning has the advantage that the machine does not have to go through resonance
either during start up or shut down of the machine.
It is also fine if the machine has to run below rated capacity.

Whether choose high tuning or not, depends on the operating frequency of the machine.
Implementing high tuning for a very high-speed machine is difficult to achieve and can
be a costly proposition.
Under such situation have to opt for a low tuned system and yet are within the
prescribed level.
This is often accomplished by using a foundation system that allows alteration of the
natural frequency of the machine-foundation-soil system when necessary by changing
its mass or stiffness.
For example, by increasing the dimensions of the foundation which will change the
mass, providing cavities in the block where mass can be added, or using commercially
available mechanical springs which will change the stiffness.
Sometimes, avoiding resonance is not possible. In such a case, you will have to make
sure that the amplitude of vibration is in the acceptable limits by incorporating dampers
in the foundation system. Dampers are instruments that absorb vibrations.

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The amplitudes of motion at operating frequencies should not exceed the limiting
amplitudes, which are generally specified by machine manufacturers. If the computed
amplitude is within tolerable limits, but the computed natural frequency is close to the
operating frequency, it is important that this situation be avoided.
The natural frequency of the foundation –soil system should not be whole number
multiple of the operating frequency of the machine to avoid resonance with the higher
harmonics.
The vibrations must not be annoying to the persons working in the shops or damaging
to the other precision machines. The nature of vibrations that are perceptible, annoying,
or harmful depends upon the frequency of the vibrations and the amplitude of motion.

METHODS OF ANALYSIS
The analysis of machine foundation is usually performed by idealizing it as a simple
System .
Figure shows a schematic sketch of a rigid concrete block resting on the ground surface
and supporting a machine.
Let us assume that the operation of the machine produces a vertical unbalanced force
which passes through the combined centre of gravity of the machine-foundation
system.
Under this condition, the foundation will vibrate only in the vertical direction about its
mean position of static equilibrium.
The vibration of the foundation results in transmission of waves through the soil.
These waves carry energy with them. This loss of energy is termed geometrical
damping . The soil below the footing experiences cyclic deformations and absorbs
‟
some energy which is termed ,material damping .
‟
The material damping is generally small compared to the geometrical damping and may

However, material damping may also become important in some cases of machine
foundation vibrations.
The problem of a rigid block foundation resting on the ground surface, (Fig. a) may
therefore be represented in a reasonable manner by a spring-mass-dashpot system
shown in Fig. b.
The spring in this figure is the equivalent soil spring which represents the elastic
resistance of the soil below the base of the foundation.
The dashpot represents the energy loss or the damping effect. The mass in Fig. b is the
mass of the foundation block and the machine.
If damping is neglected, a spring-mass system shown in Fig. c may be used to represent
the problem defined in Fig. a.
Single degree of freedom models shown in Fig. b and c may in fact be used to represent
the problem of machine foundation vibration in any mode of vibration if appropriate
values of equivalent soil spring and damping constants are used. For coupled modes of
vibration, as for combined rocking and sliding, two degree-of-freedom model is used.

Vertical Vibrations of a Machine Foundation (a) Actual Case (b) Equivalent model with damping
( c) Model without damping

All foundations in practice are placed at a certain depth below the ground surface.
As a result of this embedment, the soil resistance to vibration develops not only below
the base of the foundation but also along the embedded portion of the sides of the
foundation.
Similarly the energy loss due to radiation damping will occur not only below the
foundation base but also along the sides of the foundation.
The type of models shown in Fig. b and c may be used to calculate the response of
embedded foundations if the equivalent soil spring and damping values are suitably
modified by taking into account the behavior of the soil below the base and on the sides
of the foundation.

For designing new foundations or retrofitting existing foundations for machines,
calculate the natural frequency of the machine-foundation-soil system so as to avoid
resonance,
evaluate the amplitude of vibrations of the machine-foundation-soil system under
operating dynamic loads and frequencies and make sure that they satisfy the
acceptability criteria.
with reference to avoiding damage to machinery – it means amplitude of velocity with
the limiting value prescribed by the manufacturer and in most cases, it will be
governing criterion and will dictate your design.
with reference to avoiding damage to building structures- it can be either amplitude of
displacement or velocity as prescribed by local codes for different types of
construction, with reference to ensuring comfort of persons-it means amplitude of
displacement usually prescribed by the local codes for different frequencies of
operation, and
with reference to avoiding excessive settlement due to large number of cycles of load
applications- it means amplitude of displacement prescribed by the local codes
particularly for soil profiles consisting of loose sands and silts with high water table.

In order to calculate the natural frequency and amplitude of vibrations for a particular
machine-foundation-soil-system, you need to know the local soil profile and soil
characteristics as also the dynamic loads generated by the machine that are provided by
the manufacturer.
1. Empirical
2. Elastic half space method
3. Linear elastic weightless spring method, and

Empirical Method
On such design guideline, rather a rule of thumb was the weight of the foundation
should be at least three to five times the weight of machine being supported.
There are some empirical formulae available in literature for estimating the natural
frequency, mostly for the vertical mode of vibration.
In these formulae, it is assumed that a certain part of the soil, immediately below the
foundation, moves as a rigid body along with the foundation and is called apparent soil
mass or in-phase mass.
For example, D.D.Barken in 1962 suggested that the mass of the vibrating soil should
be between 2/3 to 3/2 times the weight of foundation and machine.
These guidelines/formulae do not take into account the nature of subsoil, type of
excitation force (harmonic/impact), contact area and mode of vibration.

In using the second and third methods, we use the concept of discrete system or lumped
parameter system,
In this system: the mass of the machine and the supporting foundation are lumped
together as one discrete rigid mass ‘m’- this is reasonable since the stiffness of
reinforced concrete and the machine is several times more than that of soil.
the soil resistance is represented by means of a spring with a constant ‘k’ ,and the
energy absorption characteristic of soil is represented by means of a dashpot with a
constant ‘c’.
Once we have these three quantities m, k, and c, using the theory of vibrations, the
natural frequency as well as the amplitude of vibration for any given type of load can
be calculated.
The formulae to calculate amplitude of vibration for either impact or cyclic type of
loading can be found in books dealing with structural dynamics.

The soil behaves essentially as a linear elastic material and the resistance to
deformation offered by it can be represented by a spring.
This is based on the fact that the magnitude of the dynamic force acting on the
foundation is very small compared to the static load, normally no more than 10%.
And, therefore, the amplitude of displacement is extremely small. Since the non-linear
behavior of the soil does not come into play, material dumping in the soil can be
neglected.
In machine foundations, however, there exists another kind of damping known as
radiation damping or geometric damping.
Every time a dynamically loaded foundation moves against soil, stress waves originate
at the contact surface in the form of surface waves and carry away some of the energy
transmitted to the soil.

This loss of energy is called radiation damping and in passing that it can be effectively
modeled by a dashpot.
Use of the elastic half space method or the linear elastic weightless spring method
enables us to calculate the value k and c.
number of ways or modes in which a machine foundation can deform.
This block foundation can deform in any one or all of six possible modes.

DEGREES OF FREEDOM OF A RIGID BLOCK FOUNDATION
A typical concrete block is regarded as rigid as compared to the soil over which it rests.
Therefore, it may be assumed that it undergoes only rigid-body displacements and
rotations.
Under the action of unbalanced forces, the rigid block may thus undergo
displacements and oscillations as follows
1. translation along Z axis
2. translation along X axis
3. translation along Y axis
4. rotation about Z axis
5. rotation about X axis
6. rotation about Y axis
Any rigid-body displacement of the block can be resolved into these six independent
displacements. Hence, the rigid block has six degrees of freedom and six natural
frequencies.
Of six types of motion, translation along the Z axis and rotation about the Z axis
can occur independently of any other motion. However, translation about the X axis (or
Y axis) and rotation about the Y axis (or X axis) are coupled motions..

Therefore, in the analysis of a block, we have to concern ourselves with four types of
motions.
Two motions are independent and two are coupled. For determination of the natural
frequencies, in coupled modes, the natural frequencies of the system in pure translation
and pure rocking need to be determined.
Also, the states of stress below the block in all four modes of vibrations are quite
different.
Therefore, the corresponding soil-spring constants need to be defined before any
analysis of the foundations can be undertaken

Modes of Vibration of a rigid block foundation

Elastic Half Space Method
This method is called the elastic half space method because the ground is assumed to be
an elastic, homogeneous, isotropic, semi-infinite body which in the Theory of Elasticity
is referred to as elastic half space.
The elastic half space theory can be used to determine the values of equivalent soil
springs and damping then make use of theory of vibrations to determine the response of
the foundation. These are known as the „the elastic half space analogs .
‟
The machine foundation is idealized as a mechanical oscillator with a circular base
resting on the surface of ground.
The Boussinesq’s solution for finding the stresses induced in soil due to a point load on
the ground surface.
In the elastic half space method, the point load is assumed to be dynamic. By
integrating the solution for a dynamic point load over a circular area, the stresses due to
a circular machine foundation is calculate.

It may be mentioned here that the equivalent soil spring and damping values depend
upon the ;
(i) type of soil and its properties,
(ii) geometry and layout of the foundation, and
(iii) nature of the foundation vibrations occasioned by unbalanced dynamic loads.
Using the theory the values k and c are calculated .
Soil properties that are required to determine k and c are the shear modulus G, mass
density ρ and Poisson’s µ.
According to the theory with the vertical vibrations of a machine foundation of radius
r₀ :

Linear Elastic Weightless Spring Method
The linear elastic weightless spring method differs from the elastic half space method in
two respects:
the soil below the foundation is considered weightless – this assumption is not valid
but it does not appreciably affect the computed response, and
damping in the soil below the foundation is neglected- this assumption is also not valid
since radiation damping exists in a machine foundation. The effect of damping is
incorporated in this method by independently estimating it using a field test.
These assumptions make it possible to represent the machine-foundation-soil-system
with and equivalent mass spring system.
There is no dashpot here since damping has been neglected. In this method, the spring
constant k for vertical vibrations is expressed as a function of the area of the foundation
Af and the coefficient of elastic uniform compression Cu is
Kz = Cu Af

INFORMATION NEEDED FOR DESIGN
1. Static weight of the machine and accessories.
2. Magnitude and characteristics of dynamic loads imposed by the machine operation
and their point of application
3. The soil profile of the site and dynamic soil properties such as dynamic shear
modulus and damping
4. Trial dimensions of the foundation. These are generally supplied by the
manufacturer.
This will give the total static weight.
5. An acceptable method of analysis i.e., a mathematical model to determine the
response of the foundation-soil system
6. A criteria for adequate design

SOIL PROFILE AND DYNAMIC SOIL PROPERTIES
Satisfactory design of a machine foundation needs information on soil profile, depth of
different layers, physical properties of soil and ground water level.
This information can be obtained by usual sub-surface exploration techniques. In
addition, one must determine dynamic shear modulus, material damping, poisons ratio
and mass density of soil for dynamic analysis of the machine foundation.
Dynamic shear modulus of a soil is generally determined from laboratory or field tests.
Material damping can be determined from vibration tests on soil columns in the
laboratory.
The values of dynamic shear modulii and damping may be estimated from empirical
estimations for preliminary design purposes. Geometrical damping is estimated from
elastic half-space theory and appropriate analogs.

Design Procedure for a Block Foundation (Reciprocating Machine-Cyclic
Loading)
Obtain machine data: Operating speed of the machine; layout of the machine and a
detailed loading diagram; unbalanced forces from the machine; and permissible
amplitude of vibrations.
Obtain soil data: Soil properties to compute the allowable bearing capacity and spring
and damping constants of the soil.
Assume a trial size of the foundation: Usually the size of the machine dictates the
minimum plan size of the block. The minimum thickness of the block is dictated by the
depth of embedment required from bearing capacity considerations plus any ground
clearance required for operational purpose.
Evaluate the natural frequency and amplitude of vibration: First determine the
natural frequency of the block of size assumed in above placed on top of soil. Then
determine the amplitude of vibrations at this natural frequency subjected to unbalanced
loads.

Check for resonance and whether amplitude is within acceptable limits:
If resonance is not likely to occur and the amplitude of vibrations is within the
permissible limits then the trial size of the block assumed at step (ii) is acceptable. If
not, change the dimensions assumed in (ii) and repeat steps (iii) and (iv) until the
design requirements are satisfied. Design is almost invariably an iterative process. Trial
dimensions are assumed based on experience.

Design Criteria for Foundation of Reciprocating Machine
The machine foundation should be isolated at all levels from the adjoining foundations
The natural frequency of the foundation –soil system should be higher then the highest
disturbing frequency and the frequency ratio should not be less then 0.4.
The amplitude of vibration should be within the permissible limits or any specified
permissible limits
For most soils, the another criterion, amplitude for low speed machines is usually taken
as 200 micron (=0.2 mm)
According to another criterion, the amplitude in mm is limited to 4/f frequencies less
then 30 hertz and 125/f2.
For higher frequencies, where f is frequencies in hertz
(cycles/sec).
Concrete block foundations should be used. However, when the soil is not suitable to
support block foundation, cellular foundation may be used.
The size of the block in plan should be larger then the bed plate of the machine. There
should be a minimum all round clearance of 15 mm.

The total width of the foundation measured at right angles to the shaft should be at least
equal to the distance between the center of the shaft and bottom of the foundation.
The eccentricity of the foundation system along X-X and Y-Y axes should not exceed
not 5% of the length of the corresponding side of the contact area.
The combined center of gravity of machine and foundation should be as much below
the top of foundation.
The depth of foundation should be sufficient to provide the required bearing capacity
and to ensure stability against rotation in the vertical plane.
The stresses in the soil below the foundation should not excess 80% of the allowable
stresses under static loads. The base pressure is limited to half the normal allowable
pressure qna in extreme case.
Where it is not practicable to design a foundation to give satisfactory dynamic
response, the transmitted vibrations may be reduced by providing anti – vibration
mounting either between the machine and foundation or between the foundation and
the supporting system.

The machine should be anchored to the foundation block using a base plate and anchor
blots. Blot holes should be backfilled with concrete and the space below the plate
should be filled with 1: 2 cement mortars.
A number of similar machines can be erected on individual pedestals on a common raft.
The analysis for such machines can be made assuming that each foundation acts
independently up the raft into sections corresponding to separate machines.

Design Procedure for a Block Foundation (Hammer- Impact Loading)
The hammer system usually consists of a frame that guides a falling, weight, known as
the tup, that strikes down on the anvil that is supported on the foundation block.
Typical tup weights range between 2.5kN and 100kN and the drop height is in the
range of 0.3 m to 2 m or more.
The anvil gets repeated blows from the tup so that the piece of metal held by the anvil
gests forged to a desired shape or is broken.
High impact energy is transmitted to the anvil form the falling tup. A part of the energy
is used in forging and the rest gets transmitted to the soil below.
To avoid breakage of the concrete foundation due to impact stresses, an elastic pad
made of felt, cork or rubber is placed between anvil and foundation block.
We have to ensure that the compressive stresses induced in the pad do not exceed the
specified value. This is an additional design criteria for block foundations for a
hammer.

You should note that unlike, the one degree freedom system with two springs and two
masses. One mass, M1, represents the mass of the anvil and spring. K1 , represents the
stiffness of the elastic pad and the other mass, M2 and spring k2 represents the mass of
the concrete foundation and the stiffness of soil.
In the design of these foundations, the damping in the soil is neglected since the period
of impact is very small compared to the period of natural frequency of vibration of
machine-foundation-soil system.
The rest of the design procedure is the same as that discussed for the block foundation
for reciprocating machines.
The equations to estimate the natural frequency of the machine-foundation-soil system,
vibration amplitudes of foundation and anvil and dynamic stresses in the pad are
different.

Reinforcement and Construction Details
The reinforcement in the concrete block should not be less than 25kg/m3
.
For machines requiring special design considering of foundations, such as machine
pumping explosive gases, the minimum reinforcement is 40 kg/m3
.
Steel reinforcement around all pits and opening shall be at least equal to 0.5 to 0.75 %
of the cross sectional area of the pit or opening.
The reinforcement shall usually consist of 12 mm bars at 200 to 250 mm spacing
extending both vertical and horizontally near all face of the foundation block. The ends
of all bars should always be hooked.
If the height of the foundation block exceeds one metre, shrinkage reinforcement shall
be placed at suitable spacing in all the three directions.
The cover should be a minimum of 75 mm at the bottom and 50 mm on sides and the
top. The concrete shall be at least M-15 with a characteristic strength of 15 N/mm2
.
The foundation block should be preferably cast in a single, continuous operation. In
case of very thick blocks (exceeding 5m), construction joints can be provided.

Transmission of vibration can be controlled and the detrimental effects
considerably reduced by isolated either the source (active isolation ) or by protecting
the receiver (passive isolation)
The following measures are generally adopted.
The machine foundation should be located away from the adjoined structures. This is
known as geometric isolation.
The amplitude if surface waves (R-waves) reduce with an increase in distance. A
considerable reduction in the amplitude is achieved by locating the foundation at a
great depth, as the R-waves also reduce considerably with an increase in depth.
Additional masses known as dampers are attached to the foundations of high frequency
machines to make it a multiple degree freedom system and to change the natural
frequency.

In reciprocating machines vibrations are considerably reduce by counter balancing the
exciting forces by attaching counterweight to the side of the crank.
Vibration are considerably reduce by placing absorbers such as rubber mounting, felts
and crock between the machine and the base .
If an auxiliary mass with a spring is attached to the machine foundation, the system
becomes a two degree freedom system. The method is especially effective when the
system is in resonance.
If the strength of the soil is increased by chemical or cement stabilization, it increases
the nature frequency of the system. The method is useful for machines of low operating
frequency.
The natural frequency of the system is modified by making structure changes in
foundation, such as connecting the adjoined foundation, changing the base area or mass
of foundation or use of attached slabs.
The propagation if wave can be reduced by providing sheet piles, screens or trenches.

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