Analytic dynamics
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The document is a detailed overview of analytic dynamics, covering foundational concepts such as Newton’s laws of motion, work and energy, and principal laws like D'Alembert's principle and Hamilton's equations. It discusses various mathematical formulations of dynamics, generalized coordinates, and types of constraints within mechanical systems. The text serves as a comprehensive reference for analytical dynamics principles and their applications.
![10
Analytic DynamicsSOLO
1.5Constraints
If the N particles are free the system has n = 3 N degrees of freedom. ( ) Nkzyxr kkkk ,,2,1,,
=
The constraints on the system can be of the following types:
(1( Equality Constraints: The general form (the Pffafian form(
( ) ( ) ( )[ ] ( ) mldttradztradytradxtra l
t
N
k
k
l
zkk
l
ykk
l
xk ,,2,10,,,,
1
==+++∑=
or
{ } maaarankmldtarda l
zk
l
yk
l
xk
l
t
N
k
k
l
k ===+⋅∑=
,,,,2,10
1
We can classify the constraints as follows:
(a( Time Dependency
(a1( Catastatic mlal
t ,,2,10 ==
(a2( Acatastatic mlal
t ,,2,10 =≠
(1( Equality Constraints
(2( Inequality Constraints](https://image.slidesharecdn.com/analyticdynamics-150812133415-lva1-app6892/85/Analytic-dynamics-10-320.jpg)
![13
Analytic DynamicsSOLO
Constraints (continue(
Displacements Consistent with the Constraints:
The real displacement consistent with the
General Equality Constraints (Pffafian form) is:
The virtual displacement consistent with the
General Equality Constraints (Pffafian form) is:
dtkdzjdyidxrd kkkk ,
++=
[ ] mldtardadtadzadyadxa l
t
N
k
k
l
k
l
t
N
k
k
l
zkk
l
ykk
l
xk ,,2,10
11
==+⋅=+++ ∑∑ ==
tkzjyixr kkkk ∆∆+∆+∆=∆ ,
[ ] mltaratazayaxa l
t
N
k
k
l
k
l
t
N
k
k
l
zkk
l
ykk
l
xk ,,2,10
11
==∆+∆⋅=∆+∆+∆+∆ ∑∑ ==
Dividing the Pffafian equation by dt and taking the limit, we obtain:
mlraa
N
k
k
l
k
l
t ,,2,1
1
=⋅−= ∑=
⋅
Now replace in the virtual displacement equationl
ta
mltrra
N
k
kk
l
k ,,2,10
1
==
∆−∆⋅∑=
⋅
Define the δ variation as:
td
d
t∆−∆=
∆
δ](https://image.slidesharecdn.com/analyticdynamics-150812133415-lva1-app6892/85/Analytic-dynamics-13-320.jpg)
![21
Analytic DynamicsSOLO
The Stationary Value of a Function and of a Definite Integral
(continue(
Examples (continue(:
Continue: The functional ( ) ( )
∫
=
2
1
,,
x
x
dx
xd
xyd
xyxFI
The necessary condition for a stationary value is
( ) ( )
( ) ( ) ( )[ ] 0
0
12
0
2
1
2
1
=−
∂
∂
+
∂
∂
−
∂
∂
=
∂
∂
+
∂
∂
=
∫
∫
=
xx
xd
yd
F
dxx
xd
yd
F
xd
d
y
F
dx
xd
xd
xd
yd
F
x
y
F
d
Id
x
x
nintegratio
partsby
x
x
ηηη
η
η
ε ε
Since this must be true for every continuous function we have( )xη
210 xxx
xd
yd
F
xd
d
y
F
≤≤=
∂
∂
−
∂
∂
Euler-Lagrange Differential Equation
By solving this differential equation, ,for which I is stationary is found. ( )xy
JOSEPH-LOUIS
LAGRANGE
1736-1813
LEONHARD EULER
1707-1783](https://image.slidesharecdn.com/analyticdynamics-150812133415-lva1-app6892/85/Analytic-dynamics-21-320.jpg)
![50
Analytic DynamicsSOLO
6.Kane’s Equations
In terms of generalized coordinates we can write:
Thomas R. Kane
1924-
Stanford University
( ) ( ) ( ) ( ) ( ) Niktqzjtqyitqxtqqrtqr iiinii .,1,,,,,,, 1
=++==
( ) Nikdzjdyidxdt
t
r
dq
q
r
tqrd iii
i
n
j
j
j
i
i .,1,
1
=++=
∂
∂
+
∂
∂
= ∑=
( )
Ni
t
r
q
q
r
td
tqrd
rv i
n
j
j
j
ii
ii .,1
,
1
=
∂
∂
+
∂
∂
=== ∑=
→
6.1
6.2
6.3
Kane and Levinson have shown that with the n generalized coordinates , is useful
to define another n variables , which are linear functions of the n :
jq
iu jq
nrZqYu r
n
j
jrjr .,1
1
=+=∑=
∆
6.4
where the matrix is invertible and[ ] { } nj
nrrjYY
,1
,1
=
=
= [ ] [ ] { } nj
nrrjWWY
,1
,1
1 =
=
−
==
njXuWq j
n
r
rrjj .,1
1
=+= ∑=
6.5
jrrjrj XandZWY ,, are functions of tandq
are called Generalized Speed (also Nonholonomic Velocities, Quasivelocities.
etc.( and are not unique.
iu](https://image.slidesharecdn.com/analyticdynamics-150812133415-lva1-app6892/85/Analytic-dynamics-50-320.jpg)
![56
Analytic DynamicsSOLO
References:
]1[Goldstein, H., Classical Mechanics, 2nd
ed., Addison-Wesley, 1981
]2[Meirovitch, L., Methods of Analytical Dynamics, Mc Graw-Hill, 1970
]3[Greenwood, D.T., Principle of Dynamics, 2nd
ed., Prentice-Hall, 1977
]4[Kane, T.R., Dynamics, 3th
ed., Stanford University, 1972
]5[Desloge, E.A., Relationship Between Kane’s Equations and the Gibbs-Appell
Equations, J. Guidance, Vol. 10, No. 1, Jan.-Feb., 1987](https://image.slidesharecdn.com/analyticdynamics-150812133415-lva1-app6892/85/Analytic-dynamics-56-320.jpg)
Analytic dynamics
- 2. 2 Analytic DynamicsSOLO 1.Background Table of Content 1.1Newton’s Laws of Motion 1.2Work and Energy 1.3The Principal Laws of Analytic Dynamics 1.4Basic Definitions 1.5Constraints 1.6Generalized Coordinates 1.7The Stationary Value of a Function and of a Definite Integral 1.8The Principle of Virtual Work 2.D’Alembert Principle 3.Hamilton’s Principle 4.Lagrange’s Equations of Motion 5.Hamilton’s Equations 6.Kane’s Equations 7.Gibbs-Appel’s Equations References
- 3. 3 Analytic DynamicsSOLO 1.1 Newton’s Laws of Motion “The Mathematical Principles of Natural Philosophy” 1687 First Law Every body continues in its state of rest or of uniform motion in straight line unless it is compelled to change that state by forces impressed upon it. Second Law The rate of change of momentum is proportional to the force impressed and in the same direction as that force. Third Law To every action there is always opposed an equal reaction. → =→= constvF 0 ( )vm td d p td d F == 2112 FF −= vmp = td pd F = 12F 1 2 21F
- 4. 4 Analytic DynamicsSOLO 1.2Work and Energy The work W of a force acting on a particle m that moves as a result of this along a curve s from to is defined by: F 1r 2r ∫∫ ⋅ =⋅= ⋅∆ 2 1 2 1 12 r r r r rdrm dt d rdFW r 1r 2r rd rdr + 1 2 F m s rd is the displacement on a real path. ⋅⋅∆ ⋅= rrmT 2 1 The kinetic energy T is defined as: 1212 2 1 2 1 2 1 2 TTrrd m dtrr dt d mrdrm dt d W r r r r r r −= ⋅=⋅ =⋅ = ∫∫∫ ⋅ ⋅ ⋅⋅⋅⋅⋅ For a constant mass m
- 5. 5 Analytic DynamicsSOLO Work and Energy (continue( When the force depends on the position alone, i.e. , and the quantity is a perfect differential ( )rFF = rdF ⋅ ( ) ( )rdVrdrF −=⋅ The force field is said to be conservative and the function is known as the Potential Energy. In this case: ( )rV ( ) ( ) ( ) 212112 2 1 2 1 VVrVrVrdVrdFW r r r r −=−=−=⋅= ∫∫ ∆ The work does not depend on the path from to . It is clear that in a conservative field, the integral of over a closed path is zero. 12W 1r 2r rdF ⋅ ( ) ( ) 01221 21 1 2 2 1 =−+−=⋅+⋅=⋅ ∫∫∫ VVVVrdFrdFrdF path r r path r rC Using Stoke’s Theorem it means that∫∫∫ ⋅×∇=⋅ SC sdFrdF 0=×∇= FFrot Therefore is the gradient of some scalar functionF ( ) rdrVdVrdF ⋅−∇=−=⋅ ( )rVF −∇=
- 6. 6 Analytic DynamicsSOLO Work and Energy (continue( and ⋅ →∆→∆ ⋅−=⋅−= ∆ ∆ = rF dt rd F t V dt dV tt 00 limlim But also for a constant mass system ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅ ⋅=⋅= ⋅+⋅= ⋅= rFrrmrrrr m rrm dt d dt dT 22 1 Therefore for a constant mass in a conservative field ( ) .0 constEnergyTotalVTVT dt d ==+⇒=+
- 7. 7 Analytic DynamicsSOLO 1.3The Principal Laws of Analytical Dynamics The basic laws of dynamics can be formulated (expressed mathematically( in several ways other that that given by Newton’s Laws. The most important are: (a( D’Alembert Principle (b( Lagrange’s Equations (c( Hamilton’s Equations (d( Hamilton’s Principle (e( Kane’s Equations (f( Gibbs-Appell’s Equations All are basically equivalent.
- 8. 8 Analytic DynamicsSOLO 1.4Basic Definitions Given a system of N particles defined by their coordinates: ( ) ( ) ( ) ( ) Nkktzjtyitxzyxrr kkkkkkkk ,,2,1,, =++== where are the unit vectors defining any Inertial Coordinate Systemkji ,, The real displacement of the particle: km ( ) ( ) ( ) Nkktdzjtdyitdxrd kkkk ,,2,1 =++= is the infinitesimal change in the coordinates along real path caused by all the forces acting on the particle. km The virtual displacements are infinitesimal changes in the coordinates; they are not real changes because they are not caused by real forces. The virtual displacements define a virtual path that coincides with the real one at the end points. ( )tzyx kkk ∆∆∆∆ ,,,
- 9. 9 Analytic DynamicsSOLO Basic Definitions (continue( ( )trk ( )1trk ( )2trk krd 1 2 F km ),,,( dttdzzdyydxxPrdr kkkkkkkk ++++=+ ),,,( tzyxP kkk ),,,( ttzzyyxxP kkkkkk ∆+∆+∆+∆+ ),,,( tzzyyxxP kkkkkk ∆+∆+∆+ tvrd kk ∆= i j k True (Dynamical or Newton) Path Virtual Path
- 10. 10 Analytic DynamicsSOLO 1.5Constraints If the N particles are free the system has n = 3 N degrees of freedom. ( ) Nkzyxr kkkk ,,2,1,, = The constraints on the system can be of the following types: (1( Equality Constraints: The general form (the Pffafian form( ( ) ( ) ( )[ ] ( ) mldttradztradytradxtra l t N k k l zkk l ykk l xk ,,2,10,,,, 1 ==+++∑= or { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1 We can classify the constraints as follows: (a( Time Dependency (a1( Catastatic mlal t ,,2,10 == (a2( Acatastatic mlal t ,,2,10 =≠ (1( Equality Constraints (2( Inequality Constraints
- 11. 11 Analytic DynamicsSOLO Constraints (continue( Equality Constraints: The general form (the Pffafian form( (continue( { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1 (b( Integrability (b1( Holonomic if the Pffafian forms are integrable; i.e.: mldt t f zd z f yd y f xd x f df N k l k k l k k l k k l l ,,2,1 1 = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∑= or ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111 == (b2( Non-holonomic if the Pffafian forms are not integrable (b2.1( Scleronomic: (b2.2( Rheonomic: ml l t f ,,2,1 0 = = ∂ ∂ or ( ) mlzyxzyxf NNNl ,,2,10,,,,,, 111 == ml l t f ,,2,1 0 = ≠ ∂ ∂
- 12. 12 Analytic DynamicsSOLO Constraints (continue( (2(Inequality Constraints: (a( Stationary Boundaries (time independent(: (b( Non-stationary Boundaries (time dependent(: ( ) mlzyxzyxf NNNl ,,2,10,,,,,, 111 =≥ ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111 =≥
- 13. 13 Analytic DynamicsSOLO Constraints (continue( Displacements Consistent with the Constraints: The real displacement consistent with the General Equality Constraints (Pffafian form) is: The virtual displacement consistent with the General Equality Constraints (Pffafian form) is: dtkdzjdyidxrd kkkk , ++= [ ] mldtardadtadzadyadxa l t N k k l k l t N k k l zkk l ykk l xk ,,2,10 11 ==+⋅=+++ ∑∑ == tkzjyixr kkkk ∆∆+∆+∆=∆ , [ ] mltaratazayaxa l t N k k l k l t N k k l zkk l ykk l xk ,,2,10 11 ==∆+∆⋅=∆+∆+∆+∆ ∑∑ == Dividing the Pffafian equation by dt and taking the limit, we obtain: mlraa N k k l k l t ,,2,1 1 =⋅−= ∑= ⋅ Now replace in the virtual displacement equationl ta mltrra N k kk l k ,,2,10 1 == ∆−∆⋅∑= ⋅ Define the δ variation as: td d t∆−∆= ∆ δ
- 14. 14 Analytic DynamicsSOLO Constraints (continue( Displacements Consistent with the Constraints (continue(: Define the δ variation as: td d t∆−∆= ∆ δ ( )trk kr δ km ),,,( dttdzzdyydxxPrdr kkkkkkkk ++++=+ ),,,( tzyxP kkk ),,,( ttzzyyxxP kkkkkk ∆+∆+∆+∆+ ),,,( tzzyyxxP kkkkkk ∆+∆+∆+ dtrrd kk ⋅ = i j k True (Dynamical or Newton) Path Virtual Path kr ∆ trr kk ∆=∆ ⋅ Then: kkk r td d trr ∆−∆= ∆ δ From the Figure we can see that δ variation corresponds to a virtual displacement in which the time t is held fixed and the coordinates varied to the constraints imposed on the system. mlra N k k l k ,,2,10 1 ==⋅∑= δ For the Holonomic Constraints: ( ) mltzyxzyxf NNNl ,,2,10,,,,,,, 111 == mlrf N k klk ,,2,10 1 ==⋅∇∑= δ
- 15. 15 Analytic DynamicsSOLO 1.6Generalized Coordinates The motion of a mechanical system of N particles is completely defined by n = 3N coordinates . Quite frequently we may find it more advantageous to express the motion of the system in terms of a different set of coordinates, say . If we take in consideration the m constraints we can reduce the coordinates to n = 3N-m generalized coordinates. ( ) ( ) ( ) ( )Nktztytx kkk ,,2,1,, = ( )T nqqqq ,,, 21 = ( ) ( ) ( ) ( ) ( ) Nkktqzjtqyitqxtqqqrtqr kkknkk ,,2,1,,,,,,,, 21 =++== Nkkdzjdyidxdt t r dq q r rd kkk k j n j j k k ,,2,1 1 =++= ∂ ∂ + ∂ ∂ = ∑= Nk t r q q r td rd rv k j n j j kk kk ,,2,1 1 = ∂ ∂ + ∂ ∂ === ∑= ⋅ In the same way Nkkzjyixt t r q q r r kkk k j n j j k k ,,2,1 1 =∆+∆+∆=∆ ∂ ∂ +∆ ∂ ∂ =∆ ∑= and Nkt t r tq q r t t r q q r trrr k j n j j kk j n j j k kkk ,,2,1 11 ==∆ ∂ ∂ −∆ ∂ ∂ −∆ ∂ ∂ +∆ ∂ ∂ =∆−∆= ∑∑ == ⋅ δ
- 16. 16 Analytic DynamicsSOLO Generalized Coordinates (continue( ( ) Nkq q r tqq q r r n j j j k jj n j j k k ,,2,1 11 = ∂ ∂ =∆−∆ ∂ ∂ = ∑∑ == δδ where tqqq jjj ∆−∆= ∆ δ The Generalized Equality Constraints in Generalized Coordinates will be: mldt t r aadq q r a dt t r aadq q r adtarda N k kl k l ti n i i k N k l k N k N k kl k l ti n i i kl k l t N k k l k ,,2,10 11 1 1 111 == ∂ ∂ ⋅++ ∂ ∂ ⋅= = ∂ ∂ ⋅++ ∂ ∂ ⋅=+⋅ ∑∑ ∑ ∑ ∑∑∑ == = = === If we define ∑ ∑∑= = ∆ = ∆ ∂ ∂ ⋅+= ∂ ∂ ⋅= N k N k kl k l t l t n i i kl k l i t r aac q r ac 1 11 & we obtain mldtcdqc l ti n i l i ,,2,10 1 ==+∑= and the virtual displacements compatible with the constraints are mlqc i n i l i ,,2,10 1 ==∑= δ
- 17. 17 Analytic DynamicsSOLO Generalized Coordinates (continue( The number of degrees of freedom of the system is n = 3N-m. However, when the system is nonholonomic, it is possible to solve the m constraint equations for the corresponding coordinates so that we are forced to work with a number of coordinates exceeding the degrees of freedom of the system. This is permissible provided the surplus number of coordinates matches the number of constraint equations. Although in the case of a holonomic system it may be possible to solve for the excess coordinates, thus eliminating them, this is not always necessary or desirable. If surplus coordinates are used, the corresponding constraint equations must be retained.
- 18. 18 Analytic DynamicsSOLO 1.7The Stationary Value of a Function and of a Definite Integral In problems of dynamics is often sufficient to find the stationary value of functions instead of the extremum (minimum or maximum(. Definition: A function is said to have a stationary value at a certain point if the rate of change in every direction of the point is zero. Examples: (1( ( ) ni u f du u f dfuuuf i n i i i n ,,2,100,,, 1 21 == ∂ ∂ →= ∂ ∂ =→ ∑= By solving those n equations we obtain for which f is stationary ( )nuuu ,,, 21
- 19. 19 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue( Examples (continue(: (2( ( )nuuuf ,,, 21 with the constraints { } marankmldua l k N k k l k ===∑= ,,2,10 1 Lagrange’s multipliers solution gives: 0 1 1 = + ∂ ∂ = ∑ ∑= = i n i m l l il i dua u f df λ By choosing the m Lagrange’s multipliers to annihilate the coefficients of the m dependent differentials we have lλ idu equationsmn mldua nia u f n l i l i m l l il i + == ==+ ∂ ∂ ∑ ∑ = = ,,2,10 ,,2,10 1 1 λ
- 20. 20 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue( Examples (continue(: (3( The functional ( ) ( ) ∫ = 2 1 ,, x x dx xd xyd xyxFI We want to find such that I is stationary, when the end points and are given. ( )xy ( )1xy ( )2xy ( )xy ( ) ( ) ( ) ( )xxyxyxy ηεδ +=+ ( )11, yx ( )22 , yx x y The variation of is( )xy ( ) ( ) ( ) ( ) ( ) ( ) ( ) 021 ==+=+= xxxxyxyxyxy ηηηεδ and ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∫∫ ++= = 2 1 2 1 ,,,, x x x x dx xd xd xd xyd xxyxFdx xd xyd xyxFI η εηεε ( ) ( ) +++== == 2 0 2 2 0 2 1 0 ε ε ε ε εε εε d d Id d d Id II
- 21. 21 Analytic DynamicsSOLO The Stationary Value of a Function and of a Definite Integral (continue( Examples (continue(: Continue: The functional ( ) ( ) ∫ = 2 1 ,, x x dx xd xyd xyxFI The necessary condition for a stationary value is ( ) ( ) ( ) ( ) ( )[ ] 0 0 12 0 2 1 2 1 =− ∂ ∂ + ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ = ∫ ∫ = xx xd yd F dxx xd yd F xd d y F dx xd xd xd yd F x y F d Id x x nintegratio partsby x x ηηη η η ε ε Since this must be true for every continuous function we have( )xη 210 xxx xd yd F xd d y F ≤≤= ∂ ∂ − ∂ ∂ Euler-Lagrange Differential Equation By solving this differential equation, ,for which I is stationary is found. ( )xy JOSEPH-LOUIS LAGRANGE 1736-1813 LEONHARD EULER 1707-1783
- 22. 22 Analytic DynamicsSOLO 1.8 The Principle of Virtual Work This is a statement of the Static Equation of a mechanical system. If the system of N particles is in dynamic equilibrium the resultant force on each particle is zero; i.e.: 0=iR 0 1 =⋅= ∑= N i ii rRW δδ Because of this, for a virtual displacement the Virtual Work of the system isir δ If the system is subjected to the constraints: { } maaarankmldtarda l zk l yk l xk l t N k k l k ===+⋅∑= ,,,,2,10 1 Then we denote the external forces on particle i by and the constraint’s forces by . The resultant force on i is: iF iF' 0' =+= iii FFR
- 23. 23 Analytic DynamicsSOLO The Principle of Virtual Work (continue( We want to find the Virtual Work of the constraint forces. There are two kind of constraints: (1(The particle i is constrained to move on a definite surface. We assume that the motion is without friction and therefore the constraint forces must be normal to the surface. The virtual variation compatible with the constraint must be on the surface, therefore. iF ir δ 0' =⋅ ii rF δ ir δ iF' (2(The particle i is acting on the particle j and the distance between them is l(t(. . iF' i j jF' ir jr ( )tl
- 24. 24 Analytic DynamicsSOLO The Principle of Virtual Work (continue( ( ) ( ) ( )tlrrrr jiji 2 =−⋅− ( ) ( ) tllrrrr jiji ∆=∆−∆⋅− ( ) llrrrr jiji = −⋅− ⋅⋅ ( ) ( ) ( ) ji rr jiji jjiiji rrrrrr trrtrrrr ji δδδδ =→=−⋅−→ →= ∆−∆− ∆−∆⋅− ≠ ⋅⋅ 0 0 ji FF '' −= If we compute the virtual variation and differential and we multiply the second equation by and add to the first we obtaint∆− Because is a real (not a generalized( force we can use Newton’s Third Law: i.e.: iF' and the virtual work of the constraint forces of this system is: ( ) 0''''' =⋅−+⋅=⋅+⋅= rFrFrFrFW iijjii δδδδδ We can generalized this by saying that: 0' 1 =⋅∑= N i ii rF δ The work done by the constraint forces in virtual displacements compatible with the constraints (without dissipation( is zero.
- 25. 25 Analytic DynamicsSOLO The Principle of Virtual Work (continue( From equation we obtain: 0' =+= iii FFR ∑∑∑∑ ==== ⋅=⋅+⋅=⋅= N i ii N i ii N i ii N i ii rFrFrFrR 1 0 111 '0 δδδδ or 0 1 =⋅= ∑= N i ii rFW δδ The Principle of Virtual Work The work done by the applied forces in infinitesimal virtual displacements compatible with the constraints (without dissipation( is zero
- 26. 26 Analytic DynamicsSOLO The Principle of Virtual Work (continue( { } mjNimaaarank mjra j zi j yi j xi N k k j k ,,2,1&,,1,, ,,2,10 1 === ==⋅∑= δ We found that the General Equality Constraint the virtual displacement compatible with the constraint must be: ir δ Let adjoin the m constraint equations by the m Lagrange’s multipliers λ j and add to the virtual work equation: 0 1 11 11 =⋅ += ⋅+⋅= ∑ ∑∑ ∑∑ = == == N i i m j j iji m j N i i j ij N i ii raFrarFW δλδλδδ There are 3N virtual displacements from which m are dependent of the constraint λj relations and 3N-m are independent. We will choose the m Lagrange’s multipliers to annihilate the coefficients of the m dependent variables: −+= = =+ ∑= iationsmNtindependenNmi mtheofbecausemi aF j m j j iji var33,,1 ,,2,1 0 1 λ λ
- 27. 27 Analytic DynamicsSOLO The Principle of Virtual Work (continue( From we obtain: 0' =+= iii FFR ∑= = m j j iji aF 1 ' λ where are chosen such thatjλ mjforaF m j j iji ,,2,10 1 ==+ ∑= λ Since , we obtain: k n k k i i q q r r δδ ∑= ∂ ∂ = 1 0 1 1 11 1 111 1 = ∂ ∂ + ∂ ∂ = = ∂ ∂ ⋅ +=⋅ += ∑ ∑ ∑∑ ∑ ∑∑∑ ∑ = = == = === = n k k m j N i k ij ij N i k i i N i n k k k i m j j iji N i i m j j iji q q r a q r F q q r aFraFW δλ δλδλδ We define: nk q r FQ N i k i ik ,,2,1 1 = ∂ ∂ =∑= ∆ nkc q r aQ m j j kj m j N i k ij ijk ,,2,1' 11 1 == ∂ ∂ = ∑∑ ∑ == = ∆ λλ nk q r ac N i k ij i j k ,,2,1 1 = ∂ ∂ =∑= ∆ Generalized Forces Generalized Constraint Forces
- 28. 28 Analytic DynamicsSOLO 2.D’Alembert Principle Newton’s Second Law for a particle of mass and a linear momentum Vector can be written as im iii vmp = D’Alembert Principle: 0' =−+ ⋅ iii pFF where and are the applied and constraint forces, respectively. iF iF' D’Alembert Principle enables us to trait dynamical problems as if they were statical. Let extend the Principle of Virtual Work to dynamic systems: 0' 1 =⋅ −+∑= ⋅N i iiii rpFF δ Assuming that the constraints are without friction the virtual work of the constraint force is zero . Then we have Generalized D’Alembert Principle: 0 1 =⋅ −∑= ⋅N i iii rpF δ 0' 1 =⋅∑= N i ii rF δ The Generalized D’Alembert Principle The total Virtual Work performed by the effective forces through infinitesimal Virtual Displacement, compatible with the system constraints are zero. 0=− ⋅ ii pF is the effective force. Jean Le Rond d’ Alembert 1717-1783 “Traité de Dynamique” 1743
- 29. 29 Analytic DynamicsSOLO 3.Hamilton’s Principle William Rowan Hamilton 1805-1865 Let write the D’Alembert Principle: in integral form0 1 =⋅ −∑= ⋅N i iii rpF δ But ( )∑∑∑ === ⋅ ⋅+ ⋅−=⋅− N i iii N i iii N i iii r td d vmrvm td d rvm 111 δδδ Let find ( )ir td d δ iiiii r td d ttvrr ∆−∆=∆−∆=δare the virtual displacements compatible with the constraints mjra N i i j i ,,2,10 1 ==⋅∑= δ ( )tri ir δ tvi ∆ ( )tPi( )tP i' ( )ttP i ∆+' ir ∆ Virtual Path True Path (P) Newtonian or Dynamic Path The Constraint Space at t mjra N i i j i ,,10 1 ==⋅∑= δ ( ) ( ) ( ) ( ) ( ) ( ) =∆=∆ =∆=∆ == 0 0 0 21 21 21 tttt trtr trtr ii ii δδ 1t 2t 0 2 1 1 =⋅ −∫∑= ⋅ t t N i iii dtrpF δ
- 30. 30 Analytic DynamicsSOLO Hamilton’s Principle (continue( Since td rd vv i Pi i == ( ) ( ) ( ) t td d vr td d vt td d r td d v t td d r td d td rd tdtd rdrd ttd rrd vvv iiiii i i iiii ttPii i ∆−∆+≈ ∆− ∆+≈ ∆+ ∆+ = = ∆+ ∆+ = ∆+ ∆+ ==∆+ ∆+ 1 1 ' ( ) ( ) tar td d tatvr td d t td d vr td d v iiiiiiii ∆+=∆+∆−∆=∆−∆=∆ δ Therefore ( ) ( ) ecommutativare td d and r td d vv td d ttavr td d iiiiii δ δδδ → →== ∆−∆=∆−∆=
- 31. 31 Analytic DynamicsSOLO Hamilton’s Principle (continue( Now we can develop the expression: tavmvvmrvm td d ram N i iii N i iii N i iii N i iii ∆⋅− ∆⋅+ ⋅−=⋅− ∑∑∑∑ ==== 1111 δδ But the Kinetic Energy T of the system is: ∑= ⋅= N i iii vvmT 12 1 ∑= ∆⋅=∆ N i iii vvmT 1 ∑∑∑ === ⋅ ⋅=⋅=⋅= N i iii N i iii N i iii vFmavmvvmT 111 Therefore Trvm td d tTTrvm td d ram N i iii N i iii N i iii δδ δδ + ⋅−= =∆−∆+ ⋅−=⋅− ∑ ∑∑ = == 1 11
- 32. 32 Analytic DynamicsSOLO Hamilton’s Principle (continue( From the integral form of D’Alembert Principle we have: ( ) ∫ ∑∫ ∑∑ ∫ ∑∫ ∑ ∫∑ ⋅+= ⋅++⋅−= = ⋅++ ⋅−= =⋅+−= === == = 2 1 2 1 2 1 2 1 2 1 2 1 111 0 11 1 0 t t N i ii t t N i ii N i t tiii t t N i ii t t N i iii t t N i iiii dtrFTdtrFTrvm dtrFTdtrvm td d dtrFam δδδδδ δδδ δ We obtained ( ) 0 2 1 2 1 1 =+= ⋅+ ∫∫ ∑= dtWTdtrFT t t t t N i ii δδδ Extended Hamilton’s Principle
- 33. 33 Analytic DynamicsSOLO Hamilton’s Principle (continue( If we develop and we can writetTTT ∆−∆= δ tvrr iii ∆−∆= δ 0 2 1 2 1 111 = ∆ ⋅+− ∆⋅+∆= ⋅+ ∫ ∑∑∫ ∑ === dttvFTrFTdtrFT t t N i ii N i ii t t N i ii δδ and because ∑= ⋅= N i ii vFT 1 02 2 1 1 = ∆− ∆⋅+∆∫ ∑= dttTrFT t t N i ii The pair and is arbitrary but compatible with the constraints: ir ∆ t∆ mjtara j t N i i j i ,,2,10 1 ==∆+∆⋅∑=
- 34. 34 Analytic DynamicsSOLO Hamilton’s Principle (continue( For a Conservative System VF ii −∇= VrVrFW N i ii N i ii δδδδ −=⋅∇−=⋅= ∑∑ == 11 We have ( ) ( ) 0 2 1 2 1 2 1 ==−=+ ∫∫∫ t t t t t t dtLdtVTdtWT δδδ where VTL −= ∆ =−∇=−== ∆ ∫ NiVFVTLdtL ii t t ,,2,1;0 2 1 δ Hamilton’s Principle for Conservative Systems Hamilton’s Principle for Conservative Systems: The actual path of a conservative system in the configuration space renders the value of the integral stationary with respect to all arbitrary variations (compatible with the constraints) of the path between the two instants and provided that the path variations vanish at those two points. ∫= 2 1 t t dtLI 1t 2t
- 35. 35 Analytic DynamicsSOLO 4. Lagrange’s Equations of Motion Joseph Louis Lagrange 1736-1813 “Mecanique Analitique” 1788 The Extended Hamilton’s Principle states: 0 2 1 1 = ⋅+∫ ∑= dtrFT t t N i ii δδ where are the virtual displacements compatible with the constraints: ir δ mjqcq q r ara n k k k i n k k N i k ij i N i i j i ,,2,10 11 11 === ∂ ∂ =⋅ ∑∑ ∑∑ == == δδδ T the kinetic energy of the system is given by: ∑ ∑∑∑ = ⋅ === ⋅⋅ = ∂ ∂ + ∂ ∂ ⋅ ∂ ∂ + ∂ ∂ =⋅= N j n i j i i j n i j i i j j N j jjj tqqT t r q q r t r q q r mrrmT 1 111 ,, 2 1 2 1 where is the vector of generalized coordinates. ( )nqqqq ,,, 21 = −∆ ∂ ∂ + ∆ ∂ ∂ +∆ ∂ ∂ + = − ∆+∆+∆+=∆ ⋅ = ⋅⋅⋅⋅ ∑ tqqTt t T q q T q q T tqqTtqqTttqqqqTT n i i i i i ,,,,,,,, 1 t T q q T q q T T n i i i i i ∂ ∂ + ∂ ∂ + ∂ ∂ = ∑=1 ( ) ( ) ∑∑ == ∂ ∂ + ∂ ∂ = ∆−∆ ∂ ∂ +∆−∆ ∂ ∂ =∆−∆= n i i i i i n i ii i ii i q q T q q T tqq q T tqq q T tTTT 11 δδδ
- 36. 36 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue( ( ) ( ) ∑∑ == ∂ ∂ + ∂ ∂ = ∆−∆ ∂ ∂ +∆−∆ ∂ ∂ =∆−∆= n i i i i i n i ii i ii i q q T q q T tqq q T tqq q T tTTT 11 δδδ But because δ and are commutative and: td d ( )ii q dt d q δδ = ∑= ∂ ∂ + ∂ ∂ = n i i i i i q dt d q T q q T T 1 δδδ This is an expected result because the variation δ keeps the time t constant. We found that , therefore∑= ∂ ∂ = n i i i j j q q r r 1 δδ ∫∑∫∑ ∑∫ ∑ ∑∫ ∑ == == == = ∂ ∂ ⋅= ∂ ∂ ⋅= ⋅ 2 1 2 1 2 1 2 1 11 11 11 t t n i ii t t n i i N j i j j t t N j n i i i j j t t N j jj dtqQdtq q r Fdtq q r FdtrF δδδδ where ForcesdGeneralizeni q r FQ N j i j ji ,,2,1 1 = ∂ ∂ ⋅=∑= ∆ Now ( ) ∫∑∑ ∫∑∫ ∑ == == − ∂ ∂ − ∂ ∂ − ∂ ∂ = + ∂ ∂ + ∂ ∂ = ⋅+= 2 1 2 1 2 1 2 1 11 0 .int 11 0 t t n i iii i i i n i t ti i partsby t t n i iii i i i t t N j jj dtqQq q T q q T td d q q T dtqQq q T q td d q T dtrFT δδδδ δδδδδ
- 37. 37 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue( 0 2 1 1 = − ∂ ∂ − ∂ ∂ ∫∑= t t i n i i ii dtqQ q T q T td d δ where the virtual displacements must be consistent with the constraints .Let adjoin the previous equations by the constraints multiplied by the Lagrange’s multipliers iqδ mkqc n i i k i ,,2,10 1 ==∑= δ ( )mkk ,,2,1 =λ 0 1 11 1 = = ∑ ∑∑ ∑ = == = n i i m k k ik m k n i i k ik qcqc δλδλ to obtain 0 2 1 1 1 = −− ∂ ∂ − ∂ ∂ ∫∑ ∑= = t t i n i m k k iki ii dtqcQ q T q T td d δλ While the virtual displacements are still not independent, we can chose the Lagrangian’s multipliers so as to render the bracketed coefficients of equal to zero. The remaining being independent can be chosen arbitrarily, which leads to the conclusion that the coefficients of are zero. It follows iqδ iqδ ( )mkk ,,2,1 =λ ( )nmiqi ,,2,1 +=δ ( )miqi ,,2,1 =δ nicQ q T q T dt d m k k iki ii ,,2,1 1 =+= ∂ ∂ − ∂ ∂ ∑= λ
- 38. 38 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue( nicQ q T q T dt d m k k iki ii ,,2,1 1 =+= ∂ ∂ − ∂ ∂ ∑= λ We have here n equations with n+m unknowns . To find all the unknowns we must add the m equations defined by the constraints, to obtain ( ) ( ) mn tqtq λλ ,,,,, 11 nicQ q T q T dt d m k k iki ii ,,2,1 1 =+= ∂ ∂ − ∂ ∂ ∑= λLagrange’s Equations: mkcqc k t n i i k i ,,2,10 1 ==+∑= Let define Generalized Constraint Forces: nicQ m k k iki ,,2,1' 1 == ∑= λ
- 39. 39 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue( If the system is acted upon by some forces which are derivable from a potential Function and some forces which are not, we can write: ( ) ( )nn qqqVrrrV ,,,,,, 2121 −=− n jF n jjj FVF +−∇= ( ) ∑∑ ∑∑ ∑∑ == == == = ∂ ∂ ⋅+ ∂ ∂ ⋅∇−= ∂ ∂ ⋅+∇−=⋅ n i ii n i i N j i jn j i j j N j n i i i jn jj N j jj qQq q r F q r Vq q r FVrF 11 11 11 δδδδ But where∑= ∂ ∂ ⋅∇= ∂ ∂ N j i j j i q r V q V 1 k z V j y V i x V V jjj j ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ Therefore: niQ q V q r F q r VQ in i N j i jn j N j i j ji ,,2,1 11 =+ ∂ ∂ −= ∂ ∂ ⋅+ ∂ ∂ ⋅∇−= ∑∑ == Generalized External Forces: Generalized External Nonconservative Forces: ni q r FQ N j i jn jin ,,2,1 1 = ∂ ∂ ⋅=∑= ∆
- 40. 40 Analytic DynamicsSOLO Lagrange’s Equations of Motion (continue( The Lagrange’s Equations nicQ q T q T dt d m k k iki ii ,,2,1 1 =+= ∂ ∂ − ∂ ∂ ∑= λ Define: ( ) ( ) ( )qVtqqTtqqL −= ∆ ,,,, Because we assume that , we have( ) i q qV i ∀= ∂ ∂ 0 Lagrange’s Equations: nicQ q L q L dt d m k k ikin ii ,,2,1 1 =+= ∂ ∂ − ∂ ∂ ∑= λ mkcqc k t n i i k i ,,2,10 1 ==+∑= We proved =−∇=−== ∆ ∫ NiVFVTLdtL ii t t ,,2,1;0 2 1 δ Hamilton’s Principle for Conservative Systems Lagrange’s Equations for a Conservative System without Constraints: ( )0,,,,2,10 =−=−∇=== ∂ ∂ − ∂ ∂ k iii ii cVTLVFni q L q L dt d If they are no constraints, from the Lagrange’s Equations, or from Euler- Lagrange Equation for a stationary solution of , we obtain: ∫= 2 1 t t dtLI
- 41. 41 Analytic DynamicsSOLO 5.Hamilton’s Equations The Lagrange’s Equations nicQ q T q T dt d m k k iki ii ,,2,1 1 =+= ∂ ∂ − ∂ ∂ ∑= λ can be rewritten as: nicQ q T tq T q qq T q qq T q T dt d m k k iki i n i i j ji j jii ,,2,1 11 222 =++ ∂ ∂ = ∂∂ ∂ + ∂∂ ∂ + ∂∂ ∂ = ∂ ∂ ∑∑ == λ therefore consist of a set of n simultaneous second-order differential equations. They must be solved tacking in consideration the m constraint equations. mkcqc k t n i i k i ,,2,10 1 ==+∑= A procedure for the replacement of the n second-order differential equations by 2n first-order differential equations consists of formulating the problem in terms of 2n Hamilton’s Equations. We define first: General Momentum: ni q T p i i ,,2,1 = ∂ ∂ = ∆ We want to find the transformation from the set of variables to the set by the Legendre’s Dual Transformation. ( )tqq ,, ( )tpq ,,
- 42. 42 Analytic DynamicsSOLO Hamilton’s Equations (continue( Legendre’s Dual Transformation. Adrien-Marie Legendre 1752-1833 Let consider a function of n variables , m variables and time t. ix iy ( )tyyxxF mn ,,,,,, 11 and introduce a new set of variables ui defined by the transformation: ni x F u i i ,,2,1 = ∂ ∂ = ∆ We can see that: ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ + ∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂ ∂ = m mnn m n nn n n dy dy dy yx F yx F yx F yx F dx dx dx x F xx F xx F x F du du du 2 1 2 1 2 1 2 11 2 2 1 2 2 1 2 1 2 2 1 2 2 1 We want to replace the variables by the new variables. We can see that the new n variables are independent if the Hessian Matrix is nonsingular. ( )nidxi ,,2,1 = ni njji xx F ,,1 ,,1 2 = = ∂∂ ∂
- 43. 43 Analytic DynamicsSOLO Hamilton’s Equations (continue( Legendre’s Dual Transformation (continue( Let define a new function G of the variables , and t. iu iy ( )tyyuuGFxuG mn n i ii ,,,,,, 11 1 =−=∑= ∆ Then: ( ) dt t F dy y F dx x F udux dt t F dy y F dx x F dxuduxdG m j j j n i i i iii n i m j j j i i n i iiii ∂ ∂ − ∂ ∂ − ∂ ∂ −+= = ∂ ∂ − ∂ ∂ − ∂ ∂ −+= ∑∑ ∑ ∑∑ == = == 11 0 1 11 But because: ( )tyyuuGG mn ,,,,,, 11 = dt t G dy y G du u G dG n i m j j j i i ∂ ∂ + ∂ ∂ + ∂ ∂ = ∑ ∑= =1 1 Because all the variations are independent we have: t F t G mj y F y G ni u G x jji i ∂ ∂ −= ∂ ∂ = ∂ ∂ −= ∂ ∂ = ∂ ∂ = ;,,1;,,1
- 44. 44 Analytic DynamicsSOLO Hamilton’s Equations (continue( Legendre’s Dual Transformation (continue( Now we can define the Dual Legendre’s Transformation from ( )tyyxxF mn ,,,,,, 11 ( ) FxutyyuuG n i iimn −= ∑=1 11 ,,,,,, to by using ni x F u i i ,,2,1 = ∂ ∂ = ni u G x i i ,,2,1 = ∂ ∂ = End of Legendre’s Dual Transformation
- 45. 45 Analytic DynamicsSOLO Hamilton’s Equations (continue( Following the same pattern to find the transformation from the set of variables to the set , we introduce the Hamiltonian:( )tqq ,, ( )tpq ,, ( )tqqTqpH n i ii ,, 1 −=∑= ∆ where ni q T p i i ,,2,1 = ∂ ∂ = Then ( )tpqHH ,, = dt t H dp p H dq q H dt t T dq q T dpq dt t T dq q T qd q T qdpdpqdH n i i i i i n i i i ii n i i i i i iiii ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ − ∂ ∂ −= = ∂ ∂ − ∂ ∂ − ∂ ∂ −+= ∑∑ ∑ == = 11 1 and 0 1 = ∂ ∂ + ∂ ∂ + − ∂ ∂ + ∂ ∂ + ∂ ∂ ∑= dt t T t H dpq p H dq q T q Hn i ii i i ii
- 46. 46 Analytic DynamicsSOLO Hamilton’s Equations (continue( If the Hessian Matrix is nonsingular, all the are independent, but not the that must be consistent with the constraints: ni njji qq T ,,1 ,,1 2 = = ∂∂ ∂ ( )nidpi ,,2,1 = ( )nidqi ,,2,1 = mjdtcdqc j t n i i j i ,,2,10 1 ==+∑= Let adjoin the previous equations by the constraint equations multiplied by the m Lagrange’s multipliers :j'λ 0''' 11 11 1 =+ = + ∑∑ ∑∑ ∑ == == = m j j ij n i i m j j ij m j j t n i i j ij dtcdqcdtcdqc λλλ We have 0'' 11 1 = + ∂ ∂ + ∂ ∂ + − ∂ ∂ + + ∂ ∂ + ∂ ∂ ∑∑ ∑ == = dtc t T t H dpq p H dqc q T q H m j j tj n i ii i i m j j ij ii λλ
- 47. 47 Analytic DynamicsSOLO Hamilton’s Equations (continue( By proper choosing the m Lagrange’s multipliers ,the remainder differentials and dt are independent and therefore we have: j'λ ii dpdq , ni c t H t T c q H q T p H q m j j tj m j j ij ii i i ,,2,1 ' ' 1 1 = − ∂ ∂ −= ∂ ∂ − ∂ ∂ −= ∂ ∂ ∂ ∂ = ∑ ∑ = = λ λ Legendre’s Dual Transformation By differentiating the General Momentum Equation and using Lagrange’s Equations we obtain: ( )∑∑ == −++ ∂ ∂ −=++ ∂ ∂ = ∂ ∂ = m j j ijji i m j j iji ii i cQ q H cQ q T q T dt d p 11 ''''' λλλ ni cQ q H p p H q m j j iji i i i i ,,2,1 1 = ++ ∂ ∂ −= ∂ ∂ = ∑= λ mkcqc k t n i i k i ,,2,10 1 ==+∑= Extended Hamilton’s Equations Constrained Differential Equations
- 48. 48 Analytic DynamicsSOLO Hamilton’s Equations (continue( For Holonomic Constraints (constraints of the form ( we can (theoretically( reduce the number of generalized coordinates to n-m and we can assume that and n represents the number of degrees of freedom of the system (this reduction is not possible for Nonholonomic Constraints(. Then: ( ) mjtqqf nj .,10,,,1 == 0== j t j i cc ni Q q H p p H q i i i i i ,,2,1 = + ∂ ∂ −= ∂ ∂ = Extended Hamilton’s Equations for Holonomic Constraints ni q V Q i i ,,2,1 = ∂ ∂ −= ( ) ( ) ( )qVtqqTtqqL −= ∆ ,,,, ni q T p i i ,,2,1 = ∂ ∂ = Extended Hamilton’s Equations for Holonomic Constraints and a Conservative System Conservative System
- 49. 49 Analytic DynamicsSOLO Hamilton’s Equations (continue( Define: Hamiltonian for Conservative Systems ( ) ( ) ( ) ( )qVtqqTtqqLqptqqH n i ii −=−=∑= ∆ ,,,,,, 1 Hamilton’s Canonical Equations for Conservative Systems with Holonomic Constraints ni q H p p H q i i i i ,,2,1 = ∂ ∂ −= ∂ ∂ = We have:
- 50. 50 Analytic DynamicsSOLO 6.Kane’s Equations In terms of generalized coordinates we can write: Thomas R. Kane 1924- Stanford University ( ) ( ) ( ) ( ) ( ) Niktqzjtqyitqxtqqrtqr iiinii .,1,,,,,,, 1 =++== ( ) Nikdzjdyidxdt t r dq q r tqrd iii i n j j j i i .,1, 1 =++= ∂ ∂ + ∂ ∂ = ∑= ( ) Ni t r q q r td tqrd rv i n j j j ii ii .,1 , 1 = ∂ ∂ + ∂ ∂ === ∑= → 6.1 6.2 6.3 Kane and Levinson have shown that with the n generalized coordinates , is useful to define another n variables , which are linear functions of the n : jq iu jq nrZqYu r n j jrjr .,1 1 =+=∑= ∆ 6.4 where the matrix is invertible and[ ] { } nj nrrjYY ,1 ,1 = = = [ ] [ ] { } nj nrrjWWY ,1 ,1 1 = = − == njXuWq j n r rrjj .,1 1 =+= ∑= 6.5 jrrjrj XandZWY ,, are functions of tandq are called Generalized Speed (also Nonholonomic Velocities, Quasivelocities. etc.( and are not unique. iu
- 51. 51 Analytic DynamicsSOLO Kane’s Equations (continue( Nonholonomic constraints are linear relations among either or the ; for m nonholonomic constraints: 6.6 6.7 where k may be n-m or n, depending on whether the nonholonomic constraints are incorporated. iu jq nmnsBuAu s mn r rsrs .,1 1 +−=+= ∑ − = ∆ If we substitute equations (6.6( in (6.5( we obtain a more general expression for :jq njXuWq j k r rrjj .,1 1 =+= ∑= Let substitute equation (6.6( in (6.3(: ( ) Ni t r X q r uW q r t r XuW q r td tqrd rv i k r n j j j i r n j rj j i i n j j k r rrj j ii ii .,1 , 1 11 1 1 = ∂ ∂ + ∂ ∂ + ∂ ∂ = = ∂ ∂ + + ∂ ∂ === ∑ ∑∑ ∑ ∑ = == = = → From this equation we can see that ∑= ∂ ∂ = ∂ ∂ n j rj j i r i W q r u v 1
- 52. 52 Analytic DynamicsSOLO Kane’s Equations (continue( 6.8 6.9 Let use now the equation of differential work: By defining we obtain: t r X q r v i n j j j ii t ∂ ∂ + ∂ ∂ =∑= ∆ 1 Nivu u v v t i k r r r i i .,1 1 =+ ∂ ∂ = ∑= ∑∑ == ⋅=⋅= N i iii N i ii rdamrdFdW 11 Equation (6.9( is now rewritten using (6.8(. On the left side we obtain: dtvu u v FdtvFrdF N i t i k r r r i i N i ii N i ii + ∂ ∂ ⋅=⋅=⋅ ∑ ∑∑∑ = === 1 111 6.10 Similarly, the right side of (6.9( becomes: dtvu u v amdtvamrdam N i t i k r r r i ii N i iii N i iii + ∂ ∂ ⋅=⋅=⋅ ∑ ∑∑∑ = === 1 111 6.11 Equations (6.10( and (6.11( are equated and terms re collected: ( ) ( ) 0 11 11 = −⋅+ −⋅ ∂ ∂ +⋅ ∂ ∂ ∑∑ ∑∑ == == dtamFvdtuam u v F u v N i iii t i k r r N i ii r i N i i r i 6.12
- 53. 53 Analytic DynamicsSOLO Kane’s Equations (continue( ( ) ( ) 0 11 11 = −⋅+ −⋅ ∂ ∂ +⋅ ∂ ∂ ∑∑ ∑∑ == == dtamFvdtuam u v F u v N i iii t i k r r N i ii r i N i i r i 6.12 The and dt are nonzero and independent and so the coefficients of each of them must be zero. Also using Newton’s Second Law: we have: krur ,1= 0=− iii amF nrZqYu r n j jrjr ,.,1 1 =+=∑= ∆ krF u v Q N i i r i r ,,1 1 =⋅ ∂ ∂ =∑= ∆ ( ) kram u v Q N i ii r i r ,,1' 1 =−⋅ ∂ ∂ =∑= ∆ krQQ rr ,,10' ==+ 6.4 6.13 6.14 6.15 Generalized Speeds Generalized Active Forces Generalized Inertia Forces Kane’s Equations
- 54. 54 Analytic DynamicsSOLO Gibbs-Appell Equations 7.1 Josiah Willard Gibbs 1839-1903 Paul Emile Appell 1855-1930 Differentiation of equation (6.8( gives: Nivu u v v t i k r r r i i .,1 1 =+ ∂ ∂ = ∑= About 100 years after Lagrange, Gibbs in 1879 and Appell in 1899; independently devise what is known the Gibbs-Appell Equations. Niv dt d u u v dt d u u v va t i k r r r i k r r r i ii .,1 11 =+ ∂ ∂ + ∂ ∂ == ∑∑ == From this equation we see that: r i r i u v u a ∂ ∂ = ∂ ∂ 7.2 If we substitute equation (7.2( in (6.14( ( ) ( ) krG u aam u am u a am u v Q r N i iii r N i ii r i N i ii r i r ,,1 2 1 ' 111 = ∂ ∂ −=⋅ ∂ ∂ −=−⋅ ∂ ∂ =−⋅ ∂ ∂ = ∑∑∑ === ∆ 7.3 ( ) kram u v Q N i ii r i r ,,1' 1 =−⋅ ∂ ∂ =∑= ∆ 6.14
- 55. 55 Analytic DynamicsSOLO Gibbs-Appell Equations (continue( ( ) ( ) krG u aam u am u a am u v Q r N i iii r N i ii r i N i ii r i r ,,1 2 1 ' 111 = ∂ ∂ −=⋅ ∂ ∂ −=−⋅ ∂ ∂ =−⋅ ∂ ∂ = ∑∑∑ === ∆ 7.3 From and krQQ rr ,,10' ==+6.15 krQG u r r ,,1 == ∂ ∂ ∑= ∆ ⋅= N i iii aamG 1 2 1 nrZqYu r n j jrjr ,.,1 1 =+=∑= ∆ krF u v Q N i i r i r ,,1 1 =⋅ ∂ ∂ =∑= ∆ Gibbs-Appell Equations Gibbs Function: Generalized Speed: Generalized Active Forces:
- 56. 56 Analytic DynamicsSOLO References: ]1[Goldstein, H., Classical Mechanics, 2nd ed., Addison-Wesley, 1981 ]2[Meirovitch, L., Methods of Analytical Dynamics, Mc Graw-Hill, 1970 ]3[Greenwood, D.T., Principle of Dynamics, 2nd ed., Prentice-Hall, 1977 ]4[Kane, T.R., Dynamics, 3th ed., Stanford University, 1972 ]5[Desloge, E.A., Relationship Between Kane’s Equations and the Gibbs-Appell Equations, J. Guidance, Vol. 10, No. 1, Jan.-Feb., 1987
- 57. August 12, 2015 57 SOLO Technion Israeli Institute of Technology 1964–1968BSc EE 1968–1971MSc EE Israeli Air Force 1970–1974 RAFAEL Israeli Armament Development Authority 1974–2013 Stanford University 1983–1986PhD AA