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ME 6590 Multibody Dynamics

Lagranges Equations for MDOF Systems with Constraints

Background

o As discussed in earlier notes, dynamic systems may be subjected to holonomic and/ornonholonomic constraints. The nature of these constraints determines how weincorporate them into Lagrange's equations. Consider the following three scenarios:

a) The constraints are holonomic, and it is easy to use them to eliminate the surplus

generalized coordinates. It is helpful here if the resulting expressions for the kinetic

and potential energies of the system arereasonable to differentiate.

b) The constraints are holonomic; however, elimination of surplus generalized

coordinates leads to very complicated expressions for the kinetic and potential

energies of the system making them difficult to differentiate.

c) The constraints are nonholonomic, and so we cannot eliminate surplus generalizedcoordinates (because the constraints are not integrable).

o In the first scenario, we may choose to eliminate the surplus coordinates and use theform of Lagrange's equations without constraints (discussed in earlier notes).

o In the second and third scenarios, we use a dependent set of generalized coordinates, andso we require a form of Lagranges equations that incorporates the constraints into the

formulation of the equations of motion. This form of Lagrange's equations is presented

below.

Lagrange's Equations with Constraints

o If the configuration of a dynamic system is to be described using n generalizedcoordinates ( 1, , )kq k n= and if the system is subjected to m independent holonomic

and/or nonholonomicconstraints of the form

0

1

0n

jk k j

k

a q a=

+ = ( 1, , )j m= (1)

then the system possesses N n m= degrees of freedom (DOF).

o In this case, the equations of motion of the system may be formulated usingLagrangesequations withLagrange multipliers

1k

m

q j jk

jk k

d K KF a

dt q q

=

= +

( 1, , )k n= (2)

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o Here, Kis thekinetic energy of the system,kq

F is thegeneralized force associated with

the generalized coordinate kq , j is the Lagrange multiplier associated with theth

j

constraint equation, and ( 1, , ; 1, , )jka j m k n= = are the constraint equation

coefficients.

o If some of the forces and torques areconservative, then we can define their contributionto the equations of motion in terms of a potential energy function. In this case, the

equations of motion can be written as

( )1

k

m

q j jk nc

jk k

d L LF a

dt q q

=

= +

( 1, , )k n= (3)

o Here, L K V= is theLagrangianof the system, Vis thepotential energy function fortheconservativeforces and torques, ( )

kq ncF is thegeneralized force associated with kq for the nonconservative forces and torques, only, j is the Lagrange multiplier

associated with the thj constraint equation, and ( 1, ; 1, )jka j m k n= = are the

constraint equation coefficients.

o The n Lagranges equations and the m constraint equations (1) form a set of n m+ differential/algebraic equations for the n m+ unknowns the n generalizedcoordinates ( 1, , )kq k n= and the m Lagrange multipliers ( 1, , )j j m = .

o The equations are differential in the generalized coordinates andalgebraic in theLagrange multipliers. The Lagrange multipliers are related to theforces andmomentsrequired tomaintain the constraints. Note that, it is essential that K,

kqF , and jka be

written only in terms of kq and kq and no other variables.

Example: Equations of Motion of the Simple Pendulum

Approach #1: Using as the generalized coordinate

o The simple pendulum is a single degree of freedom(SDOF) system, and using as the generalized

coordinate, we can write the Lagrangian as

2 212

cos( )L mL mgL = +

Using Lagrange's equations without constraints, we find the equation of motion to be

sin( ) 0g

L + = (4)

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Approach #2: Using ( ),x y as the generalized coordinates

o Using two generalized coordinates for an SDOF system, we must use Lagrangesequations with a constraint. In this case, we can write aconfiguration constraint as

2 2 2x y L+ =

o Differentiating, we can write the constraint in the form of Eq. (1) and identify thecoefficients ( 1; 0,1, 2)jka j k= =

0xx yy+ = and 10 11 120, ,a a x a y= = = (5)

o Lagrange's equations with constraints may then be written as1k

k k

d L La

dt q q

=

( 1,2)k = (6)

o The Lagrangian may be written in terms of the generalized coordinates and theirderivatives as

2 212

( )L m x y mgy= + +

o Using Eq. (6) and the definitions of the constraint equation coefficients in Eq. (5), wefind the following two differential/algebraic equations

0

0

mx x

my mg y

=

=

(7)

o Differentiating the constraint Eq. (5) again, and combining it with Eq. (7) gives thefollowingcomplete set ofthree differential/algebraic equations for( ), ,x y .

2 2

0

0

0

mx x

my mg y

xx yy x y

=

=

+ + + =

(8)

o Alternatively, we can use the first two of the above equations to eliminate theLagrangemultiplier . For example, if we multiply the first equation by y and the second

equation by x andsubtract the two equations, we can reduce the first two equations to

a single equation. Then we have onlytwo coupled differential equations of motion

2 2

0

0

yx xy gx

xx yy x y

=

+ + + =

(9)