lagrange’s equations of motion with constraint forces · meam 535 university of pennsylvania 1...

MEAM 535

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Lagrange’s Equations of Motion with Constraint Forces

Kane’s equations do not incorporate constraint forces

MEAM 535


Review: Linear Algebra

A is m × n of rank r   The row space, Col (AT), dimension = r   Col (AT) is spanned by:

  The null space, N (A), dimension = n – r

http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/VideoLectures/detail/lecture14.htm

€

r1 =

a11

a12

a1n

, r2 =

a21

a22

a2n

, , rm =

am1

am2

amn

.

r1 r2

rm

€

x ∈ N A( )

N (A)

Col (AT) orthogonal

Rn

Ax=0

MEAM 535


Review: Linear Algebra A is m × n of rank r

  The column space, Col (A), dimension = r   Col (A) is spanned by columns

  The left null space, N (AT), dimension = m – r

yTA =0

c1 c2

cn

€

y ∈ N AT( )

€

c1 =

a11

a21

am1

, c2 =

a12

a22

am2

, , cn =

a1n

a2n

amn

.

N (AT)

Col(A) orthogonal

dim Col(A) = r dim Col(AT) = r dim N(A) = n – r dim N(AT) = m – r

Rm

MEAM 535


Application to Velocity Constraints

C is m × n of rank m   The row space, Col (CT), dimension = m   Col (CT) is spanned by:

  The null space, N (C), dimension = n – m

€

r1 =

c11

c12

c1n

, r2 =

c21

c22

c2n

, , rm =

cm1

cm2

cmn

.

r1 r2

rm

€

˙ q ∈ N C( )

N (C)

Col (CT) orthogonal

Rn

€

Cm×n ˙ q n×1 = 0

N (C) set of admissible velocities (that don’t violate constraints)

Physical Interpretation of N(C)

Col (CT) is the set of constraint forces orthogonal to admissible velocities!

= Col (Γ)

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Example 2: Rolling Disk (Simplified)

(x, y) φ

θ radius R

C

τd

τs

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Example: Rolling Disk Simplified

(x, y) φ

θ radius R

C

τd

τs

C Γ 2×4 4×2 €

˙ q n×1 = Γ n× p up×1

€

Cm×n Γ n× p = 0

Two equations of motion

Nonholonomic constraints provide two additional equations

€

Cm×n ˙ q n×1 = 0

MEAM 535


Example 2: Rolling Disk (Simplified)

(x, y) φ

θ radius R

C

€

′ Q * =

−m˙ q 1−m˙ q 2−Ia˙ q 3−It˙ q 4

, ′ Q =

00τ d

τ s

τd

τs

PT Γ

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Application to Generalized Forces

Γ is n× p of rank p   The column space, Col (Γ), dimension = p   The left null space, N (ΓT), dimension = n – p = m

PTΓ =0

c1 c2

cp

€

P ∈ N ΓT( )

N (ΓT)

Col(Γ) orthogonal

Rn

€

c1 =

Γ11

Γ21

Γn1

, c2 =

Γ12

Γ22

Γn2

, , c p =

Γ1p

Γ2p

Γnp

.

Physical Interpretation of N(ΓT) set of admissible velocities (that don’t violate constraints)

= Col (Γ)

set of constraint forces

= N(ΓΤ) = Col (CT)

= N(C)

MEAM 535


Lagrange’s Equations with Multipliers

Pk

p Equations of Motion

m Constraints

m Columns of CT span the null space of ΓT

There exist a vector of m constants (multipliers) λ, such that

P lies in the null space of ΓT

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Example: Rolling Disk Simplified

(x, y) φ

θ radius R

C

τd

τs

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p+m+m equations in p+m+m unknowns

p+m equations in p+m unknowns

Rolling Disk Simplified: Comparison

(x, y) φ

θ radius R

C

τd

τs

C Γ m×n n×p

p eq

uatio

ns

n eq

uatio

ns

m e

quat

ions

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Multipliers are Constraint Forces!

m Columns of CT span the null space of ΓT

1. CTλ are generalized forces (associated with the derivatives of generalized coordinates)

2. The m constants (multipliers) are coefficients for vectors that are orthogonal to the allowable directions of motion

(x, y) φ

θ radius R

C

τd

τs

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Constraint Forces for Holonomic Systems

What if you choose more generalized coordinates than necessary?   n is the number of generalized coordinates (more than necessary)   p is the number of degrees of freedom (i.e., only p gen. coords. necessary)   n > p

Notice the parallel with nonholomic systems!   n speeds, but only p independent speeds

€

′ Q k + ′ Q k*( )Wkj

k=1

n

∑ = 0


m Constraints €

˙ q n×1

= W[ ]n× pup×1+ X

n×1

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Example: Particle in circular hoop

θ R x

y

where

Pr =d

dt

!!L!r

"! !L

!r= mr !mr"2 + mg sin "

P! =d

dt

!!L!"

"! !L

!"= mr2" + mgr cos ".

Because of the hoop, there are constraints:

C(q)q = 0 (1)

or

[1 0]

#

$r

"

%

& = 0,

so that

CT (q) =

#

$1

0

%

& (2)

The equations of motion are given by:

P = CT #,

or

Pr = 1.#

P! = 0.#,

where # is the Lagrange multiplier.

From (1), r = r = 0. substituting into the equations of motion we get:

!mr"2 + mg sin " = # (3)

mr2" + mgr cos " = 0. (4)

From (3), it is clear that # is the outward pointing normal force acting on the particle.

The Kane Lagrange equations of motion are obtained by recognizing that

q = !u =

#

$0

1

%

& " (5)

and writing

PT ! =0 ,

or

mr2" + mgr cos " = 0 (6)

2

LHS of Lagrange’s equations of motion for unconstrained problem

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Example Generalized speed:

  u=dθ/dt Velocities

 

Generalized Active Forces   -Fa1

  τa3

Generalized Active Inertial Forces   -m AaP = -m x2dot a1

B

P

x

Find the constraint forces at the pin joint Q

Q

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Example (continued)

θ d e

φ

p=1

n=3

€

˙ q n×1

= W[ ]n× pup×1+ X

n×1

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Constraint Forces for Holonomic Systems


m Constraints €

˙ q n×1

= W[ ]n× pup×1+ X

n×1

There exist a vector of m constants (multipliers) λ, such that

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Example: Normal Force at P Generalized speed:

  u=dθ/dt Velocities

 

Generalized Active Forces   -Fa1

  τa3

Generalized Active Inertial Forces  

B

P

x

What if we relax the constraint that keeps the piston moving horizontally?

MEAM 535


Example (continued)

n=1

p=1, n=2

€

˙ q n×1

= W[ ]n× pup×1+ X

n×1

θ y φ r l

C

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Example: Normal Force at P Generalized speeds

 

Partial Velocities  

Generalized Active Forces   -Fa1   τa3

Generalized Active Inertial Forces    

B

P

θ y φ r l

x

lagrange’s equations of motion with constraint forces · meam 535 university of pennsylvania 1...

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