Holonomic and Nonholonomic Constraints

8/3/2019 Holonomic and Nonholonomic Constraints

1/20

MEAM 535

University of Pennsylvania 1

Holonomic and Nonholonomic Constraints


2/20

MEAM 535


Holonomic Constraints

Constraints on the position

(configuration) of a system of particles

are called holonomic constraints.

Constraints in which time explicitly

enters into the constraint equationare

called rheonomic.

Constraints in which time is notexplicitly present are called

scleronomic.

Note: Inequalities do not constrain the position in

the same way as equality constraints do.

Rosenberg classifies inequalities as nonholonomic

constraints.

Particle is constrained to lie on a

plane:

A x1 +B x2 + C x3 +D = 0

A particle suspended from a string

in three dimensional space.

(x1a)2 +(x2b)

2 +(x3c)2r2 = 0

A particle on spinning platter

(carousel)

x1

= a cos(t+ );x2

= a sin(t+ )

A particle constrained to move on a

circle in three-dimensional space

whose radius changes with time t.x1 dx1 +x2 dx2 + x3 dx3 - c

2 dt= 0


3/20

MEAM 535


Holonomic Constraint

x1

x2

x3 Configuration space

f(x1,x2,x3)=0


4/20

MEAM 535


Scleronomic and Rheonomic

tScleronomic

f(x1,x2)=0

x1

f(x1,x2)=0

x2

Configuration space

x1

x2

Rheonomic

f(x1,x2, t)=0

f(x1,x2 , t1)=0

f(x1,x2 , t2)=0

f(x1,x2 , t3)=0

f(x1,x2 , t4)=0


5/20

MEAM 535


Nonholonomic Constraints

A particle constrained to move on a

circle in three-dimensional space

whose radius changes with time t.

x1

dx1

+x2

dx2

+ x3

dx3

- c2 dt= 0

The knife-edge constraint

Definition 1

All constraints that are not

holonomic

Definition 2

Constraints that constrain the

velocities of particles but not their

positions

We will use the second definition.

0cossin 3231 = xxxx &&

u

1

2

3

C


6/20

MEAM 535


When is a constraint on the motion nonholonomic?

Velocity constraint

Or constraint on instantaneous motion

Question

Can the above equation can be reduced

to the form:

f(x1,x2, ...,xn-1, t) = 0

P dx + Q dy +Rdz=0

3 dimensional case0... 112211 =++++ nnn axaxaxa &&&

=

R

Q

P

v

0... 112211 =++++ dtadxadxadxa nnn

vPfaffian Form


7/20

MEAM 535


When is a scleronomic constraint on motion in a three-dimensional

configuration space nonholonomic?Velocity constraint

Or constraint in the Pfaffian formP dx + Q dy +Rdz=0 (1)

QuestionCan the above equation can be reduced

to the form:

f(x,y,z)=0

0=++ zRyQxP &&&

If it is an exact differential, there

must exist a functionf, such that

z

fR

y

fQ

x

fP

=

=

= ,,

.,,z

Q

y

R

x

R

z

P

x

Q

y

P

=

=

=

Or,

Can we at least say when the differential

form (1) an exact differential?

A sufficient condition for (1) to be

integrable is that the differential

form is an exact differential.

The necessary and sufficient

conditions for this to be true is that

the first partial derivatives ofP, Q,

andR with respect tox,y, andz

exist, and

df= P dx + Q dy +Rdz Recall Stokes Theorem!


8/20

MEAM 535



configuration space nonholonomic? If (2) is an exact differential, theremust exist a functiong, such that

The necessary and sufficient

conditions for this to be true is that

the first partial derivatives ofP, Q,

andR with respect tox,y, andz

exist, and

z

gR

y

gQ

x

gP

=

=

= ,,

( ) ( )

( ) ( )

( ) ( ).

,

,P

z

Q

y

R

xR

zP

x

Q

y

=

=

=

Constraint in the Pfaffian form

P dx + Q dy +Rdz=0 (1)

Question

Can the above equation can be reduced

to the form:

f(x,y,z)=0

For the constraint to be integrable, it is

necessary and sufficient that there existan integrating factor(x,y,z), such that,

P dx + Q dy + Rdz=0 (2)

be an exact differential.


9/20

MEAM 535



configuration space nonholonomic?( ) ( )

( ) ( )

( ) ( ).

,

,P

z

Q

y

R

xR

zP

x

Q

y

=

=

=

.

,

,

=

=

=

y

R

z

QQ

zR

y

zP

xRR

xP

z

y

P

x

QQ

xP

y

vv =

=

R

Q

P

v

0= vv

Necessary and sufficient condition for

(2) to be holonomic, provided v is a

well-behaved vector field and


10/20

MEAM 535


Examples

1. sinx3 dx1 - cos x3 dx2 = 0

=

0

cos

sin

3

3

x

x

v

2. 2x2x3 dx1 +x1x3 dx2 + x1x2 dx3 = 0

=

21

31

322

xx

xx

xx

vx1 (2x2x3 dx1 + x1x3 dx2 + x1x2 dx3) = 0

d((x1)2x2x3) = 0

0321 = xxx &&&3.


11/20

MEAM 535


Holonomic

0= vv

0= v

P dx + Q dy +Rdz=0

Other nonholonomic constraints

Holonomic

Nonholonomic

Nonholonomic constraints in 3-D

A 3


12/20

MEAM 535


Extensions: 1. Multiple Constraints

dx2x3 dx1 = 0

and

dx3x

1dx

2= 0

Are the constraint equations non holonomic?

Individually: YES!

Together:

+== cdxkexkex

xx

1232

21

2

21

,

dx3x1 dx2 = dx3x1 (x3 dx1) = 0

MEAM 535


13/20

MEAM 535


Extensions 2: Constraints in > 3 generalized speeds

n dimensional configuration space

m independent constraints (i=1,..., m)

01

==

n

j

ijij dtbdxa

or

=

=n

j

ijij bua1

0

MEAM 535


14/20

MEAM 535


(c) Rolling disk.

Eliminate u4 and u5 in terms ofu1, u2and u3.

( ) ( )

0

2332211

*

*

*

=

+++=

+=

bbbbv

vv

Ruuu

PC

CA

CACAPA

Rolling constraint

0=PA

v

x

y

z

P

q2

q3q1

q5

q4

C

b1

b2

b3 C*

e1

e2

e3

a1

a2

a3

0sincossinsin

0tansincos

215214

3221514

=+

=++

qquqqu

RuqRuququ

m=2 constraints

n=5 generalized coordinates

MEAM 535


15/20

MEAM 535


Frobenius Theorem: Generalization to n dimensons

n dimensional configuration space

m independent constraints (i=1,..., m)

The necessary and sufficient condition for the existence of m independent

equations of the form:

fi(x

1,x

2, ...,x

n) = 0, i=1,..., m.

is that the following equations be satisfied:

where uk and wl are components of any two n vectors that lie in the null space of

the mxn coefficient matrix A = [aij]:

= =n

jjij dxa

10

= ==

n

k lk

n

l l

ik

k

il wux

a

x

a

1 1

0

,0,011 == ==

n

jjij

n

jjij waua

v

u

w

MEAM 535


16/20

MEAM 535


Generalized Coordinates and Speeds

Holonomic Systems

Number of degrees of freedom of a systemin any reference frame

the minimum number of variables tocompletely specify the position of every

particle in the system in the chosenreference

The variables are called generalizedcoordinates

There can be no holonomic constraint

equations that restrict the values thegeneralized coordinates can have.

q1, q2, ..., qn denote the generalizedcoordinates for a system with n degrees offreedom in a reference frameA.

n generalized coordinates specify the

position (configuration of the system)

The number of independent speeds is also

equal to n

In a system with n degrees of freedom in a

reference frame A, there are n scalar

quantities, u1, u2, ..., un (for that reference

frame) called generalized speeds. They that

are related to the derivatives of the

generalized coordinates by :

where the nxn matrix Y = [Yij

] is non

singular and Z is a nx1 vector.

( ) ( )tqqZqtqqYu in

jjiji ,...,,,...,, 21

1

21=

+= &

MEAM 535


17/20

MEAM 535


Example 1

Generalized Coordinates

q1 , q2 , q3 , q4 , q5

Generalized Speeds

AC= u1 b1 + u2 b2 + u3 b3

u4 = derivative ofq4

u5 = derivative ofq5

x

y

z

P

q2

q3q1

q5

q4

C

b1

b2

b3

C*

Locus of the

point of contact Q on

the planeA

=

5

4

3

2

1

5

4

3

2

1

qq

q

q

q

uu

u

u

u

&&

&

&

&

Y

MEAM 535


18/20

MEAM 535


Example 2

Generalized coordinates inA

s,

Generalized speeds

Generalized speeds and derivatives of

generalized coordinates

Bug on the turntable

b1

b2s

dt

dyu

dt

dxu

=

=

2

1

( ) ( )tqqZqtqqYu in

jjiji ,...,,,...,, 21

1

21=

+= &

Appears in

rheonomic constraints

a2

a1

MEAM 535


19/20

MEAM 535


Nonholonomic Constraints are Written in Terms of Speeds

m constraints in n speeds

m speeds are written in terms of the n-m

(p) independent speeds

Define the number of degrees of

freedom for a nonholonomic system in a

reference frameA asp, the number of

independent speeds that are required to

completely specify the velocity of any

particle belonging to the system, in the

reference frameA.

( ) ( ) 0,...,,,...,, 211

21 =+=

tqqDutqqC i

n

jjij

( ) ( ) 0,...,,,...,, 211

21 =+= =

tqqButqqAu i

p

kkiki

MEAM 535


20/20

MEAM 535


Example 3

Number of degrees of freedom

nm = 2 degrees of freedom

Generalized coordinates (x1,x2,x3)

Speeds

forward velocity along the axis of

the skate, vf

the speed of rotation about the

vertical axis, and the lateral (skid) velocity in the

transverse direction, vl

=

=

=

=

l

f

v

v

u

u

u

x

x

x

q

q

q

3

2

1

3

2

1

3

2

1

, uq

u

1

2

3

C

ZqYu += &

=

=

0

0

0

,

0cossin

100

0sincos

33

33

ZY

xx

xx

Documents

Holonomic and Nonholonomic Constraints