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Holonomic and Nonholonomic Constraints
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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
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Holonomic Constraint
x1
x2
x3 Configuration space
f(x1,x2,x3)=0
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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
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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
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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
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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!
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When is a scleronomic constraint on motion in a three-dimensional
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.
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When is a scleronomic constraint on motion in a three-dimensional
configuration space nonholonomic?( ) ( )
( ) ( )
( ) ( ).
,
,P
z
Q
y
R
xR
zP
x
Q
y
=
=
=
.
,
,
=
=
=
y
R
z
zR
y
zP
xRR
xP
z
y
P
x
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
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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.
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Holonomic
0= vv
0= v
P dx + Q dy +Rdz=0
Other nonholonomic constraints
Holonomic
Nonholonomic
Nonholonomic constraints in 3-D
A 3
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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
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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
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(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
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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
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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=
+= &
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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
q
q
q
uu
u
u
u
&&
&
&
&
Y
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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
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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
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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