24
MEAM 535 University of Pennsylvania 1 Principle of Virtual Work Aristotle Galileo (1594) Bernoulli (1717) Lagrange (1788)
MEAM 535
University of Pennsylvania 1
Principle of Virtual Work
AristotleGalileo (1594)Bernoulli (1717)Lagrange (1788)
MEAM 535
University of Pennsylvania 2
Virtual WorkKey Ideas
Virtual displacementSmallConsistent with constraintsOccurring without passage of time
Applied forces (and moments)Ignore constraint forces
Static equilibriumZero acceleration, orZero mass
O
rPi
Fi(a)
e2
e1
e3
[ ]∑=
δ⋅=δN
i
Pai
iW1
)( rF n generalized coordinates, qj
∑= =
δ
∂∂⋅=δ ∑
n
jj
N
i j
Pa
i qq
Wi
1 1
)( rF
MEAM 535
University of Pennsylvania 3
ExampleB
P
r
l
θφ
m F
G=τ/2rQ
x
Applied forcesF acting at PG acting at Q
Constraint forces?
Single degree of freedomGeneralized coordinate, θ
Motion of particles P and Q can be describedby the generalized coordinate θ
MEAM 535
University of Pennsylvania 4
Static Equilibrium Implies Zero Virtual Work is DoneForces
Forces that do workApplied ForcesExternal Forces
Forces that do no workConstraint forces Fi
(a)
Ri
[ ] 0)( =+ ia
i RF
Implies sum of all forces on each particle equals zero
[ ] 01
)( =+∑=
N
ii
ai RF [ ] 0.
1
)( =δ+∑=
i
N
ii
ai rRF
Static Equilibrium
MEAM 535
University of Pennsylvania 5
The Key Idea
Constraint forces do zero virtual work!
[ ] 0.1
)( =δ+∑=
i
N
ii
ai rRF
0
Why?
MEAM 535
University of Pennsylvania 6
Constraints: Two Particles Connected by Rigid Massless Rod
(x1 , y1)
(x2 , y2)e
F1 R1
F2
F1
R2
F2
(x1 – x2)2 +(y1 – y2)2 = r2
R1 = -R2 = αe( )( ) ( )( ) 021212121 =δ−δ−+δ−δ− yyyyxxxx
( ) ( )[ ] ( ) ( )[ ] 02
22121
1
12121 =
δδ
−−−
δδ
−−yx
yyxxyx
yyxx
( ) 021 =δ−δ⋅ rre ( )0
.....
21
212211
=δ−δα=δα−δα=δ+δ
rrerererRrR
MEAM 535
University of Pennsylvania 7
Rigid Body: A System of Particles
A rigid body is a system of infinite particles.
The distance between any pair of particles stays constant through its motion.
Each pair of particles can be considered as connected by a massless, rigid rod.
The internal forces associated with this distance constraint areconstraint forces.
The internal forces do no virtual work!
MEAM 535
University of Pennsylvania 8
Contact Constraints and Normal Contact ForcesRigid body A rolls and slides on rigid body B
A
B
P1
P2
n
contactnormal
AvP2 .n = BvP1 .n
O
r1
r1
Contact Kinematics CvP2 .n = CvP1 .nδr2 .n = δr1 .n
Contact Forces
N1 = -N2 = αn
N2 N1
T1
T2
( )0
.....
21
212211
=−=−=+
rrnrnrnrNrN
δδαδαδαδδ
C
MEAM 535
University of Pennsylvania 9
Normal and Tangential Contact Forces
1. Normal contact forcesNormal contact forces are constraint forces Equivalently, normal forces do no virtual work
2. Tangential contact forcesIf A rolls on B (equivalently B rolls on A)
then, tangential contact forces are constraint forcesIn general (sliding with friction), tangential forces will contribute to virtual work
AvP2 = BvP1
AvP2 = BvP1
( )0
.....
21
212211
≠δ−δβ=δβ−δβ=δ+δ
rrtrtrtrTrT
N2 N1
T1
T2
t
MEAM 535
University of Pennsylvania 10
StatementA system of N particles (P1, P2,…, PN) is in static equilibrium if and only if the virtual work done by all the applied (active) forces though any (arbitrary) virtual displacement is zero.
A holonomic system of N particles is in static equilibrium if and only if all the generalized (active) forces are zero.
Only “applied” or “active” forces contribute to the generalized forceThe jth generalized force is given by
01 1
)( =∑= =
δ
∂∂⋅=δ ∑
n
jj
N
i j
Pa
i qq
WirF
∑
∑
=
=
∂∂⋅=
∂∂⋅=
N
i j
Pa
i
N
i j
Pa
ij
q
i
i
1
)(
1
)(
&
&rF
rFWhy?
MEAM 535
University of Pennsylvania 11
Velocity PartialsIn any frame A
Define the jth velocity partial
The jth generalized force is
( )
dtdq
qdtdq
qdtdq
qt
dtd
tqqq
n
n
PPPP
PPA
nPP
iiii
ii
ii
∂∂
++∂∂
+∂∂
+∂∂
=
=
=
rrrr
rv
rr
K
K
2
2
1
1
21 ,,,,
nPn
PPP
PA qqqt
iiii
i &K&& vvvrv ++++∂∂
= 2211
j
P
j
PPj qq
iii
&
&
∂∂
=∂∂
=rrv
n speeds
[ ]∑=
⋅=N
i
Pj
aij
iQ1
)( vF
a1
a3
a2
O
A
Pi
rPi dtd i
iPA
PA rv =
MEAM 535
University of Pennsylvania 12
Example Illustrating Partial VelocitiesThree Degree-of-Freedom Robot Arm
REFERENCE POINT
l1
l2
l3θ3
θ2
θ1
φ(x,y)
x
y
( ) ( )( ) ( )
( )321
321321211
321321211
sinsinsincoscoscos
θ+θ+θ=φθ+θ+θ+θ+θ+θ=θ+θ+θ+θ+θ+θ=
lllylllx
( ) ( )( ) ( )
( )321
123321312212111
123321312212111
θ+θ+θ=φ
θ+θ+θ+θ+θ+θ=
θ+θ+θ−θ+θ−θ−=
&&&&
&&&&&&&
&&&&&&&
clclclyslslslx
33
22
11
θ=
θ=
θ=
&
&
&
uuu
differentiating
MEAM 535
University of Pennsylvania 13
Example (continued)Equations relating the joint velocities and the end effector velocities
The three partial velocities of the point P (omitting leading superscript A) are columns of the “Jacobian” matrix
( ) ( )( ) ( ) 123321312212111
123321312212111
clclcly
slslslx
θ+θ+θ+θ+θ+θ=
θ+θ+θ−θ+θ−θ−=&&&&&&&
&&&&&&&
REFERENCE POINT
l1
l2
l3θ3
θ2
θ1
φ(x,y)
x
y
P1v
P2v
P3v
P
in matrix form
( ) ( )( ) ( )
+++
−+−++−=
3
2
1
12331233122123312211
12331233122123312211
uuu
clclclclclclslslslslslsl
yx&
&
P1v P
2v P3v
MEAM 535
University of Pennsylvania 14
Example 1Generalized speed:
u=dθ/dtVelocities
Generalized Active ForcesF = -Fa1
No friction, gravity
B
P( )11 cos
sin avφφ+θ
−=RPA
φφ=θθφφ−θθ−=
φ=θφ+θ=&&&&& coscos;sinsin
sinsin;coscoslrlrx
lrlrx
θφθ
=φ &&coscos
lr
r
l
θφ
m F
G=2τ/rQ
( )211 cossin2
aav θ+θ−−=rQA
( )21 cossin2 aaG θ+θ−τ
=r
QPQ 111 vGvF ⋅+⋅=( )φ
φ+θ+τ=
cossin
1FRQ
x
MEAM 535
University of Pennsylvania 15
Example 2
Chθ
φ
2l
O
P
Q
C
P
Q
AssumptionsNo friction at the wallGravity (center of mass is at midpoint, C)Massless string, OP
homogeneous rod, length 3l
l
MEAM 535
University of Pennsylvania 16
Equivalent System of Forces
Fi
iFj
O
rjri
A system of forces acting on a rigid body can be replaced by
A resultant force F
A moment about a convenient reference point O
∑=
=r
ii
1FF
∑=
×=r
iiiO
1FrM
C'O
C
F
OM
A couple is a set of forces whose resultant force is zero, but the resultant moment is non zero.
MEAM 535
University of Pennsylvania 17
Resultant Moment Depends on Reference PointResultant force is independent of origin (reference point)
Resultant moment is dependent on the origin
F
OM
F
O
∑=
×+=r
i
POOP
1FrMM
O
MP
P
MEAM 535
University of Pennsylvania 18
Generalized Forces for Rigid BodiesGeneralized force Velocity partials
But,
Velocity partials can be rewritten
nPn
PPP
PA qqqt
iiii
i &K&& vvvrv ++++∂∂
= 2211
[ ]∑=
⋅=N
i
Pj
aij
iQ1
)( vF
iBAPAPA i ρ×ω+= vv
Pi
Fj
Pj
ri rj
Pρi
ρj
[ ]
( )
[ ]j
n
j j
iBAPA
OPBAOP
jn
j j
PAPPA
tt
qqt
iii
&&
&&
∑∂
ρ×ω+∂+
×∂ω∂
+∂
∂=
∑∂
∂+
∂∂
=
=
=
1
1
v
rr
vrv
Fi
O
MEAM 535
University of Pennsylvania 19
Generalized Forces for Rigid BodiesGeneralized force Velocity partials
Angular velocity partial
Generalized force can be rewritten
( )
iBj
APj
Aj
iBA
j
PAPj
Aqq
i
ρ×ω+=
∂ρ×ω∂
+∂∂
=
v
vv&&
[ ] ( )[ ]∑ ρ×ω⋅+∑ ⋅===
N
ii
Bj
Ai
N
i
PjijQ
11FvF
( )[ ]∑ ω⋅×ρ=
N
i
Bj
Aii
1F
Bj
AP
PjjQ ω⋅+⋅= MvF
( )j
BABj
Aq&∂ω∂
=ω
[ ]∑=
⋅=N
i
Pj
aij
iQ1
)( vF
Pi
Fj
Pj
ri rj
Pρi
ρj
Fi
O
MEAM 535
University of Pennsylvania 20
ExampleGeneralized speed:
u=dθ/dtVelocities
Generalized Active Forces-Fa1
τa3
rl
θ φm
τF
B
P
( )11
31
cossin av
a
φφ+θ
−=
=ωRPA
BA
( )φ
φ+θ+τ=
cossin
1FRQ
φφ=θθφφ−θθ−=
φ=θφ+θ=&&&&& coscos;sinsin
sinsin;coscoslrlrx
lrlrx
θφθ
=φ &&coscos
lr
x
MEAM 535
University of Pennsylvania 21
ExampleGeneralized coordinates
(θ1, θ2)Generalized speeds
(u1, u2)Velocity Partials
Generalized forces
l1
l2
θ1
θ2τ2
τ1
PMz
(Fx, Fy)
C2
C1
y
x
Pj
ACj
ACj
ABj
ABj
A vvv ,,,, 2121 ωω
dtd
u jj
θ=
( )( ) 22
121
32122
21323231
Bj
Az
Pj
Ayx
Cj
A
Cj
ABj
ABj
Aj
MFFgm
gmQ
ω⋅+⋅++⋅−
⋅−ω⋅τ+ω⋅τ−τ=
avaava
vaaaa
MEAM 535
University of Pennsylvania 22
Example (continued)
l1
l2
θ1
θ2τ2
τ1
P
Mz
(Fx, Fy)C2
C1
y
x
Generalized forces
Velocities
Generalized Forces
( )( ) 22
121
32122
21323231
Bj
Az
Pj
Ayx
Cj
A
Cj
ABj
ABj
Aj
MFFgm
gmQ
ω⋅+⋅++⋅−
⋅−ω⋅τ+ω⋅τ−τ=
avaava
vaaaa
( ) ,, 3213121 aa uuu BABA +=ω=ω
( )( ) ( )( )( ) ( )( )212121122121111
21212112221
121111
12111121
2
1
uucslucsl
uucslucsl
ucsl
PA
CA
CA
++−++−=
++−++−=
+−=
aaaav
aaaav
aav
( ) ( )( ) z
yx
Mclclgmcglm
clclFslslFQ
++−−
+++−τ=
12221
11211121
122111221111
( ) ( ) ( ) zyx MclgmclFslFQ +−+−τ= 12221
212212222
MEAM 535
University of Pennsylvania 23
Conservative Holonomic SystemsAll applied forces are conservativeOrThere exists a scalar potential function such that all applied forces are given by:
The virtual work done by the applied forces is:
( )tqqq na
i ,,, 21)( Kφ∇−=F
∑= =
δ
∂∂⋅=δ ∑
n
jj
N
i j
Pa
i qq
Wi
1 1
)( rF
∂∂∂∂∂∂
∂φ∂
∂φ∂
∂φ∂
−=∂∂⋅
j
i
j
i
j
i
iiij
Pa
i
qzqyqx
zyxq
irF )(
jj q
Q∂φ∂
−=
δφδ∂φ∂
−=δ −=∑=
n
jj
jq
qW
1
MEAM 535
University of Pennsylvania 24
StatementA conservative, holonomic system of N particles (P1, P2,…, PN) is in static equilibrium if and only if the change in potential energy though any (arbitrary) virtual displacement is zero.
0=δφ
