MEAM 535

University of Pennsylvania 1

Principle of Virtual Work Aristotle Galileo (1594) Bernoulli (1717) Lagrange (1788)

1. Start with static equilibrium of holonomic system of N particles

2. Extend to rigid bodies

3. Incorporate inertial forces for dynamic analysis

4. Apply to nonholonomic systems

MEAM 535

University of Pennsylvania 2

Principle of Virtual Work A 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 system of N particles (P1, P2,…, PN) with n degrees of freedom is in static equilibrium if and only if all the n generalized forces are zero.

OR

MEAM 535

University of Pennsylvania 3

Extension to Rigid Bodies Need the concept of moments and couples

F i

i F j

O

rj ri

A couple is a set of forces whose resultant force is zero, but the resultant moment is non zero.

The moment about a convenient reference point O:

Suppose

the couple C is given by:

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University of Pennsylvania 4

Resultant Moment Depends on Reference Point Resultant force is independent of origin (reference point)

Resultant moment is dependent on the origin

F

O M

F

P

O

MP €

MP =MO + rPO ×F

O

Note that a couple is independent of the choice of reference point

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University of Pennsylvania 5

Equivalent System of Forces

F i

i F j

O

C ' O

C

F

O M

rj ri

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

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University of Pennsylvania 6

Wrench Resultant moment is dependent on the origin

F

O M

F

P’

O

MP’

O

Can always choose P so MP is parallel to F.

The system of forces (and couples) can be reduced to a wrench, a net force vector and a couple parallel to the force!

F

P

MP

€

MP =MO + rPO ×F

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University of Pennsylvania 7

Generalized Forces for Rigid Bodies with N forces Generalized force Velocity partials

Fi

Pi Fj

Pj

O

ri rj

P ρi ρj

Choose a reference point, P Velocity partials can be rewritten

Relate velocities of Pi to velocity of P

Define partial angular velocity

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University of Pennsylvania 8

Generalized force Velocity partials

Fi

Pi Fj

Pj

O

ri rj

P ρi ρj

Generalized force can be rewritten

Generalized Forces for Rigid Bodies with N forces (cont’d)

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University of Pennsylvania 9

Fi

Pi Fj

Pj

ri rj

P ρi ρj

Generalized force can be rewritten

Generalized Forces for Rigid Bodies with N forces and M couples

Ci

Cj

Without couples

With M couples

Resultant of all couples

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University of Pennsylvania 10

Example Generalized speed:

  u=dθ/dt Generalized Active Forces

  -Fa1

  τa3

Partial Velocities

B

P

x

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Example Generalized coordinates

  (θ1, θ2) Generalized speeds

  (u1, u2) Velocity Partials

Generalized forces

l1

l2

θ1

θ2 τ2

τ1

P Mz

(Fx, Fy)

C2

C1

y

x

Robot arm subject to a force at P and moment about P, motor torques at joints 1 and 2, and gravity.

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University of Pennsylvania 12

Example (continued) Generalized forces

Velocities

Generalized Forces

l1

l2

θ1

θ2 τ2

τ1

P

Mz

(Fx, Fy) C2

C1

y

x

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University of Pennsylvania 13

Example Revisited (different speeds) Generalized speeds   (u1, u2)

Velocities

Generalized Forces

l1

l2

φ1

φ2 τ2

τ1

P

Mz

(Fx, Fy) C2

C1

y

x

Note change