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WORK, VIRTUAL WORK,

AND THE PRINCIPLE OF VIRTUAL WORKObjectives:

a) Understand the definitions of work and virtual work.

b) Apply principle of virtual work to solve statics problems.

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APPLICATION

During operation the scissors lifthas one degree of freedom.

By using the principle of virtual

work we can determine the

hydraulic force required to lift the

platform without dismembering

the mechanism.

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WORK OF A FORCE AND

WORK OF A COUPLE MOMENT

Work of a force on a particle is defined as the force

multiplying the displacement of the particle along the

direction of the force, i.e.,

dU=F dr (vector form) or dU= F ds cos

Work is a scalar quantity with unit Joule in SI system

or ftlb in FPS.

Any general differential displacement of a body can

be considered as a combination of translation and

rotation. When the body subjected to a couple, the

work corresponds to translation is zero, while the

work corresponds to rotation is

dU= F (r/2)d+ F (r/2)d= F r d=M d

ordU=M d (vector form)

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PRINCIPLE OF VIRTUAL WORK

For a body under static equilibrium, the virtual work is defined by

external forces multiplying the virtual movement along the

direction of the external forces, i.e.,

U=F r or U=M .

The virtual work for a particle under

equilibrium can be expressed asU= F r

= ( Fx i + Fyj + Fzk ) (x i + yj + zk)

= Fx x + Fy y + Fz z

= 0

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From the free body diagram of the

beam, the virtual work can be

written as:

U=Ay y - P y

=Ay (l ) - P (l/2)

= 0

Since 0, Ay =P /2

EXAMPLE

Given: A rigid beam is subjected to load

P in the transverse direction.

Find: The support reaction at point A

using the principle of virtual work.

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DEGREE OF FREEDOM FOR A SYSTEM OF

CONNECTED RIGID BODIES

Degree of freedom for a system is the number ofindependent

coordinates required to define the location of all members of the

system

For the links with sliding block

system, the only coordinate

required is , the location of the

block can be determined from:

b2 = a2 +x2 - 2ax cos

For the two-bar linkage, theindependent coordinates are 1 and

2.

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PRINCIPLE OF VIRTUAL WORK FOR A SYSTEM OF

CONNECTED RIGID BODIES

The principle of virtual work for a system of frictionlessly-

connected rigid bodies can be stated as follows:

A system of connected rigid bodies is in equilibriumprovided the virtual work doneby all the external forces and

couples acting on the system is zero for each independent

virtual displacement of the system.For a system with n degree of freedom, we can write n

independent virtual work equations, by taking one virtual

displacement along the independent coordinate axis, while the

remaining n-1 independent coordinates are held fixed.

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STEPS FOR SOLVING PROBLEMS

USING THE PRINCIPLE OF VIRTUAL WORK

1. Draw the free body diagram and define the independent

coordinates for the system

2. For each independent virtual movement, sketch the deflected

position of the system on the free body diagram (while the

movements for other independent coordinates for the system

remain zero)

3. Write the virtual work equation corresponds to each

independent virtual movement. Factor out the common

virtual movement from the equation

4. Solve for the unknown forces, couple moments, or

equilibrium position from the virtual work equations.

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EXAMPLE

Given: The two-bar linkage, assuming the weight of the links

are neglected.

Find: The statically equilibrium position.

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EXAMPLE (continued)

Fig. 11-10b Fig. 11-10c

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EXAMPLE (continued)

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GROUP PROBLEM SOLVING

Given: The mechanism is in equilibrium when = 45.

Find: The horizontal force Cx.

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GROUP PROBLEM SOLVING (continued)

Fig. 11-9b

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GROUP PROBLEM SOLVING (continued)