School

Texas A&M UniversityDepartment

Aerospace EngineeringCourse Code

AERO 306Professor

John WhitcombLecture

20This

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Chapter 1

Analysis of Mechanisms

Using Virtual Work

What is Virtual Work

&

Why is it useful?

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IntroductionītoīVirtualīWorkī

This course relies heavily on something called "Virtual Work" to derive equilibrium

equations. Until now you derived equilibrium equations by imposing the conditions that

the total linear momentum and angular momentum are each zero. When you expressed

the equilibrium equations in terms of forces and moments, you did not consider

kinematicsā¦ that is, displacements. That is not to say that you did not eventually express

the forces and moments in terms of displacements through constitutive relations. You had

to whenever you wished to solve any statically indeterminate problem or if you wished to

determine the deformed shape. For virtual work you must consider both forces and

displacements. At first this might seem like an unnecessary complication to have to

consider displacements when deriving the equilibrium equations. Hang in there and you

will soon see that the pain is worth the gain.

Most of the problems you will see in this book can be solved using the concepts that have

been introduced in previous courses. It is logical to ask, āWhy are we learning about

something with this unusual nameā¦ Virtual Workā. The reason is that the Virtual Work

method forms the basis of the most powerful structural analysis methodology we have

today: finite elements. Using finite elements, we can solve complicated problems that you

cannot begin to solve by trying to solve the differential equations directly. Much of the

course will examine fairly simple problems. This is to minimize the amount of new

material you have to digest while learning the new technique. However, the strategies we

will study are just as valid for solving problems that geniuses could not solve a few

decades ago. You will see later in the course that the primary reason that Virtual Work is

so powerful is because it can easily take advantage of the computational power of

computers.

This new strategy requires the calculation of a special type of work (virtual work). For

there to be work (even virtual work) performed, there must be a force and motion. This

means we must consider both force terms and kinematics (i.e. motion). Before we jump

into Virtual Work, let's first review Real Work

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Review of Real Work

work = force * (displacement in direction of force). If the force is variable, we must

integrate

cosWFdSF dS

ļ±

ļ½ļļ½

ļ²ļ²

ļ¶

ļ¶

Examples

y

x

B

A

ļ¶

F

ļ¶

S

If constant, work going from

A to B

ļ¶

ļ¶ļ¶

F

FS

ļ½

ļ½ļ

ĖĖ

Suppose

ĖĖ

Then

F

ai b j

Scidj

Wacbd

ļ½

ļ«

ļ½ļ«

ļ½ļ«

ļ¶

ļ¶

NOTE: Work can be negativeā¦ if the

force and the displacement are in

opposite directions.

Now letās consider the deformation of a linear spring.

F

uuo

WFduFku

Wkuduku

u

u

ļ½ļ½

ļļ½ ļ½

ļ²

ļ²

0

1

20

2

0

0

0

but

k

F

,u Force vs. Displacement

If we assume the spring is elastic, there is no dissipation, and the work is stored as āstrain

energyā. We will not consider inelastic systems, which dissipate energy via friction,

plasticity, and other mechanisms.

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