Ch.11 Virtual Work
2
2/18/2014 1 11. Virtual Work HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Engineering Mechanics – Statics 11.01 Virtual Work Chapter Objectives • To introduce the principle of virtual work and show how it applies to finding the equilibrium configuration of a system of pin-connected members • To establish the potential-energy function and use the potential energy method to investigate the type of equilibrium or stability of a rigid body or system of pin-connected members HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Engineering Mechanics – Statics 11.02 Virtual Work § 1 . Definition of Work - The work done by a force = = : work : force that done the work : ∠( , ) : displacement - The work done by a couple = : couple that do the work : small angle of rotation HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Engineering Mechanics – Statics 11.03 Virtual Work § 1 . Definition of Work - Virtual work Consider an imaginary or virtual movement of a body in static equilibrium, which indicates a displacement or rotation that is assumed and does not actually exist • The virtual work done by a force having a virtual displacement = • The virtual work done by a couple having a virtual rotation in the plane of the couple forces = HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Engineering Mechanics – Statics 11.04 Virtual Work § 2 . Principle of Virtual Work - If a body is in equilibrium, then the algebraic sum of the virtual work done by all the forces and couple moments acting on the body, is zero for any virtual displacement of the body = 0 - Example Imagine the ball to be displaced downwards a virtual amount • the weight does positive virtual work + • the normal force does negative virtual work − For equilibrium, the total virtual work must be zero = − = − = 0 Since ≠ 0, then = as required by applying ∑ =0 HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Engineering Mechanics – Statics 11.05 Virtual Work § 2 . Principle of Virtual Work - Example Imagine the beam is given a virtual rotation about point • the only forces that do work are and The virtual work equation for this case = − ′ = − (/2) = ( − /2) = 0 Since ≠ 0 then = /2 Excluding , notice that the terms in parentheses actually represent the application of ∑ =0 HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien Engineering Mechanics – Statics 11.06 Virtual Work
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Ch.11 Virtual Work
Transcript of Ch.11 Virtual Work
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2/18/2014
1
11. Virtual Work
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.01 Virtual Work
Chapter Objectives
To introduce the principle of virtual work and show how itapplies to finding the equilibrium configuration of a system of
pin-connected members
To establish the potential-energy function and use the potentialenergy method to investigate the type of equilibrium or stability
of a rigid body or system of pin-connected members
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.02 Virtual Work
1.Definition of Work
- The work done by a force
= =
: work
: force that done the work
: ( , )
: displacement
- The work done by a couple
=
: couple that do the work
: small angle of rotation
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.03 Virtual Work
1.Definition of Work
- Virtual work
Consider an imaginary or virtual movement of a body in static
equilibrium, which indicates a displacement or rotation that is
assumed and does not actually exist
The virtual work done by a force having a virtualdisplacement
=
The virtual work done by a couple having a virtualrotation in the plane of the couple forces
=
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.04 Virtual Work
2.Principle of Virtual Work
- If a body is in equilibrium, then the algebraic sum of the virtual
work done by all the forces and couple moments acting on the
body, is zero for any virtual displacement of the body = 0
- Example
Imagine the ball to be displaced downwards a virtual amount
the weight does positive virtual work +
the normal force does negative virtual work
For equilibrium, the total virtual work must be zero
= = = 0
Since 0, then = as required by applying = 0HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.05 Virtual Work
2.Principle of Virtual Work
- Example
Imagine the beam is given a virtual rotation about point
the only forces that do work are and
The virtual work equation for this case
= = (/2) = ( /2) = 0
Since 0 then = /2
Excluding , notice that the terms in parentheses actuallyrepresent the application of = 0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.06 Virtual Work
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2/18/2014
2
3.Principle of Virtual Work for a System of Connected Rigid Bodies
- Utilize the principle of virtual work
To determine active forces that maintain the system inequilibrium
To determine the equilibrium positions
To relate the work done by conservative forces with potentialenergy
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.07 Virtual Work
3.Principle of Virtual Work for a System of Connected Rigid Bodies
- Example 11.1 Determine the angle for equilibrium of thetwo-member linkage. Each member has a mass of 10
Solution
Free-body diagram
Virtual displacements
Fixed pin support
Position coordinate : ,: = 2 1 = 2
= 0.51 = 0.5
Virtual work equations
= + + = 0
98.1 0.5 +98.1 0.5 +25 2 = 0
= 198.1
50= 63.00
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.08 Virtual Work
3.Principle of Virtual Work for a System of Connected Rigid Bodies
- Example 11.2 Determine the required
force needed to maintain equilibriumof the scissors linkage when = 600.The spring is unstretched when =300. Neglect the mass of the links
Solution
Free-body diagram
= = 5000 0.3 0.3300
= 1500 750 ()
Virtual displacements
Position coordinate : , : = 0.3 = 0.3
= 0.9 = 0.9HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.09 Virtual Work
3.Principle of Virtual Work for a System of Connected Rigid Bodies
Virtual displacements
Position coordinate : , : = 0.3 = 0.3
= 0.9 = 0.9
Virtual work equations
= + = 0
1500 750 0.3
+ 0.9 = 0
0.9 + 225 450 = 0
= 500 250
when = 600,
= 500600 250
= 183
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.10 Virtual Work
3.Principle of Virtual Work for a System of Connected Rigid Bodies
- Example 11.3 If the box has a mass of 10, determine thecouple moment needed to maintainequilibrium when = 600. Neglect themass of the members
Solution
Free-body diagram
Virtual displacements
= 0.45 +
= 0.45
Virtual work equations
= 10 9.81 = 0
( 44.145) = 0
= 44.145600 = 22.1
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.11 Virtual Work
3.Principle of Virtual Work for a System of Connected Rigid Bodies
- Example 11.4 The mechanism supports the 50 cylinder.Determine the angle for equilibrium if the
spring has an unstretched length of 2when = 00. Neglect the mass of themembers
Solution
Free-body diagram
= = 200 2 2 = 400(1 )
Virtual displacements
= ; = 3; = 2
=; = 3; =2
Virtual work equations
= + 50 = 0
= 34.90
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 11.12 Virtual Work
