# What is Virtual work: Definition and 70 Discussions

In mechanics, virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.
Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies.

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1. ### Forces involved in principle of virtual work

The answer is as such: There’s only one way for the system to move: the rectangle can deform into a parallelogramso that the left horizontal arm moves up, and the right horizontal arm moves down by thesame amount. Then the total virtual work done on the scale by the weights is zero, so thesystem...
2. ### A Virtual work of constraint forces in Hamilton’s principle

Goldstein 2ed pg 36 So in the case of holonomic constraints we can move back and forth between Hamiltons principle and Lagrange equations given as ##\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_{j}}\right)-\frac{\partial L}{\partial q_{j}}=0## But the Lagrange equations were...
3. ### I Virtual work and constraint forces

Suppose I'm considering a system of N particles that are constrained in their possible motions and so there are less that 3N generalized coordinates. Suppose now I perform a virtual displacement on one particle, which due to some constraints might force some other particles to more virtually...
4. ### A Principle of Virtual Work and the forces that DO NOT do work

In this 2D figure below, I can place: a motor at O a motor at J gravity on each link I can use Hamilton's principle, modified to the principle of virtual work and I can compute the motion of the linkage system. I do not have to account for these force FOUR forces (in this planar problem)...
5. ### Virtual Work and Static Equilibrium

Hi there, this question has already been discussed in this forum, however I do not know how to proceed further and if my reasoning is correct. I start by imagining a downard displacement of the 1kg weight. As a consequence of this, block A moves upward and to the right. Also, block B moves...
6. ### Principle of Virtual Work to the FE method

(I do understand there are many ways to formulate the FE method, but I wish to understand this one.) I understand that Hamilton's Principle is just that: a principle ( a law the recapitulates the equations of motion) I also understand that the Principle of Virtual Work (PVW) is a reformulation...
7. ### A Hamilton's principle and virtual work by constraint forces

Found a question on another website, I have the exact same question. Please help me Goldstein says : I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. How does the fact that "the same Hamilton's principle holds for both holonomic and...
8. ### Using virtual work to find the angle of a bar supported by two planes

I imagined the bar slipping along a virtual displacement, with the top and the bottom slipping by equal amounts. Since the planes are orthogonal, I took the components of these displacements: the lower end is displaced Lsinα in the direction of the right plane and by Lcosα in the direction of...
9. ### Torque Balancing and Virtual Work: Solving for Equations in Force Balancing

So I tried using force balancing. I have attached files of my solution in my notebook Torque balancing about O which gives me equation 1 And I used property of triangle, for equation 2 But i can't seem to get the right answer Also, I was wondering if it could be done by concept of virtual work.
10. ### Virtual work in a pulley system

If 1kg make a displacement of 1unit upward then A make a 1/2unit down because 1/sin 30.after that i have no idea what do I will really need help. Thanks!
11. ### D'Alembert's principle and the work done by constraint forces

From what I understand, constraint forces do no work because they are perpendicular to the allowed virtual displacements of the system. However, if you consider an unbalanced Atwood machine, in which both masses are accelerating in opposite directions, you'll find that the tension force of the...
12. ### Uncovering the Motivation behind the Principle of Virtual Work

Hello May I ask: "What is the guiding philosophy of the Principle of Virtual Work?" I do understand it an how to use it, in classical mechanics. (And I will openly admit that I do not entirely understand what I am looking for, or what I am asking.) But what is happening at this moment in...
13. ### Virtual work in Atwood's machine

The first chapter in Goldstein's Classical Mechanics ends with 3 examples about how to apply Lagrange's eqs. to simple problems. The second example is about the Atwood's machine. The book says that the tension of the rope can be ignored, but I don't understand why. The two masses can move...
14. ### Virtual work = 0 implies net torque = 0

Homework Statement Homework Equations ## \vec \tau = \Sigma \vec r_i \times \vec F_i ## The Attempt at a Solution ## \vec \tau = \Sigma \vec r_i \times \vec F_i ## ## \delta W = \Sigma F_i \delta x_i ## Let's take our Cartesian coordinate system such that the axis of rotation...
15. ### Principle of virtual work for continuous systems

I always thought that the principle of virtual work (PVW) is valid for all structures, including continuous structures (like bars, beams, plates, etc.). However, in his book 'Fundamentals of Structural Mechanics', Hjelmstad states that the PVW is only valid for discrete systems with N particles...
16. D

### Potential Energy and Force Between Capacitor Plates

Hello guys, came across some Analytical Physics lectures (http://www.physics.rutgers.edu/ugrad/227_f11/classes/lect08.pdf) that got me confused. It is described in the lecture the problem of determining the force between capacitor plates for a constant voltage (ie. connected to a battery). For...
17. ### Virtual work and D'alembert's principle

I can't for the life of me figure out what virtual work or D'alemberts principle mean and what the intuition behind them is. As far as I'm concerned D'alemberts principle is just a restatement of Newton's second law but considering the work instead of just the forces. What am I missing? I'm...
18. ### I Confusion in variation derivative

This link shows us how to derive Hamilton's generalised principle starting from D'Alembert's principle. While I had no trouble understanding the derivation I am stuck on this particular step. I can't justify why ## \frac{d}{dt} \delta r_i = \delta [\frac{d}{dt}r_i] ##. This is because if I...
19. ### Calculating force between capactor plates using virtual work

Homework Statement Use “virtual work” to calculate the attractive force between conductors in the parallel plate capacitor (area A, separation z). That is, use conservation of energy to determine how much work must be done to move one plate by an infinitesimal amount, and then use the value of...
20. ### Why does the Principle of Virtual Work work without higher math?

The last chapter of most introductory textbooks on STATICS introduces VIRTUAL WORK. It is rarely taught (I studied the syllabi of colleagues). I understand the Principle of Virtual Work, having researched and studied the Calculus of Variations, Hamilton's Principle, the Lagrangian and related...
21. ### Calculating weights on a slope using virtual work.

Homework Statement A cart on an inclined plane is balanced by the weight w. All parts have negligible friction. Find the weight W of the cart using the principle of virtual work. Picture of set-up is attached.Homework Equations ∫Fdh = 0 for virtual displacement dh in statics F_p = W cos...
22. ### On the proof of the Principle of Virtual Work

I'm having some trouble understanding the proof of the Principle of Virtual Work for deformable bodies. I'll give below the proof that I've read, and, next, I'll remark what I'm not understanding. The first thing to remember before going through the proof is that the virtual work done by a...
23. ### How to Solve Support Reaction (Virtual Work) (Past Paper)?

Hey guys, I was revising using some past papers for my structural mechanic module when I realized that I don't know how to do this problem: https://physicsforums-bernhardtmediall.netdna-ssl.com/data/attachments/83/83930-3dbc2a331721a2497ccdff8c950df951.jpg If it was for the deflection of node...
24. ### How to Solve Support Reaction (Virtual Work)?

Hey guys, I was revising using some past papers for my structural mechanic module when I realized that I don't know how to do this problem: https://physicsforums-bernhardtmediall.netdna-ssl.com/data/attachments/83/83583-240450b2eac78c09f31d35a531383268.jpg If it was for the deflection of node...
25. ### Prove forces of semiholonomic constraints do no virtual work

I do not see how (2.34) shows that forces of semi-holonomic constraints do no work in the displacements ##\delta q_k## between the varied path and the actual path. Starting from (2.31), I seem to be able to prove that such forces do do work, in contrary to what is claimed in the paragraph...
26. ### Virtual work principle: equilibrium state of a system

Homework Statement We have a system (see attached graphics) of two rods with length ##l## and mass ##m## and some external force ##F##. The coefficients ##k## of the two springs are given. The springs have no deformation when ##\varphi =0##. a) Find generalized force of the system. b) Determine...
27. ### Why is there no virtual work done by a rolling friction?

Why is there no virtual work done by a rolling friction? In pure rolling, is the virtual displacement parallel or perpendicular to the surface? I believe it can't be perpendicular because the object is not allowed to lose contact with the surface. But if it's parallel, then there must be a...
28. ### Virtual Work to find equilibrium condition

Homework Statement Find the tension T needed to hold the cart shown (pic included) in equilibrium, no friction. Using virtual work, and force components. (I don't care about signs, just looking for the magnitude of tension with quick reasoning) (not homework, just studying virtual work)...
29. ### Virtual Work - Determining Relations

Homework Statement Hello, I'm having problems determining the relationships between delta a and delta c . I don't see how how delta C = 4/9 delta A [/B] http://imgur.com/a/B6eTx Thank you.
30. ### Find the Tension in a Flexible Chain Resting on a Cone

Homework Statement A loop of flexible chain, of total weight W, rests on a smooth, frictionless right circular cone of base radius r and height h. The chain rests in a horizontal circle on the cone, whose axis is vertical. Find the tension in the chain. Homework Equations Virtual work, but...
31. ### Virtual Work in a Pulley System

Homework Statement The system shown in Fig. 2-6 is in static equilibrium. Use the principle of virtual work to find the weights A and B. Neglect the weight of the strings and the friction in the pulleys. Quick aside: This is straight out of the Exercises for the Feynman Lectures on Physics, in...
32. ### Virtual Work & Quadcopter Torques: Exploring Rotational Dynamics

Here is what we know from virtual work: $$\delta W=\sum_{i=1}^N{\vec F_i\cdot\delta\vec r_{i}}$$ Where ##N## is the number of bodies in the system. I am considering a quadcopter, modeled as a rigid body so it is just one body and we have: $$\delta W=\vec F\cdot\delta\vec r$$ My question...
33. ### Virtual work in finite plane bending of Euler-Bernoulli beam

Please refer to the following image, which shows a portion of the deformed centerline of a beam in its equilibrium configuration with a uniformly distributed load. The stress resultants are the axial forces T, transverse shears Q, and bending moments M at sections 1 and 2, with the rotations...
34. ### Why Does Virtual Work Seem Incorrect in This Statics Problem?

1. The problem statement, all variables and /known data Help! I'm stuck on an exceedingly simple statics problem, number 2.24 in the New Millennium edition of exercises for the Feynman lectures. The problem consists of an inclined plane (inclination angle 30 degrees) on wheels with a...
35. ### Virtual work (internal = external)

Homework Statement How can one show/prove that for a beam (hinged supports on both ends) subjected to bending due to a uniformly distributed load over its entire length, the virtual work of internal forces is equal to the virtual work of external forces? Given are the length of the beam (L)...
36. ### Principle of Virtual Work and static equilibrium

Homework Statement The system shown in Fig. 2-6 is in static equilibrium. Use the principle of virtual work to find the weights A and B. Neglect the weight of the strings and the friction in the pulleys. Homework Equations Conservation of gravitational potential energy. The Attempt at a...
37. ### A problem about virtual work principle

hello, I want to apply the virtual work principle to a continuous mechanical system to derive equilibrium equation, naemly, a long thin bar with one end fixed in the wall and other end applied with pressure P. but I can not derive the correct equilibrium equations. I hope some expert could give...
38. ### A problem about virtual work principle for continuous system

dear all, the virtual work pinciple can be used to derive the equilibrium equations for the mechanical systems. however, when I want to apply it to a continuous system, I found it can not give out the simple equilibrium equations. there should be something wrong with my thinking. I expect some...
39. ### Virtual Work Method on a Truss

Homework Statement I want to calculate the horizontal deflections at joints A and F (using Virtual Work method taught in Structural Engineering). Cross-Sectional Area (A) and E values are provided in the problem, and the attachment has all relevant dimensions and applied loads.Homework...
40. ### Troubleshooting a Virtual Work Problem

https://www.physicsforums.com/attachment.php?attachmentid=65907&stc=1&d=1390369310 I've been stuck on doing my revision on this problem. The equation of the virtual work, from what I deduce is [du = 0] -Pdx + mg dh However the answer given is...
41. ### Hamilton's Principle and the Principle of Virtual Work

I understand that accepting Hamilton's principle will yield identical results as accepting Newton's laws. However, simply accepting that the integral of the difference between kinetic and potential energies is an extrema seems not intuitively obvious. The textbook that I used for my classical...
42. ### When do I need to use virtual work in writing the equations of motion?

I'm studying for our comprehensive exam . I just need to clarify something. So the equation of motion for lagrangian dynamics is \frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{i}} = \frac{\partial L}{\partial {q}_{i}} However, in my notes there are example which uses the principle of virtual...
43. ### Virtual work of constraint forces

On a rigid body we usually use the formula δL=F*δP to calculate virtual work. My problem is about the force. This kind of force exists only before the contact. If I imagine a movement δP of the constrained body outside ,in the free space, I will have δL≥0 but as soon as P moves the force F...
44. ### Principal Virtual Work Theories - Cylindrical Pipe Application

Dear All, I did post this issues in the physics forum, however there is somebody ask me to post it in the engineering forum. With this I re-post it here. Does anyone here familiar with a theory called Principal Virtual Work (Equilibrium theory) ? I read somewhere that with this theory, i...
45. ### Principal Virtual Work Theories - Cylindrical Pipe Application

Dear All, Does anyone here familiar with a theory called Principal Virtual Work (Equilibrium theory) ? I read somewhere that with this theory, i could relate(for pipeline application - collapse & buckling) the components for ovalisation, external pressure and hoop strain. Appreciate if...
46. ### Calculate Vertical Displacement at Point C with Virtual Loads in Truss

Hi I am trying to calculate the vertical displacement at point C for the following diagram https://www.physicsforums.com/attachments/57161 So far i have calculated the forces in each member and i now need to calculate the Virtual forces! I applied a virtual load at point c of 1 KN. I also...
47. ### Calculating Bending Moment and Using Virtual Work Principle for Cantilever Beam

Homework Statement see attachment Homework Equations The Attempt at a Solution I think #1 is to find the bending moment equation, then use the principle of virtual work. I have no idea how to find the bending moment equation and which principle of virtual work to use. Any...
48. ### Equation of Virtual Work VS Potential Energy

For linear elastic materials (3D General)... The virtual work equation looks exactly like the equation for potential energy except that in the potential energy equation, the internal work term has a (1/2) coefficient. Why are these different?
49. ### Computing Virtual Work: Assumed Modes Method & Generalized Forces

Hi all, I've been trying to understand the vibration of a cantilever beam subjected to a forcing function using Lagrange's equation, but have got stuck at the virtual work part. I would appreciate your inputs here. Using the assumed modes method, the transverse deformation is written as...
50. ### Virtual work and kinetic energy

Question about analytical mechanics. Suppose we have a system with N degrees of freedom, and hence N generalized coordinates {q_i}, \ i=1...N. A virtual displacement is defined as an infinitesimal change in a generalized coordinate without producing a change in the generalized velocities...