Virtual Work

Learning Objectives

  • Select a generalized coordinate for a constrained system.
  • Derive compatible virtual displacements from geometry.
  • Sum force and couple work contributions with a consistent sign convention.
  • Verify equilibrium and compare it with direct force or moment equilibrium.
  • Detect invalid or singular mechanism configurations.
Compatible virtual displacement of a constrained systemA small compatible virtual motion relates force and moment work through one generalized coordinate while ideal constraint reactions contribute no virtual work.PδθδrΣ F · δr + Σ M δθ = 0
Compatible virtual displacement of a constrained system
A small compatible virtual motion relates force and moment work through one generalized coordinate while ideal constraint reactions contribute no virtual work.

Virtual Displacement

A virtual displacement is an imagined infinitesimal displacement that is compatible with the system constraints at a fixed instant.

Principle of Virtual Work for Equilibrium

The total virtual work of external forces and couples vanishes for an equilibrium configuration.

δW=∑iFi⋅δri+∑jMjδθj=0\delta W = \sum_i \mathbf{F}_i\cdot\delta\mathbf{r}_i + \sum_j M_j\delta\theta_j = 0

Variables

SymbolDescriptionUnit
δW\delta WTotal virtual workN·m
δri\delta\mathbf{r}_iCompatible virtual displacement of a force application pointm
δθj\delta\theta_jCompatible virtual rotationrad

Equilibrium Method

Virtual work is an alternative equilibrium method. It does not imply acceleration, actual motion, or energy conservation over a finite path.

Constraint Compatibility

Virtual displacements cannot be chosen independently in a constrained mechanism. Derive every displacement from the selected generalized coordinate before summing work.

Worked Example Summary

For a lever with a 200 N200\ \text{N} load at a 0.5 m0.5\ \text{m} arm and an input at a 2.0 m2.0\ \text{m} arm, compatible rotation gives δsP=2.0δθ\delta s_P=2.0\delta\theta and δsW=0.5δθ\delta s_W=0.5\delta\theta. From P(2.0δθ)−200(0.5δθ)=0P(2.0\delta\theta)-200(0.5\delta\theta)=0, the required input is P=50 NP=50\ \text{N}, matching direct moment equilibrium.

Simulation 1 Instructions

Change the input and load arms. Compare the virtual-work result with direct moment equilibrium.

Virtual Work Equilibrium Suite

Concept and model scope

Use one admissible virtual rotation and compare virtual work directly with moment equilibrium.

Simulation purpose: Use constraint-compatible infinitesimal motions to test equilibrium without treating virtual work as finite-path energy conservation.

Model scope: Ideal rigid mechanisms and inextensible rope compatibility. Constraint reactions are absent from the work equation only when the admissible virtual motion makes their virtual work zero. Pulley efficiency modifies actual input force after the ideal rope-compatibility relation is established.

Verification: Check derivative signs, singular states, and the direct equilibrium counterpart. The lever uses opposing moments, the pulley keeps its n:1 rope-length ratio independent of efficiency, and the scissor coordinates are derived from the same physical member length shown in the diagram.

Controls
Applied load

Applied load

External load entering the equilibrium work equation. It is not an energy input over a finite motion.

120 N
Input arm

Input arm

Distance from the lever pivot to the input-force application point.

1.8 m
Load arm

Load arm

Perpendicular moment-arm radius from the pivot to the load application point for the shown lever.

0.8 m
compatible equilibrium
WPrW = 0.8 mrP = 1.8 m
Required input
53.333 N
Virtual-work residual
0.00e+0 N·m

Virtual-work residual

Uses a unit virtual rotation; zero is equivalent to direct moment equilibrium.

Input displacement / rad
1.800 m/rad
Load displacement / rad
0.800 m/rad
PrP δθ−WrW δθ=0⟺PrP=WrWP r_P\,\delta\theta-W r_W\,\delta\theta=0\quad\Longleftrightarrow\quad P r_P=W r_W

Equation concept

Virtual work here is an equilibrium statement for admissible infinitesimal motion; direct force or moment equilibrium provides the independent check.

Simulation 1 Concept Question

Why does the common virtual rotation cancel from the lever equation?

Simulation 2 Instructions

Change the number of supporting rope segments and efficiency. First compare the ideal input force with the n:1n{:}1 rope displacement ratio, then observe that efficiency changes the actual required input force without changing rope-length compatibility.

Virtual Work Equilibrium Suite

Concept and model scope

Let rope-length compatibility set the ideal displacement ratio, then apply efficiency separately to the actual input force.

Simulation purpose: Use constraint-compatible infinitesimal motions to test equilibrium without treating virtual work as finite-path energy conservation.

Model scope: Ideal rigid mechanisms and inextensible rope compatibility. Constraint reactions are absent from the work equation only when the admissible virtual motion makes their virtual work zero. Pulley efficiency modifies actual input force after the ideal rope-compatibility relation is established.

Verification: Check derivative signs, singular states, and the direct equilibrium counterpart. The lever uses opposing moments, the pulley keeps its n:1 rope-length ratio independent of efficiency, and the scissor coordinates are derived from the same physical member length shown in the diagram.

Controls
Applied load

Applied load

External load entering the equilibrium work equation. It is not an energy input over a finite motion.

120 N
Supporting rope segments

Supporting rope segments

Positive integer count of ideal rope segments supporting the moving block. Rope-length compatibility gives δs_in=nδy.

4
Mechanical efficiency

Mechanical efficiency

Actual output-work/input-work efficiency. It increases the actual required input force when below 1, but it does not alter the ideal rope-length displacement ratio.

0.90
ideal compatibility resolved
1234Wequivalent moving block · n = 4 supporting segments
Actual required input
33.333 N

Actual required input

Includes mechanical efficiency only after the ideal compatibility relation is solved.

Ideal input force
30.000 N
Rope displacement ratio
4:1

Rope displacement ratio

Purely kinematic: the free end moves n times the moving-block displacement.

Ideal work residual
0.00e+0 N·m

Ideal work residual

Evaluated for a 1 m reference virtual load displacement; efficiency is intentionally excluded from this ideal compatibility residual.

δsin=n δy,Pidealδsin−Wδy=0,Pactual=Pidealη\delta s_{in}=n\,\delta y,\qquad P_{ideal}\delta s_{in}-W\delta y=0,\qquad P_{actual}=\frac{P_{ideal}}{\eta}

Equation concept

The ideal virtual-work balance uses the rope-length constraint only. Efficiency represents loss and therefore changes the actual required input force without changing δs_in=nδy.

Simulation 2 Concept Question

Why must the free end of the rope move farther when the load force is reduced?

Simulation 3 Instructions

Use the scissor mechanism and select the link angle as the generalized coordinate. Observe the singular behavior near a flat configuration.

Virtual Work Equilibrium Suite

Concept and model scope

Generate horizontal and vertical virtual motions from the same link geometry and expose the flat-configuration singularity.

Simulation purpose: Use constraint-compatible infinitesimal motions to test equilibrium without treating virtual work as finite-path energy conservation.

Model scope: Ideal rigid mechanisms and inextensible rope compatibility. Constraint reactions are absent from the work equation only when the admissible virtual motion makes their virtual work zero. Pulley efficiency modifies actual input force after the ideal rope-compatibility relation is established.

Verification: Check derivative signs, singular states, and the direct equilibrium counterpart. The lever uses opposing moments, the pulley keeps its n:1 rope-length ratio independent of efficiency, and the scissor coordinates are derived from the same physical member length shown in the diagram.

Controls
Applied load

Applied load

External load entering the equilibrium work equation. It is not an energy input over a finite motion.

120 N
Link length

Link length

Physical length of each crossing scissor member. It sets both x=ℓ cosθ and y=ℓ sinθ and therefore both compatible derivatives.

1.8 m
Generalized link angle

Generalized link angle

Angle of each scissor member above the horizontal. At θ=0°, dx/dθ=0 and a horizontal input cannot balance a nonzero vertical load by first-order virtual work.

40 deg
compatible mechanism
ℓ = 1.8 mθ = 40°
Required horizontal input
143.010 N
Virtual-work residual
0.00e+0 N·m
dx/dθ
-1.157 m/rad
dy/dθ
1.379 m/rad
x=ℓcos⁡θ,y=ℓsin⁡θ,P(−dxdθ)−Wdydθ=0x=\ell\cos\theta,\quad y=\ell\sin\theta,\quad P\left(-\frac{dx}{d\theta}\right)-W\frac{dy}{d\theta}=0

Equation concept

Virtual work here is an equilibrium statement for admissible infinitesimal motion; direct force or moment equilibrium provides the independent check.

Simulation 3 Concept Question

Why does the required horizontal input become very large as the scissor mechanism approaches a flat position?

Simulation 4 Instructions

Inspect dx/dθdx/d\theta and dy/dθdy/d\theta to see how small displacements remain compatible with the mechanism geometry.

Virtual Work Equilibrium Suite

Concept and model scope

Inspect signed dx/dθ and dy/dθ from one generalized coordinate instead of assigning independent displacements.

Simulation purpose: Use constraint-compatible infinitesimal motions to test equilibrium without treating virtual work as finite-path energy conservation.

Model scope: Ideal rigid mechanisms and inextensible rope compatibility. Constraint reactions are absent from the work equation only when the admissible virtual motion makes their virtual work zero. Pulley efficiency modifies actual input force after the ideal rope-compatibility relation is established.

Verification: Check derivative signs, singular states, and the direct equilibrium counterpart. The lever uses opposing moments, the pulley keeps its n:1 rope-length ratio independent of efficiency, and the scissor coordinates are derived from the same physical member length shown in the diagram.

Controls
Link length

Link length

Physical length of each crossing scissor member. It sets both x=ℓ cosθ and y=ℓ sinθ and therefore both compatible derivatives.

1.8 m
Generalized link angle

Generalized link angle

Angle of each scissor member above the horizontal. At θ=0°, dx/dθ=0 and a horizontal input cannot balance a nonzero vertical load by first-order virtual work.

40 deg
compatible derivatives
ℓ = 1.8 mθ = 40°
x coordinate
1.379 m
y coordinate
1.157 m
dx/dθ
-1.157 m/rad
dy/dθ
1.379 m/rad

dy/dθ

Signed derivatives come from the same generalized coordinate; they are not independent choices.

δx=dxdθδθ,δy=dydθδθ\delta x=\frac{dx}{d\theta}\delta\theta,\qquad \delta y=\frac{dy}{d\theta}\delta\theta

Equation concept

Compatibility is a geometric derivative relation at one instant. It is not a statement of energy conservation over an actual finite motion.

Simulation 4 Concept Question

What error occurs if δx\delta x and δy\delta y are assigned arbitrary independent values?

Minimum Input Force for a Prescribed Moment

Required force when the force direction forms angle alpha with the position vector.

P=Mrequiredrsin⁡αP=\frac{M_{\mathrm{required}}}{r\sin\alpha}

Variables

SymbolDescriptionUnit
PPRequired input forceN
MrequiredM_{\mathrm{required}}Required balancing momentN·m
rrDistance from pivot to force application pointm
α\alphaAngle between the position vector and forcedeg or rad

Simulation 5 Instructions

Rotate the input-force direction and locate the orientation that minimizes the required force.

Virtual Work Equilibrium Suite

Concept and model scope

Rotate the input force, observe the effective perpendicular moment arm, and verify that 90° minimizes the required magnitude.

Simulation purpose: Use constraint-compatible infinitesimal motions to test equilibrium without treating virtual work as finite-path energy conservation.

Model scope: Ideal rigid mechanisms and inextensible rope compatibility. Constraint reactions are absent from the work equation only when the admissible virtual motion makes their virtual work zero. Pulley efficiency modifies actual input force after the ideal rope-compatibility relation is established.

Verification: Check derivative signs, singular states, and the direct equilibrium counterpart. The lever uses opposing moments, the pulley keeps its n:1 rope-length ratio independent of efficiency, and the scissor coordinates are derived from the same physical member length shown in the diagram.

Controls
Applied load

Applied load

External load entering the equilibrium work equation. It is not an energy input over a finite motion.

120 N
Input arm

Input arm

Distance from the lever pivot to the input-force application point.

1.8 m
Load arm

Load arm

Perpendicular moment-arm radius from the pivot to the load application point for the shown lever.

0.8 m
Force-to-arm angle

Force-to-arm angle

Acute angle between the force line and lever arm. The effective perpendicular moment arm is r sinα; α=0° is singular and α=90° is optimal.

40 deg
moment balance resolved
WPPerpendicular force direction gives the maximum moment arm and minimum force.rW = 0.8 mrP = 1.8 mα = 40°
Required input at selected α
82.972 N
Minimum possible input
53.333 N at 90°
Effective moment arm
1.157 m

Effective moment arm

The perpendicular lever arm is rP sinα and is maximized when the force is perpendicular to the position vector.

Moment residual
0.00e+0 N·m
PrPsin⁡α=WrW,Pmin=WrWrP  at  α=90∘P r_P\sin\alpha=W r_W,\qquad P_{min}=\frac{W r_W}{r_P}\;\text{at}\;\alpha=90^\circ

Equation concept

Virtual work here is an equilibrium statement for admissible infinitesimal motion; direct force or moment equilibrium provides the independent check.

Simulation 5 Concept Question

Why is the minimum input force obtained when the force is perpendicular to the position vector?

Virtual Work Procedure

  1. Identify all constraints and the system degrees of freedom.
  2. Choose one convenient generalized coordinate.
  3. Express every compatible displacement and rotation in terms of that coordinate.
  4. Assign a positive direction and use it consistently.
  5. Sum the virtual work of external forces and couples.
  6. Set the total to zero and solve for the unknown equilibrium quantity.
  7. Check the result against direct equilibrium when practical.
Virtual-Work Equilibrium Workflow

Use one or more independent compatible virtual displacements so ideal workless constraints disappear correctly and each generalized equilibrium condition is enforced.

Virtual-Work Equilibrium WorkflowUse one or more independent compatible virtual displacements so ideal workless constraints disappear correctly and each generalized equilibrium condition is enforced.. Identify constraints and independent degrees of freedom → Choose independent generalized coordinates qi; Choose independent generalized coordinates qi → Construct arbitrary compatible virtual displacements δqi; Construct arbitrary compatible virtual displacements δqi → Are omitted constraint reactions ideal and workless for the chosen virtual motion?; Are omitted constraint reactions ideal and workless for the chosen virtual motion? — No → Include non-workless reactions or revise coordinates; Are omitted constraint reactions ideal and workless for the chosen virtual motion? — Yes → Write virtual work of active forces and couples: δW = ΣF·δr + ΣMδθ; Include non-workless reactions or revise coordinates → Choose independent generalized coordinates qi; Write virtual work of active forces and couples: δW = ΣF·δr + ΣMδθ → Set the coefficient of each independent δqi to zero; Set the coefficient of each independent δqi to zero → Solve the resulting generalized equilibrium equations; Solve the resulting generalized equilibrium equations → Direct equilibrium, signs, units, and constraint conditions consistent?; Direct equilibrium, signs, units, and constraint conditions consistent? — Yes → Virtual-work solution verified; Direct equilibrium, signs, units, and constraint conditions consistent? — No → Recheck signs and algebra; Recheck signs and algebra → Solve the resulting generalized equilibrium equations

Identify constraints and independent degrees of freedom → Choose independent generalized coordinates qi; Choose independent generalized coordinates qi → Construct arbitrary compatible virtual displacements δqi; Construct arbitrary compatible virtual displacements δqi → Are omitted constraint reactions ideal and workless for the chosen virtual motion?; Are omitted constraint reactions ideal and workless for the chosen virtual motion? — No → Include non-workless reactions or revise coordinates; Are omitted constraint reactions ideal and workless for the chosen virtual motion? — Yes → Write virtual work of active forces and couples: δW = ΣF·δr + ΣMδθ; Include non-workless reactions or revise coordinates → Choose independent generalized coordinates qi; Write virtual work of active forces and couples: δW = ΣF·δr + ΣMδθ → Set the coefficient of each independent δqi to zero; Set the coefficient of each independent δqi to zero → Solve the resulting generalized equilibrium equations; Solve the resulting generalized equilibrium equations → Direct equilibrium, signs, units, and constraint conditions consistent?; Direct equilibrium, signs, units, and constraint conditions consistent? — Yes → Virtual-work solution verified; Direct equilibrium, signs, units, and constraint conditions consistent? — No → Recheck signs and algebra; Recheck signs and algebra → Solve the resulting generalized equilibrium equations

  • Identify constraints and independent degrees of freedom: terminator
  • Choose independent generalized coordinates qi: process
  • Construct arbitrary compatible virtual displacements δqi: process
  • Are omitted constraint reactions ideal and workless for the chosen virtual motion?: decision
  • Include non-workless reactions or revise coordinates: process
  • Write virtual work of active forces and couples: δW = ΣF·δr + ΣMδθ: process
  • Set the coefficient of each independent δqi to zero: process
  • Solve the resulting generalized equilibrium equations: process
  • Direct equilibrium, signs, units, and constraint conditions consistent?: decision
  • Recheck signs and algebra: process
  • Virtual-work solution verified: terminator
Key Takeaways
  • Virtual work uses compatible infinitesimal motion to express equilibrium.
  • Constraint reactions that do no virtual work can be eliminated from the equation.
  • Sign consistency is essential.
  • Singular configurations can require unbounded idealized input force.
  • Virtual work is not a dynamics or finite-energy simulation.