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.

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.

Advanced engineering statics simulation

Virtual Work Equilibrium Suite

Five distinct, constraint-compatible virtual-work models; efficiency is separated from geometry.

Use one virtual rotation so both force-point displacements are compatible with the rigid lever.

Applied load
120 N
N
20–400

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Input arm
1.8 m
m
0.3–3.5

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Load arm
0.8 m
m
0.2–2.5

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

compatible virtual motion
WPδθ
Required input
53.333 N
Virtual-work residual
0.00e+0 N
Should be near zero
dx/dθ
-2.314 m/rad
dy/dθ
2.758 m/rad
δW=∑Fi⋅δri+∑Miδθi=0\delta W=\sum \mathbf{F}_i\cdot\delta\mathbf{r}_i+\sum M_i\delta\theta_i=0

All virtual displacements come from one admissible generalized coordinate.

Concept question: Predict the required input force.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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. Compare input force with the corresponding input-to-load displacement ratio.

Advanced engineering statics simulation

Virtual Work Equilibrium Suite

Five distinct, constraint-compatible virtual-work models; efficiency is separated from geometry.

Use the rope-length constraint δsin=nδy; efficiency changes required force, not the kinematic displacement ratio.

Applied load
120 N
N
20–400

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Supporting rope segments
4
1–10

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Mechanical efficiency
0.90
0.50–1.00

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

compatible virtual motion
δsinδyEach supporting segment shortens by δy
Required input
33.333 N
Virtual-work residual
1.42e-14 N
Should be near zero
Rope displacement ratio
4:1
Geometry only
Efficiency
90.0%
Changes force, not rope constraint
δsin=n δy,ηPδsin−Wδy=0\delta s_{in}=n\,\delta y,\qquad \eta P\delta s_{in}-W\delta y=0

The number of supporting segments fixes displacement compatibility. Efficiency accounts for loss in the work balance.

Concept question: Predict the required input force.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.

Advanced engineering statics simulation

Virtual Work Equilibrium Suite

Five distinct, constraint-compatible virtual-work models; efficiency is separated from geometry.

Derive horizontal and vertical virtual motions from one generalized angle.

Applied load
120 N
N
20–400

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Link length
1.8 m
m
0.3–3.5

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Generalized link angle
40 deg
deg
5–90

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

compatible virtual motion
δxδyOne coordinate θ = 40° determines both motions
Required input
143.010 N
Virtual-work residual
0.00e+0 N
Should be near zero
dx/dθ
-2.314 m/rad
dy/dθ
2.758 m/rad
|dy/dx|=2757759995.228
δW=∑Fi⋅δri+∑Miδθi=0\delta W=\sum \mathbf{F}_i\cdot\delta\mathbf{r}_i+\sum M_i\delta\theta_i=0

All virtual displacements come from one admissible generalized coordinate.

Concept question: Predict the required input force.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.

Advanced engineering statics simulation

Virtual Work Equilibrium Suite

Five distinct, constraint-compatible virtual-work models; efficiency is separated from geometry.

Inspect dx/dθ and dy/dθ before applying force equilibrium.

Link length
1.8 m
m
0.3–3.5

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Generalized link angle
40 deg
deg
5–90

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

compatible virtual motion
δxδyOne coordinate θ = 40° determines both motions
Required input
kinematic only
Virtual-work residual
0.00e+0 N
Should be near zero
dx/dθ
-2.314 m/rad
dy/dθ
2.758 m/rad
|dy/dx|=2757759995.228
δW=∑Fi⋅δri+∑Miδθi=0\delta W=\sum \mathbf{F}_i\cdot\delta\mathbf{r}_i+\sum M_i\delta\theta_i=0

Compatibility is derived first; arbitrary independent dx and dy values would violate the mechanism constraint.

Concept question: Predict dy/dθ for the selected link length and angle.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.

Advanced engineering statics simulation

Virtual Work Equilibrium Suite

Five distinct, constraint-compatible virtual-work models; efficiency is separated from geometry.

Rotate the applied force relative to a lever arm and maximize its perpendicular moment arm.

Applied load
120 N
N
20–400

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Input arm
1.8 m
m
0.3–3.5

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Load arm
0.8 m
m
0.2–2.5

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Force-to-arm angle
40 deg
deg
5–90

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

compatible virtual motion
PWeffective arm = r sin α
Required input
82.972 N
Virtual-work residual
0.00e+0 N·m
Should be near zero
dx/dθ
-2.314 m/rad
dy/dθ
2.758 m/rad
PrPsin⁡α−WrW=0P r_P\sin\alpha-Wr_W=0

All virtual displacements come from one admissible generalized coordinate.

Concept question: Predict the required input force.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.
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.
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