Virtual Displacements and the Principle of Virtual Work

The principle of virtual work is the most heavily executed idea of eighteenth-century mechanics — not by physicists, but by software. The finite element method, which sizes aircraft wings, bridge decks, hip implants, and phone frames, is the principle verbatim: a structure is in equilibrium when the internal stresses do virtual work equal to that of the applied loads for every admissible virtual displacement field, and FEM simply restricts "every" to a large finite family of shape functions. Each row of the resulting stiffness system is one virtual-work statement; a million-element model is a million disciplined applications of equation (5). When a 2026 CAD package reports a stress margin overnight, it has been doing Bernoulli's thought experiment at gigahertz rates. The other stronghold is robotics. The statics of a robot arm asks: what motor torques $\tau_j$ hold a payload wrench $\mathbf{F}$ at the end-effector? The answer shipped in every manipulator controller is $\tau = J^{\mathsf{T}}\mathbf{F}$, where $J$ is the kinematic Jacobian — and that formula is equation (7): the joint torque conjugate to $q_j$ is the applied force dotted into $\partial\mathbf{r}/\partial q_j$. Gravity compensation in surgical robots and the whole-body balance controllers of current humanoid robots evaluate exactly these generalized forces hundreds of times per second, holding equilibrium by continuously solving $Q_j = 0$ — virtual work as a real-time control law rather than an exam technique.

Thread a bead on a curved, frictionless wire. The wire can push the bead only perpendicular to itself — that is what frictionless means — while every displacement the bead is allowed to make points along the wire. Perpendicular push, tangential freedom: the wire's force is structurally incapable of doing work on any motion the wire permits. That orthogonality is not an accident of the bead; it is the defining geometry of an ideal constraint, and the principle of virtual work is the art of asking only questions to which that force contributes nothing.

The word virtual signals a thought experiment. We are not asking what the system does next — that is dynamics. We are asking what it could do, consistently with its constraints, right now: an exploratory probe of configuration space. Equilibrium is the statement that no admissible probe direction lets the applied forces do positive work — the system has nowhere profitable to go. Instead of balancing three force components at every joint of a linkage, you write one number — the total virtual work — for each independent way the mechanism can flex, and demand that it vanish. The bookkeeping collapses from vectors-and-reactions to exactly as many scalar equations as the system has degrees of freedom. The subtle move — the one that makes the method exact rather than approximate — is freezing time. For a stationary wire, a virtual displacement and a small real displacement look identical, and you may blur the two. But let the constraint itself move — a bead on a rotating hoop, a pendulum whose support point is being cranked — and the real displacement acquires a second part: the piece the moving constraint drags the particle through. That dragged piece is generally parallel to the constraint force, so the constraint force does real work on it. The virtual displacement is defined precisely to strip the drag away: freeze the hoop, freeze the crank, and displace only along what the frozen geometry allows.

Keep one economic image in mind throughout: virtual work is a scalar gatekeeper. Constraint forces are the prices the geometry charges to enforce itself — the sibling article on constraint forces computes them one free-body diagram at a time. The principle of virtual work is the accounting system that never asks for the prices at all, because it only ever proposes trades the enforcer is indifferent to.

The whole subject rests on one distinction, so we build it carefully. A holonomic system's configuration is a point $q = (q_1,\dots,q_f)$, and each particle's position is a known map $\mathbf{r}_i = \mathbf{r}_i(q_1,\dots,q_f,t)$ — the explicit $t$ appears whenever a constraint moves on its own schedule (a rotating hoop, a lengthening pendulum). During a real interval $\mathrm{d}t$ the particle moves for two reasons: because the coordinates changed, and because the constraint map itself changed underneath it. Differentiating the map says exactly that:

where: $\mathrm{d}\mathbf{r}_i$ — the real displacement of particle $i$ in time $\mathrm{d}t$; the first sum is motion within the constraint geometry; the last term is the drag of the moving constraint itself. For a bead on a hoop rotating at $\omega$, that last term is the $\omega\times\mathbf{r}$ sweep — often the largest piece of the actual velocity. A virtual displacement keeps the first mechanism and amputates the second by decree. It is defined as any infinitesimal change of configuration taken at one frozen instant, $\delta t = 0$:

where: $\delta q_j$ — free, independent variations of the generalized coordinates ($f$ of them, one per degree of freedom); no $(\partial\mathbf{r}_i/\partial t)$ term appears, by definition. Read physically: $\delta\mathbf{r}_i$ is tangent to the constraint surface as it stands right now — a probe of where the frozen geometry would let the system go. When constraints are scleronomic (no explicit $t$), equations (1) and (2) describe the same set of directions and the distinction evaporates; the moment a constraint moves, they part company, and everything that follows depends on using (2), not (1). Next, split every force in the problem into the two kinds the subject cares about:

where: $\mathbf{F}_i^{(a)}$ — the applied (impressed) forces: gravity, springs, motors, loads — the forces with formulas, the ones you care about; $\mathbf{F}_i^{(c)}$ — the constraint forces: normal reactions, rod stresses, string tensions — formula-less enforcers of geometry. The classification is operational, not fundamental: sliding friction, though it comes from a contact, must be booked as applied, because it fails the test that comes next. That test is the ideal-constraint condition, the load-bearing hypothesis of all analytical mechanics:

where: the sum runs over all particles and all constraint forces, dotted into any admissible virtual displacement. This is not a small technical lemma — it is a physical claim about which constraints deserve the name ideal, and it is provable for the standard inventory: smooth surfaces, rigid bodies, inextensible strings, frictionless pins, rolling without slipping. The proof is the first derivation below, and the frozen-time definition (2) is exactly what makes it survive moving constraints.

Now the payoff. Take a system at rest and in equilibrium: every particle satisfies $\mathbf{F}_i = \mathbf{0}$, hence $\mathbf{F}_i^{(a)} + \mathbf{F}_i^{(c)} = \mathbf{0}$. Dot each equation into a virtual displacement, sum over particles, and apply (4) — the constraint forces exit without leaving an address:

where: $\delta W$ — the total virtual work of the applied forces only. This is the principle of virtual work: a system with ideal constraints is in equilibrium precisely when the applied forces do no net work under any virtual displacement. The logic runs both ways — the forward direction is the two lines above; the converse (if $\delta W = 0$ for all probes, no admissible motion can start) is what makes the principle a test for equilibrium rather than a mere consequence. One scalar condition per independent probe direction, and the probes number exactly $f$, the degrees of freedom. Before generalizing, extract the special case that ran civilization's machinery for two millennia. An ideal machine — lever, pulley system, screw, gear train — is a one-degree-of-freedom constraint connecting an input point to an output point. Apply (5) with an input force $F_{\mathrm{in}}$ driving against an output load $F_{\mathrm{out}}$:

where: $\delta s_{\mathrm{in}}, \delta s_{\mathrm{out}}$ — the virtual displacements of the input and output points along their forces, linked by the machine's geometry. Mechanical advantage is a displacement ratio: what you gain in force you pay in distance, and you can read the price off the kinematics without analyzing a single internal reaction. A screw jack with pitch $p = 5.0\ \mathrm{mm}$ and a handle sweeping a circle of radius $0.35\ \mathrm{m}$ trades $2\pi(0.35) = 2.20\ \mathrm{m}$ of hand travel for $5.0\ \mathrm{mm}$ of lift — mechanical advantage $2.20/0.0050 = 440$, which is why $19\ \mathrm{N}$ of fingertip effort holds up the $8300\ \mathrm{N}$ corner of a car.

For anything richer than a one-knob machine, translate (5) into generalized coordinates. Substitute the virtual displacement (2) into the principle and swap the order of summation — the work sorts itself by knob:

where: $Q_j$ — the generalized force conjugate to $q_j$: the work the applied forces deliver per unit twist of that one knob, all other knobs held fixed. Its dimensions are whatever makes $Q_j\,\delta q_j$ an energy — a torque if $q_j$ is an angle, an ordinary force if $q_j$ is a length. This is the object that will later sit on the right-hand side of Lagrange's equations; here it makes statics one line long:

where: each generalized force must vanish separately, because the $\delta q_j$ are independent — you may wiggle any one knob while freezing the rest. Equation (8) is the principle of virtual work in its most usable form: $f$ scalar equations, one per degree of freedom, with every constraint force already gone.

Which tool answers which question

1 · Block-and-tackle: lifting an engine with one hand (confidence builder)

A mechanic lifts an $80\ \mathrm{kg}$ engine block with a block-and-tackle in which the load hangs from $n = 4$ supporting rope strands. Find the required pull, and how much rope must be hauled to raise the engine $0.50\ \mathrm{m}$. Treat pulleys and rope as ideal.

Step 1. Set up the one virtual probe. The system has one degree of freedom: let the load rise by $\delta h$. Inextensibility of the single rope means the four supporting strands must each shorten by $\delta h$, so the free end pays out $\delta s = 4\,\delta h$ of rope. The tensions and pulley-axle reactions are ideal constraint forces — they never enter.

Step 2. Write the virtual work of the applied forces (the pull $F$ and the weight $W = mg$): $\delta W = F\,(4\,\delta h) - W\,\delta h = (4F - W)\,\delta h$.

Step 3. Demand $\delta W = 0$ for arbitrary $\delta h$: $F = W/4 = (80 \times 9.81)/4 = 784.8/4 = \boxed{196\ \mathrm{N}}$ — about the weight of a 20 kg suitcase, one-handed.

Step 4. The distance price, from the same kinematics: $\Delta s = 4 \times 0.50 = \boxed{2.0\ \mathrm{m}}$ of rope for half a metre of lift. Force divided by four, distance multiplied by four — equation (6) audited: $196 \times 2.0 = 392\ \mathrm{J}$ in, $784.8 \times 0.50 = 392\ \mathrm{J}$ out.

2 · The A-frame and the cord that holds it (exam level)

Two uniform rods, each of mass $m = 3.0\ \mathrm{kg}$ and length $L = 1.2\ \mathrm{m}$, are hinged together at the top and stand like an A-frame on a frictionless floor, each rod making $\theta = 30°$ with the vertical. A light horizontal cord ties the two feet together. Find the cord tension — without ever computing the hinge force or the floor reactions.

Step 1. Choose the probe. With the cord momentarily cut and replaced by its tension $T$ as an applied force at each foot, the frame has one degree of freedom: the splay angle $\theta$. The floor normals are vertical while the feet move horizontally (workless); the top hinge is an internal ideal constraint (workless); gravity and the two tension forces are the only work-doers.

Step 2. Kinematics of the probe. Each rod's centre of mass sits at height $y_{\mathrm{cm}} = (L/2)\cos\theta$, so $\delta y_{\mathrm{cm}} = -(L/2)\sin\theta\,\delta\theta$ for each rod. The foot separation is $s = 2L\sin\theta$, so $\delta s = 2L\cos\theta\,\delta\theta$.

Step 3. Virtual work. Gravity (both rods): $\delta W_{\mathrm{grav}} = -2mg\,\delta y_{\mathrm{cm}} = +mgL\sin\theta\,\delta\theta$ — splaying lowers the mass, gravity likes it. The cord: tension opposes the feet separating, $\delta W_{\mathrm{cord}} = -T\,\delta s = -2TL\cos\theta\,\delta\theta$. Setting the total to zero: $mgL\sin\theta = 2TL\cos\theta$, so

Step 4. Numbers: $T = \tfrac{1}{2}(3.0)(9.81)\tan 30° = 14.715 \times 0.5774 = \boxed{8.5\ \mathrm{N}}$.

Step 5. Cross-check by Newton (the long way). One rod, torques about the top hinge: the floor normal on its foot is $N = mg$ (symmetry: each foot carries one rod's weight), giving $mgL\sin\theta - mg\tfrac{L}{2}\sin\theta - TL\cos\theta = 0$, i.e. $T = \tfrac{mg}{2}\tan\theta$ — identical, but only after introducing and then eliminating $N$ and shepherding the hinge force. Virtual work never met either. Note the scaling: as $\theta \to 90°$ (frame flattening), $T \to \infty$ — a nearly flat A-frame shreds its cord, which is why stepladder spreaders are stiff bars, not strings.

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §1.4 (D'Alembert's Principle and Lagrange's Equations) · Landau & Lifshitz, Mechanics, 3rd ed., §1-2 · Morin, Introduction to Classical Mechanics, Ch. 6 · Taylor, Classical Mechanics, Ch. 7 · Lanczos, The Variational Principles of Mechanics, Ch. IV Three results worth keeping in view: the virtual-displacement formula $\delta\mathbf{r}_i = \sum_j (\partial\mathbf{r}_i/\partial q_j)\delta q_j$ carries no time term by construction; the 'cancellation of dots' identity $\partial\mathbf{r}_i/\partial q_j = \partial\dot{\mathbf{r}}_i/\partial\dot q_j$ is used in the inertial-term reduction on the road to Lagrange's equations; and the lever law $W_1\ell_1=W_2\ell_2$ serves as a concrete worked check. No empirical constants are claimed. Worked-example numbers on this page ($F = 196\ \mathrm{N}$ and $2.0\ \mathrm{m}$ of rope for the tackle; $T = 8.5\ \mathrm{N}$ for the A-frame; the screw-jack advantage of 440) are illustrative arithmetic computed from the stated formulas.

Builds on: Constraints and Their Classification · Generalized Coordinates and Configuration Space · Degrees of Freedom. Leads to: d'Alembert's Principle · The Euler-Lagrange Equations · Generalized Forces and Generalized Momenta · The Lagrangian Function.