---
# System prepended metadata

title: Method of virtual work
tags: [statics, Mechanical Mechanics]

---

# Method of virtual work
"When an object or structure is in equilibrium, the total virtual work done by all external and internal forces on it on any virtual displacement is zero."
$$dU = F\cdot dr = F\ ds\ cos(\theta)$$

$$dU = M\ d\theta$$
***
$$\sum W_V = \sum F\cdot du = 0$$
* virtual work : $W_V$
$$\sum(T\cdot dL) = \sum(P\cdot du)$$
* Internal force : $T$
* External force : $P$
$$U_{min} = U_{Strain}+U_{External}$$
* Strain Energy : $U_{Strain}$
* Potential Energy of External Forces : $U_{External}$
## Efficiency
$$\eta = \frac{\text{output work}}{\text{input work}}$$

ideal machine : $\eta = 1$
in general : $\eta<1$
## Energy
$$U_g = Wh = mgh$$

$$K = \frac12mv^2$$

$$U_k = \frac12kx^2$$

## Stability of equilibrium
$$U_g = Wh = mgy$$

$dU_g/d\theta = 0$ : for equilibrium
$d^2U_g/d\theta^2 > 0$ : stable equilibrium
$d^2U_g/d\theta^2 < 0$ : unstable equilibrium
$d^2U_g/d\theta^2 = d^3U_g/d\theta^3 = ... = 0$ : neutral equilibrium