- Virtual displacement
A virtual displacement delta mathbf {r}_i "is an assumed infinitesimal change of system coordinates occurring while time is held constant. It is called virtual rather than real since no actual displacement can take place without the passage of time."cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |language=English |isbn=0-03-063366-4 |chapter=Energy Methods] rp|263
The
total differential of any set of system position vectors, mathbf {r}_i, that are functions of other variables, lbrace q_1, q_2, ..., q_m brace, and time, t may be expressed as follows:rp|264:d mathbf{r}_i = frac {partial mathbf {r}_i}{partial t} d t + sum_{j=1}^m frac {partial mathbf {r}_i} {partial q_j} d q_jIf, instead, we want the virtual displacement (virtual differential displacement), thenrp|265:delta mathbf{r}_i = sum_{j=1}^m frac {partial mathbf {r}_i} {partial q_j} delta q_j
This equation is used in
Lagrangian mechanics to relategeneralized coordinates , q_j, tovirtual work , delta W, andgeneralized forces , Q_j.In
analytical mechanics the concept of a virtual displacement, related to the concept ofvirtual work , is meaningful only when discussing a physical system subject to constraints on its motion.Fact|date=September 2007 A special case of aninfinitesimal displacement (usually notated dmathbf{r}), a virtual displacement (notated delta mathbf{r}) refers to an infinitesimal change in the position coordinates of a system such that the constraints remain satisfied.Fact|date=September 2007For example, if a bead is constrained to move on a hoop, its position may be represented by the position coordinate heta, which gives the
angle at which the bead is situated. Say that the bead is at the top. Moving the bead straight upwards from its height z to a height z + dz would represent one possibleinfinitesimal displacement, but would violate the constraint. The only possible virtual displacement would be a displacement from the bead's position, heta to a new position heta + delta hetaFact|date=September 2007 (where delta heta could be positive or negative).It is also worthwhile to note that virtual displacements are spatial displacements exclusively -
time is fixed while they occur. When computing virtual differentials of quantities that are functions ofspace andtime coordinates, no dependence ontime is considered (formally equivalent to saying delta t = 0).ee also
*
d'Alembert principle
*Virtual work References
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