Virtual Work for a System of Connected Rigid Bodies

Virtual work is a powerful method for analyzing systems with multiple connected rigid bodies in equilibrium. When the system is in equilibrium, virtual work equals zero, allowing you to calculate resulting forces during virtual displacement. This approach simplifies complex calculations by using a single independent coordinate to specify the system configuration, making it ideal for principle of virtual work problem solving.

Virtual displacement represents an imaginary, infinitesimal change in the system's configuration used to analyze equilibrium. By marking the deflected position resulting from positive virtual displacement on a free-body diagram, you can express each coordinate in terms of an independent variable. This allows calculation of virtual work done by applied and reaction forces without actually moving the system.

A free-body diagram identifies which forces perform work during virtual displacement and which do not. By measuring position coordinates from a fixed point and directing them toward forces that do work, you can determine virtual displacements for each force. This systematic approach ensures accurate calculation of total virtual work and prevents errors from including forces that contribute no work.

Virtual work is positive when a force or couple moment acts along the direction of positive virtual displacement. Virtual work is negative when the force or moment opposes the positive virtual displacement direction. This sign convention is critical for correctly setting up the equilibrium equation where total virtual work equals zero.

A one-degree-of-freedom system has connected rigid bodies whose complete configuration can be specified using only one independent coordinate. This constraint greatly simplifies analysis because all positions and displacements can be expressed in terms of that single variable. Most practical machines with connected links, like toggle vises, function as one-degree-of-freedom systems.

In a toggle vise, define coordinates in terms of angle theta and express all positions relative to this angle. Calculate virtual work done by the applied force and reaction force on the wooden block. Set total virtual work to zero at equilibrium, then differentiate and solve to find the reaction force equation in terms of the applied force and geometry.

Forces and moments that act perpendicular to virtual displacement do no work and can be ignored. In a toggle vise example, reaction forces at support points and normal forces perpendicular to motion contribute zero work. Focusing only on forces aligned with displacement simplifies calculations while maintaining accuracy for equilibrium analysis.