Principle of Virtual Work: Problem Solving

The principle of virtual work states that if a system is in equilibrium, the total work done by all forces during a virtual displacement equals zero. This principle is used to solve equilibrium problems by considering hypothetical displacements and calculating the work done by internal and external forces. It is particularly beneficial for solving problems involving complex systems with many interconnected parts.

Position coordinates of internal and external forces must be expressed in terms of a common coordinate parameter, typically an angle, distance, or displacement relevant to the problem. Once expressed, these coordinates are differentiated to find the virtual displacements corresponding to each force. This allows calculation of the virtual work done by each force acting on the system.

External forces do positive work when they act in the direction of their virtual displacement, while internal forces do negative work when they oppose their virtual displacements. In a scissors linkage system with a spring, the external force moves with its displacement doing positive work, while the spring force opposes the displacement doing negative work.

First, identify unknown forces and moments in the system. Then evaluate the work done by these forces during a hypothetical virtual displacement. Next, write the virtual-work equation considering virtual displacements and corresponding work by each force. Finally, solve the equation to determine unknown forces and moments. This systematic approach applies to virtual work for a system of connected rigid bodies.

The virtual-work equation is set equal to zero at equilibrium. By determining position coordinates of forces relative to a fixed point and calculating spring compression and corresponding spring force, you can differentiate to find virtual displacements. Solving the virtual-work equation yields the deflection angle needed to maintain the scissors linkage system's equilibrium.

Virtual displacements are very small hypothetical displacements that the system could undergo without violating constraints. They are not actual movements but mathematical constructs used to calculate work done by forces. This hypothetical approach allows engineers to analyze equilibrium conditions and determine unknown forces without requiring the system to physically move.

Both internal and external forces contribute to the total virtual work during a hypothetical displacement. External forces applied to the system and internal forces like spring forces are evaluated for their work contributions. At equilibrium, the algebraic sum of work done by all forces equals zero, allowing determination of system behavior and unknown force values.