The equations of motion of a four-bar linkage with principal vectors and virtual work

Abstract

The motion of a four-bar linkage is considered with the goal to study the use of principal vectors to formulate the equations of motion and to get insight. Firstly, kinematic relations for the positions, velocities and accelerations are derived. Then, the motion of the centre of mass of the system is described with the aid of principal points and principal vectors, for which the mass of one link is replaced with equivalent masses. The condition of dynamic force balance is that the centre of mass is stationary. It is shown that the motion of the centres of mass of the links can be described in terms of the principal vectors. The equations of motion and the expressions for the force and moment on the base are derived with the aid of the principle of virtual work, which directly give conditions for dynamic force and moment balance. The equations of motion show a clear structure in their coefficients. The expression for the reaction force becomes simple, but the expression for the reaction moment remains rather complicated.