Question

Find the equations of motion using D’alembert principle.

Answer 1

D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that is to be

D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that is to be delta W = (Q_1 + Q_1*) delat q_1 + .... + (Q_m + Q_m*) delta q_m = 0, for any set of virtual displacements delta qj. This condition yields m equations, Q_j + Q_j* = 0, j = 1, ...., m, which can also be written as, d/dt partial differential T/partial differential q_j = Q_j, j = 1, ...., m. ..... (1) The result is a set of m equations of motion that define the dynamics of the rigid body system. Where, T is the Kinetic energy and Q_j is the generalized force.

for any set of virtual displacements δq_j. This condition yields m equations,

which can also be written as,

.....................(1)

The result is a set of m equations of motion that define the dynamics of the rigid body system. Where, T is the Kinetic energy and $Q_{j}$ is the generalised force. Here, if we assume velocity independent potential, we can write generalised Force as gradient of a potential V.

Therefore, we can re-write (1) as,

Where, L=T-V known as the Lagrangian. And the above equation is called as the Lagrangian equation of motion.

Let us consider the system as a pendulum made of a spring with a mass m on the end . The spring is arranged to lie in a straight line. The equilibrium length of the spring is . Let the spring have length + x(t), and let its angle with the vertical be θ(t). Now, Let us find the Kinetic Energy and Potential Energy of the system. Here, the generalised coordinates are displacement of spring 'x' and angle of the pendulum θ.