The dynamic system shown below consists of a cart of mass M carrying a simple pendulum L. The cart moves vertically. It is constrained by a spring k and damper c, and is subject to an external force f. At time t=0 the spring k is in its free length.

• . Derive the governing equations for this system in terms of the angular displacement of the pendulum ?(?), and the linear displacement of the cart ?(?).
• . Derive the expression for the tension force of the pendulum in terms of ?(?) and ?(?).

Image transcription text

M g

There exists a horizontal reaction force from the surfaces acting on the cart M that can be represented as a single force. Instead of representing the interaction force at the pivot as its horizontal and vertical components, consider a single tension force T of the pendulum.  Instead of describing rotation of the pendulum about an accelerating pivot, consider only motion of its bob in the horizontal and vertical directions separately.

1 M L m g