Mechanical Vibration Homework#1

MENG-470 Due on 17/11/1432

(1) What are the generalized coordinates (DOFs) you will define in order to study the dynamics

of the system? SHOW THESES COORDINATES ON THE FIGURE ABOVE

(2) Formulate the kinetic energy of the system.

(3) Formulate the potential energy of the system.

(4) Use Lagrange’s equations to derive the equations of motion of the system. WRITE THE

SYSTEM’S EQUATIONS OF MOTION IN MATRIX FORM

(1) Write down the equation of motion of the “stopping-system”.

(2) In order to solve the above equation, what initial conditions you will specify?

(3) Determine the maximum displacement of the railroad car after engaging the springs and

the damper.

(4) Determine the time taken to reach the maximum displacement.

Problem1:

The dynamical system shown below consists of two elastic springs, each with stiffness �, an

inverted pendulum with mass � and length � and a movable cart with mass 5�.

Problem2:

A railroad car of mass 2000kg traveling at a speed v=10m/s is stopped at the end of the tracks

by a “stopping-system” which can be modeled as a spring-damper system, as shown in the

figure below. The stiffness of the spring k=40N/mm and the damping constant c=20 Ns/mm.