Homework 7 (2 Problems)

1. A 2-DOF system is shown in Figure 1. The mass of m₂ is acted by the external force f(t) =

fo sin(wt) and the system is also excited by x (t) = xo sin(wt) through the left massless plate.

The system parameters are given as m₁ = 4m₂ = 16 kg, c = 200 N s/m, k₂ = 2k₂ =

100 kN/m, fo = 10 kN, xo = 1 m, and w = 200 rad/s. The initial conditions are given as

x₁ (0) = 0.5 m, x₂(0) = 0 m, ₁ (0) = 0 m/s, and x₂(0) = 20 m/s. For the given information,

solve the following problems:

(a) The coordinates of x₂ and x₂ are the displacements of m, and m₂, respectively. Draw the free-

body diagrams (FBDs) of m, and m₂, and derive EOMS by applying the Newton's 2nd Law of

Motion to the FBDs. Write the EOMS in a matrix form in terms of mass matrix, M, damping

matrix, C, and stiffness matrix, K.

(b) Find the free vibration responses in terms of 4 unknown coefficients, four eigenvalues, and four

eigenvectors. In your solution, clearly show what the eigenvalues and eigenvectors are. Do not

use an order reduction to make the eigenvalue problem to a standard form.

(c) Find the expression of the forced vibration responses in terms of the mass matrix, M, damping

matrix, C, and stiffness matrix, K.

(d) By applying the initial conditions to the total solutions (i.e., the summation of the free and

forced vibration responses), find the expression for the vector of the unknown coefficients in

the free vibration responses.

(e) Write a Python code to plot the free, forced, and total displacements from t=0 s to t = 0.3 s.

Submit your Python code.

x(1)

für

Massless plate

m₁

HX1

Figure 1-2-D05 Surtom

m₂

HX₂

f(t)