mechanical engineering vibration HW

Homework No. 11 – Due Friday, 4/17

1. An underdamped oscillator of mass m = 10 kg is excited by a shock input force shown in the figure. (a) Set up by hand the convolution integrals for the response x(t) over the entire time range. (b) Calculate and plot the response and then determine the maximum response from the graph for the following two cases: Case (I): ωn = 10 rad/s and ζ = 0.1. Plot the response for 10 s. Case (II): ωn = 0.8 rad/s and ζ = 0.1. Plot the response for 25 s. (c) Use “ode45” in MATLAB to solve the response numerically and plot the response for both cases.

Ans: (b) 0.319 m for Case (I), 76.3 m for Case (II) 2. Consider a car going over a speed bump at a velocity υ as shown in the figure. The mass of the car is m = 1000 kg and the suspension system provides stiffness k = 400 kN/m and damping ζ = 0.3. The speed bump can be approximated by a sinusoidal that has amplitude Y = 0.25 m and length L = 0.6 m. (a) Set up by hand the convolution integrals for the response x(t) over the entire time range. (b) Calculate and plot the response for 1 s and then determine from the graph the maximum displacement of the car as it passes the bump for the following two cases: Case (I): υ = 5 km/hr. Case (II): υ = 20 km/hr (c) Use “ode45” in MATLAB to solve the response numerically and plot the response for both cases. Ans: (b) 0.310 m for case (I), 0.244 m for case (II) 3. A slab is connected to a spring and laid on top of a disk as shown. The uniform disk rolls without slipping on a fixed rail and there is no slippage between the disk and the slab. The small oscillation of the system is defined by the slab’s displacement x measured from the equilibrium position, and the equation of motion of the system is

derived as: 4 4 10 3 9 9

mx cx kx P+ + =  .

The system parameters are m = 10 kg, k = 300 N/m, c = 45 N∙s/m, and r = 0.1m.The input excitation P(t) is shown in the figure with t1 = 4 s, t2 = 6 s, t3 = 12 s, and F0 = 80 N. (a) Calculate and plot the velocity response of the slab vs(t) for 15 s, and determine from the graph the maximum response vsmax. (b) Calculate and plot the displacement response xG(t) at the center of the disk for 15 s, and determine from the graph the maximum response xGmax. (c) Calculate and plot the response of the force FT(t) transmitted to the left wall for 15 s, and determine from the graph the maximum transmitted force FTmax. (d) Use “ode45” in MATLAB to solve the response numerically and plot the three responses vs(t), xG(t), and FT(t). Ans: (a) 1.905 m/s (b) 0.1576 m (c) 145.1 N

F (N)

t (s)

300

100

2 5

m

L

y Y

x m

k c

υ

2r

r

x

c

k

m G

k

rail

m P

4. Consider the following nonlinear differential equation:

2

2 1 0 1

b x x x

x ζ

  + − − = 

−   

(a) Recast the equation in state space using the following state variables: y1 = x and y2 = x . (b) Use “ode45” in MATLAB to solve the nonlinear equation numerically for the following parameters: ζ = 0.2 and b = 0.75, with initial conditions x(0) = −0.8, and (0) 0.3x = . Plot the solution x(t) from t = 0 to t = 40 s. (c) Use “NDSolveValue” in Mathematica to solve this nonlinear oscillation numerically and plot the response for 40 s.