Mechanical engineering vibration test HELP!!!

ME 3504 Dynamic Systems – Vibrations

TEST 2 – April 20, Spring 2020

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– You must work on the test on your own. Remember to include any codes you used in the solutions. 1. (20 points) The small sphere of mass m is mounted on the uniform slender rod of mass M = 4m and length L. The system is pinned at O and supported by a spring as shown, and the equilibrium position of the assembly is at vertical where θ = 0. (a) Use the energy method to derive the equation of motion for small angle θ of oscillation. (b) Use the parameters m = 1 kg and L = 3 m to determine the minimum value of k for the system to have stable oscillatory response. 2. (17 points) A 5 kg surveillance camera mounted on two identical supporting rods undergoes horizontal oscillation during an earthquake. The horizontal base motion y can be assumed to be harmonic with 0.05 m amplitude and 20 Hz frequency. Each supporting aluminum rod (E = 70 GPa and ρ = 2700 kg/m3) has a length of 0.8 m and the damping of the system is ζ = 0.02. Determine the required diameter of the rod if the bending stress in the rod induced by the oscillation is not to exceed 50 MPa. The diameter has to be at least 50 mm for safety concerns. Include the mass of the rod in your design: meq = m + 2 × 0.2mrod where mrod is the mass of one rod. Note: The rods have clamped- clamped ends. 3. (16 points) A 20 kg mechanical equipment is driven by a harmonic excitation that has force amplitude of F0 = 200 N and excitation frequency of 6 Hz. The isolator of the system provides a stiffness value k = 12.5 kN/m. If the oscillation of the equipment is not to exceed 10 mm and the force transmitted to the supporting frame is to be limited to 180 N, determine the acceptable range of the isolator’s damping coefficient c. 4. (15 points) In a hammer testing an oscillator of mass m = 8 kg, damping c = 16 N∙s/m, and stiffness k = 200 N/m is subjected to impulse excitations that can be expressed as ( ) 10 ( ) 40 ( 5) 60 ( 8) NF t t t tδ δ δ= − − + − Plot the response for 12 seconds and determine from the graph the maximum response. 5. (15 points) The equation of motion of a mechanical system is given as mx cx kx cy+ + =   where y is the input motion. For the harmonic input motion y(t) = Y sinωt, apply complex response solution approach to show that the amplitude X of the

response is 2 2 2 1/ 2 2

[(1 ) (2 ) ] r

X Y r r

ζ ζ

= × − +

where ζ is the damping ratio and r is the frequency ratio r = ω/ωn.

NOTE: You must show the step by step solution by hand.

O

θ

k

2L/3

m

M = 4m

y

x

6. (17 points) A 5 kg oscillator with a stiffness of k = 320 N/m and damping coefficient c = 16 Ns/m is driven by a periodic excitation F(t) shown in the figure. Plot the steady state response for 12 s using 100 terms in the Fourier series solution. Also determine from the graph the maximum steady state response.

F (N) 60

1 3

4

6

7

t (s)

1.5 7.5 4.5