Final Exam Instructions: You have 120 minutes to complete this exam. This is a closed-book, closed-notes exam. You are NOT allowed to use a calculator with communication capabilities during the exam. Usage of mobile phones and other electronic communication devices is NOT permitted during the exam. Only work included on the FRONT of a page will be graded. Problem 1 (20 pts) 2 (20 pts) 3 (20 pts) 4 (20 pts) 5 (20 pts) Total Score Name: Exam 1 Page 1 of 13
Problem 1 (20 points): A block of mass M is suspended vertically via a spring of stiffness k and a dashpot with damping coefficient c. A pendulum of length L and mass m is suspended from the block at point O. Let x describe the position of O, where the springs are unstretched when x = 0. Find: (a) The kinetic energy associated with the SYSTEM; (b) The potential energy associated with the SYSTEM (You must clearly indicate your datum to receive credit); and (c) The Rayleigh dissipation function associated with the SYSTEM. Hint: I G = 1 12 ml2 and M m Exam 1 Page 2 of 13
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Problem 2 (20 points): Given: A string of length L and mass density ρ is vibrating transversely. The string is supported between two fixed ends, such that it has an internal tension of P. Find: (a) An expression for the jth natural frequency of the system; and (b) An expression for the corresponding jth mode shape. Hint: The differential equation governing the system is given by: P 2 u(x, t) x 2 = ρ 2 u(x, t) t 2, where u(x, t) represent the transverse displacement of the string. Exam 1 Page 4 of 13
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Problem 3 (20 points): Given: Box B (of mass M) is given a prescribed displacement of x B (t) = b sin Ωt. A component A (of mass m) is placed inside the box and connected to it by two springs, each with stiffness k. Assume that the inside of the box is smooth and that x corresponds to the displacement of component A. Find: (a) Draw a free-body diagram for component A; (b) Find the equation of motion (EOM) for component A; (c) Determine the natural frequency of the system; (d) Find the particular solution associated with the movement of component A; and (e) State if this particular solution is in-phase or out-of-phase with the forcing, if Ω = 2ω n. Exam 1 Page 6 of 13
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Problem 4 (20 points): Given: A diesel engine, weighing W = 3000 N, is supported on a pedestal mount. It has been observed that the engine induces vibration into the surrounding area through its pedestal mount at an operating speed of Ω = 6000 rpm. The magnitude of the exciting force is F 0 = 250 N, and the amplitude of motion of the absorber mass is to be limited to x max = 2 mm. Find: (a) Draw a free-body diagram of the system; and (b) Find the vibration absorber design (mass and stiffness) needed to satisfy the design criteria. Hints: You should design your absorber such that the motion of the pedestal goes to zero. You may ignore the impact of both gravity and damping in this problem. Exam 1 Page 8 of 13
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Problem 5 (20 points): (Part A: 10 points) Given: The transverse displacement of a thin beam of length L is governed by the equation of motion: EI 4 w(x, t) x 4 + ρa 2 w(x, t) t 2 = f(x, t), where w(x, t) denotes the transverse displacement of the beam, EI specifies the flexural rigidity of the beam, and ρa denotes the mass per unit length of the beam. In the course of analysis, you approximate the dynamics of this beam using the single mode approximation: w(x, t) = φ(x)z(t) to recover an ordinary differential equation for the system of the form: m eff z + k eff z = F eff. Find: (a) An expression for the effective mass m eff ; (b) An expression for the effective stiffness k eff ; and (c) An expression for the effective force F eff. Fact: Use the methods of modal projection taught in class to receive full credit. You DO NOT need to find the mode shape φ(x). Exam 1 Page 10 of 13
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(Part B: 10 points) Circle the correct answer below. A system with a damping ratio ζ = 0 is said to be: (a) Undamped (b) Underdamped (c) Critically damped (d) Overdamped A system with a damping ratio ζ > 1 is said to be: (a) Undamped (b) Underdamped (c) Critically damped (d) Overdamped An N degree of freedom system with no repeated natural frequencies will have: (a) 1 resonance (b) N 1 resonances (c) N resonances (d) An infinite number of resonances A continuous system with no repeated natural frequencies will have: (a) 1 resonance (b) N 1 resonances (c) N resonances (d) An infinite number of resonances An anti-resonance in a multi-degree of freedom system always leads to very large motion at that frequency. (a) True (b) False A rigid body mode has a corresponding natural frequency of ω = k/m, where m and k represent the non-zero effective mass and effective stiffness of the system respectively. (a) True (b) False Exam 1 Page 12 of 13
In a rigid body mode all coordinates must have unequal modal displacements: (a) True (b) False A benefit of using a convolution integral solution is that the initial condition portion of the total response is uncoupled from the forced part. (a) True (b) False The maximum amplitude a damped vibration system can have will always be at the resonance frequency. (a) True (b) False In vibration isolation problems, a designer wants transmissibility to be minimized. (a) True (b) False Exam 1 Page 13 of 13