hwk10 - ME563 – Fall 2011 Purdue University West...

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Unformatted text preview: ME563 – Fall 2011 Purdue University West Lafayette, IN Homework Set No. 10 Assignment date: Friday, November 4 Due date: Friday, November 11 Please attach this cover sheet to your completed homework assignment. Name PUID Problem 10.1 Problem 10.2 Problem 10.3 TOTAL Homework 10.1 A fixed-fixed string having a tension of P and mass/length ρ is excited by a force/length of f (x, t) = f0 g (t), as shown below with f0 being constant in both x and t. If g (t) = sinΩt, the particular solution for the EOM for this system at x = L/3 can be written as u(L/3, t) = U (Ω)sinΩt a) Determine an expression for U (Ω) using modal uncoupling. This result will be in terms of an infinite summation. b) Identify the modes that do NOT contribute to U (Ω). c) Make a hand sketch of U (Ω) vs. Ω for Ω up through the first four resonance frequencies that appear in the response at x = L/3. (Note that not all modes will contribute to this response.) Clearly indicate all frequencies of resonance and anti-resonance in this frequency range. () f0 g t () u x ,t x=0 x = L /2 x =L Homework 10.2 Consider the single-DOF model for a suspension system for an automobile moving with a constant speed v along a roadway with a spatially-varying harmonic function of vertical roughness. It is your task to design a suspension system with the following criteria: • The amplitude of the vertical motion of the body no greater than 0.05y0 for a design speed v = vdesign . • The transmissibility of the vertical force to the body is no greater than 1.1 for an automobile speed that corresponds to the undamped natural frequency of your suspension design. Ignore gravity in your dynamic design. Assume that the following parameters are known: m, y0 , λ and v = vdesign . Your design will constitute appropriate values for c and k in terms of these known parameters. HINT: Recall from lecture that the transmissibility for displacement and transmitted force in a base-excited system (such as this system) are the same. You need not re-derive the transmissibility function; you may use the results directly from lecture once you have derived the system’s EOM. v m () k yz c x z y0 ! Homework 10.3 An undamped single-DOF has a base excitation of z (t) = z0 sinΩt. Unfortunately, an operating excitation frequency of Ω = Ωoper produces resonance. You are asked to design an absorber system of the type shown below right under the constraint that M ≤ 0.2m. a) Determine an appropriate set of values for M and K for which perfect vibration absorption occurs Ω = Ωoper and with the constraint that you achieve a maximum possible separation between the resulting natural frequencies of the system with the absorber. Express your design results in terms of m and k of the original system. b) What is separation between the resulting natural frequencies of the system with your absorber design? () () x zt k y x zt k m original system K m K M system with absorber ...
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