hwk08 - ME563 – Fall 2011 Purdue University West...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ME563 – Fall 2011 Purdue University West Lafayette, IN Homework Set No. 8 Assignment date: Friday, October 14 Due date: Friday, October 21 Please attach this cover sheet to your completed homework assignment. Name PUID Problem 8.1 Problem 8.2 Problem 8.3 TOTAL Homework 8.1 The EOMs for an undamped two-DOF system are given by: 100 2 −1 0 0 ¨ 0 1 0 ￿ + −1 2 −1 ￿ = 0 sinΩt x x 001 0 −1 1 1 ￿ The particular solution of these EOMs is to be written as: ￿ P (t) = XsinΩt. x 1. Determine all resonance and anti-resonance frequencies for X1 (Ω), X2 (Ω) and X3 (Ω). Also, determine the values of X1 (0), X2 (0) and X3 (0). 2. Make hand sketches of |X1 (Ω)|, |X2 (Ω)| and |X3 (Ω)| vs. Ω. These sketches should clearly show the resonance and anti-resonance frequencies, as well as the values for X1 (0), X2 (0) and X3 (0) found above. You may use Matlab to determine the resonance frequencies. All other work (including your sketches) should be done by hand. Homework 8.2 Consider a damped two-DOF with the following EOMs: ￿￿ ￿￿ ¨ + [C ]￿ + [K ]￿ = 2 sinΩt + 0 cosΩt ˙ [M ]￿ x x x 0 1 (N ewtons) The real and imaginary parts of three elements of the complex transfer matrix H (Ω) vs. Ω are shown on the following pages. Determine the steady-state responses xP 1 (t) and xP 2 (t) corresponding to Ω = 0.5 rad/sec. Write your solutions as xP j (t) = Xj sin(Ωt − φj ). 50 40 real(H11) mm/N 30 20 10 0 10 20 30 40 0 0.5 1 1.5 rad/sec 2 2.5 3 0 0.5 1 1.5 rad/sec 2 2.5 3 20 10 0 imag(H11) mm/N 10 20 30 40 50 60 70 80 50 40 real(H12) mm/N 30 20 10 0 10 20 30 40 0 0.5 1 1.5 rad/sec 2 2.5 3 0 0.5 1 1.5 rad/sec 2 2.5 3 20 10 0 imag(H12) mm/N 10 20 30 40 50 60 70 80 50 40 real(H22) mm/N 30 20 10 0 10 20 30 40 0 0.5 1 1.5 rad/sec 2 2.5 3 0 0.5 1 1.5 rad/sec 2 2.5 3 20 10 0 imag(H22) mm/N 10 20 30 40 50 60 70 80 Homework 8.3 The damped single-DOF shown below experiences a T-periodioc excitation, g (t). The excitation g (t) is defined as being equal to the harmonic function shown below for −T /4 < t < T /4 and equal to zero elsewhere over −T /￿< t < T /2. Since g (t) is T-periodic, it can be written as the following 2 Fourier series: g (t) = g0 + ∞ [gcj cosj Ωt + gsj sinj Ωt], where Ω = 2π /T . j =1 1. Determine the Fourier coefficients g0 , gcj and gsj for g (t). 2. If the steady-state response is written in its Fourier series as xP (t) = X (0) + φj ), determine the response Fourier amplitudes X (j ) . ￿∞ j =1 X (j ) sin(j Ωt− 3. Numerically evaluate kX (j ) /A for j = 1, 2, 3, 4. Use damping ratio and natural frequency values of ζ = 0.1 and ωn = π /T , respectivley. ...
View Full Document

This document was uploaded on 12/23/2011.

Ask a homework question - tutors are online