hwk09_w_cover - ME563 – Fall 2011 Purdue University West...

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Unformatted text preview: ME563 – Fall 2011 Purdue University West Lafayette, IN Homework Set No. 9 Assignment date: Friday, October 28 Due date: Friday, November 4 Please attach this cover sheet to your completed homework assignment. Name PUID Problem 9.1 Problem 9.2 Problem 9.3 TOTAL Homework 9.1 Consider the EOM for a forced, damped single-DOF system: 2 x + 2ζωn x + ωn x = ¨ ˙ f ( t) m Starting with the general form of the convolution integral solution derived in lecture (see equations (IV.12) and (IV.13) in Section IV.3 of the lecture notes), derive the convolution integral for an underdamped (0 < ζ < 1) system. Note that the final form of the underdamped convolution integral solution is found on page IV-36 of the lecture notes; you can use this to check your answer). Homework 9.2 Consider the EOM for a forced, undamped single-DOF system: 2 x + ωn x = ¨ f ( t) m In this problem, you are asked to use the convolution integral approach to produce a qualitative sketch for the response of this system with the excitation shown below and zero initial conditions corresponding to: a) T = π /ωn b) T = 2π /ωn for 0 < t < 4T . In your sketches, clearly identify the following: • the exact times for the zero crossing of x(t) • the exact times and corresponding response values for local maxima x(t) • the exact times and corresponding response values for local minima x(t) Homework 9.3 Consider the cantilevered beam shown below where the beam has negligible mass and a flexural rigidity of EI . Three particles, each of mass m, are attached to the beam at locations shown. The transverse deflections for the particles will described by the generalized coordinates y1 (t), y2 (t) and y3 (t). A transverse force F (t) is applied to the beam as shown in the figure. Although we generally write our EOMs for multi-DOF systems in terms of the mass and stiffness matrices: ¨ ￿ [M ]￿ + [K ]￿ = f (t) y y (1) we will write our EOMs for this problem in terms of the mass and flexibility matrices: ¨ ￿ [D]￿ + ￿ = [A]f (t) yy (2) where [D] = [A][M ] and [A] = [K ]−1 = flexibility matrix. We will do so here to take advantage of some previously-derived results for the flexibility matrix for this problem. ￿ a) Develop the mass matrix [M ] and the forcing vector f (t) for the ￿ (t) coordinates. y b) Write down the [D] = [A][M ] matrix. Note that the [D] matrix is symmetric. Do not derive the flexibility matrix [A]; you may directly use the results of Example I.6.1 from the course lecture notes. ￿ c) Note that the eigenvalue problem for equation (2) above using ￿ (t) = Y e−iωt is: y ￿2 ￿ ￿ −ω [D ] + [ I ] Y = ￿ 0 where [I ] is the 3 × 3 identity matrix. Determine the natural frequencies ωj and and modal ˆ ˆ ￿ ￿ ￿ vectors Y (j ) . Normalize the modal vectors such that Y (j )T [D]Y (j ) = 1. d) Using the modal coordinate transformation: ￿ ( t) = y 3 ￿ˆ ￿ Y ( j ) pj ( t) j =1 write down the uncoupled EOMs for the modal coordinates p1 (t), p2 (t) and p3 (t). Rank order the magnitudes of the modal forcing functions. e) (10 bonus points) Suppose that the forcing on the beam is F (t) = F0 δ (t), where δ (t) is the Dirac delta function. Determine the response of the system ￿ (t) corresponding to zero initial y conditions. ...
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This document was uploaded on 12/23/2011.

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