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sample_finals - ME 563 Fall 2004 Final Exam Problem No 1 20...

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Unformatted text preview: ME 563 - Fall 2004 Final Exam Problem No. 1 - 20 points The three—DOF system shown below: 2fo sith 5 fo sith has natural frequencies of: col -0.7 rad/sec wz = 1.0 rad/sec (03 = 1.6 rad/sec and a mass normalized modal matrix of: 0.3384 0.7242 0.8801 - ;_ ~(I) ~(2) ~(3) [P]- 0.5276 0.5832 '0'4673JJE [9 «_p g ] 0.7791 -0.3681 0.0835 Find the particular solution for x, corresponding to an excitation frequency of £2 = 1.3 rad/sec if f0 = 0.40 newtons. E ME 563 - Fall 2004 Final Exam Problem No. 3 - 20 points Consider a forcing f (t) a 2 |sin6t|. a) What is the least fundamental period of f( t)? (The least fundamental frequency is the shortest time over which the function repeats.) b) Determine the Fourier series for fl t). f(t) W t Jt/6 Jc/3 n/Z 2n/3 ME 563 - Fall 2004 Final Exam Problem No. 4 - 20 points An undamped single-DOF system: mjr' + kx - f (I) is given the excitation f( t) shown below: f(t) e 2fo fo T 2T t W for the re5ponse of the system corresponding to initial conditions of x(0) - x0 and x(0) - 0. Clearly indicate the solution that is valid for the time ranges of: 0 OstsT ° TstsZT ° r22T You are NOT asked to evaluate these integrals. ME 563 - Fall 2000 Name Final Exam Problem No. 3 A damped three-DOF system has the following set of EOM’s; [MB + [CB +[K]x - fusith (0-! 3L0 —1 l J [C] - BIK] where and [i = 0.1 sec. The undamped natural frequencies (0- (j = 1,2,3) and mass-normalized modal vectors l - 1,2,3) have been determined, with the following results: 001 - 1.85rad/ sec 002 - 4.24rad/ sec (03 :- 4.85rad/ sec 0.372 0.707 0602 if” - [0.6021/ $73 in) - l 0 l/ JR; é”) =l-0.3721/JE [0.372] [—0.707] [ 0.602 J For this problem: a) Write down the uncoupled modal differential equations of motion. In these equations, explicitly indicate the modal damping ratios, the modal natural frequencies and the modal forcings. b) If the particular solution of the original EOM’s is written as: xpilt) - Disin(S2t - ¢i) ; j = 1, 2, 3 make sketches of D1 and D2 as functions of the exciting frequency 9. c) Interpret your sketch in b) above in terms of the modal contributions to the response. ME 563 - Fall 2000 Name FfinalExanI Problem No. 4 The following undamped, single-DOF system: x(t) l—’ ft () k is given a forcing f(t) shown below and initial conditions of x(O) - x(O) - 0. For this system, let k =101r2newtons/ meter and m - 10kg . f(t) t (secs) For this system: a) Write down the convolution integral solutions for: i) t = 1 sec ii) t = 4 sec iii) t = 6 sec DO NOT EVALUATE THESE INTEGRALS. b) Based on the graphical convolution approach, determine the times at which i) the first five maxima occur in x(t) ii) the first five minima occur in x(t) Multiple copies of f(t) are provided on the next sheet that you can use to assist you in making sketches for determining these times for maximum/ minimum responses (this sketching sheet will NOT be graded). c) Make a sketch of x(t) on the axes provided above for a time range that includes all of the times listed above in b). ME 563 - Fall 2000 Name Final Exam Problem No. 5 As shown in Figure (i) below. a loading f(x,t) acts on a wing of an aircraft, with the loading acting over a short period of time T = 0.2 seconds. f(x.t) F(t) F(t) fo x = 0 x = L 0 T/2 T g t Figure (i) Figure (ii) Figure (iii) Suppose that the wing is modeled as a single-DOF system (as shown in Figure (ii) above) where k = 10,000 newtons / meter, m = 100 kg and the forcing F(t) is as shown above in Figure (iii) with f0 - 120 newtons. a) Treating F(t) as a SHOCK loading, determine an upper bound AND a lower bound on the value of zmax for the response of the single-DOE model shown above. b) Treating F(t) as an IMPACT loading, determine an 112m on the value of zmax for the response of the single-DOF model shown above. c) Does the wing serve as an effective shock isolator in transmitting the applied force to the support of the wing? (You may need to refer to the lecture notes to recall the definition of effective shock isolation.) IGNORE THE INFLUENCE OF GRAVITY IN YOUR CALCULATIONS. ME563 - Fall 1998 Name Final Exam Problem No. 1 The mass matrix [M] and the flexibility matrix [A] ( = [K].1 , where [K] is the stiffness matrix) is given below for a 5-DOF undamped system. a) Determine a lower bound on the lowest natural frequency for the system. b) Derive the form of the Rayleigh quotient for an undamped system in terms of its flexibility matrix and mass matrix. c) Using [M] and [A] in the Rayleigh quotient found in b) to find an M bound on the lowest natural frequency for the system. [10 1 [2 2 2 2 2] | 10 I I2 4 4 4 4| [M]-: 10 :(kg) [A]- :2 4 6 6 6:x10'4(meter/ N) | 10 I I2 4 6 8 8' l 10] [2 4 6 8 9] ME563 - Fall 1998 Name Final Exam Problem N o. 3 An undamped, single-DOF oscillator experiences a base motion of y(t), with y (t) as shown below. The center of mass motion of the homogeneous cylinder is to be described by the coordinate x(t), where x(t) is measured relative to the base. The oscillator is given initial conditions of: x(0) - 5((0)-o. a) Write down the convoluti%n integral solution of x(t) for 2a < t < 3a. DO NOT EVALUATE THIS INTEGRAL. b) Say that we know that a = n V3m/2k. Without evaluating any integrals, make a sketch of the response x(t) for 0 < t < 5a on the axes provided below. In your sketch, clearly identify the times corresponding to: local maximum of x(t), local minima of x(t) and all zero crossings of x(t). y(t) x(t) |—>|‘—>I k m I !/ no slip y(t) ME563 - Fall 1998 Name Final Exam Problem No. 5 IQNORE QRAVITY IN YOQR WORK WITH THI§ PROBLEM. a) A constant force f0 = 100 N is applied to a particle of mass M that is supported by a beam of length L and having negligible mass compared to the particle. As a result of the application of this force, the tip of the beam deflects by an amount of 0.01 meters. If M = 10 kg, what is the natural frequency (on of this system? f0 I‘— L A El negligible mass A harmonically-varying force (with a magnitude f0 = 100 N) is now applied to the tip of the beam with the frequency of the force (unfortunately) tuned to the natural frequency of the beam/ particle system. To eliminate the resonant response of the beam, a second particle B (having a mass of 2M) is attached to the tip of the beam with a spring of stiffness K. i) What must be the stiffness of this spring in order to completely eliminate the vibrations in the beam? ii) What is the amplitude of the motion of the newly-attached particle B? f0 coswt negligible mass ...
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