《振动与波动力学》部分答案.doc

PAGE PAGE 1 Answers (Courtesy of Cheng Feifei. I used some of her typing) 2.3 2.3 A given system of unknown mass m and spring constant k was observed to oscillate harmonically in free vibration with the natural period . When a mass M=0.9kg was added to the system the new rose to . determine the system parameters m and k. ANSWER The period of the original system . (1) It changes with an added mass M, to (2) Comparison of with (2) yields . Therefore m=1.6 kg , 1.6×10N/m. 2.18 A 2.18 A simple pendulum is immersed in viscous fluid so that there is a resisting force of magnitude acting on the bob, where the c is the coefficient of viscous damping, L the length of the pendulum and the angular displacement. In the case in which the mass of the bob has the value m=1kg and the length is L=1m the period of small-amplitude oscillations was observed to be T=2.02s. Determine the damping coefficient c. ANSWER The equation of motion is expressed by using the theorem of angular momentum with respect to the support point, as . For a minor vibration in neighborhood of the equilibrium, i.e. =0, the forgoing can be rewritten as , where . On the other hand, the period of the under-damped system is , therefore, Substitution of and into the forgoing results in the damping ratio As a result, the damping coefficient becomes c = From the observation of the response of an underdamped single-degree-of- freedom system, it was determined that the maximum displacement amplitude during the second cycle is 75% of the first. Calculate the damping factor From the observation of the response of an underdamped single-degree-of- freedom system, it was determined that the maximum displacement amplitude during the second cycle is 75% of the first. Calculate the damping factorand determine the maximum displacement amplitude during cycle 4.5 as a fraction of the first. ANSWER The logarithmic decrement is solved through the following formula , wh