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Homework answers / question archive / in this assignment, you will model the dynamics of a vibrating system, analyze its behaviour and design a system to damp away the vibrations

in this assignment, you will model the dynamics of a vibrating system, analyze its behaviour and design a system to damp away the vibrations


in this assignment, you will model the dynamics of a vibrating system, analyze its behaviour and design a system to damp away the vibrations. ALLLLLL ) The vibrations are expected to become violent as the fore as the forcing frequency reaches a certain Consider a system where a motor, designed to run at a constant speed, is mounted on a non- heleeee Primary value. This violent resonance condition can be controlled by adding a small mass to the rigid cantilever beam as shown in Figure 1. The rotating part of the motor has an eccentricity system. Suppose the vibration absorber represented by the parameters K, and M, has been (off-centre effect) which gives rise to a sinusoidal force of frequency, wy. This sinusoidal force which can be written as f(t) = Asinwyt, will be exerted on the cantilever on which the motor M added and the system can be modelled as shown in Figure 3. Suppose the equations of motion describing the behaviour of the system in Figure 3 are as follows : is mounted. A denotes the amplitude of the force. = /(t) - Ku(t) - Ki(w(t) - z(0)) 2 - Kily(t) - z(t)) Figure 3: Model of the system with the vibration absorber. vibration absorber to reduce or remove the vibrations. Find the transfer function, Ga(s), between the cantilever displacement Y'(s) and the force, New ripd cantilevel beans In order for the designs to differ from one student to another, you should generate values of M F(s), generated by the rotation of the motor, taking into consideration the vibration ab- Figure 1: Motor mounted on a non-rigid cantilever beam. and K using your matriculation number as follows sorber. Mass, M is (C' + 1) kg The cantilever beam can be modeled as a spring-mass system as shown in Figure 2. For this . Spring constant, K, is 2250(D + 1) aves like a spring with spring constant K. The force model, the cantilever is of mass Af, and behaves like a spring with generated by the motor causes the cantilever to vibrate in a certain manner. One way to reduce where C and D may be derived by reading the last 4 digits of your student matriculation or remove the vibrations on the cantilever is to attach a vibration absorber to the cantilever number. Equate the first two digits to C and the remaining two to D. For example if you so that the dynamical properties of the cantilever, together with the vibration absorber, are last 4 digits of your matriculation number atriculation number is 1234, then choose C = 12 and D = 34. With altered. The vibration absorber should be designed with a certain mass and material property. he values of Af and K determined in this manner, you may proceed to answer the following 6) What conditions can you impose on Kj and My such that G2(s) takes the form of The model for the system with the vibration absorber is shown in Figure 3. You may assume questions. that the mass of the motor has no effect on the bending moment of the cantilever. C(Mis + K1 ) sing Newton's second law of motion, derive the differential equation governing the motion G2(s) = 7.2 ( 82 + wal ) (8 2 + wn 2 ) of the cantilever (Figure 2) as it vibrates due to the force generated by the motor. Essentially, obtain the relationship between the displacement of the cantilever, v where C is a constant. generated by the rotation of the motor. You may assume that the tension of the spring at any time t is equal to Ky(t). M 7) Design the parameters of the vibration absorber so that the vibrations can be reduced to Figure 2: Model of the motor mounted on cantilever. zero. Explain how you arrive at your answer. This assignment explores the modelling of such a system and leads up to the design of the (8) Sketch the amplitude response (IG2(jw)| versus w) of the new system. 2) Derive the transfer function, G;(s) (between Y(s) and F(s)), of the cantilever. Find the poles of Gi(s). ") Assume that the cantilever is at equilibrium with y(0") = 0 and y(0") = 0, before the motor was started at t - 0. Calculate the response of the cantilever when the vibration from the motor has a frequency, wy = 10 rad/s. Assume the force to be sinusoidal of amplitude 4) Explain what happens to the cantilever when the vibration from the motor is of frequency

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