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2.004 MODELING DYNAMICS AND CONTROL II Spring 2003 Problem Set No. 8 Problem 1. Three equal masses m slide without friction on a rigid horizontal rod. Six identical springs with spring constant k are attached to the masses as shown in the sketch below. The free vibrations of the system are described by three natural modes of vibration and their corresponding natural frequencies of vibration. Your problem is to identify as many natural modes and natural frequencies as possible. Problem 2. Torsional vibration. Probably the most common mechanical power trans-mission device is the simple shaft. A driving motor or engine applies a torque τ to one end of the shaft and the other end applies the torque τ to the driven element: a propeller, a machine tool, a pump, an electric generator, etc. The shaft is twisted by the torque, but if the driver can maintain a rotation rate ω, then the power transmitted along the rotating, twisted shaft is π = τω. In the ideal case both the torque τ and the speed ω are constant and there is a steady flow. In actuality the characteristics of both drivers and driven ele-ments are such that the power flow fluctuates throughout each revolution. One of the worst offenders for fluctuating torque is the internal combustion engine. A sequence of explosions produces a fluctuating torque, which is just about the opposite of a steady constant torque. A consequence of fluctuating torque is fluctuating twist in the shaft: torsional vibration. Historically, torsional vibration was a major technical challenge during the first quarter of the twentieth century. Otto and Diesel engines were being widely adopted in cars and 1ships, and there was an epidemic of shaft failures due to torsional vibration. This was a particularly insidious type of vibration because the twisting vibration was superposed on the steady average rate of rotation ω and was unobservable by the crude instrumentation available at that time. Professor Edward Miller, one time head of the Mechanical Engineer-ing Department at MIT, used to tell the story of how he once detected a severe torsional vibration before the shaft failed in fatigue, when he happened to notice, during the intial test of a pump driven by a Diesel Engine, that the shaft connecting the engine to the pump was glowing “cherry red”. Engineering has advanced sufficiently, since then, that torsional vibration is no longer a mystery. In most cases, standard design procedures are available to ensure that the levels of torsional vibration remain within acceptable limits. In this problem, you are asked to study a particular system consisting of an electric mo-tor driving a pump through a shaft whose torsional stiffness (remember this from 2.001?) is τ GIZK == ∆θ L where G is the shear modulus, IZ is the polar moment of inertia of the shaft cross-sectional area, and L is the length of the shaft. The mass moment of inertia of the motor rotor is Im and the mass moment of inertia of the pump impellor is Ip. A schematic of the system is shown in Fig.1 2It is desired to find the natural modes and natural frequencies of the undamped system and the forced response when the motor is known, for the case where Im =20lb-in-sec2 Ip =40lb-in-sec2 K = 1, 000, 000 in-lb/rad The rotational angles of the motor and pump are designated as θm and θp, respectively. (a) Obtain equations of motion for the generalized coordinates θm and θp during free, undamped motion. (b) Construct the eigenvalue problem for the natural modes and natural frequencies. (c) Solve the eigenvalue problem and obtain the modal matrix and the diagonal matrix of the squares of the natural frequencies. (d) Consider the steady-state response of the system when the motor torque has the form τm = τ0 + τaltsin Ωt where τ0 is the constant average torque produced by the motor, and Ω is the constant average rotational speed of the motor. The quantity τalt is the amplitude of a small alternating torque due to the variation in magnetic field around the circumference of the gap between the rotor and stator. Find the steady-state responses of the angles θm and θp. (e) What is the average power delivered to the pump by the motor? (f) The shaft connecting the motor and the pump was designed very conservatively to transmit the power predicted in (e), never-the-less it was found that, when the system was installed, there was a very early fatigue failure of the shaft. If you were called in to consult on this failure, could you suggest a reason for the failure? (g) Could you suggest a remedy for this problem which would not require major changes in the hardware? 3Problem 3. Crane Dynamics. The crane shown is used to move loads from one place on a construction site to another. vcarriage vload θ m Lg Consider horizontal transport (L is constant). The crane operator controls the carriage velocity vcarriage but the load velocity is not necessarily the same as the carriage velocity because of the pendulum action. A major problem for crane operators is to control the car-riage velocity in such a way that the load is moved smoothly and rapidly to its destination without being left swinging back and forth through such a large angle that it takes several minutes to decay down to a level that can be safely handled. (a) Using the angle θ as a generalized coordinate, derive a linearized equation of motion for the response θ(t) due to the input vcarriage(t). (b) Obtain a differential equation relating the load velocity vload(t) to the carriage velocity vcarriage(t). (c) Find the transfer function from the carriage velocity vcarriage(s) to the angle θ(s). (d) Find the transfer function from the carriage velocity vcarriage(s) to the load velocity vload(s). 4� A desirable time history for the load velocity is sketched below. There is a short start interval of duration T in which the load velocity smoothly accelerates upto speed V0.At the end of this period both the carriage and the load are travelling at velocity V0 with the angle θ = 0. Then there is a short stop interval in which the load velocity smoothly de-celerates to zero speed and both the carriage and the load come to rest with the angle θ =0. vload T V0 T t Start Stop The analytical description of the load velocity during the start interval is V0 t vload = 2(1 − cosπT ) if the origin of t is placed at the beginning of the start


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MIT 2 004 - Problem Set 8

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