DSC HW 3: Assigned 6/25/11, Due 7/2/12 Page 1

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1 DSC HW 3: Assigned 6/25/11, Due 7/2/12 Page 1 Problem 1 (Motor-Fan): A motor and fan are to be connected as shown in Figure 1. The torque-speed characteristics of the motor and fan are plotted on the same graph. Figure 1: Motor-fan and characteristics. (a) Draw a bond graph model of this system, neglecting any storage of energy (i.e., include only sources, loads, and ideal power conversion). (b) Determine the speed of the motor that maximizes its allowable power, and find this power. (c) Determine the speed and torque of the fan for maximum power transfer as well as the pulley ratio that achieves them. Neglect belt losses (and stretching), so power is conserved.

2 DSC HW 3: Assigned 6/25/11, Due 7/2/12 Page 2 Problem 2 (Mass-Damper): You have been asked to model and design a device to damp the motion of a large mass, and the system shown in Figure 2 has been recommended to you. A preliminary bond graph of this system after the mass engages the damper has been provided. Figure 2: (a) instrumented circular shaft (b) design of damped-shaft sensor (a) Assign causality to the bond graph and derive the state equation(s). Since you don t yet know the damper force, assume F d = Φ d (V d ), where V d is the velocity of the damper piston. Let the effective piston area be A p. Derive the state equation(s) in terms of the mass velocity, V m. (b) Assume that you can model the flowrate of fluid between chambers 1 and 2 through any orifice using the relation, Q = C o A o 2/ρ P 1 P 2, where C o is a constant (that depends on Reynold s number, etc.), A o is the orifice area, and ρ is the fluid density 1. Simplify this notation by letting, K o = C o A o 2/ρ, and let P P 1 P 2. Show that Q = K o P, or P = K 1 o Q Q. Determine the damper force constitutive relation using this information. (c) Use linearization techniques (ref. Chapter 4 of BP notes) to recommend an equivalent linear damper model and find the linear damping parameter, b l. Use sketches of what the functions would look like (i.e., graphs of force vs. velocity). (d) Sketch what you think the response of the mass/damper system would look like after the mass engages the damper at initial velocity V o. What would the response look like if you used the equivalent linear damper. 1 See, for example, Fox and McDonald, Introduction to Fluid Mechanics, 4th edition, p. 379

3 DSC HW 3: Assigned 6/25/11, Due 7/2/12 Page 3 Problem 3 (Torque-meter): A torque meter is made from a circular shaft that is rigidly mounted at one end and has strain gauges attached to it as shown in Figure 3(a) below. In a certain application it is desired to measure reversing and time-varying torques, T (t), while filtering out some parts of the torque (due to vibration, noise). To filter out the unwanted part of the torque, the system in (b) is proposed. In the following modeling and analysis, assume the rotational inertia of the gears and shafts can be neglected. Figure 3: (a) instrumented circular shaft (b) design of damped-shaft sensor (a) Assume that the instrumented circular shaft will be designed so that the torque-deflection relation is linear. Find the relation and identify the torsional stiffness, K. (b) Construct a bond graph model for the system and clearly label the elements. Assume all elements have linear constitutive relations. Assume that the gear ratios are all 1:1. (c) Assign causality and show that this is a first-order system. Derive the state equation. (d) What is the system time constant in terms of system parameters? (e) If a step torque is applied to the input shaft, sketch the voltage response you d expect to see at the output from the strain-gauge amplifier as a function of time. Also sketch the torque on the damper. Explain the trends you have indicated. Problem 4 (J-Estimator): Build a model that considers an unknown rotational inertia, J, attached to the torque meter of Problem 2. Explain whether and how this system could be used to estimate J. Provide a detailed explanation of the basis for your design. Discuss any problems/shortcomings that would introduce error in your estimate. Hint: It is assumed that you will use 1st and/or 2nd order system models to help you design/analyze this concept.

4 DSC HW 3: Assigned 6/25/11, Due 7/2/12 Page 4 Problem 5 (Motion sensor): The system shown in Figure 4 is a motion sensor in which a permanent magnet moving relative to a coil generates a voltage, v o, proportional to the ground motion, indicated here by V g (t). In this sensor, there is damping material and flexible elements that must be chosen to achieve the most favorable performance, and these components are mounted between the seismic mass, m, and the rigid case. The mass, m, includes the mass of the permanent magnet as well. Figure 4: Model of seismometer Note that the mass of the case should not be considered since the case is assumed to be rigidly mounted to the moving ground. Assume also that the case is very rigid. (a) With the electrical port free (open), build a bond graph and apply causality and derive state equations. Develop an expression for the voltage, e o, and express it as an output equation of this system. (b) Consider now that a voltage amplifier is connected to measure e o. An ideal voltage amplifier has a very high input impedance and is designed to draw negligible current from a circuit. Model this with an element that will correctly include this type of input to the system. Neglect resistance or inductance on the electrical side for now and develop state equations and an expression for the output voltage. Hint: the voltage amplifier specifies the current (i.e., flow), not the voltage (effort). Write these equations in linear state space form. (c) Repeat the previous step, but now consider the system with a current amplifier connected. The current amplifier has a negligible voltage drop associated with its function of amplifying a current. Develop the state space equations including an expression for the output current. (d) Repeat step (b), but now include some resistance in the output circuit. Develop state equations and an expression for the output current. Convert the state equations of (d) into an n-th order ODE.

5 DSC HW 3: Assigned 6/25/11, Due 7/2/12 Page 5 Problem 6 (Towed ship): A fishing boat weighing 32,200 lbf is to be towed by a much larger ship (define boat mass, m b ). The tow cable is linearly elastic and elongates 0.40 ft for each 1,000 lbf of tension in it (define stiffness, k c ). The wave and viscous drag on the fishing boat can be assumed to be linearly proportional to its velocity, and equal to b d = 3, 500 lbf-sec/ft. (a) Develop a bond graph model of this system. The model should only be of second order. Assume all elements are linear, including the cable which will be assumed to have no initial slack. (b) At time t = 0, the large tow ship starts moving with constant velocity V o. Find an expression for the fishing boat displacement, x, as a function of time. Express in terms of the physical variables in this problem. (c) If V o = 5 ft/sec, what is the maximum force in the cable, F c,max, and at what time, t max, does it occur? Do not plug in numerical variables until after an expression for F c,max is derived. (d) What is the elongation of the tow cable, x c,, due to the drag of the fishing boat at t =? (e) It is desired to change the stiffness of the cable so the fishing boat will approach the velocity of the tow ship as fast as possible without oscillating. What should the cable stiffness be in this case? (f) If the tow cable were 0.15 times the length of the cable whose stiffness is given above, what would be the peak velocity obtained by the fishing boat, and at what time would this occur? (g) Briefly explain the consequences of any initial slack in the cable. How would you model the cable stiffness in this case? Sketch the cable characteristic in this case.

6 DSC HW 3: Assigned 6/25/11, Due 7/2/12 Page 6 Problem 7 (Level recorder): The system shown below is used to record sea-level elevations and is called a tide recorder. Fluctuations in the level of the ocean surface are transmitted through the line to the tank where the water surface elevation is recorded by an electrical or mechanical recorder. It is usually desired that the tide recorder filter out high-frequency variations in the water surface, such as waves, and pass the low-frequency variations due to diurnal tides. As in Problem 5, do not plug in any numerical values until after expressions are completely derived symbolically. In a particular application, it is required that the variations in the tank level, h, due to the waves be less than 1% of the variation due to tides and that the tide amplitudes be correct to 1% accuracy. It is known that the variation in P 1 (t) due to waves has an amplitude of 17.2 kpa (34.4 kpa peak-to-peak) and a period of 12 sec; the variation due to tides has an amplitude of 25.9 kpa (51.8 kpa peak-to-peak) and a period of 12 hours. (a) Formulate a model to help you evaluate the performance of this system as it is subjected to harmonic forcing of wave-induced pressure fluctuations. Begin by developing a bond graph with appropriate constitutive relations for the necessary elements. Derive state equations (there should be two) and transform them into a single 2nd order differential equation in the variable h and in the standard form. You should be able to identify expressions for the natural frequency and the system damping. (b) Determine whether the system will function properly (i.e., according to the specifications cited above) if it s geometry and fluid properties are: L = 6 m, A = 0.09 m 2, D = m µ = viscosity = N-sec/m 2 ρ = density = N-sec 2 /m 4 (c) Generate a table and state (qualitatively) the effect on the variations of h due to the waves when each of the following modification are made individually: (i) an increase in L, (ii) an increase in D, (iii) an increase in A.

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