Modeling Physical Systems: Problems

Chapter 3 Modeling Physical Systems: Problems i V k x moving part stationary external circuit Figure 3.1: A relay. 1. Relays are electromechanical switches that open and close another circuit. To start your car, a relay engages the starter and battery when you turn the ignition key. In figure 3.1, voltage V or current i activates the relay s coil, and the resultant electromagnetic force pulls the relay contactors together. Construct a bond graph/word bondgraphthatshowshowpowerflowsinthisdevice. Includethesource, energizing coil, losses in the coil s resistance, electromechanical conversion, and mechanical elements of the relay, such as mass and the retraction spring. Indicate the direction 1

2 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS of all power flows. fan blades motor switch stator motor rotor stator motorfan shaft air flow 110 V AC Figure 3.2: Table fan with motor, shaft, and fan blades. 2. A table fan consists of power cord (with resistance), switch, electric motor, shaft, and fan blades. Rotation of the rotor causes the fan blades to move air for cooling. Construct a complete bond graph of the table fan. Include all elements, correct structure, power flows, causality, and labels on bonds sufficient to extract the state equations. 3. Construct a word bond graph thatdescribes conversion ofwind power into electrical power, via the windmill shown in figure 3.3. 4. Derive an expression for the electrical energy stored in the coil of figure 3.1, if the coil is energizedby current i. Selectan appropriate element (R, I,orC) to modelthe coil, and determine if the element is linear or nonlinear. Flux linkage λ and current i through the coil are linked via λ = n 2 A x/µ air l/µ Fe i (3.1) where n is the number of turns in the coil, A and l are the cross sectional area and circumferential length of the relay s flux return path, µ air and µ Fe are magnetic permeabilities of air and iron, and x is the size of the contactor gap.

3 wind propeller wind generator electrical Figure 3.3: Windmill with propeller, generator, and power cable. 5. In the amusement park ride Brain Centerfuge, cars A and A rotate about point P at 10 RPM, see figure 3.4. With passengers aboard, each car weighs about 500 kg. Estimate the kinetic energy stored in the rotor under normal operating conditions. 6. In an elevator system shown in figure 3.5, a circuit (power source V and line resistance R) energizes an electric motor with gearbox which drives a pulley and cable system. A cable spools over a winch, runs around pulleys atop the elevator car, beneaththeplatform, andatopthecounterweights, andterminates beneaththetop floor platform. (The counterweights balance the elevator car such that the total potential energy between the two is always constant). Rotation of the winch pulls the cable, which simultaneously lifts the elevator and lowers the counterweight. Construct a word bond graph for the elevator system. Indicate the direction of all power flows. 7. Suppose the elevator in figure 3.5 is initially at rest, but the motor draws 25 amperes current at 200 volts for 5 seconds. If the elevator car and counterweight each weigh 1000 kg, what is the maximum speed the elevator can attain? The counterweights balance the elevator car so that the total potential energy between counterweights and elevator car is constant. Hint: Assume no power or energy losses and apply P = de/dt.

4 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS 20 m A 20 cm P A' 20 cm Figure 3.4: Amusement park ride. 8. The engine in an airplane twirls its propeller. The propeller induces airflow over the wing, which generates lift on the wings and thrust on the plane. Starting with the engine, construct a word bond graph for the airplane system, i.e., show where the power goes. Include inertia of the propeller, mechanical losses, and airflow over the wings, fluid losses, motion of the plane, and altitude of the plane. Indicate the direction of all power flows. 9. Suppose the airplane is initially at rest, and the engine produces 400 hp at 3,600 rpm for 5 minutes. If the airplane weighs 8000 lb, what is the maximum speed the airplane can attain? Also, what is the maximum altitude? Note: 1hp 0.75kW, 1kg 2.2lb. Hint: Assume no losses, all power into the relevant quantity, and apply P = de/dt. 10. An important effect in the dynamics of an antifriction (ball) bearing involves compression of the balls. When a force F = Cδ 3/2 presses two balls together, the centers approach each other by the amount δ = x o x 0, see figure 3.6. Derive an expression for the energy storedor power dissipated during the compression, select an appropriate element (R, I, or C) to model the compression process, and determine if the element is linear or nonlinear. 11. A Belleville spring (or washer) with has an approximately parabolic force F = F(δ) versusdeflection δ relationshowninfigure3.7. Thespring,initiallyunloaded, is compressed 0.1 cm. Roughly estimate the energy stored or dissipated during the compression process, select an appropriate modeling element (R, I, or C), and determine if the element is linear or nonlinear. 12. The flux linkage λ of a nonlinear inductor depends upon its current i according

5 R V motor gearbox winch/spool pulley cable fixed to platform pulley cable counterweights pulley elevator car guide rail elevator car counterweight guide rail ground Figure 3.5: An elevator. to λ(i) = (2/π)Li o arctan[i/i o ], where i o is a constant current, and L is a constant. Assumingthecurrentinitiallyzero,afterenergizingtheinductorto i = i o, determine the total kinetic energy stored. 13. A crane lifts objects by pulling a cable over pulleys. The cable winds around a spool on the crane s winch. As the spool rotates, cable is brought in (lifting) or let out (lowering). The diameter D on the spool about which the cable winds and the rotational inertia J of the winch vary with the amount of cable present on the spool. Without cable, the spool has diameter D s and inertia J s. Derive an expression for the nonlinear rotational inertia J = J(θ) of the winch s spool as a function of the angular displacement θ of the spool. Assume the cable has diameter d and linear density η [kg m 1], and packs perfectly over the width w of the spool. HINT: Estimate J from J s and the layers of cable on the spool. 14. Friction force between bodies in sliding contact depends in a nonlinear manner on the relative speed of sliding v between the bodies. As figure 3.8 shows, the friction force F µ = { F d (F s F d )e v /v d} sgn(v) varies between static friction force F s and dynamic friction force F d. Here v d is a characteristic velocity. The

6 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS x o F F x F F Figure 3.6: Balls under compression. 2000 F F F (N) 1000 0 0.1 δ (cm) Figure 3.7: Belleville spring or washer. sign function sgn(v) equal to 1 when its argument is positive and 1 when its argument is negative appears because friction forces always oppose relative motion. Select an appropriate bond graph element to model this friction law, label the effort and flow on the bonds, assign a causality appropriate to this law, and estimate the instantaneous power. Why is this element dissipative?. 15. The diode pictured in figure 3.9 responds to voltage V with the current i shown in the graph. Determine the bond graph element that best models this diode. 16. Construct a complete bond graph for the circuit of figure 3.10. Include all pertinent bond graph elements, and indicate direction of all power flows. Then use the bond graph to answer the following questions: a.) How many energy storage elements are present in this system? b.) How many independent energy storage elements are present in this system? c.) What is the dynamic order of the system?

7 Friction force F µ F d v d F s F s v d F d v Figure 3.8: Simple constitutive relation for friction force. v i i v Figure 3.9: Diode. 17. Repeat the instructions of problem 16 for the circuit of figure 3.11. 18. Repeat the instructions of problem 16 for the circuit of figure 3.12. 19. Shown in figure 3.13 is a passive network bandpass filterwhich rejects high and low frequencies, but passes intermediate frequencies. This is an example of a πfilter design: the legs of the π have capacitances C 2 and inductances L 2 in parallel, while the bar of the π has capacitance C 1 and inductance L 1 in series. Develop a bond graph for this circuit. 20. A block (m = 10 kg) initially at rest is pulled by force F(t) = 100N, see figure 3.14. As the block slides over the flat, motion is resisted by tangential force T acting between the block and flat. For the sliding block, construct a bond graph and determine the block speed v, for: a) a lubricated surface, where T = bv, and b = 50Nm 1 s; b) a dry surface, where T = µmg, with dynamic friction coefficient µ = 0.5 and g = 9.8ms 2. 21. Develop a bond graph for the two degree of freedom (DOF) system consisting of two sliding masses m 1 and m 2 and two springs k 1 and k 2 forced by f(t), see

8 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS R 1 V C 1 R 2 I L Figure 3.10: Electric circuit. R 1 R 2 V C 1 n I C 2 L 2 L 1 Figure 3.11: Electric circuit. figure 3.15. Assume the friction (force F µ = µ d mg, where µ d is the coefficient of dynamic friction) between the mass and the sliding surface. 22. A gearbox with input shaft, layshaft, and output shaft is driven by an input torque, see figure 3.16. The gear on the input shaft has 20 teeth, and its mating gear on the layshaft has 25 teeth. The other gear on thelayshaft has 20 teeth,and its matinggear on the output shaft has 25 teeth. Construct a complete bond graph for the gearbox system. Assume the shafts to be rigid, i.e., no storage of energy in shaft torsion, account for all inertia, and include losses in the bearings. 23. The drive train on an automobile with manual transmission consists of motor, flywheel, clutch, gears, and shafts, see figure 3.17. The clutch is essentially two plates

9 R 1 L V R 2 C 1 n Figure 3.12: Electric circuit. C 1 L 1 R load e in L 2 C 2 L 2 C 2 e out Figure 3.13: A passive Pifilter bandpass circuit. with dry friction between; here friction forces transmit torque between plates. At the end of the drive train is the vehicle s inertia. Construct a complete bond graph for the drive train system. Assume the shafts to be rigid, i.e., no storage of energy in shaft torsion, account for all inertia, and include losses in the bearings and the clutch. At the end of the drive train is the vehicle s inertia. Construct a complete bond graph for the drive train system. 24. A motor propels a motorcycle. Attached to the motors crankshaft is a flywheel (not shown) and a friction clutch/ gearbox. The gearbox changes gear ratios. On the output shaft of the clutch/gearbox is a sprocket, which by a chain drive rotates the rear wheel via another sprocket on the rear wheel. Rolling of the rear wheel movesthe motorcycleforward. Construct a complete bondgraph for themotorcycle system, to describe the vehicles forward motion propelled by the motor. Include

10 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS v m F(t) Figure 3.14: Sliding block. y 1 y 2 k 1 k 2 f(t) m 1 m 2 friction Figure 3.15: Sliding masses. effects ofinertia ofthe motorscrankshaft andflywheel, effects ofthe friction clutch, the gearbox, transmission of power by the chain drive and sprockets, inertia of the driven rear wheel, and losses in wheel bearings. Also convert the rotational power of the rear wheel into translational power of forward motion. Include effects of motorcycle and rider mass, and losses due to wind. 25. The conical tank shown in figure 3.19 has radius r = r(z) = z, where z marks distance from the bottom of the tank. If v is the fluid volume in the tank, determine the potential energy stored for a fluid level z = h. 26. Construct a complete bond graph for the system of fluid tanks fed by sources u 1, u 2, and u 3 and depicted in figure 3.20. Assume long constricted pipes. Determine the dynamic order of the system. Then extract the state equations. 27. An electric car is powered by a battery of constant voltage, which energizes a motor with resistance R m and inductance L m on its input circuit. The lossless motor, with torque to current T = Ki and back emf to motor speed V m = Kω, drives

11 layshaft input torque gears output torque input shaft output shaft Figure 3.16: Gearbox with input shaft, layshaft, and output shaft. friction vehicle inertia motor flywheel clutch gears Figure 3.17: Automobile transmission system schematic. the transmission with friction clutch, gears, and shafts. The automobile with mass M moves when its wheels with diameter D rotate. Assume all shafts rigid, and account for bearing friction and inertia of wheels a.) Bond graph the electric car system. Include all important elements, bonds, power flow arrows, causality, and labels on bonds sufficient to extract state equations. b.) What is the order of this dynamic system? 28. During braking, the regenerative brakes on a hybrid automobile converts the car s kinetic energy into electrical power, which is then routed and stored in the battery. Attached to anddrivenby each ofthe fourwheels (figure 3.22showsonly two wheels) is a motorgenerator with back emf to motor speed V m = KΩ and torque to current T = Ki. Bond graph the regenerative brake system and state the order of this dynamic system. The automobile has mass M and the wheels have diameter D. Assume all shafts and axles rigid, but account for bearing friction and inertia of wheels and motors. Here R m and L m represent motor/generator resistance and inductance, respectively.

12 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS wheel sprocket sprocket motor clutch chain road Figure 3.18: Motorcycle Drive Train. z r(h) = h h Figure 3.19: A conically shaped fluid tanks, with sources. 29. Construct a complete bond graph for the elevator system shown in figure 3.5, given the conditions of problem 6. 30. The tubular loudspeaker system shown in figure 3.23 is activated by voltage E(t) applied to an Nturn voice coil of diameterd. Magnetic force F = (BNπD)I, induced by permanent magnet flux B interacting with voice coil current I, causes displacement x(t) of the plunger/cone assembly. The electrical to mechanical power conversion also induces a back emf E b = (BNπD)dx/dt in the coil. Cone motions x(t) then impels sound (acoustic) flow v(t) = Adx/dt down the acoustic tube. Here A is the speaker cone area. (a) Construct a bond graph for the tubular loudspeaker system. Include voice coil resistance and inductance; the mechanical mass, stiffness and damping inherent in the plunger/cone assembly; and fluidic long pipe effects down the length of the acoustic tube. Assume lossless electrical to mechanical, and mechanical to hydraulic power conversions.

13 u 1 h 1 = v 1 /A 1 h 2 = v 2 /A 2 h R 12 R 3 = v 3 /A 3 23 u 3 I 12 I 23 u 2 R 14 I 14 h 4 = v 4 /A 4 Figure 3.20: A system of interconnected fluid tanks, with sources. automobile battery R m i motor friction clutch L m gears vehicle wheels diameter D Figure 3.21: An automobile with electric drive. (b) Labelthebondgraph,includingsourcesandenergystorageelements: express effortsandflowsonbondstoallenergystorageelementsintermsofrespective energy variables. 31. Arackpinionmechanismdepictedinfigure3.24controlsmotionsofthecarriageona machinetool. ADCmotorenergizedbyacurrentsource i s (t) drivesthemechanism. Figure 3.24 also showsthe nominal values ofthe system s components. Construct a bondgraphforthe system,including power flows,labels onenergy storageelements, and causal strokes. Include motor inertia J m, motor damping B m, motor constant k m, shafttorsionalstiffness k s, gear(pinion)inertia J p, andlumpedbarandcarriage mass m b m c. The speed at the end of the shaft is Ω 1, the shaft torque T s, the

14 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS automobile battery vehicle wheels diameter D motor/generator R m i L m Figure 3.22: An automobile with regenerative braking. load velocity v 2, the supply current i s. 32. Amechanicalspeedometermeasuresvehiclespeed. Rotationsofashaftconnectedto thevehiclewheelstwistacable; theoppositeendofthecableinducesaninertiainside the speedometer to rotate. Torque from the inertia deflects a spring. Deflections of a readout needle connected to the spring are approximately proportional to the vehicle s speed. (a) Sketch the speedometer design. Show all important elements. Indicate all input and output variables. (b) Constructabondgraphofthespeedometersystem,includingcomponents/effects as bond graph elements, power flows, causal strokes, and label bonds to all sources and energy storage elements. Finally, state the system order. 33. Bond graph the pumping/storage system that consists of a circuit connected to an electric motor that drives a pump via shaft 1 mounted on bearings, pulley 1 (with inertia), vbelt (assume no slip and/or stretch), pulley 2 (with inertia), and shaft 2 mounted on bearings. The pump forces water from the lake through a long pipe into a storage tank. Include all important components/effects as bond graph elements, indicate the direction of all power flows, label bonds off all sources and Energy StorageElements (intermsofenergy variables), show causality, andstatethesystem order. 34. Assuminglinearelements,extractstateequationsforthewaterturbinedrivensystem shown in figure 34, and then put them into matrix form. Note: Flow source Q(t)

15 ACOUSTIC IMPEDANCE MATCHING TUBE speaker cone (area A) I E(t) SOUND WAVES v(t) = A dx/dt plunger/cone motions x(t) permanent magnet voice coil (Nturns & diameter D) Figure 3.23: Loudspeaker mounted onto a tube for acoustic impedance matching. is injected into the tank. The turbine, which relates shaft speed N to pipe flow Q p via N = µq p, is lossless. You may neglect the inertance of the short pipe. 35. Construct a complete bond graph for the system shown in figure 35. Then extract the state equations. Note: Voltage source V (t) energizes the lossless electric motor (with i = µt, where T is the motor torque) which induces shaft rotation. The shaft with inertia J transmits power to the pump, which causes volumetric flow rate Q = βw (w = shaft speed). Q s (t) is a flow source.

16 CHAPTER 3. MODELING PHYSICAL SYSTEMS: PROBLEMS current supply i s (t) compliant shaft k s = 8500 [Nm/rad] pinion J p = 0.0025 [Nmsec 2 /rad] r p = 10.0 [cm] v 2 Ω 1 lathe guide DC motor J m = 0.0075 [Nmsec 2 /rad] B m = 0.03 [Nmsec/rad] k m = 1.0 [Nm/amp] rack (rigid) m b = 2.0 [Nsec 2 /m] load m c = 60.0 [Nsec 2 /m] Figure 3.24: Schematic representation of motordriven rackpinion mechanism. R L pulley with inertia J 1 and radius R 1 V C motor e i belt drive: small mass & no slip pump tank 1 Q out (t) bearings with b pulley with inertia J 2 and radius R 2 water supply long pipe with linear Ip and Rp Q(t) tank short pipe with constriction resistance R lossless turbine: N = µ Q p J b

17 R bearings Q s (t) V motor e i J b pump storage tank L pipe shaft shaft inertia water supply