problem 1 (motor-fan): a motor and fan are to be connected …longoria/me344/hw/hw3_su12_s… ·...

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

Problem 1 (Motor-Fan): A motor and fan are to be connected as shown in Figure 1. Thetorque-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., includeonly sources, loads, and ideal power conversion).

Solution: Model the motor torque-speed curve by a Thevenin equivalent (in bond graphform), the pulley-belt-pulley combination as an ideal transformer and the fan characteristicsas an ideal rotational resistive element. The bond graph is,

(b) Determine the speed of the motor that maximizes its allowable power, and find thispower.

Solution: By sketching maximum power lines that intersect the motor torque-speed curve,you can approximate the speed of the motor for maximum power. This is approximately at1750 rpm (see figure below). At that point, the torque is about 9.1 in-lbf, therefore,

Po,max = T · ω = (9.1inlbf) · (1750rpm)= (9.1inlbf) · (1750rpm) m

39.37in4.448Nlbf

2πrev

min60 sec

= 188.1W = 0.253hp

(c) Determine the speed and torque of the fan for maximum power transfer as well as the

R.G. Longoria, Summer 2012 ME n344, UT-Austin


pulley ratio that achieves them. Neglect belt losses (and stretching), so power is conserved.

Solution: First, solve the problem graphically. Extend the maximum power line (hyperbola)so it intersects the fan characteristic. This will be the function plotted as a torque, T (ω) =Pmax/ω, where Pmax = 188.1W . The intersection is the operating point at which the motordelivers maximum power to the fan. The torque and speed at this point are about 14 in-lbfand 1150 rpm, respectively. To find the pulley ratio needed, use the transformer relation,ω2 = m · ω1, so that, m = ω2/ω1 = 1150/1750 ≈ 2/3.

Compare this to using the numerical data in Matlab and ginput() to locate intersections,and the values found previously are fairly close.



Problem 2 (Mass-Damper): You have been asked to model and design a device to dampthe motion of a large mass, and the system shown in Figure 2 has been recommended toyou. A preliminary bond graph of this system after the mass engages the damper has beenprovided.

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’tyet know the damper force, assume Fd = Φd(Vd), where Vd is the velocity of the damperpiston. Let the effective piston area be Ap. Derive the state equation(s) in terms of the massvelocity, Vm.

Solution:

(b) Assume that you can model the flowrate of fluid between chambers 1 and 2 throughany orifice using the relation, Q = Co · Ao ·

√2/ρ

√| P1 − P2 |, where Co is a constant

(that depends on Reynold’s number, etc.), Ao is the orifice area, and ρ is the fluid density1.Simplify this notation by letting, Ko = Co · Ao ·

√2/ρ, and let ∆P ≡| P1 − P2 |. Show that

Q = Ko

√∆P , or ∆P = K−1o Q | Q |. Determine the damper force constitutive relation using

this information.

Solution: Note the typo: ∆P = K−2o Q | Q | (Ko should be raised to 2nd power). Since,Fd = Ap∆P , and Q = ApVd, then Fd = (A3

p/K2o )Vd | Vd |.

(c) Use linearization techniques (ref. Chapter 4 of BP notes) to recommend an equivalent

1See, for example, Fox and McDonald, Introduction to Fluid Mechanics, 4th edition, p. 379



linear damper model and find the linear damping parameter, bl. Use sketches of what thefunctions would look like (i.e., graphs of force vs. velocity).

Solution: Use secant linearization. For −Vdm ≤ Vd ≤ +Vdm, where Vdm is the maximumvelocity value expected, the slope of the secant line is bd = (A3

p/K2o )Vdm, so the linearized

force is Fd = bdVd, where the − represents values for the expected range of operation.

(d) Sketch what you think the response of the mass/damper system would look like afterthe mass engages the damper at initial velocity Vo. What would the response look like if youused the equivalent linear damper.

Solution: For the linear case, you can show the system will have a time constant, τ = bd/m,so if the mass is moving with initial velocity Vo, then Vm(t) = Voexp[−t/τ ].



Problem 3 (Torque-meter): A torque meter is made from a circular shaft that is rigidlymounted at one end and has strain gauges attached to it as shown in Figure 3(a) below. Ina certain application it is desired to measure reversing and time-varying torques, T (t), whilefiltering out some parts of the torque (due to vibration, noise). To filter out the unwantedpart 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-deflectionrelation is linear. Find the relation and identify the torsional stiffness, K.

Solution: For a round shaft in torsion, T = GIpθ/L, where G is the shear modulus, Ip is thepolar area moment of inertia, and L is the shaft length, so stiffness is defined by T = Ksθ,so Ks = GIp/L.

The solutions for the following steps are summarized on the next page.

(b) Construct a bond graph model for the system and clearly label the elements. Assumeall 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 expectto see at the output from the strain-gauge amplifier as a function of time. Also sketch thetorque on the damper. Explain the trends you have indicated.



Solution:



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 beused to estimate J . Provide a detailed explanation of the basis for your design. Discuss anyproblems/shortcomings that would introduce error in your estimate. Hint: It is assumed thatyou will use 1st and/or 2nd order system models to help you design/analyze this concept.

Solution:



Problem 5 (Motion sensor): The system shown in Figure 4 is a motion sensor in whicha permanent magnet moving relative to a coil generates a voltage, vo, proportional to theground motion, indicated here by Vg(t). In this sensor, there is damping material andflexible elements that must be chosen to achieve the most favorable performance, and thesecomponents 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 berigidly 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 derivestate equations. Develop an expression for the voltage, eo, and express it as an outputequation of this system.

(b) Consider now that a voltage amplifier is connected to measure eo. An ideal voltageamplifier has a very high input impedance and is designed to draw negligible current froma circuit. Model this with an element that will correctly include this type of input to thesystem. Neglect resistance or inductance on the electrical side for now and develop stateequations and an expression for the output voltage. Hint: the voltage amplifier specifies thecurrent (i.e., flow), not the voltage (effort). Write these equations in linear state spaceform.

(c) Repeat the previous step, but now consider the system with a current amplifier con-nected. The current amplifier has a negligible voltage drop associated with its function ofamplifying a current. Develop the state space equations including an expression for theoutput current.

(d) Repeat step (b), but now include some resistance in the output circuit. Develop stateequations and an expression for the output current.

Convert the state equations of (d) into an n-th order ODE.



Solution:



Solution: Note that part (d) should have read: “Repeat step (c), but now ...”. It wouldnot make sense to repeat step (b) because in that case you get an output voltage not current(and you can show that adding resistance makes no difference in the output voltage). Addingcoil resistance changes the output current relation to the acceleration as before.



Problem 6 (Towed ship): A fishing boat weighing 32,200 lbf is to be towed by a muchlarger ship (define boat mass, mb). The tow cable is linearly elastic and elongates 0.40 ft foreach 1,000 lbf of tension in it (define stiffness, kc). The wave and viscous drag on the fishingboat can be assumed to be linearly proportional to its velocity, and equal to bd = 3, 500lbf-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 initialslack.(b) At time t = 0, the large tow ship starts moving with constant velocity Vo. Find anexpression for the fishing boat displacement, x, as a function of time. Express in terms ofthe physical variables in this problem.(c) If Vo = 5 ft/sec, what is the maximum force in the cable, Fc,max, and at what time,tmax, does it occur? Do not plug in numerical variables until after an expression for Fc,maxis derived.(d) What is the elongation of the tow cable, xc,∞, due to the drag of the fishing boat att =∞?(e) It is desired to change the stiffness of the cable so the fishing boat will approach thevelocity of the tow ship as fast as possible without oscillating. What should the cablestiffness 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 thisoccur?(g) Briefly explain the consequences of any initial slack in the cable. How would you modelthe cable stiffness in this case? Sketch the cable characteristic in this case.



Solution:(1) 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 initialslack. Ans. The bond graph and derivation of 2nd order ODE for boat velocity, Vm areshown below:

For this case, 2ζωn = B/m, so ζ = B/2√mK1 = 3500/(2 ·

√1000 · 2500) = 1.11, so this

system is over-damped.

(2) At time t = 0, the large tow ship starts moving with constant velocity Vo. Find anexpression for the fishing boat displacement, x, as a function of time. Express in terms ofthe physical variables in this problem. Ans. Refer to BP notes: for over-damped systemwith step response, Vo, and initial condition, Vm(0). Define ‘fast’ and ‘slow’ time constants,τf = −1/s2 and τs = −1/s1, where s1 and s2 are both real roots and s1 < s2. Then,Vm(t) = Vo[1− Y (t)], where,

Y (t) =τs

τs − τfe−t/τs − τf

τs − τfe−t/τf .

Then the boat displacement is, x2(t) =∫ t0Vm(t)dt. After integrating,

x2(t) = Vot− A(1− exp[−t/τs]) +B(1− exp[−t/τf ]),



where A = τ 2s /(τs − τf ) and B = τ 2f /(τs − τf ).

(3) If Vo = 5 ft/sec, what is the maximum force in the cable, Fc,max, and at what time,tmax, does it occur? Do not plug in numerical variables until after an expression for Fc,max isderived. Ans. The force in the cable is F1 = K1x1, where x1 = V (t)−p/m. With V (t) = Vo,x1 = VoY (t), which when integrated gives, x1(t) = VoA(1− exp[−t/τs])−B(1− exp[−t/τf ]).At steady-state, F1,max = K1Vo(τ

2s − τ 2f )/(τs − τf ), which is reached after about 4τs. For the

numerical values given, F1,max = 17500 lbf.

(4) What is the elongation of the tow cable, xc,∞, due to the drag of the fishing boat at t =∞?Ans. You can show from letting t → ∞ in x1(t) above that, x1,ss = Vo(τ

2s − τ 2f )/(τs − τf ),

which for the parameters in this problem gives a value of 7 feet.

(5) It is desired to change the stiffness of the cable so the fishing boat will approach thevelocity of the tow ship as fast as possible without oscillating. What should the cablestiffness be in this case? Ans. Make ζ = (B/2m) ·

√m/K = 1, which gives, K = 3063

lbf/ft. You can show that the time constant is about 0.57 seconds, so steady-state is reachedin about 4τ = 2.3 seconds.

(6) 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 thisoccur? Ans. Making this change gives a stiffer cable with K = 16, 667 lbf/ft. This leads to aζ of 0.43, indicating the system becomes underdamped and ωn = 4.08 rad/sec. The responsefor an underdamped system will have a peak at Tp = π/ωd, where ωd = ωn

√1− ζ2. This

gives Tp = 0.852 seconds. From the general solution for a step response (see BP, equation4.137),

Vm − VoVm(0)− Vo

=e−ζωnt√1− ζ2

sin(ωdt+ φd),

where φd = tan−1(√

1− ζ2/ζ). Setting t = Tp, the peak velocity is, Vm,p = Vo(1 −exp[−ζπ/

√1− ζ2] sin(π + φd)/

√1− ζ2) = Vo(1.224) = 6.12 ft/sec.

(7) Briefly explain the consequences of any initial slack in the cable. How would you modelthe cable stiffness in this case? Sketch the cable characteristic in this case. Ans. The cablewould not transmit force until it grew taut. If the extension in the cable is less than zero,then the cable needs to be modeled with a nonlinear stiffness model. It would not be possibleto apply the linear analysis tools used above.



Problem 7 (Level recorder): The system shown below is used to record sea-level el-evations and is called a tide recorder. Fluctuations in the level of the ocean surface aretransmitted through the line to the tank where the water surface elevation is recorded byan electrical or mechanical recorder. It is usually desired that the tide recorder filter outhigh-frequency variations in the water surface, such as waves, and pass the low-frequencyvariations due to diurnal tides. As in Problem 5, do not plug in any numerical values untilafter expressions are completely derived symbolically.

In a particular application, it is required that the variations in the tank level, h, due to thewaves be less than 1% of the variation due to tides and that the tide amplitudes be correctto 1% accuracy. It is known that the variation in P1(t) due to waves has an amplitude of17.2 kPa (34.4 kPa peak-to-peak) and a period of 12 sec; the variation due to tides has anamplitude 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 subjectedto harmonic forcing of wave-induced pressure fluctuations. Begin by developing a bond graphwith appropriate constitutive relations for the necessary elements. Derive state equations(there should be two) and transform them into a single 2nd order differential equation inthe variable h and in the standard form. You should be able to identify expressions for thenatural frequency and the system damping.(b) Determine whether the system will function properly (i.e., according to the specificationscited above) if it’s geometry and fluid properties are:

L = 6 m, A = 0.09 m2, D = 0.0508 mµ = viscosity = 0.012 N-sec/m2

ρ = density = 54.013 N-sec2/m4

(c) Generate a table and state (qualitatively) the effect on the variations of h due to thewaves when each of the following modification are made individually: (i) an increase in L,(ii) an increase in D, (iii) an increase in A.



Solution: (a) Ans. First, model the line with lumped fluid inertia and resistance, assuminglinear elements. The system model is shown below, state equations are derived, which canbe used to come up with a 2nd order ODE in h with H(t) as a forcing function. This linear2nd order ODE can be used to find a transfer function between h and H.

The fluid inertia relates fluid momentum, Γ, to flowrate, Q, Γ = IfQ, where If = ρLp/Ap,Lp is the line (or pipe) length, and Ap is the area. The fluid resistance for laminar flow in apipe is, Rf = 128µLp/πD

4. Finally, the elevation tank is a simple hydraulic capacitor, withpressure at the base, P = V/Cf , and Cf = A/ρg.

(b) Ans. The following geometry and fluid properties were given:

L = 6 m, A = 0.09 m2, D = 0.0508 mµ = viscosity = 0.012 N-sec/m2

ρ = density = 54.013 N-sec2/m4

These fluid properties are not really valid. It’s likely these are numbers for ‘English’ unitsthat I did not convert to SI units! You should have caught this! It’s likely you have notdone enough calculations with fluids lately to remember that water has a density of about1000 kg/m3. Finding this error would have led you to check the viscosity number as well.



Keeping the ‘bad values’, you find the following:

More realistic values for the fluid properties of sea water would be a density of 1069 kg/m3

and absolute viscosity of 1.88·10−3 Pa·s. Using these values would lead to an underdampedsystem with ζ of 0.057 (ωn stays the same). For this case you’d find that the ratio of heightresponse of waves to tides is about 10 percent (worse).

(c) Generate a table and state (qualitatively) the effect on the variations of h due to thewaves when each of the following modification are made individually: (a) an increase in L,(b) an increase in D, (c) an increase in A.

Ans. The dependence of the response due to waves relative to changes in L, D, and A isbetter understood with reference to using the damping ratio and where the forcing frequencysits relative to the natural frequency, as shown on the attached graph.


problem 1 (motor-fan): a motor and fan are to be connected …longoria/me344/hw/hw3_su12_s… ·...

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