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physics
classical dynamics of particles

System Dynamics 3rd edition William Palm III - Solutions

Use the impedance method to obtain the transfer function Vo(s)/Vs(s) for the circuit shown in Figure.
Use the impedance method to obtain the transfer function Vo(s)/Vs(s) for the circuit shown in Figure.
Use the impedance method to obtain the transfer function Vo(s)/Vs(s) for the circuit shown in Figure.
Draw a block diagram of the circuit shown in Figure. The inputs are v1 and v2- The output is i2.
Draw a block diagram of the circuit shown in Figure. The inputs are v1 and v2 - The output is v3.
Obtain the transfer function Vo(s)/Vi{s) for the op-amp system shown in Figure.
The Wheatstone bridge, like that shown in Figure, is used for various measurements. For example, a strain gage sensor utilizes the fact that the resistance of wire changes when deformed. If the sensor is one resistance leg of the bridge, then the deformation can be determined from the voltage v1.
Obtain the transfer function Vo(s)/Vi(s) for the op-amp system shown in Figure.
Obtain the transfer function Vo(s)/Vi(s) for the op-amp system shown in Figure.
Obtain the transfer function Vo(s)/Vi, (s) for the op-amp system shown in Figure.
Obtain the transfer function Vo(s)/Vi(s) for the op-amp system shown in Figure
(a) Obtain the transfer function Ɵ(s)/Vi(s) for the D'Arsonval meter, (b) Use the final value theorem to obtain the expression for the steady-state value of the angle θ if the applied voltage vi, is a step function.
(a) Obtain the transfer function Ω(s)/TL(s) for the field-controlled motor of Example 6.5.2. (b) Modify the field-controlled motor model in Example 6.5.2 so that the output is the angular displacement θ, rather than the speed ω, where ω = θ. Obtain the transfer functions Ɵ(s)/Vf (s) and
Modify the motor model given in Example 6.5.2 to account for a gear pair between the motor shaft and the load. The ratio of motor speed to load speed ωL is N. The motor inertia is lm and the motor damping is cm. The load inertia is IL, and the load damping is cL. The load torque TL. acts directly
The derivation of the field-controlled motor model in Section 6.5 neglected the elasticity of the motor-load shaft. Figure shows a model that includes this elasticity, denoted by its equivalent torsional spring constant kT. The motor inertia is I1, and the load inertia is l2. Derive the
Figure is the circuit diagram of a speed-control system in which the dc motor voltage va is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not as sensitive to
The parameter values for a certain armature-controlled motor are KT = Kb = 0.2 N.m/A c = 5 x 10-4 N.m-s/rad Ra = 0:8Ω The manufacturer's data states that the motor's maximum speed is 3500 rpm, and the maximum armature current it can withstand without demagnetizing is 40 A. Compute the no-load
The power supply of the circuit showed in Figure supplies a voltage of 9V. Compute the current i and the power P that must be supplied.
The parameter values for a certain armature-controlled motor are KT = Kb = 0.05 N.m/A Ra = 0.8 Ω La = 3x 10-3 H I = 8 x 10-5 kg.m2 Where I includes the inertia of the armature and that of the load investigate the effect of the damping constant c on the motor's characteristic roots and on its
The parameter values for a certain armature-controlled motor are KT = Kb = 0.2 N . m/A c = 5 x 10-4 N-m-s/rad Ra = 0.8 Ω La 4 x l0-3 H I = 5 x 10-4 kg-m2 Where c and I include the effect of the load Obtain the step response of ia(t) and ω(t) if the applied voltage is va = 10V. Obtain the step
The following measurements were performed on a permanent magnet motor when the applied voltage was va = 20 V. The measured stall current was 25 A. The no-load speed was 2400 rpm and the no-load current was 0.6 A. Estimate the values of Kb, KT, Ra, and c.
A single link of a robot arm is shown in Figure. The arm mass is m and its center of mass is located a distance L from the joint, which is driven by a motor torque Tm through spur gears. Suppose that the equivalent inertia felt at the motor shaft is 0.215 kg.m2. As the arm rotates, the effect of
A conveyor drive system to produce translation of the load is shown in Figure. Suppose that the equivalent inertia felt at the motor shaft is 0.05 kg.m2, and that the effect of Coulomb friction in the system produces an opposing torque of 3.6 N.m at the motor shaft. Neglect damping in the system.
Consider the accelerometer model in Section 6.7. Its transfer function can be expressed as Y(s)/ Z(s) = -s2/(s2 + (c/m)s + k/m) Suppose that the input displacement is z(t) = 10 sin 120/ mm. Consider two cases, in SI units: (a) k/m = 100 and c/m = 18 (b) k/m = 106 and c/m = 1800. Obtain the
An electromagnetic microphone has a construction similar to that of the speaker shown in Figure 6.7.2, except that there is no applied voltage and the sound waves are incoming rather than outgoing. They exert a force fs on the diaphragm whose mass is m, damping is c, and stiffness is k. Develop a
Consider the speaker model developed in Example 6.7.1. The model, whose transfer function is given by equation (3) in that example, is third order and therefore we cannot obtain a useful expression for the characteristic roots. Sometimes inductance L and damping c are small enough to be ignored. If
The parameter values for a certain armature-controlled motor are KT = Kb = 0.2 N.m/Ac = 5 x 10-4 N.m.s/radRb = 0.8ΩLa = 4 x 10-3 HI = 5 x 10-4 kg.m2 Where c and I include the effect of the load the load torque is zero. Use MATLAB to obtain a plot of the step response of ia(t) and ω(t) if the
Consider the motor whose parameters are given in Problem 48. Use MATLAB to obtain a plot of the response of ia(t) and ω(t) if the applied voltage is the modified step va(t) = 10(1 - e-100t) V. Determine the peak value of ia(t).In Problem 48KT = Kb = 0.2 N.m/Ac = 5 x 10-4 N.m.s/radRb = 0.8ΩLa = 4
Obtain the model of the voltage v1, given the current is, for the circuit shown in Figure.
Consider the circuit shown in Figure. The parameter values are R = 103 Î©, C = 2 x 10-6 F, and L = 2 x 10-3 H. The voltage v1 is a step input of magnitude 5 V, and the voltage v2 is sinusoidal with frequency of 60 Hz and amplitude of 4 V. The initial conditions are zero. Use MATLAB to obtain a
The parameter values for a certain armature-controlled motor are KT = Kb = 0.2 N.m/A c = 3 x 10-4 N.m.s/rad Ra = 0.8 Ω La = 4xl0-3H I = 4 x I0-4 kg-m2 The system uses a gear reducer with a reduction ratio of 3:1. The load inertia is 10-3 kg.m2, the load torque is 0.04 N.m, and the load damping
The parameter values for a certain armature-controlled motor areKT = Kb = 0.05 N.m/Ac = 0Ra = 0.8 Î©La = 3 x l0-3 HI = 8 x l0-5 kg.m2Where I include the inertia of the armature and that of the load the load torque is zero. The applied voltage is a trapezoidal function defined as follows.Use
A single link of a robot arm is shown in Figure. The arm mass is m and its center of mass is located a distance L from the joint, which is driven by a motor torque Tm through spur gears. Suppose that the equivalent inertia felt at the motor shaft is 0.215 kg.m2. As the arm rotates, the effect of
Consider the circuit shown in Figure. The parameter values are R = 104 Î©, C = 2 x 10-6 F, and L = 2 x 10-3 H. The voltage v1 is a single pulse of magnitude 5 V and duration 0.05 s, and the voltage v2 is sinusoidal with frequency of 60 Hz and amplitude of 4 V. The initial conditions are
Consider the circuit shown in Figure. The parameter values are R = 2 x 104 Î© and C = 3 x 10-6 F. The voltage vs is vs(t) = 12us (t) + 3 sin 120Ï€tV. The initial conditions are zero. Use Simulink to obtain a plot of the responses vo(t) and v1(t).
The parameter values for a certain armature-controlled motor are KT = Kb = 0.2 N.m/A c = 5 x 10-4 N.m.s/rad Ra = 0.8 Ω La = 4 x 10-3 H I = 5 x 10-4 kg.m2 Where c and / include the effect of the load. The load torque is zero. a. Use Simulink to obtain a plot of the step response of the motor
The parameter values for a certain armature-controlled motor areKT = Kb = 0.05 N.m/Ac = 0Ra= 0.8 Î©La = 3 x 10-3 HI = 8 x 10-5 kg.m2Where I include the inertia of the armature and that of the load the load torque is zero. The applied voltage is a trapezoidal function defined as follows.A
(a) Obtain the model of the voltage vo, given the supply voltage vs, for the circuit shown in Figure.(b) Suppose vs(t) = Vus(t). Obtain the expressions for the free and forced responses for v"(t).
(a) Obtain the model of the voltage vo, given the supply voltage vs for the circuit shown in Figure(b) Suppose vs(t) = Vus(t). Obtain the expressions for the free and forced responses for vo(t).
(a) The circuit shown in Figure is a model of a solenoid, such as that used to engage the gear of a car's starter motor to the engine's flywheel. The solenoid is constructed by winding wire around an iron core to make an electromagnet. The resistance R is that of the wire, and the inductance L is
For the hydraulic system shown in Figure, given A1 = 10 in.2, A2 = 30 in.2, and mg = 60 lb, find the force f1 required to lift the mass m a distance x2 = 6 in. Also find the distance x1 and the work done by the force f1.
Consider the cylindrical tank shown in Figure. Derive the dynamic model of the height h, assuming that the input mass flow rate is qm(t).
Consider the tank shown in Figure. Derive the dynamic model of the height h, assuming that the input mass flow rate is qm (t).
Air flows in a certain cylindrical pipe 1 m long with an inside diameter of 1 mm. The pressure difference between the ends of the pipe is 0.1 atm. Compute the laminar resistance, the Reynolds number, the entrance length, and the mass flow rate. Comment on the accuracy of the resistance calculation.
Derive the expression for the linearized resistance due to orifice flow near a reference height h.
Consider the cylindrical container treated in Example 7.4.3. Suppose the outlet flow is turbulent. Derive the dynamic model of the system (a) In terms of the gage pressure p at the bottom of the tank (b) In terms of the height h.
A certain tank has a bottom area A = 20 m2. The liquid level in the tank is initially 5 m. When the outlet is opened, it takes 200 s to empty by 98%. a. Estimate the value of the linear resistance R. b. Find the steady-state height if the inflow is q = 3 m3/s.
A certain tank has a circular bottom area A = 20 ft2. It is drained by a pipe whose linear resistance is R = 150 m-1sec-1. The tank contains water whose mass density is 1.94 slug/ft3 a. Estimate how long it will take for the tank to empty if the water height is initially 30 ft. b. Suppose we dump
The water inflow rate to a certain tank was kept constant until the water height above the orifice outlet reached a constant level. The inflow rate was then measured, and the process repeated for a larger inflow rate. The data are given in the table. Find the effective area CdAo" for the tank's
In the system shown in Figure, a component such as a valve has been inserted between the two lengths of pipe. Assume that turbulent flow exists throughout the system. Use the resistance relation 7.3.7.(a) Find the total turbulent resistance,(b) Develop a model for the behavior of the liquid height
The cylindrical tank shown in Figure 7.4.3 has a circular bottom area A. The mass inflow rate from the flow source is qmi(t), a given function of time. The flow through the outlet is turbulent, and the outlet discharges to atmospheric pressure pa. Develop a model of the liquid height h.
Refer to the water storage and supply system shown in Figure 7.1.2. The cylindrical tank has a radius of 11 ft, and the water height is initially 5 ft. Find the water height after 5 hr if 1000 gallons per minute are pumped out of the well and 800 gallons per minute are withdrawn from the tank. 1
In the liquid level system shown in Figure, the resistances R1 and R2 are linear, and the input is the pressure source ps. Obtain the differential equation model for the height h, assuming that > D.
The water height in a certain tank was measured at several times with no inflow applied. See Figure 7.4.3. The resistance R is a linearized resistance. The data are given in the table. The tank's bottom area is A = 6 ft2, a. Estimate the resistance R. b. Suppose the initial height is known to be
Derive the model for the system shown in Figure. The flow rate mi is a mass flow rate and the resistances are linearized.
(a) Develop a model of the two liquid heights in the system shown in Figure. The inflow rate qmi(t) is a mass flow rate,(b) Using the values R1 = R, R2 = 3R, A1 = A, and A2 = 4A, find the transfer function H2(s)/Qmi(s).
Consider Example 7.4.6. Suppose that R1 = R, R2 = 3R, A1 = A, and A2 = 2A. Find the transfer function H1(s)/Qmi(s) and the characteristic roots.
Design a piston-type damper using an oil with a viscosity at 20°C of μ = 0.9 kg/(m.s). The desired damping coefficient is 2000 N.s/m.
For the damper shown in Figure 7.4.7, assume that the flow through the hole is turbulent, and neglect the term m. Develop a model of the relation between the force f and , the relative velocity between the piston and the cylinder.
An electric motor is sometimes used to move the spool valve of a hydraulic motor. In Figure the force f is due to electric motor acting through a rack-and-pinion gear. Develop a model of the system with the load displacement y as the output and the force f as the input. Consider two cases:(a) m1 =
In Figure the piston of area A is connected to the axle of the cylinder of radius R, mass in, and inertia I about its center. Develop a dynamic model of the axle's translation x, with the pressures p1 and p2 as the inputs.
Figure shows a pendulum driven by a hydraulic piston. Assuming small angles Î¸ and a concentrated mass in a distance L1 from the pivot, derive the equation of motion with the pressures p1 and p2 as inputs.
Consider the piston and mass shown in Figure 7.1.4a. Suppose there is dry friction acting between the mass m and the surface. Find the minimum area A of the piston required to move the mass against the friction force μmg, where μ = 0.6, mg = 1000 N, p1 = 3 x 105 Pa, and p2 = 105 Pa.
Figure shows an example of a hydraulic accumulator, which is a device for reducing pressure fluctuations in a hydraulic line or pipe. The fluid density is (, the plate mass is m, and the plate area is A. Develop a dynamic model of the pressure p with the pressures p1 and p2 as the given inputs.
Design a hydraulic accumulator of the type shown in Figure. The liquid volume in the accumulator should increase by 30in.3 when the pressure p increases by 1.5 lb/in.2. Determine suitable values for the plate area A and the spring constant k.
Consider the liquid-level system treated in Example 7.4.10 and shown in Figure 1 A.M. The pump curve and the line for the steady-state flow through both valves are shown in Figure. It is known that the bottom area of the tank is 2 m2 and the outlet resistance is R2 = 400 l/(m/s).(a) Compute the
Consider the V-shaped container treated in Example 7.2.2, whose cross section is shown in Figure. The outlet resistance is linear. Derive the dynamic model of the height h.
Consider the V-shaped container treated in Example 7.2.2, whose cross section is shown in Figure. The outlet is an orifice of area Ao and discharge coefficient Cd. Derive the dynamic model of the height h.
Consider the cylindrical container treated in Problem 7.8. In Figure the tank is shown with a valve outlet at the bottom of the tank. Assume that the flow through the valve is turbulent with a resistance R. Derive the dynamic model of the height h.
A certain tank contains water whose mass density is 1.94 slug/ft3. The tank's circular bottom area is A = 100 ft2. It is drained by an orifice in the bottom. The effective cross-sectional area of the orifice is CdAo = 0.5 ft2. A pipe dumps water into the tank at the volume flow rate qv. a. Derive
(a) Derive the expression for the fluid capacitance of the conical tank shown in Figure. The cone angle Î¸ is a constant and should appear in your answer as a parameter,(b) Derive the dynamic model of the liquid height h. The mass inflow rate is qmi(t). The resistance R is linear.
(a) Determine the capacitance of a spherical tank of radius R, shown in Figure.(b) Obtain a model of the pressure at the bottom of the tank, given the mass flow rate qmi.
Obtain the dynamic model of the liquid height h in a spherical tank of radius R, shown in Figure. The mass inflow rate through the top opening is qmi and the orifice resistance is Ro.
In Figure the piston of area A is connected to the axle of the cylinder of radius R, mass in. and inertia I about its center. Given p1- p2 = 3 x 105 Pa, A = 0.005 m2, R = 0.4 m, m = 100 kg, and I = 7 kg.m2, determine the angular velocity Ï‰(t) of the cylinder assuming that it starts from rest.
A rigid container has a volume of 20 ft3. The air inside is initially at 70"F. Find the pneumatic capacitance of the container for an isothermal process.
Figure shows two rigid tanks whose pneumatic capacitances are C1 and C2. The variables Î´pi, Î´p1, and Î´p2 are small deviations around a reference steady-state pressure pss. The pneumatic lines have linearized resistances R1 and R2. Assume an isothermal process. Derive a model of the
(a) Compute the thermal capacitance of 250 ml of water, for which p = 1000 kg/m3 and cp = 4.18 x 103 J/kg.C. 1ml = 10-6m3 (b) How much energy does it take to raise the water temperature from room temperature (20°C) to 99°C (just below boiling).
A certain room measures 15 ft by 10 ft by 8 ft.(a) Compute the thermal capacitance of the room air using cp = 6.012 x 103 ft-lb/slug- oF and ( = 0.0023 slug/ft3,(b) How much energy is required to raise the air temperature from 68°F to 72°F neglecting heat transfer to the walls, floor, and ceiling?
Liquid initially at 20oC is pumped into a mixing tank at a constant volume flow rate of 0.5 m3/s. See Figure 7.6.1. At time t = 0 the temperature of the incoming liquid suddenly is changed to 80oC. The tank walls are perfectly insulated. The tank volume is 12 m3, and the liquid within is well-mixed
The copper shaft shown in Figure consists of two cylinders with the following dimensions: L1 = 10 mm, L2 = 5 mm, D1 =2 mm, and D2 = 1.5 mm. The shaft is insulated around its circumference so that heat transfer occurs only in the axial direction,(a) Compute the thermal resistance of each section of
A certain radiator wall is made of copper with a conductivity k = 47 lb/sec-°F at 212oF. The wall is 3/16 in. thick and has circulating water on one side with a convection coefficient h1 = 85 lb/sec-ft-°F. A fan blows air over the other side, which has a convection coefficient h2 = 15
A particular house wall consists of three layers and has a surface area of 30 m2. The inside layer is 10 mm thick and made of plaster board with a thermal conductivity of k = 0.2 W/(m.oC). The middle layer is made of fiberglass insulation with k = 0.04 W/(m.°C). The outside layer is 20omm thick
A certain wall section is composed of a 12 in. by 12 in. brick area 4 in. thick. Surrounding the brick is a 36 in. by 36 in. concrete section, which is also 4 in. thick. The thermal conductivity of the brick is k = 0.086 lb/sec- F. For the concrete, k = 0.02 lb/sec-°F. (a) Determine the thermal
Refer to Figure 7.1.4a, and suppose that p1 - p2 = 10 lb/in.2, A = 3 in.2, and mg = 600 lb. If the mass starts from rest at x(0) = 0, how far will it move in 0.5 sec, and how much hydraulic fluid will be displaced?
Water at 120°F flows in an iron pipe 10 ft long, whose inner and outer radii are 1 /2 in. and ¾ in. The temperature of the surrounding air is 70°F. (a) Assuming that the water temperature remains constant along the length of the pipe, compute the heat loss rate from the water to the air in the
Consider the water pipe treated in Example 7.7.4. Suppose now that the water is not flowing. The water is initially at 120oF. The copper pipe is 6 ft long, with inner and outer radii of 1 /4 in. and 3/8 in. The temperature of the surrounding air is constant at 70°F. Neglect heat loss from the ends
A steel tank filled with water has a volume of 1000 ft3. Near room temperature, the specific heat for water is c = 25.000 ft-lb/slug-oF, and its mass density is ( = 1.94 slug/ft3. a. Compute the thermal capacitance C1 of the water in the tank. b. Denote the total thermal resistance (convective and
Consider the tank of water discussed in Problem 7.52. A test was performed in which the surrounding air temperature To was held constant at 70oF. The tank's water temperature was heated to 90° and then allowed to cool. The following data show the tank's water temperature as a function of time. Use
The oven shown in Figure has a heating element with appreciable capacitance C1. The other capacitance is that of the oven air C. The corresponding temperatures are T1 and T2, and the outside temperature is To. The thermal resistance of the heater-air interface is R1 that of the oven wall is R2.
A simplified representation of the temperature dynamics of two adjacent masses is shown in Figure. The mass with capacitance C2 is perfectly insulated on all sides except one, which has a convective resistance R2. The thermal capacitances of the masses are C1 and C2, and their representative
A metal sphere 25 mm in diameter was heated to 95°C, and then suspended in air at 22°C. The mass density of the metal is 7920 kg/m3, its specific heat at 30°C is cp = 500 J/(kg.°C), and its thermal conductivity at 30°C is 400 W/(m.oC). The following sphere temperature data were
A copper sphere is to be quenched in an oil bath whose temperature is 50°C. The sphere's radius is 30 mm, and the convection coefficient is h = 300 W/(m2.°C). Assume the sphere and the oil properties are constant. These properties are given in the following table. The sphere's initial
Consider the quenching process discussed in Problem 7.57. Suppose the oil bath volume is 0.1 m3 Neglect any heat loss to the surroundings and develop a dynamic model of the sphere's temperature and the bath temperature. How long will it take for the sphere temperature to reach I30°C?
Consider Example 7.7.1. The MATLAB left division operator can be used to solve the set of linear algebraic equations AT = b as follows: T = A\b. Use this method to write a script file to solve for the three steady-state temperatures T1, T2, and T3, given values for the resistances and the
Pure water flows into a mixing tank of volume V = 300 m3 at the constant volume rate of 10 m3/s. A solution with a salt concentration of si, kg/m3 flows into the tank at a constant volume rate of 2 m3/s. Assume that the solution in the tank is well mixed so that the salt concentration in the tank
Fluid flows in pipe networks can be analyzed in a manner similar to that used for electric resistance networks. Figure (a) shows a network with three pipes, which is analogous to the electrical network shown in part (b) of the figure. The volume flow rates in the pipes are q1, q2, and q3. The
The equation describing the water height h in a spherical tank with a drain at the bottom is π(2rh - h2) dh/dt = -CdAo√(2gh) Suppose the tank's radius is r = 3 m and that the circular drain hole has a radius of 2 cm. Assume that Cd = 0.5, and that the initial water height is h(0) = 5 m. Use g =
The following equation describes a certain dilution process, where y(t) is the concentration of salt in a tank of fresh water to which salt brine is being added. dy/dt + 2/(10 +2t) y = 4 Suppose that y(0) = 0. a. Use MATLAB to solve this equation for y(t) and to plot y(t) for 0 ≤ f ≤ 10. b.
A tank having vertical sides and a bottom area of 100 ft2 is used to store water. To fill the tank, water is pumped into the top at the rate given in the following table. Use MATLAB to solve for and plot the water height h(t) for 0 ‰¤ t ‰¤ 10 min.
A cone-shaped paper drinking cup (like the kind used at water fountains) has a radius R and a height H. If the water height in the cup is h, the water volume is given by V = 1/3 π (R/H)2 h3 Suppose that the cup's dimensions are R = 1.5 in. and H = 4 in. a. If the flow rate from the fountain into

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