QUESTION 1 Create a block diagram of a DC motor that includes electrical dynamics, viscous friction, and Coulomb friction. QUESTION 2 A D/A converter and an A/D converter have the following parameters: e max = 10 V, e min = 10 V, and n = 4. Complete the following: a. Simulate D/A output voltages for command voltages between 15 and 15 V. Compare the e c results to the approximate equation eout = RDACround, where e out is the output RDAC voltage, R DAC is the D/A converter resolution, and e c is the command voltage. b. Simulate A/D sensed voltages for input voltages between 15 and 15 V. Compare the e in results to the approximate equation es = RADCround, where e s is the sensed RADC voltage, R ADC is the A/D converter resolution, and e in is the input voltage.
QUESTION 3 For a four bit D/A converter with minimum and maximum voltages of 10 V and 10 V, respectively, plot the output voltages versus digital numbers from 0 to 15 for the following conditions: a. bit 1 is missing b. bit 2 is missing c. bit 3 is missing d. bit 4 is missing Compare the results to the case where no bits are missing. QUESTION 4 A tachometer with a gain 0.5 V/(rev/s) is used to sense the angular velocity of a roller (). If the A/D converter has a voltage range of ±10 V and 8 bits, what is the range of roller speeds that may be sensed and what is the resolution with which the angular velocity may be sensed? Page 2
QUESTION 5 For a four bit A/D converter with minimum and maximum voltages of 10 V and 10 V, respectively, plot the input digital numbers for input voltages from 10 to 10 V for the following conditions: a. bit 1 is missing b. bit 2 is missing c. bit 3 is missing d. bit 4 is missing Compare the results to the case where no bits are missing. QUESTION 6 For the system shown in, the D/A converter has a range of 5 V to 5 V. Determine the minimum number of bits for an unsigned D/A converter such that the DC motor angular velocity can be altered by increments of at least 0.1 rad/s. 10 0.1s + 1 Page 3
QUESTION 7 For the system shown in, the D/A converter has a range of 10 V to 10 V and 4 bits. Calculate the DC motor steady state angular velocity for each digital input to the D/A converter. The parameter ω m is in units of rpm. 10 0.1s + 1 QUESTION 8 A hydraulic motor is controlled via a computer process. The hydraulic motor angular velocity is sensed via a tachometer with a gain K tach = 0.1 V/(rad/s) that is wired to an A/D converter, which has minimum and maximum voltages of 10 and 10 V, respectively. The computer processor sends out a voltage via a D/A converter, which has minimum and maximum voltages of 10 and 10 V, respectively. The output voltage drives the valve with a steady state gain of K v = 2 mm/v. The steady state gain between the hydraulic motor angular velocity and the valve is K ω = 40 (rad/s)/mm. Complete the following: Determine the minimum number of bits of the A/D converter such that increments of at most ω res = 0.1 rad/s can be sensed. Page 4
Determine the minimum number of bits such that the computer processor can change the hydraulic motor angular speed by increments of at most ω res = 0.1 rad/s. QUESTION 9 A DC motor has the following parameters: J m = 4 10 5 kg m 2, R = 5.5 Ω, K t = 5.9 10 2 N m/a, K v = 5.9 10 2 V/(rad/s), and K a = 1.2. An incremental rotary encoder is attached to the DC motor with n counts/revolution. Create a simulation and plot the actual and measured angular position versus time on one graph and the actual and estimated angular velocity versus time on another graph. Use the measured angular positions to estimate the angular velocity. The initial angular position is 0.75 rad and V c (t) = 0.8sin(20t) V. Use n = 25 and n = 1000. QUESTION 10 For the system shown in, a DC motor with a mass moment of inertia of J m is rigidly attached to flywheel 1 with a mass moment of inertia of J 1. Flywheel 1 is attached to flywheel 2 with a mass moment of inertia of J 2. The flywheels are connected with a long thin rod whose inertia may be ignored. The rod may be modeled as a spring and damper. Determine the two second order differential equations that relate the angular displacements of flywheel 1 and flywheel 2 to the command voltage (e c ). The DC motor has an angular position and an angular velocity of θ m and ω m, respectively. Flywheel 1 has an angular position and an angular velocity of θ 1 and ω 1, respectively. Flywheel 2 has an angular position and an angular velocity of θ 2 and ω 2, respectively. Page 5
: System Schematic. QUESTION 11 For the DC motor with tachometer feedback (), complete the following: Symbolically determine the transfer function Ω E m c ( s) ( s), time constant, and steady state gain. Symbolically determine a set of difference equations required to simulate the ω m, e tach, e i, and e a. The DC motor has the following parameters: J m = 2 10 3 kg m 2, R a = 8 Ω, K t = 0.8 N m/a, K v = 0.8 V/(rad/s), K a = 40, and K tach = 10 3 V/rpm. Create a simulation and plot ω m, e tach, e i, and e a as functions of time for e c (t) = 10 V. Page 6
K a 1 1 K v τ m s + 1 K tach QUESTION 12 For a hydraulic rotational motor with mechanical viscous friction, complete the following: a. Ignoring leakage effects, symbolically derive the differential equation relating the motor angular velocity to the valve displacement and the Coulomb friction torque. Determine the steady state gains between the motor angular velocity and the valve displacement and between the motor angular velocity and the Coulomb friction torque. Determine the natural frequency and damping ratio. b. The hydraulic motor has the following parameters: J m = 0.6 kg m 2, B m = 4 N m/(rad/s), V 0 = 0.5 m 3, β = 10 9 N/m 2, D m = 10 4 m 3 /rad, K c = 10-8 m 5 /N s, and K q = 30 m 2 /s. Simulate the system for the valve displacement shown in and a Coulomb friction torque with magnitude 50 N m. Plot the motor angular position, angular velocity, and angular acceleration, which are zero at time 0. Page 7
x v 5 mm 0 mm 0 s 1 s t QUESTION 13 A motor is rotating at an angular velocity of 122 rad/s. The motor angular velocity is sensed via a tachometer with a gain K tach = 2.5 10 2 V/(rad/s) that is wired to an A/D converter. The A/D converter range is 10 V to 10 V and there are 10 bits. Determine the measured angular velocity (i.e., the number, in rad/s, in the computer processor) and the absolute value of the error between the actual and measured angular velocity. QUESTION 14 For the valve controlled piston in, determine the symbolic transfer function from the piston position to the valve position. Ignore leakage effects. The nominal volume in each chamber is V 0 and the bulk modulus is β. Determine a set of difference equations required to simulate the piston position. Simulate the system for the valve position time history shown in Figure 2. Plot the valve position, piston position, load flow, and load pressure versus time. The Page 8
system parameters are m = 0.8 kg, A = 10 3 m 2, B m = 10 N/(m/s), V 0 = 0.1 m 3, β = 10 9 N/m 2, K c = 10 8 m 5 /N s, and K q = 0.25. x p m A x v v 1 p 1 q 1 v 2 p 2 q 2 return p s : Valve Controlled Piston Schematic. Page 9
x v 0.01 m 0 m 0 s 0.1 s 0.2 s 0.3 s 0.4 s t -0.01 m Figure 2: Valve Position Time History. QUESTION 15 For the hydraulic motor shown in, symbolically determine the transfer functions from the motor angular velocity to the input force (f) and to the disturbance torque (T f ) and from the motor angular position to the input force and to the disturbance torque. Assume there is no internal or external leakage and there is no viscous friction acting on the servo valve or the motor shaft. Page 10
m J m T f v 1 p 1 q 1 v 2 p 2 q 2 m f p s x v QUESTION 16 For the solenoid by Vaughan and Gamble [1996] in the handout, m a = 0.101 kg, b a = 6 N/(m/s), k a = 1.9 10 4 N/m, f L = 0 N, τ = 2 ms, and R s = 2.95 Ω. The nonlinear friction is modeled as Coulomb friction with a magnitude of 2 N. Complete the following: a. Determine a set of difference equations required to simulate the solenoid. b. Create a numerical simulation and simulate the solenoid for e s (t) = 7 V. On separate graphs plot e L (t), f a (t), I s (t), I d (t), I r (t), v a (t), x a (t), and λ s (t) versus time. QUESTION 17 Question Page 11
QUESTION 18 A voice coil schematic is shown in. When the current is positive it flows out of the open circles and into the closed circles. The magnets are part of the stator and the coils are part of the armature, which has a mass is m. Assume nonlinear friction does not act on the voice coil armature. Complete the following tasks: a. Symbolically determine a set of first order differential equations describing the voice coil dynamics. b. Assume the armature voltage is related to the command voltage by e a (t) = K a e c (t). Symbolically determine the steady state gain relating the armature velocity to the command voltage and the armature velocity to the load force. c. The system parameters are K a = 2.4, R = 5.4 Ω, L = 1.12 10 2 H, K t = 9.35 N/A, K v = 9.35 V/(m/s), m = 8.39 10 2 kg, and b = 15.8 N/(m/s). The load torque and the initial conditions are zero. Simulate the system for a triangular command voltage signal with an amplitude of 4 V and a frequency of 50 Hz. Run the simulation for 5 cycles and plot the motor angular velocity, current, motor torque, and command voltage on separate graphs. A triangular signal may be generated by the function A(2/π)sin 1 (sin(2πft)), where A is the amplitude and f is the frequency in Hz. Page 12
R brush N wire coil L + e a (t) S f L (t) - N b QUESTION 19 A schematic of a field series wound DC motor is shown in. Assume the motor voltage is e(t) = K a e c (t) and a load torque acts against the motor rotor. The product of the motor constant and the field flux is K m I f (t). Complete the following: a. Symbolically determine a set of first order differential equations describing the motor dynamics. b. Symbolically determine the steady state motor torque as a function of rotor angular velocity and command voltage assuming the load torque and nonlinear friction torque are zero. The system parameters are K a = 2, R a = 0.4832 Ω, R f = 84.91 Ω, L a = 6.763 10 3 H, Page 13
L f = 13.39 H, J m = 0.2053 kg m 2, B m = 7.032 10 3 N m/(rad/s), and K m = 0.7096 H. Plot the steady state motor torque versus motor angular velocity for e c = 2, 5, and 8 V. c. Assume the nonlinear torque is modeled by Coulomb torque with a magnitude of T C = 5.282 10 4 N m. The load torque and initial conditions are zero. Simulate the system for a square command voltage signal with an amplitude of 4.5 V and a frequency of 1 Hz. Run the simulation for 5 cycles and plot the motor angular velocity, motor torque, current, and command voltage on separate graphs. A square signal may be generated by the function Asgn(sin(2πft)), where A is the amplitude and f is the frequency in Hz. L f R a R f L a B m + e(t) - + e b (t) - J m Page 14
QUESTION 20 A schematic of a field shunt wound DC motor is shown in. Assume the motor voltage is e(t) = K a e c (t) and a load torque acts against the motor rotor. The product of the motor constant and the field flux is K m I f (t). Complete the following: a. Symbolically determine a set of first order differential equations describing the motor dynamics. b. Symbolically determine the steady state motor torque as a function of rotor angular velocity and command voltage assuming the load torque and nonlinear friction torque are zero. The system parameters are K a = 2, R a = 0.4832 Ω, R f = 84.91 Ω, L a = 6.763 10 3 H, L f = 13.39 H, J m = 0.2053 kg m 2, B m = 7.032 10 3 N m/(rad/s), and K m = 0.7096 H. Plot the steady state motor torque versus motor angular velocity for e c = 2, 5, and 8 V. c. Assume the nonlinear torque is modeled by Coulomb torque with a magnitude of T C = 5.282 10 4 N m. The load torque and initial conditions are zero. Simulate the system for a sinuaoidal command voltage signal with an amplitude of 2 V and a frequency of 5 Hz. Run the simulation for 5 cycles and plot the motor angular velocity, motor torque, armature current, and field current on separate graphs. Page 15
R f R a L f L a B m + e(t) - + e b (t) - J m QUESTION 21 sgn ( ) () If the load flow rate is given by () () () and the flow pressure coefficient. q t = kx t p x t p t, determine the flow gain L v s v L Page 16