Manufacturing Equipment Control

Transcription

1 QUESTION 1 An electric drive spindle has the following parameters: J m = kg m 2, R a = 8 Ω, K t =.5 N m/a, K v =.5 V/(rad/s), K a = 2, J s = kg m 2, and K s =.3. Ignore electrical dynamics and all sources of Coulomb and viscous friction. Complete the following: a. Design a proportional plus integral plus derivative controller to regulate the spindle angular position such that the closed loop system is characterized by three overdamped poles with time constants of.45,.5, and.55 s. b. Determine the difference equations required to simulate the system. c. Simulate the closed loop spindle system for the reference input given in Figure 1. The value between 1 and 3 s is π rad and the sinusoidal section has an amplitude of π/2 rad. A constant disturbance torque of 1.2 N m acts on the spindle after 2 s. The spindle angular position is measured via an encoder with 2 counts/rev and quadrature encoding is utilized. The command voltage is sent to the motor amplifier via a D/A converter with 1 bits and a voltage range of ±1 V. The motor current range is ±3 A. Plot the actual and reference spindle angular positions on one plot, the measured spindle angular position error on another plot, the command voltage on another plot, and the current on another plot. All initial conditions are zero and assume only the spindle angular position can be measured.

2 5 4 θ r (rad) time (s) Figure 1 QUESTION 2 An empirical model of a powder feeder is given in equations (1) and (2). The unit of motor angular velocity is rpm and the unit of nozzle powder flow rate is gpm. Complete the following: a. Ignoring the nonlinear friction, design a general tracking controller such that the error dynamics are described by three time constants τ 1, τ 2, and τ 3 and it can robustly reject constant disturbances. b. Using Euler s method, determine a set of difference equations to simulate the powder feeder and simulate the controller if it is implemented in a Smith Predictor Corrector structure. c. The powder feeder has the following parameters: τ m =.121 s, k m = 158 rpm/v, ω f = 98.8 rpm, τ p =.61 s, k p = gpm/rpm, and t d = 1.98 s. The minimum and maximum command voltages are 2 and 2 V, respectively. The desired closed loop time constants Page 2

3 are τ 1 =.8 s, τ 2 =.85 s, and τ 3 =.9 s. Simulate the closed loop powder feeder system for a reference powder flow rate m r (t) = 2.5sign(2πsin(.5t))+12.5 and two cycles. Plot the reference and nozzle mass flow rates on one graph, and the error, command voltage, and motor angular velocity on separate graphs. All initial conditions are zero and assume e c (t) = for t <. () t + ( t) = sgn ( ) + k e ( t) τ & ω ω ω ω (1) m m m f m m c ( ) + ( ) = ω ( ) τ mt & mt k t t (2) p p m d QUESTION 3 A hydraulic rotational axis has the following parameters: J m =.6 kg m 2, J r = 1 kg m 2, V =.2 m 3, β = 1 9 N/m 2, D m = 1 2 m 3 /rad, K r =.2, K c = 1 8 m 5 /(N s), and K q = 3 m 2 /s. Complete the following: a. Determine the difference equations required to simulated the system. b. Simulate the closed loop hydraulic rotational axis system using a proportional controller with a reference angular displacement of ω ref = π rad. Use a sample period of 1 5 s and a proportional gain of 1.5 m/rad. On separate graphs plot the axis angular displacement and the valve displacement versus time. All initial conditions are zero. c. Repeat part (b) using a proportional gain of 2 m/rad. Page 3

4 QUESTION 4 An electric linear axis has the following parameters: J m = kg m 2, R a = 5 Ω, K t =.8 N m/a, K v =.8 V/(rad/s), K a = 4, J l = kg m 2, p = 2 mm/rev, K l =.5, and m = 1 kg. Complete the following: a. Design a proportional controller such that the steady state error during a ramp motion is less than 1. mm when there is a constant disturbance force of 1 N. b. Determine the difference equations required to simulate the system. c. Simulate the closed loop linear axis system with the proportional controller for the reference input given in Figure 1. Plot the actual and reference axis positions on one plot and the command voltage on another plot. Design a proportional controller and redo the simulations for a maximum steady state error of 1. mm..1.8 x r (m) time (s) Figure 1 Page 4

5 QUESTION 5 An electric drive spindle has the following parameters: J m = kg m 2, R a = 8 Ω, K t =.5 N m/a, K v =.5 V/(rad/s), K a = 2, J s = kg m 2, and K s =.3. Ignore electrical dynamics and all sources of viscous and nonlinear friction. Complete the following: a. Design a general tracking controller that will regulate the spindle angular velocity and reject sinusoidal disturbances with frequencies ω. The closed loop system should be characterized by three overdamped poles with time constants of.6,.7, and.8 s. b. Determine the difference equations required to simulate the system. c. Simulate the closed loop spindle system for the reference input given in Figure 1. A disturbance torque of 1.2sin(ω(t.5)) N m, where ω = π rad/s, acts on the spindle after.5 s. The spindle angular velocity is measured via a tachometer with a gain of 1 (rad/s)/v. The tachometer voltage is sent to an A/D converter with 1 bits and a voltage range of ±5 V. The command voltage is sent to the motor amplifier via a D/A converter with 1 bits and a voltage range of ±1 V. The motor current range is ±3 A. On separate graphs, plot the actual and reference spindle angular velocities, measured spindle angular velocity error, command voltage, and current versus time. All initial conditions are zero and assume only the spindle angular velocity can be measured. Page 5

6 ω r (rad/s) time (s) Figure 1 QUESTION 6 An electric drive linear axis has the following parameters: J m = 1 3 kg m 2, R a = 7.5 Ω, K t =.9 N m/a, K v =.9 V/(rad/s), K a = 5.5, m = 1 3 kg, J l = kg m 2, p = m/rad, and K l = 1. Ignore electrical dynamics and all sources of Coulomb and viscous friction. Complete the following: a. Design a proportional controller such that the steady state position error for any ramp input is at most 1 mm. b. Design a proportional plus derivative controller such that the closed loop system dynamics are characterized by two underdamped poles with a damping ratio of.65 and a natural frequency of 2 rad/s. c. Determine the difference equations required to simulate the linear axis for the controllers designed in parts (a) and (b). Page 6

7 d. Simulate the closed loop linear axis system for the controllers designed in parts (a) and (b) and the reference input given in Figure 1. A constant disturbance force of 5 N acts on the linear axis after 1 s. The axis position is measured via an encoder with a resolution of 1 µm and the command voltage is sent to the motor amplifier via a D/A converter with 12 bits and a voltage range of ±1 V. The current range is ±7 A. On separate graphs plot the actual and reference axis positions, measured axis position error, command voltage, and current versus time. All initial conditions are zero and assume only the axis position can be measured..1.8 x r (m) time (s) Figure 1 Page 7

8 QUESTION 7 A spindle has the following parameters: J m = kg m 2, R a = 8 Ω, K t =.5 N m/a, K v =.5 V/(rad/s), K a = 2, J s = kg m 2, and K s =.3. Ignore electrical dynamics and all sources of viscous and nonlinear friction. Complete the following: a. Design a general tracking controller that will regulate the spindle angular velocity and reject ramp load torques. The closed loop system should be characterized by three overdamped poles with time constants of.6,.7, and.8 s. b. Determine the difference equations required to simulate the system. c. Simulate the closed loop spindle system for the reference input given in Figure 1. A disturbance torque of.5(t.5) N m acts on the spindle after.5 s. The spindle angular velocity is measured via a tachometer with a gain of 1 (rad/s)/v. The tachometer voltage is sent to an A/D converter with 1 bits and a voltage range of ±5 V. The command voltage is sent to the motor amplifier via a D/A converter with 1 bits and a voltage range of ±1 V. The motor current range is ±3 A. On separate graphs plot the actual and reference spindle angular velocities, measured spindle angular velocity error, command voltage, and current versus time. All initial conditions are zero and assume only the spindle angular velocity can be measured. Page 8

9 ω r (rad/s) time (s) Figure 1 QUESTION 8 A model of a linear axis system is given below in equation (1). Complete the following: a. For the proportional plus integral plus derivative control law in equation (2), determine the closed loop transfer functions from the linear axis velocity to the reference spindle speed and from the linear axis velocity to the load force. b. Repeat part (a) for the control law in equation (3). c. Describe the differences between the corresponding closed loop transfer functions. () ( ) ( ) ( ) τ && x t + x& t = Ke t K f t (1) a c L L () = () + ( τ) τ () c P I x D t e t K x t K e d K x& t (2) () = () + ( τ) τ + () c P x I x D x t e t K e t K e d K e& t (3) Page 9

10 QUESTION 9 A model of a spindle system is given below in equation (1). Complete the following: a. For the proportional plus integral control law in equation (2), determine the closed loop transfer functions from the spindle angular velocity to the reference spindle angular velocity and from the spindle angular velocity to the load torque. b. Repeat part (a) for the control law in equation (3). c. Describe the difference between the corresponding closed loop transfer functions. () t + ω ( t) = Ke ( t) K T ( t) τω& (1) s s s c L L () () ( ) c P s I t e t = K ω t + K e τ dτ (2) () () ( ) c P I t ω e t = K eω t + K eω τ dτ (3) QUESTION 1 QUESTION 11 A linear axis has the following parameters: J m = 1 3 kg m 2, R a = 7.5 Ω, K t =.9 N m/a, K v =.9 V/(rad/s), K a = 5.5, m = 1 3 kg, J l = kg m 2, p = m/rad, and K l = 1. Ignore electrical dynamics and all sources of nonlinear and viscous friction. Complete the following: Page 1

11 a. Design a general tracking controller that will regulate the linear axis position and reject constant disturbances. The closed loop system should have poles with time constants of.5,.6, and.7 s. b. Determine the difference equations required to simulate the system. c. Simulate the closed loop linear axis system for the reference input given in Figure 1. A constant load force of 5 N acts on the linear axis after 1 s. The axis position is measured via an encoder with a resolution of 1 µm and the command voltage is sent to the motor amplifier via a D/A converter with 12 bits and a voltage range of ±1 V. The current range is ±7 A. On separate graphs plot the actual and reference axis positions, measured axis position error, command voltage, and current versus time. All initial conditions are zero and assume only the axis position can be measured..1.8 x r (m) time (s) Figure 1 Page 11

12 QUESTION 12 A linear axis has the following parameters: J m = 1 3 kg m 2, R a = 7.5 Ω, K t =.9 N m/a, K v =.9 V/(rad/s), K a = 5.5, m = 1 3 kg, J l = kg m 2, p = m/rad, and K l = 1. Ignore electrical dynamics and all sources of viscous and nonlinear friction. Complete the following: a. Design a general tracking controller to regulate the linear axis position such that the closed loop system dynamics are characterized by two underdamped poles with a damping ratio of.8 and a natural frequency of 23 rad/s. b. Design a proportional plus integral plus derivative controller to regulate the linear axis position such that the closed loop system dynamics are characterized by an overdamped pole with a time constant of 6 ms and two underdamped poles with a damping ratio of.8 and a natural frequency of 23 rad/s. c. Determine the difference equations required to simulate the system for the controllers designed in parts (a) and (b). d. A. Simulate both closed loop linear axis systems for a ramp reference trajectory with a constant acceleration interpolator. The initial position is m, the final position is.1 m, the velocity is 1 mm/s, and the acceleration is 5 mm/s 2. There is no disturbance torque. The axis position is measured via an encoder with a resolution of 1 µm and the command voltage is sent to the motor amplifier via a D/A converter with 12 bits and a voltage range of ±1 V. The current range is ±7 A. On separate graphs plot the actual and reference axis positions, measured axis position error, command voltage, the current versus time. All initial conditions are zero and assume only the axis position can be measured. Page 12

13 QUESTION 13 The schematic of a closed loop powder feeder system is shown in Figure 1. The output shaft is a screw with a cross sectional area of A = 21 mm 2 and a pitch of l = mm. The screw pushes powder out of the hopper only when it rotates in the positive direction. The powder travels through the distributor and down the tubes and is deposited from the nozzle. The nozzle mass flow rate is related to the hopper mass flow rate by equation (1). The DC motor has an internal gear box such that the output shaft speed is 1/218.4 of the motor shaft speed. Assume the output shaft mass moment of inertia and all sources of friction acting on the output shaft are negligible. The DC motor has the following parameters: J m = kg m 2, R = 4.62 Ω, K t = N m/a, K v = V/(rad/s), B m = N m/(rad/s), and K a = The motor Coulomb friction magnitude is T f = N m. The DC motor current range is ±5.19 A and the powder density is ρ = 72 kg/m 3. Assume the powder flow rate is measured with a sensor having a gain of 5 (gpm)/v, where gpm = gram per minute, and sent to the controller via an A/D converter with a voltage range of ±1 V and 12 bits. The command voltage is sent to the DC motor via a D/A converter with a voltage range of ±1 V and 12 bits. The sample period is Δt = 1 ms. Complete the following: a. Design a general tracking controller that will reject constant disturbances such that the closed loop poles are characterized by two overdamped poles with time constants of 6 and 7 ms. b. The controller will be implemented in the Smith Predictor structure. Determine the difference equations required to simulate the command voltage and the signal e 1. c. Simulate the powder feeder closed loop system for the reference powder flow rate time history shown in Figure 2. On one graph plot the hopper and reference powder flow rates Page 13

14 versus time. On another graph plot the nozzle and reference powder flow rates versus time, and on separate graphs plot the command voltage and the current versus time. All initial conditions are zero. reference mass flow rate controller A/D D/A voltage hopper motor distributer shaft flow rate sensor tubes nozzle part Figure m r (gpm) time (s) Figure 2 Page 14

15 ( s) ( ) Mn 1 ts 1 d = e = e M s τ s s+ 1 h n 1.6s (1) QUESTION 14 A linear axis model is given in equation (1) where x(t) is the linear axis position in mm, e c (t) is the command voltage in V, and F f (t) is the nonlinear friction force in mm/s. The function F f (t) is given in equation (2) where v(t) is the linear axis velocity in mm/s. Complete the following: a. Design a general tracking controller to regulate the linear axis position. b. Code the controller in a Simulink subsystem having two inputs, reference position and measured position in mm, and one output, command voltage in V. c. Tune and implement your controller in the Linear Axis Rapid Development System. Use a triangular reference trajectory with an amplitude of 5 mm and a frequency of 2 Hz. d. Discuss your results. ( ) + ( ) = ( ) ( ).8&& x t x& t 1.72e c t F f t (1) ( ) vt ( ) () ( ) v( t) if > Ff () t = if v t = if < (2) Page 15