# Manufacturing Equipment Modeling

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1 QUESTION 1 For a linear axis actuated by an electric motor complete the following: a. Derive a differential equation for the linear axis velocity assuming viscous friction acts on the DC motor shaft, leadscrew, and linear axis (i.e., B m 0, B l 0, and b a 0) and that the motor has velocity feedback such that the input voltage to the amplifier is e(t) = e c (t) K tach ω m (t). Include the axis disturbance force, but do not include motor electrical dynamics or Coulomb friction torques acting on the motor shaft or leadscrew. b. Determine the time constant for this system. c. For a constant command voltage with magnitude E c and a constant disturbance load with magnitude F L, determine the steady state velocity. d. Determine the steady state gains from axis velocity to command voltage and from axis velocity to load force. e. Draw a block diagram of the electric linear axis system. QUESTION 2 For a spindle actuated by an electric motor complete the following: a. Derive a differential equation for the spindle angular velocity assuming viscous friction acts on the motor and spindle and that the motor has velocity feedback such that the input

2 voltage to the amplifier is e(t) = e c (t) K tach ω m (t). Include the spindle load torque, but do not include motor electrical dynamics or nonlinear friction torques. b. Determine the spindle system time constant. c. For a constant command voltage with magnitude E c and a constant load torque with magnitude T L, determine the steady state angular velocity. d. Determine the steady state gains from spindle angular velocity to command voltage and from spindle angular velocity to load torque. e. Draw a block diagram of the electric spindle system. QUESTION 3 Two trials are run for a spindle system. The outputs are given in Figure 1. Approximate the spindle system model parameters (i.e., τ, K, and K d ) for the model given in equation (1). () t + ω ( t) = Ke ( t) K T ( t) τω (1) s s c d d Page 2

3 spindle speed (rad/s) T d = 0 N m T d = 25 N m time (s) Figure 1: Spindle System Output with e c (t) = 8 V. QUESTION 4 For a rotational axis actuated by a hydraulic motor complete the following: a. Derive differential equations relating rotational axis angular velocity and load pressure to valve displacement, disturbance torque, and motor shaft Coulomb friction. Include motor shaft and fluid viscous friction, fluid Coulomb friction, leakage effects, and rotational axis disturbance torque. b. For a constant command voltage, disturbance torque, and motor Coulomb friction torque, determine the steady state angular velocity and load pressures. c. Determine the steady state gains from axis angular velocity to command voltage, disturbance torque, and motor Coulomb friction. d. Determine the steady state gains from load pressure to command voltage, disturbance torque, and motor Coulomb friction. Page 3

4 e. Draw a block diagram of the hydraulic rotational axis system. QUESTION 5 A spindle has a mass moment of inertia of J s = kg m 2 and is connected to a DC motor having the following parameters: J m = kg m 2, R = 5 Ω, K t = 0.7 N m/a, K v = 0.7 V/(rad/s), and K a = 30. The maximum command voltage is e c = 10 V. Complete the following: a. Select the gear gain K s such that the maximum steady state spindle speed, without disturbance, will be ω ss 250 rad/s and the maximum change in spindle speed due to a disturbance torque of 10 N m will be 10 rad/s. b. Determine the time constant. QUESTION 6 For a linear axis actuated by an electric motor complete the following a. Derive a differential equation relating linear axis velocity to command voltage, disturbance force, DC motor shaft Coulomb friction, and leadscrew Coulomb friction. Include motor shaft, leadscrew, and linear axis viscous friction and axis disturbance force, but do not include motor electrical dynamics. b. Determine the linear axis system time constant. c. For a constant command voltage, disturbance force, and Coulomb friction torques, determine the steady state velocity. Page 4

5 d. Determine the steady state gains from axis velocity to command voltage, disturbance force, motor Coulomb friction, and leadscrew Coulomb friction. e. Draw a block diagram of the electric linear axis system. QUESTION 7 A hydraulic rotational axis has the following parameters: J m = 0.6 kg m 2, J r = 10 kg m 2, V 0 = 0.2 m 3, β = 10 9 N/m 2, D m = 10 2 m 3 /rad, K r = 0.2, K c = 10 8 m 5 /(N s), and K q = 3 m 2 /s. Complete the following: a. Using the Euler method, derive first order difference equations for the axis angular position, axis angular velocity, and load pressure. b. For a constant valve displacement of 10 mm and a disturbance torque of 10 4 sin(100t) N m, where t is time in s, create a numerical simulation and, on separate graphs, plot the axis angular position, axis angular velocity, load pressure, and load flow versus time. All initial conditions are zero. QUESTION 8 A hydraulic rotational axis system 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), K q = 30 m 2 /s, J r = 10 3 kg m 2, B r = 1 N m/(rad/s), and K r = 0.7. Complete the following: a. Using the Euler method, derive first order difference equations for the axis angular position, axis angular velocity, and load pressure. Page 5

6 b. For the valve displacement shown in Figure 1 and a constant disturbance torque of 50 N m, create a numerical simulation and, on separate graphs, plot the axis angular position, axis angular velocity, load pressure, and load flow versus time. All initial conditions are zero. x v 5 mm 0 mm 0 s 1 s t Figure 1 QUESTION 9 An electric drive spindle has the following parameters: J m = 10 2 kg m 2, R = 5 Ω, K t = 0.5 N m/a, K v = 0.5 V/(rad/s), K a = 30, L = 0.1 H, J s = 4 kg m 2, and K s = 0.1. The current is limited to ±10 A. Complete the following: a. Symbolically derive a differential equation relating spindle angular velocity to command voltage and disturbance torque, a transfer function relating spindle angular velocity to command voltage, and a transfer function relating spindle angular velocity to disturbance torque. Ignore all sources of viscous friction and Coulomb friction acting on the motor shaft. Page 6

7 b. Calculate the time constants numerically. c. Using the Euler method, symbolically derive difference equations to simulate the spindle angular velocity and current. d. Simulate the electric spindle system for e c (t) = 4 + sin(5t) V and T d = 0. On separate graphs, plot the spindle angular velocity and current versus time. All initial conditions are zero. QUESTION 10 An electric linear axis has the following parameters: J m = 10 3 kg m 2, R a = 7.5 Ω, K t = 0.9 N m/a, K v = 0.9 V/(rad/s), K a = 55, L a = H, J l = kg m 2, K l = 1, p = 20/(2000π) m/rad, m = 1000 kg. The current is limited to ±10 A. Complete the following: a. Symbolically derive a differential equation relating axis position to command voltage and load force, a transfer function relating axis position to command voltage, and a transfer function relating axis position to load force. Ignore all sources of viscous and nonlinear friction. b. Calculate the time constants numerically. c. Using the Euler method, symbolically derive difference equations to simulate the axis position, axis velocity, and armature current. d. Simulate the system for a square command voltage signal with an amplitude of 2 V and a frequency of 0.25 Hz, and and f L = 0. Run the simulation for 2 cycles and plot the axis position, axis velocity, motor torque, and armature current versus time on separate Page 7

8 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. All initial conditions are zero. QUESTION 11 A hydraulic rotational axis has the following parameters: J m = 0.6 kg m 2, J r = 10 kg m 2, V 0 = 0.2 m 3, and β = 10 9 N/m 2. Select the gear gain, volumetric displacement, flow gain, and flow pressure coefficient such that that system has a damping ratio of 2, a natural frequency of 100 rad/s, the angular velocity will change by 0.6 rad/s for a change in valve displacement of 10 mm, and the angular velocity will change by 1 rad/s for a change in disturbance torque applied to the rotational axis of 2500 N m. Neglect viscous damping in the motor and rotational axis. QUESTION 12 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, symbolically determine a differential equation relating the powder flow rate to the command voltage. b. Ignoring the nonlinear friction, symbolically determine a transfer function relating the powder flow rate to the command voltage. c. The powder feeder has the following parameters: τ m = s, k m = 158 rpm/v, ω f = 98.8 rpm, τ p = s, k p = gpm/rpm, and t d = 1.98 s. Determine a set of difference equations, using Euler s method, to simulate the powder feeder. Simulate the powder Page 8

9 feeder for a command voltage e c (t) = 2sin(t). Plot the motor angular velocity and nozzle mass flow rate on separate graphs. All initial conditions are zero and e c (t) = 0 for t < 0. d. Ignoring the nonlinear friction, plot the magnitude and phase frequency plots for the transfer function derived in part b. () 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 13 For a spindle actuated by an electric motor complete the following: a. Derive a differential equation relating spindle angular velocity to command voltage, disturbance torque, and DC motor shaft Coulomb friction. Include motor shaft and spindle viscous friction and spindle disturbance torque, but do not include motor electrical dynamics. b. Determine the spindle system time constant. c. For a constant command voltage, disturbance torque, and Coulomb friction torque, determine the steady state angular velocity. d. Determine the steady state gain from spindle angular velocity to command voltage, disturbance torque, and motor Coulomb friction. e. Draw a block diagram of the spindle system. Page 9

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