Dynamic Modeling. For the mechanical translational system shown in Figure 1, determine a set of first order

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1 QUESTION 1 For the mechanical translational system shown in, determine a set of first order differential equations describing the system dynamics. Identify the state variables and inputs. y(t) x(t) k m b nonlinear friction QUESTION 2 provides the output data (i.e., voltage in V) for a circuit where the input is a constant voltage of 10 V. Use empirical techniques to determine a differential equation describing the dynamics of this system.

2 voltage (V) time (s) QUESTION 3 provides system output data for a step input with a magnitude of 4. Use empirical techniques to determine a mathematical model for this system. Page 2

3 2.5 2 output time (s) QUESTION 4 For the system shown in, the force through the spring is f s (t) = kx 3 (t) where x(t) is the displacement across the spring. The model parameters are m = 2 kg, k = 2 N/m 3, and b = 2.5 N/(m/s). If the equilibrium position is x, determine the equilibrium velocity and force symbolically. Derive the symbolic linearized equations. For equilibrium positions of x = 0 m and x = 0.5 m, determine the equilibrium force and the numerical linearized equations. Page 3

4 QUESTION 5 A machining force process, including servomechanism dynamics, is given in equation (1) where F(t) is the machining force and f(t) is the commanded feed. For an equilibrium machining force of F, determine the equilibrium commanded feed and the linearized equations. 2 2 α () t 2 ζω F& ( t) F( t) ω = ω Kf ( t) F & (1) n n n QUESTION 6 The mathematical model of the height of fluid in a tank is given in equation (1). Determine the equilibrium height for an input mass flow rate of system about the equilibrium condition. m in. Determine the linearized equation for the 1 1 h& () t = m () [ () ] in t ρgh t p A (1) Aρ RAρ Page 4

5 QUESTION 7 In the circuit in, resistor 1 obeys the nonlinear relationship e1( t) R1ln{ I1( t) } = where e 1 (t) is the voltage across the resistor and I 1 (t) is the current through the resistor. For an equilibrium voltage across the capacitor of e C, symbolically determine the equilibrium input voltage, e, and the linearized differential equation that relates the voltage across the capacitor to the input voltage. R 1 e(t) - R 2 C QUESTION 8 A machining force process, including servomechanism dynamics, is given in equation (1) where F(t) is the machining force, d(t) is the depth of cut, and f(t) is the commanded feed. Both d(t) Page 5

6 and f(t) may be adjusted. For nominal depth of cut and feed, respectively, of d and f, symbolically determine the equilibrium machining force F and the linearized equation(s). β α ( t) F( t) Kd ( t) f ( t) τ F& = (1) QUESTION 9 For the mechanical rotational system shown in, determine a set of first order differential equations describing the system dynamics. Identify the state variables and inputs. QUESTION 10 For the electrical system shown in, determine a set of first order differential equations describing the system dynamics. Identify the state variables and inputs. Page 6

7 R 1 R 2 R 3 e(t) - C 1 C 2 C 3 QUESTION 11 For the electrical system shown in, determine a set of first order differential equations describing the system dynamics. Identify the state variables and inputs. R 1 R 2 R 3 e(t) C 1 C 2 L - Page 7

8 QUESTION 12 For the electrical system shown below, complete the following: a. Determine a set of state equations describing the system dynamics. b. Determine the transfer function relating E o (s) to E i (s). C 2 R e i (t) C 1 - e o (t) QUESTION 13 For the electrical system shown below, complete the following: a. Determine a set of state equations describing the system dynamics. b. Determine the transfer function relating E o (s) to E i (s). Page 8

9 QUESTION 14 For the electrical system shown below, complete the following: a. Determine a set of state equations describing the system dynamics. b. Determine the transfer function relating E o (s) to E i (s). R 3 R 4 e i (t) R 1 - C 2 - e o (t) C 1 R 2 Page 9

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Mechanical Translational Systems

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