Mechatronics. MANE 4490 Fall 2002 Assignment # 1

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1 Mechatronics MANE 4490 Fall 2002 Assignment # 1 1. For each of the physical models shown in Figure 1, derive the mathematical model (equation of motion). All displacements are measured from the static equilibrium position. 2. A load inertia I 2 is driven through gears by a motor with inertia I 1, as shown in Figure 2. The shaft inertias are I 3 and I 4 ; the gear inertias are I 5 and I 6. The gear ratio is 5:1 (the motor shaft has the greater speed). The motor torque is T 1, and the viscous damping coefficient is c = 1.2 lb-ft-sec/rad. Neglect elasticity in the system, and use the following inertia values (sec 2 -ft-lb/rad): I 1 = 0.01, I 2 = 0.5, I 3 = 0.001, I 4 = 0.005, I 5 = 0.02, I 6 = (a) Derive the mathematical model for the motor shaft speed ω 1 with T 1 as the input; (b) Derive the mathematical model for the load shaft speed ω 2 with T 1 as the input. 3. A load inertia I 5 is driven through a double-gear pair by a motor with inertia I 4, as shown in Figure 3. The shaft inertias are negligible. The gear inertias are I 1, I 2, and I 3. The speed ratios are ω 1 /ω 2 = 2 and ω 2 /ω 3 = 5. The motor torque is T 1 and the viscous damping coefficient c = 4 lb-ft-sec/rad. Neglect elasticity in the system, and use the following inertia values (sec 2 -ft-lb/rad): I 1 = 0.1, I 2 = 0.2, I 3 = 0.4, I 4 = 0.3, I 5 = 0.7. Derive the mathematical model for the motor shaft speed ω 1 with T 1 as the input. 4. In the pulley system shown in Figure 4, assume that the cable is massless and inextensible. The input is the applied force f and the output is the displacement x 1. The pulleys are frictionless and there is no slip between the cable and the pulleys. (a) Assume the pulley masses are negligible and derive the system s equation of motion; (b) Suppose the mass of pulley 2 is negligible, but pulley 1 has a mass m p and an inertia about its center of mass I p. Derive the system s equation of motion. 5. Determine the mathematical model for each of the levered systems shown in Figure 5, with force f as the input. Assume small displacements. Neglect pivot friction. The center of gravity of the lever is at the pivot. (a) the lever is rigid and massless; (b) the lever is rigid and has an inertia I relative to the pivot. 6. A dynamic vibration absorber consists of a mass and an elastic element that is attached to another mass in order to reduce its vibration. Figure 6 is a representation of a vibration absorber attached to the cantilever support. For a cantilever beam with a force at its end, k = Ewh 3 /4L 3 where L = beam length, w = beam width, and h = beam thickness. (a) Obtain the equation of motion for the system. The force f is a specified force acting on the mass m, and is due to the rotating unbalance of the motor. The displacements x and x 2 are measured from the static equilibrium positions when f = 0. (b) Obtain the transfer functions x/f and x 2 /f. 7. Develop the equivalent rotational model of the rack-and-pinion gear system shown in Figure 7. The applied torque T is the input variable, and the angular displacement θ is the output variable. Neglect any twist in the shaft. Bearings are frictionless. The pinion gear mass moment of inertia about its CG (geometric center) is I p. 1

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4 8. A cylinder has a radius R, a mass M, and a mass moment of inertia I G about its mass center. It rolls on a semicircular track of radius r without slipping. Its axle moves in a frictionless slot in a link. The link has a mass m and a mass moment of inertia I O about frictionless pivot point O. Derive the equation of motion of the system by: (a) Newton s Method and (b) Energy Method. 9. Find the equation of motion of the pendulum shown which consists of a concentrated mass m c a distance L c from frictionless pivot point O, attached to a rod of length L R, mass m r, and mass moment of inertia I RG about its mass center G. Discuss the case where the rod s mass is small compared to the concentrated mass. 4

5 10. A physical model of the spring-pendulum dynamic system, demonstrated and discussed in class, is shown below. k l + r θ m The equations of motion for the system are: ( ) ( ) 2 mr m + r θ + kr + Ft mg cosθ= 0 + r θ+ 2r θ+ gsinθ= 0 Using the numerical values for the model parameters discussed in class, develop a MatLab/Simulink simulation diagram to solve this mathematical model. Simulate two sets of initial conditions: θ= rad r = m θ= rad r = m Submit the simulation diagram and the plots of r vs. t and θ vs. t for each set of initial conditions. Discuss your results. 5

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