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OMU324- H.W#4 Solutions - OMU 324 System Dynamics&...

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Unformatted text preview: OMU 324 System Dynamics & Control Homework #4 Solutions HACETTEPE UNIVERSITY Mechanical Engineering Department System Dynamics & Control Homework #4 Due Date: 05.04.2011 1) A mode}. of a commercial rotting machine for metals processing is shown at right. The model. comprises a cylinder ' fit) of mass in and radius r that spins about ' ¢ a A! a a an axle. The cylinder ions without slip on the lower surface. As seen from the . hydratiiic actuator figure, a damper b, a spring k, and a hydi‘auiic actuator are attached to the axle lioosing. The actuator applies a. force f on the cyiinder atong x direction. a) Obtain the governing differential equations of motion. b) Obtain the state space representation of the system. X (s) c) Obtain the transfer function. HS) Note: I is the moment of inertia of the cylinder. 2) f(t): force which is input of the system 131,152 ,b3: damping coefficients for friction. 64 : damping coefficient of damper. X1, X2 : diSplacements which are outputs of the system. ml , m2 : Mass of the block and sheii. a) Draw the free body diagrams for the biock and sheik and obtain the governing differential equations of motion. b) Obtain the state space representation of the system. 0) Determine the following transfer functions. X169) F(s) ’ X2“) F(s) G109)fl 62(5): 3) Consider the system shown in the figure below. An ammmre controlled DC servomotor drives a load consisting of the moment of ilieifia IL. The torque developed by the motor is T. The angular displacements of the motor rotor and the load element are an and B, respectively The gear ratio is n: 92, Obtain the transfer function 9.2.9:). R :Armature resistance L :Arrnature inductance i : Armature current 1' f : Field cun‘ent T : Servomotor Torque T =K.z‘ eb= 5—4»?- dt I: (I. . r. 62 T (Motor Torque) 8,. : voltage input to armature e!) : back eiectromotor force (emf) 91 : angular displacement of servomotor J : Moment of inertia Take J] , J2 =0 for gear trains. The system shown in the figure is combined translational and rotational mechanical elements. The mass and spring are connected to the disk by a flexible cable. Actually, the spring might be used to represent the stretching of the cable. The mass M is subjected to the external force )2 (I) . Find the state variable equations for the system, treating (r) and the weight of the mass as inputs. J : Moment of inertia B: Rotational damper 9 : Rotational displacement Kl , K2 : Constants of shaft and spring 5) é'l'qg The fluid system which is combined two tanks is shown in the figure. The steady state head of tanks are E and El: respectively. The steady state inflow and outflow rates are The H2 (5) £2(S) ' capacitance of tanks are C1 and C2. Derive the transfer function 6) 3: 1 Steady state temperature of inlet liquid 90 : Steady state temperature of outflow liquid G : Mass flow rate of liquid M :Mass of liquid contained in the heating chamber C ‘. Specific heat of iiquid R : Thermal Resistance C : Thermal capacitance H : Steady state heat input Heater Thermal system shown in the figure contains heating chamber, heater and mixer. (9,: , 60 and h are the changing of inlet temperature, outflow temperature and energy given by the heater. . . . . 0 Obtain the heat balanoing equation and transfer function 0“) . f{(s) H‘W #14 fine;qu ® 9.? 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