Problem 6.1 Draw the linear graph for the system shown in Fig. P6.1 and derive the transformer or transducer constants equations.

Fig. P6.1 DC motor driving an inertial load, J _{ L }

Problem 6.2 Draw the linear graph for the system shown in Fig. P6.2 and derive the transformer or transducer constants equations.

Fig. P6.2 Hybrid fluid-translational mechanical system

Problem 6.3 Draw the linear graph for the system shown in Fig. P6.3 and derive the transformer or transducer constants equations.

Fig. P6.3 Rotational mechanical system

Problem 6.4 Draw the linear graph for the system shown in Fig. P6.4 and derive the transformer or transducer constants equations.

Fig. P6.4 Hybrid fluid-translational mechanical system

Problem 6.5 Draw the linear graph for the system shown in Fig. P6.5 and derive the transformer or transducer constants equations.

Fig. P6.5 Rotational mechanical system

Problem 6.6 Draw the linear graph for the system shown in Fig. P6.6 and derive the transformer or transducer constants equations.

Fig. P6.6 Electromechanical system

Problem 6.7 Draw the linear graph for the system shown in Fig. P6.7 and derive the transformer or transducer constants equations.

Fig. P6.7 Translational mechanical system

Problem 6.8 Draw the linear graph for the system shown in Fig. P6.8 and derive the transformer or transducer constants equations.

Fig. P6.8 Electromechanical system. The cross on the top linkage is the centerline of the solenoid’s magnetic force.

Problem 6.9 A rotational mechanical system is shown in the Fig. P6.9 . An angular velocity source, Ω (t ), acts on the input shaft of a fluid coupling. The output shaft of the fluid coupling drives a pinion with N _{1} teeth, which is engaged in a rack with mass M and N _{ 2 } teeth per meter. The gear has N _{3} teeth and inertia J _{2} . The bearings on the input shaft collectively have damping b _{1} . The bearings on the output shaft collectively have damping b _{2} .

6.9.a Draw the linear graph of the existing system and determine the transformer equations.

6.9.b Draw the equivalent linear graph, if the gear set is eliminated, and inertias J _{1} and J _{2} , the bearings b _{2} , and the torsional shaft with spring constant K are replaced by equivalent elements attached to the input shaft.

6.9.c Calculate the equivalent elemental parameters for the equivalent system of part b .

Fig. P6.9 Hybrid rotational-translational mechanical system

Problem 6.10 A mechanical system consisting of a torque source and a rack and pinion is shown in Fig. P6.10 . The pinion has \( {N_1} = 8 \) teeth. The rack has \( {N_2} = 4\,{\rm{teeth/in}} \) . The torque source acts on a pinion with mass moment of inertia \( J = 0.05\,{\rm{kg}} \cdot {{\rm{m}}^2} \) . The pinion is engaged with a rack of mass \( M = 1\,{\rm{kg}} \) . The rack slides on a lubricating film with damping \(b = 6\,{\rm{N}} \cdot {\mathop{\rm s}\nolimits} /{\rm{m}}\) .

Check their units.

The system is at rest at time \( t = {0^ - } \) , when a torque \( T( t ) = 60\,{\rm{N}} \cdot {\rm{m}}\,{u_s}( t ) \) is applied.

6.10.b Solve the system equations.

6.10.c Plot the responses from \( t = 0 \) to \( t = 6\,\tau\) using Mathcad or MATLAB.

Fig. P.6.10 Hybrid rotational-translational mechanical system

Problem 6.11 An electromechanical schematic of a DC motor is shown in Fig. P6.11a . The motor’s resistance is \( R = 4\,{\rm{\mathit\Omega }} \) . The relationship between the motor current and the motor torque is \( {T_M} = {K_T}{i_M} \) , where \( {K_T} = 8\,{\rm{N}} \cdot {\rm{m/A}} \) . The motor’s mass moment of inertia is \( {J_M} = 0.3\,{\rm{kg}} \cdot {{\rm{m}}^2} \) . The motor turns a flywheel with mass moment of inertia \( {J_L} = 2\,{\rm{kg}} \cdot {{\rm{m}}^2} \) and damping \( b = 0.1\,{\rm{N}} \cdot {\rm{m}} \cdot {\mathop{\rm s}\nolimits} /{\rm{rad}} \) .

and check their units.

The motor was running in steady-state at time \( t = 0 \) under the previously applied step input, when the pulse shown in Fig. P6.11b was applied.

6.11.b Solve the system equations.

6.11.c Plot the responses from time \( t = 0 \) to \( t = 6\,\tau\) sec using Mathcad or MATLAB.

Fig. P6.11 a Schematic of a DC motor. b Voltage v (t ) applied to the motor

Problem 6.12 A mechanical system is shown in Fig. P6.12a . A force source acts on a lever with lengths \( {L_A} = 1\,{\rm{m}} \) , \( {L_B} = 2\,{\rm{m}} \) , and \( {L_C} = 1\,{\rm{m}} \) . Connected to the lever are a spring, \( K = 80\,{\rm{N/m}} \) , and a damper, \( b = 4\,{\rm{N}} \cdot {\rm{s/m}} \) .

Check their units.

The system is in steady-state under the action of a previously applied step input of 20 N, before the force input, F (t ), plotted in Fig. P6.12b , is applied.

Fig. P6.12 a Translational mechanical system. b Force input F (t )

Problem 6.13 The rotational mechanical system shown in Fig. P6.13 a consists of a velocity source acting on the input shaft of rotational damper \( {b_1} = 100\,{\rm{N}} \cdot {\rm{m}} \cdot {\rm{s}} \) (a drag cup). The output shaft of damper b _{1} drives a pinion, \( {N_1} = 12 \) teeth meshed with two gears with \( {N_2} = 24 \) and \({N_3} = 36\) teeth. The gears, in turn, drive the input shaft of rotational damper \( {b_2} = 100\,{\rm{N}} \cdot {\rm{m}} \cdot {\rm{s}} \) and rotational inertia \( J = 25\,{\rm{kg}} \cdot {{\rm{m}}^2} \) .

Check their units.

The system was in steady-state under a previous step input of Ω (t ) = 5 rad/sec before the angular velocity input, plotted in Fig. P6.13a , was applied.

Fig. P6.13 a Rotational system. b Angular velocity input

Problem 6.14 A translational mechanical system is shown schematically in Fig. P6.14 . The system consists of a force source\( F( t ) \) acting on a pivoted beam with lengths \( {L_{\,1}} = 36\,{\rm{in}} \) , \( {L_{\,2}} = 48\,{\rm{in}} \) , and \( {L_{\,2}} = 36\,{\rm{in}} \) . Attached to the beam are damper \( {b_{\,1}} = 200\,{\rm{N}} \cdot {\rm{sec/m}} \) , spring \( K = 4,000\,{\rm{N/m}} \) , and mass \( M = 20\,{\rm{kg}} \) . There is a lubricating film under mass M with damping \({b_2} = 10\,{\rm{N}} \cdot {\rm{s/m}}\) .

Check their units.

The system was de-energized for time \( t < 0 \) , before the input force \( F( t ) = 1,000\,{\rm{N}}\,{u_s}( t ) \) acted on the system.

Fig. P6.14 Translational mechanical system

Problem 6.15 A model of an engine lathe is shown in Fig. P6.15a . The motor and gear box is modeled as a torque source driving a rotational inertia \( J = 2.5\,{\rm{kg}} \cdot {{\rm{m}}^2} \) . The motor drives a lead (or power) screw, which is supported by bearings in the tail stock. The lead screw is modeled as having negligible torsional compliance. The lead screw bearings have damping \( {b_1} = 10\,{\rm{N}} \cdot {\rm{m}} \cdot {\rm{s/rad}} \) . The lead screw pitch is 2 threads per inch. The lead screw engages a lead nut attached to the load carriage. The lead nut moves the load carriage with mass \( M = 150\,{\rm{kg}} \) along the lubricated ways of the lathe. The lubricating film between the load carriage and the lathe’s ways has a damping coefficient \({b_2} = 60\,{\rm{N}} \cdot {\rm{s/m}}\)

Check their units.

The system was at rest before the input torque T (t ), plotted in Fig. P6.15b , was applied.

Fig. P6.15 a Schematic model of an engine lathe. b Torque input T (t )

Problem 6.16 A mechanical system is shown in Fig. P6.16a . The angular velocity source drives a compliant shaft with spring constant \( K = 100\,{\rm{N}} \cdot {\rm{m}} \cdot {\rm{s/rad}} \) . The shaft drives a pinion with \( {N_1} = 10 \) teeth. The pinion is engaged in a rack. The rack has \( {N_2} = 4\,{\rm{teeth/in}} \) . The rack is attached to mass \( M = {\rm 10 kg}\) . The rack and mass slide on a lubricating film with damping \(b = 2\,{\rm{N}} \cdot {\mathop{\rm s}\nolimits} /{\rm{m}}\) .

Check their units.

The system is in steady-state under a step input of angular velocity of − 40 rad/sec and time \( t = {0^ - } \) , when the angular velocity pulse shown in Fig. P6.16b is applied.

6.16.b Solve the system equations.

6.16.c Plot the responses from \( t = 0 \) to \( t = 2{\mathop{\rm s}\nolimits} \, + 6\,\tau\) using Mathcad or MATLAB.

Fig. P.6.16 Hybrid rotational-translational mechanical system

Problem 6.17 A fluid translational mechanical system is shown in Fig. P6.17a . The system consists of a pump modeled as a pressure source, two fluid resistances \( {R_{\,1}} = 10\,{\rm{MPa}} \cdot {\mathop{\rm s}\nolimits} /{{\rm{m}}^3} \) and \( {R_2} = 100\,{\rm{MPa}} \cdot {\mathop{\rm s}\nolimits} /{{\rm{m}}^3} \) , an accumulator (fluid capacitor) C = 5.5 L/MPa, an hydraulic cylinder with a piston of diameter \( D = 6\,{\rm{in}} \) , and mass \( M = 2\,{\rm{kg}} \) .

Check their units.

The system was in steady-state under a previous step input of 2,000 psi, when the input plotted in Fig. P6.17b was applied at time \( t = 0 \) .

6.17.b Solve the system equations.

6.17.c Plot the responses from \( t = 0 \) to \( t = 6\,\tau\) using Mathcad or MATLAB.

Fig. P6.17 a Fluid translational mechanical system. b Pressure input

Problem 6.18 A translational system consisting of a force source F (t ), a lever, two springs, and two dashpots is shown in Fig. P6.18a . The parameter values are tabulated.

Check their units.

The system is at de-energized at time \( t = 0 \) when the system is subjected to input force F (t ), plotted in Fig. P6.18b .

Table P6.18 Parameter values

Fig. P6.18 a Translational mechanical system. b Input force F (t )

Problem 6.19 A rotational mechanical system is shown schematically in Fig. P6.19 . The system is driven by an angular velocity source which is connected to the input shaft, by a fluid coupling with damping b _{1} . The output shaft of the fluid coupling connects to a compliant shaft with spring constant K , which turns the pinion of a gear set. The pinion has N _{1} teeth. Gear N _{2} drives a shaft which rotates mass moment of inertia J and is supported by a bearing with damping b _{2} .

Check their units.

The system is at de-energized at time \( t = 0 \) when the system is subjected to the step input torque of 800 N-m.

Table P6.19 Parameter values

Fig. P6.19 Rotational mechanical system

Problem 6.20 An electromechanical schematic of a DC motor is shown in Fig. P6.20 . The motor is powered by a voltage source v (t ). The motor’s resistance is \( R = 4\,{{\mathit\Omega }} \) . The motor’s bearings and brushes have damping \( {b_1} = 0.4\,{\rm{N}} \cdot {\rm{m}} \cdot {\rm{rad/s}} \) . The relationship between the motor current and the motor torque is \( {T_M} = {K_T}{i_M} \) , where \( {K_T} = 8\,{\rm{N}} \cdot {\rm{m/A}} \) . The motor turns a compliant shaft with spring constant \( K = 500\,{\rm{N}} \cdot {\rm{m/rad}} \) . The shaft turns a flywheel with mass moment of inertia \( J = 2\,{\rm{kg}} \cdot {{\rm{m}}^2} \) and damping \( {b_2} = 0.1\,{\rm{N}} \cdot {\rm{m}} \cdot {\mathop{\rm s}\nolimits} /{\rm{rad}} \) .

Check their units.

The motor de-energized when a step input \( v( t ) = 48\,{\rm{VDC}}\,{u_s}( t ) \) was applied.

6.20.b Solve the system equations.

6.20.c Plot the responses from time \( t = 0 \) to \( t = 6\,\tau\) sec using Mathcad or MATLAB.

Fig. P6.20 DC motor driving an inertial load through a compliant shaft