Problem 10.1 Fig P10.1 shows a translational mechanical system. Mass M is supported by frictionless rollers. A dashpot with damping b and a spring with spring constant K are attached between the mass and ground. The mass is acted upon by a sinusoidal force, F (t ). The parameter values are M = 3, b = 2, and K = 100

Fig. P10.1 Translational spring–mass–damper system acted on by the force, F (t )

10.1.a Derive the system equation for:

i. The velocity of the mass.

ii The force acting through the spring.

iii The force acting to accelerate the mass.

10.1.b Derive the transfer function for the system equation of part a and plot the poles and zeros (if any) of transfer function on the s -plane.

10.1.c Determine the steady-state response of the transfer functions of part b for:

i \(F( t ) = 10\,\,{\rm{N}}\,\,\sin ( {3\,t} )\)

ii \(F( t ) = 10\,\,{\rm{N}}\,\,\sin ( {6t} )\)

iii \(F( t ) = 10\,\,{\rm{N}}\sin ( {12t} )\)

10.1.d Formulate the state-space representation of the system with the output variables:

i The velocity of the mass.

ii The force acting through the spring.

iii The force acting to accelerate the mass.

10.1.e Solve the state and output equations of part d for the sinusoidal inputs of part c for the duration, \(t \approx 10\,\tau\) , where \(\tau= \left| {\frac{1}{{{\sigma _{\min }}}}} \right|\) .

Problem 10.2 Fig. P 10.2 shows a rotational mechanical system. The flywheel has rotational inertia J. The flywheel’s shaft is supported by a hydrodynamic bearing with damping b and attached to a torsion spring with spring constant K. The other end of the torsion spring is rigidly attached to the machine frame, which is ground. The system is driven a sinusoidal torque, T (t ). The parameter values are J = 5, b = 4, and K = 1,000

Fig. P10.2 Translational spring–mass–damper system acted on by the force, F (t )

10.2.a Derive the system equation for:

i The angular velocity of the flywheel.

ii The torque acting through the spring.

iii The torque acting to accelerate the flywheel.

10.2.b Derive the transfer function for the system equation part a and plot the poles and zeros (if any) of transfer function on the s -plane.

10.2.c Determine the steady-state response of the transfer functions of part b for:

i \(T( t ) = 10\,\,{\rm{N}} \cdot {\rm{m}}\,\,\sin ( {7\,t} )\)

ii \(T( t ) = 10\,\,{\rm{N}} \cdot {\rm{m}}\,\,\sin ( {14t} )\)

iii \(T( t ) = 10\,\,{\rm{N}} \cdot {\rm{m}}\,\,\sin ( {21t} )\)

10.2.d Formulate the state-space representation of the system with the output variables:

i The angular velocity of the flywheel.

ii The torque acting through the spring.

iii The torque acting to accelerate the flywheel.

10.2.e Solve the state and output equations of part d for the sinusoidal inputs of part c for the duration, \(t \approx 10\,\tau\) , where \(\tau= \left| {\frac{1}{{{\sigma _{\min }}}}} \right|\)

Problem 10.3 Fig. P. 10.3 shows an RLC electric circuit with a voltage source. The input is a sinusoidal voltage, v (t ). The parameter values are \(L = 0.01\,\,{\rm{H}}\) , \(C = 4 \times {10^{ - 6}}\,\,{\rm{F}}\) , and \(R = 0.1\,\,\Omega\) .

Fig. P10.3 RLC electric circuit

10.3.a Derive the system equation for:

i The current through the inductor.

ii The voltage across the capacitor.

iii The current from the voltage source.

10.3.b Derive the transfer function for the system equation of part a and plot the poles and zeros (if any) of transfer function on the s -plane.

10.3.c Calculate the ideal, undamped, natural frequency, ω _{ n } , and the damping ratio, ζ , of the circuit.

10.3.d Determine the steady-state response of the transfer functions of part b using the natural frequency determined in part c for:

I \(v( t ) = 24\,\,{\rm{VDC}}\,\,\sin \left( {\frac{{{\omega _n}}}{2}\,t} \right)\)

ii \(v( t ) = 24\,\,{\rm{VDC}}\,\,\sin ( {{\omega _n}\,t} )\)

iii \(v( t ) = 24\,\,{\rm{VDC}}\,\,\sin ( {2{\omega _n}\,t} )\)

10.3.e Formulate the state-space representation of the system with the output variables:

i The current through the inductor.

ii The voltage across the capacitor.

iii The current from the voltage source.

10.3.f Solve the state and output equations of part e for the sinusoidal inputs of part c for the duration, \(t \approx 10\,\tau\) , where \(\tau= \left| {\frac{1}{{{\sigma _{\min }}}}} \right|\) .

Problem 10.4 Fig. P10.4 shows a fluid system with a pump modeled as a pressure source, two fluid resistances, a fluid inertance, and a fluid capacitance (accumulator). The parameter values are:

$${{R}_{\,1}}=1\,\,\frac{\text{MPa}}{{{\text{m}}^{\text{3}}}}\,{{R}_{2}}=20\,\,\frac{\text{MPa}}{{{\text{m}}^{\text{3}}}},\, I = 5\,\,\frac{{{\rm{MPa}} \cdot {{\mathop{\rm sec}\nolimits} ^2}}}{{{{\rm{m}}^3}}},\,{\rm{and}}\, C=4\,\,\frac{{{\text{m}}^{\text{3}}}}{\text{MPa}}$$

.

Fig. P10.4 Schematic of a fluid system:

10.4.a Derive the system equation for

i The volume flow rate through the fluid inertance.

ii The pressure in the fluid capacitance.

iii The volume flow rate into to the fluid capacitance.

10.4.b Derive the transfer function for the system equation of part a and plot the poles and zeros (if any) of transfer function on the s -plane.

10.4.c Calculate the ideal, undamped, natural frequency, ω _{ n } , and the damping ratio, ζ , of the fluid system.

10.4.d Determine the steady-state response of the transfer functions of part b using the natural frequency determined in part c for

i \(p( t ) = 3,000\,\,{\rm{psi}}\,\,\sin \left( {\frac{{{\omega _n}}}{2}\,t} \right)\)

ii \(p( t ) = 3,000\,\,{\rm{psi}}\sin ( {{\omega _n}\,t} )\)

iii \(p( t ) = 3,000\,\,{\rm{psi}}\,\,\sin ( {2{\omega _n}\,t} )\)

10.4.e Formulate the state-space representation of the system with the output variables:

i The volume flow rate through the fluid inertance.

ii The pressure in the fluid capacitance.

iii The volume flow rate into to the fluid capacitance.

10.4.f Solve the state and output equations of part e for the sinusoidal inputs of part c for the duration, \(t \approx 10\,\tau\) , where \(\tau= \left| {\frac{1}{{{\sigma _{\min }}}}} \right|\) .

Problem 10.5 An electromechanical schematic of a DC motor is shown in Fig. P10.5 . The motor’s resistance is \(R = 4\,\,{\rm{\Omega }}\) . 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 \(b = 0.1\,\,{\rm{N}} \cdot {\rm{m}} \cdot \sec /{\rm{rad}}\) .

Fig. P10.5 Schematic of a DC motor with an inertial load and driven by a voltage source

and check their units.

10.5.b Derive the transfer function for the system equation of part a and plot the poles and zeros (if any) of transfer function on the s -plane.

10.5.c Determine the steady-state response of the transfer functions of part b for:

i \(v( t ) = 24\,\,VDC\sin ( {3.5t} )\)

ii \(v( t ) = 24\,\,VDC\sin ( {7t} )\)

iii \(v( t ) = 24\,\,VDC\sin ( {14t} )\)

10.5.d Formulate the state-space representation of the system with the output variables:

10.5.e Solve the state and output equations of part d for the sinusoidal inputs of part c for the duration, \(t \approx 10\,\tau\) , where \(\tau= \left| {\frac{1}{{{\sigma _{\min }}}}} \right|\) .

Problem 10.6 The rotational mechanical system shown in Fig. P10.6 consists of angular velocity source, Ω (t ), driving a fluid coupling with damping, \(b = 80\,\,{\rm{N}} \cdot {\rm{m}} \cdot \sec {\rm{/rad}}\) . The output shaft of the fluid coupling drives a pinion with N _{1} teeth. The pinion with \({N_1} = 10\) teeth is meshed with Gear Two with \({N_2} = 20\) teeth and Gear Three with \({N_3} = 30\) teeth. Gear Two is mounted on a compliant shaft with spring constant, \(K = 600\,\,{\rm{N}} \cdot {\rm{m/rad}}\) . The other end of the compliant shaft is attached rigidly to the machine frame. Gear Three is mounted on a rigid shaft which carries rotational inertia, \(J = 6\,\,{\rm{kg}} \cdot {{\rm{m}}^2}\) , on ideal, frictionless bearing.

Fig. P10.6 Rotational mechanical system

10.6.a Derive the system equation for:

i The angular velocity of inertia, J .

ii The torque acting through the compliant shaft.

iii The torque applied by the angular velocity source.

10.6.b Derive the transfer function for the system equation of part a and plot the poles and zeros (if any) of transfer function on the s -plane.

10.6.c Determine the steady-state response of the transfer functions of part b for:

i \(\Omega ( t ) = 10\,\,\frac{{{\rm{rad}}}}{{\rm{s}}}\sin ( {0.1t} )\)

ii \(\Omega ( t ) = 10\,\,\frac{{{\rm{rad}}}}{{\rm{s}}}\sin ( {0.5t} )\)

iii \(\Omega ( t ) = 10\,\,\frac{{{\rm{rad}}}}{{\rm{s}}}\sin ( t )\)

10.6.d Formulate the state-space representation of the system with the output variables of part a .

10.6.e Solve the state and output equations of part d for the sinusoidal inputs of part c for the duration, \(t \approx 10\,\tau\) , where \(\tau= \left| {\frac{1}{{{\sigma _{\min }}}}} \right|\)

Problem 10.7 The rotational-translational mechanical system, shown schematically in Fig. P10.7 , consists of an angular velocity source, Ω(t), acting on a compliant shaft modeled as a torsion spring with spring constant, \(K = 120\,\,{\rm{N}} \cdot {\rm{m/rad}}\) . The spring drives a pinion with \({N_1} = 12\) teeth, engaged in a rack with \({N_2} = 3\) teeth per inch. The rack is attached to mass, \(M = 5\,\,kg\) . The rack and mass both slide on a lubricant film with damping, \(b = 3\,\,N \cdot m \cdot \sec {\rm{/}}rad\) .

Fig. P10.7 Hybrid rotational-translational mechanical system

10.7.a Derive the system equation for:

i The torque acting through the compliant shaft, K .

ii The velocity of mass, M .

iii The force acting to accelerate mass M .

10.7.b Derive the transfer function for the system equation of part a and plot the poles and zeros (if any) of transfer function on the s -plane.

10.7.c Determine the steady-state response of the transfer functions of part b for:

i \(\Omega ( t ) = 10\,\,\frac{{{\rm{rad}}}}{{\rm{s}}}\sin ( {24t} )\) , ii \(\Omega ( t ) = 10\,\,\frac{{{\rm{rad}}}}{{\rm{s}}}\sin ( {48t} )\)

iii \(\Omega ( t ) = 10\,\,\frac{{{\rm{rad}}}}{{\rm{s}}}\sin ( {96t} )\)

10.7.d Formulate the state-space representation of the system with the output variables of part a .

10.7.e Solve the state and output equations of part d for the sinusoidal inputs of part c for the duration,\(t \approx 10\,\tau\) , where. \(\tau= \left| {\frac{1}{{{\sigma _{\min }}}}} \right|\)

Problem 10.8 Plot the log magnitude and phase-angle Bode plots, using MATLAB or Mathcad, for transfer functions:

i \(G(s) = \frac{{50}}{{s(s + 25)}}\)

ii \(G(s) = \frac{{11}}{{s( {s + 3} )( {s + 16} )}}\)

iii \(G( s ) = \frac{{320}}{{( {s + 4} )( {{s^2} + 3s + 55} )}}\)

iv \(G(s) = \frac{{4s(s + 10)}}{{(s + 20)(4{s^2} + 8s + 64)}}\)

v \(G(s) = \frac{1}{{{{(s + 1)}^2}({s^2} + s + 2)}}\)

Problem 10.9 Plot the log-magnitude and phase-angle Bode plots, using MATLAB or Mathcad, for transfer functions:

i \(G(s) = \frac{{28( {8{s^2} + 96s + 1,280} )}}{{s( {{s^2} + 112s + 262} )}}\)

ii \(G(s) = \frac{{239{s^2} + 5497s}}{{0.5{s^3} + 11.8{s^2} + 386s + 1260}}\)

iii \(G(s) = \frac{{8( {60{s^2} + 7800s} )}}{{{s^3} + 320{s^2} + 19,900s + 168,000}}\)

iv \(G(s) = \frac{{(s + 3)}}{{({s^2} + 5s + 4)({s^2} + 4s + 8)}}\)

v \(G(s) = \frac{{8(3{s^2} + 0.6s + 3)}}{{s(2s + 50)({s^2} + 20s + 1200)}}\)

Problem 10.10 For transfer functions below:

Problem 10.11 For transfer functions below:

Problem 10.12 Plot the Nyquist polar plot, using MATLAB or Mathcad, for the transfer functions:

i. \(G(s) = \frac{{50}}{{s(s + 25)}}\)

ii. \(G(s) = \frac{{11}}{{s( {s + 3} )( {s + 16} )}}\)

iii. \(G( s ) = \frac{{320}}{{( {s + 4} )( {{s^2} + 3s + 55} )}}\)

iv. \(G(s) = \frac{{4s(s + 10)}}{{(s + 20)(4{s^2} + 8s + 64)}}\)

v. \(G(s) = \frac{1}{{{{(s + 1)}^2}({s^2} + s + 2)}}\)

Problem 10.13 Plot the Nyquist polar plot, using MATLAB or Mathcad for the transfer functions:

i. \(G(s) = \frac{{28( {8{s^2} + 96s + 1,280} )}}{{s( {{s^2} + 112s + 262} )}}\)

ii. \(G(s) = \frac{{239{s^2} + 5497s}}{{0.5{s^3} + 11.8{s^2} + 386s + 1260}}\)

iii. \(G(s) = \frac{{8( {60{s^2} + 7800s} )}}{{{s^3} + 320{s^2} + 19,900s + 168,000}}\)

iv. \(G(s) = \frac{{(s + 3)}}{{({s^2} + 5s + 4)({s^2} + 4s + 8)}}\)

v. \(G(s) = \frac{{8(3{s^2} + 0.6s + 3)}}{{s(2s + 50)({s^2} + 20s + 1200)}}\)

Fig. P10.14 Bode plots of an unknown system

Fig. P 10.15 Bode plots of an unknown system

Fig. 10.16 Bode plots of an unknown system

Problem 10.17 Estimate the transfer function that produced the Nyquist plot in Fig. P10.17 .

Fig P. 10.17 Nyquist polar plot of an unknown system