Problems 1, 2, and 3 are to be solved “manually ,” meaning, that you may use your calculator to perform arithmetic operations but not linear algebra. After you have solved the problems manually, you are encouraged to check your manual linear algebra computations using your calculator so that you become familiar with its capabilities.

Problem 7.1 Evaluate the following matrix expressions manually

$$ 7.1.\text{a}\quad \left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\\end{matrix} \right]+\left[ \begin{matrix} 5 & 6 \\ 7 & 8 \\\end{matrix} \right]=\quad 7.1.\text{b}\quad 8\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\\end{matrix} \right]-\left[ \begin{matrix} 5 & 6 \\ 7 & 8 \\\end{matrix} \right]= $$

Problem 7.2 Evaluate the following matrix expressions manually

$$ 7.2.\text{a}\quad \left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\\end{matrix} \right]\left[ \begin{matrix} 5 & 6 \\ 7 & 8 \\\end{matrix} \right]=\quad 7.2.\text{b}\quad \left[ \begin{matrix} 5 & 6 \\ 7 & 8 \\\end{matrix} \right]\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\\end{matrix} \right]= $$

$$ 7.2.\text{c}\quad \left[ \begin{matrix} 1 & 2 \\ 3 & 4 \\\end{matrix} \right]\left[ \begin{matrix} 5 \\ 6 \\\end{matrix} \right]=\quad 7.2.\text{d}\quad \left[ \begin{matrix} 3 & 4 \\\end{matrix} \right]\left[ \begin{matrix} 5 \\ 6 \\\end{matrix} \right]= $$

$$ 7.2.\text{e}\quad \left[ \begin{matrix} 5 \\ 6 \\\end{matrix} \right]\left[ \begin{matrix} 3 & 4 \\\end{matrix} \right]= $$

Problem 7.3 Calculate the inverses of the following matrices manually , as the transpose of the cofactors of the matrix divided by the determinant of matrix. No credit will be awarded for inverting the matrices using a graphical algorithm. Check by multiplying the matrix by its inverse manually .

$$ 7.3.\text{a}\quad {{\left[ \begin{matrix} 2 & 3 \\ 4 & 5 \\\end{matrix} \right]}^{-1}}=\quad 7.3.\text{b}\quad {{\left[ \begin{matrix} 2 & 0 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\\end{matrix} \right]}^{\,-1}}= $$

Problem 7.4 A translational mechanical system consisting of a mass M sliding on a lubricating fluid film with damping b . A spring K is attached between the mass and ground. The mass is acted upon by an applied force F (t )(Table 7.6).

Fig. P7.4 Second order translational mechanical system

Table P7.4 Parameter values

7.4.a Derive the second-order system equation for the force acting through the spring and express it in standard form.

7.4.b Derive the state equations for this system.

7.4.c Derive the output equations for:

i The velocity of the mass M.

ii The force acting through spring K .

iii The angular velocity difference across spring K .

7.4.d Express the state and output equations in matrix form.

7.4.e For Cases I and II, calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω _{n} , the damped natural frequency ω _{d} , and the damping ratio ζ .

7.4.f The system is de-energized when it is acted on by a step input of 10 N. Solve the state equations and the output equations using Mathcad or MATLAB for Cases I and II. Plot the responses of the output variables.

Problem 7. 5 A rotational mechanical system consisting of a spring, an inertia, and a damper acted upon by an applied torque is shown in the schematic Fig. P7.5 .

Fig. P7.5 Second order rotational mechanical system

Table P7.5 Parameter values

7.5.a Derive the second-order system equation for the torque in the spring and express it in the standard form.

7.5.b Derive the state equations for this system.

7.5.c Derive the output equations for:

i The angular velocity of the inertia J.

ii The torque acting through the compliant shaft spring K .

iii The angular velocity difference across the compliant shaft spring K .

iv The angular velocity difference across the fluid coupling b.

7.5.d Express the state and output equations in matrix form.

7.5.e For Cases I and II, calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω _{ n } , the damped natural frequency ω _{d} , and the damping ratio ζ

7.5.f The system is de-energized when it is acted on by a step input of 10 rad/sec. Solve the state equations and the output equations using Mathcad or MATLAB for Cases I and II. Plot the responses of the output variables.

Problem 7. 6 A translational mechanical system consisting of a spring, two masses, and two dampers acted upon by an applied force is shown in the schematic Fig. P7.6 .

Fig. P7.6 Third-order mechanical system

Table P7.6 Parameter values

7.6.a Derive the state equations for this system.

7.6.b Derive the output equations

i The velocity of mass M _{1} .

ii The force acting through the spring K .

iii The velocity of mass M _{2} .

iv The force acting through damper b _{1} .

v The force acting through damper b _{2} .

7.6.c Express the state and output equations in matrix form.

7.6.d Calculate the system’s eigenvalues for Cases I and II. If the system is underdamped, determine the ideal, undamped natural frequency ω _{ n } , the damped natural frequency ω _{d} , and the damping ratio ζ.

7.6.e The system was de-energized when a step input of 10 was applied. For Cases I and II, solve the state equations and the output equations using Mathcad or MATLAB. Plot the responses of the output variables.

Problem 7. 7 A fluid system consisting of a fluid power unit modeled as a pressure source, three fluid resistances, a fluid accumulator with capacitance C , a fluid inertance I , and a hydraulic piston/cylinder driving a mass M is shown is shown in the schematic, Fig. P7.7 . Note that the mass and the fluid inertance are dependent energy storage elements.

Fig. P7.7 Hybrid fluid-translational mechanical system

Table P7.7 Parameter values

7.7.a Derive the state equations for this system and check their units

7.7.b Derive output equations for:

i The pressure in the fluid accumulator.

ii The velocity of mass M .

iii The volume flow rate through the fluid inertance.

iv The volume flow rate from the pressure source.

v The force acting to accelerate mass M .

vi The pressure drop across the fluid inertance.

7.7.c Express the state and output equations in matrix form.

7.7.d Calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω _{ n } , the damped natural frequency ω _{d} , and the damping ratio ζ.

7.7.e The system was de-energized when a step input of 100 was applied. Solve the state equations and the output equations using Mathcad or MATLAB. Plot the responses of the output variables.

Problem 7.8 A hybrid fluid-translational rotational system is shown in Fig. P7.8 . The fluid power unit is modeled as a pressure source, which discharges into a hydraulic line with fluid resistance R . The hydraulic piston/cylinder has diameter D . Its piston rod is attached to the linkage. A dashpot with damping b and a spring K are also attached to the linkage. The linkage has rotational inertia J .

Fig. P7.8 Fluid-translational mechanical system

Table P7.8 Parameter values

7.8.a Derive the state equations for this system and check their units.

7.8.b Derive output equations for:

i The volume flow rate from the source.

ii The force applied by the piston rod.

iii The force in spring K .

iv The angular velocity of the linkage.

v The velocity difference across the spring.

7.8.c Express the state and output equations in matrix form.

7.8.d Calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω _{ n } , the damped natural frequency ω _{d} , and the damping ratio ζ.

7.8.e The system was de-energized when a step input of 2,000 was applied. Solve the state equations and the output equations using Mathcad or MATLAB. Plot the responses of the output variables.

Problem 7. 9 A translational mechanical system is shown schematically in Fig. P7.9 . The system consists of a force source F (t ) acting on a lever, to which are attached a damper b _{1} and two springs, K _{1} and K _{2} . Spring K _{2} is attached to mass M , which slides on a rail with a lubricating film, damping b _{2} . The system is de-energized before the input step force of 1,000 N acts on the system.

Fig. P7.9 Translational mechanical system

Table P7.9 Parameter values

7.9.a Derive the state equations for this system and check their units.

7.9.b Derive output equations for:

i The force acting to accelerate mass M .

ii The velocity of mass M .

iii The force acting through spring K _{1} .

iv The force acting through spring K _{2} .

v The velocity of the point of application of the input force F (t ).

7.9.c Express the state and output equations in matrix form.

7.9.d Calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω _{ n } , the damped natural frequency ω _{d} , and the damping ratio ζ.

7.9.e Solve the state equations and the output equations using Mathcad or MATLAB. Plot the responses of the output variables.

Problem 7.10 An electromechanical system is shown in Fig. P7.10 . The input to the system is a voltage which drives a DC motor through wires with electrical resistance R . The relationship between the motor’s current and torque is T _{ M } = αi _{ M } . Flywheel J _{1} is attached to the motor’s shaft and is supported by bearings with damping b _{1} . Coupled to the motor’s shaft is a compliant shaft modeled as torsion spring K . The compliant shaft drives flywheel J _{2,} which is supported by bearings with damping b _{2} .

Fig. P7.10 Hybrid electromechanical system

Table P7.10 Parameter values

7.10.a Derive the state equations for this system and check their units.

7.10.b Derive output equations for:

i The current from the source.

ii The torque acting to accelerate angular velocity of rotational inertia J _{1} .

iii The angular velocity of rotational inertia J _{1} .

iv The angular velocity of rotational inertia J _{2} .

v The torque acting through spring K .

7.10.c Express the state and output equations in matrix form.

7.10.d Calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω _{ n } , the damped natural frequency ω _{d} , and the damping ratio ζ.

7.10.e The system was de-energized before a step input of 48 VDC was applied. Solve the state equations and the output equations using Mathcad or MATLAB. Plot the responses of the output variables.

Problem 7. 11 A rotational system show is shown in Fig. 7.11 . An angular velocity source drives the shaft attached to the pulley with diameter D _{1} . The belt over pulley D _{1} drives the pulley with diameter D _{2,} which is attached to the same shaft as the pulley with diameter D _{3} . That shaft is compliant, with spring constant K , and drives the flywheel with rotational inertia J _{1} and damping b _{1} . The belt over pulley D _{3} drives the pulley with diameter D _{4,} which is attached to a shaft connect to the fluid coupling with damping b _{2} . The output shaft of the fluid coupling drives the flywheel with rotational inertia J _{2} and damping b _{1} .

Fig. P.7.11 Rotational system

Table P7.11a Parameter Values

Table 7.11b Torsion spring stiffness K values

7.11.a Derive the state equations for this system and check their units.

7.11.b Derive output equations for:

i The torque acting through spring K _{.}

ii The angular velocity of flywheel J _{1} .

iii The torque applied by the angular velocity source.

iv The torque acting to accelerate flywheel J _{2} .

v The angular velocity of flywheel J _{2} .

7.11.c Express the state and output equations in matrix form.

7.11.d Calculate the system’s eigenvalues for Cases I, II, and III using the damping coefficient values of Table 7.11b . If the system is underdamped, determine the ideal, undamped natural frequency ω_{ n } , the damped natural frequency ω_{d} , and the damping ratio ζ.

7.11.e The system was de-energized before a step input of 1,500 RPM was applied. Solve the state equations using Mathcad or MATLAB.

7.11.f Solve and plot the responses of the output variables using Mathcad or MATLAB.

Problem 7. 12 A rotational mechanical system is shown schematically in Fig. P7.12 . The system consists of an angular velocity source Ω (t ) acting on the input shaft of a fluid coupling with damping b _{1} . The output shaft of the fluid coupling drives inertia J _{1,} which drives a compliant shaft with torsional spring constant K . The compliant shaft drives a pinion with N _{1} teeth. The pinion engages a gear with N _{2} teeth. The gear shaft drives rotational inertia J _{2} . The gear shaft bearings have damping b _{2} .

Fig. P7.12a A shaft-driven rotational system with a compound gear

Table 7.12a Parameter values

Table 7.12b Fluid coupling damping b _{1} values

Table 7.12c Gear Teeth Numbers

7.12.a Derive the state equations and check their units.

7.12.b Derive output equations for the following variables:

i The angular velocity of the pinion.

ii The torque acting through the compliant shaft.

iii The torque acting to accelerate inertia J _{2} .

iv The torque acting through the fluid coupling b _{1} .

7.12. c Write the state equations and output equations in vector-matrix form.

7.12.d Calculate the system’s eigenvalues for Cases I, II, and III. If the system is underdamped, determine the ideal, undamped natural frequency ω_{ n } , the damped natural frequency ω_{d} , and the damping ratio ζ.

The system is de-energized before the input pulse of angular velocity shown in Fig. 7.12b is applied.

Fig. P7.12b Angular velocity input

Problem 7. 13 A rotational-translational mechanical system is shown schematically in Fig. P7.13 . The system consists of a torque source acting on the input shaft of a pinion with N _{1} teeth. The pinion engages a gear with N _{2} teeth and rotational inertia J . The gear drives a compliant shaft modeled as a torsional spring with spring constant K . The shaft turns a power screw with a linear pitch of n threads per inch. The power screw threads through the crosshead. The crosshead translates when the power screw rotates. The viscous friction between the rotating power screw and the translating crosshead is modeled as rotational damping b _{1} . The crosshead has mass M and slides on two rails. The combined viscous friction of the crosshead sliding on the two rails is modeled as translational damping b _{2} .

Fig. P7.13 A hybrid rotational-translational system

Table 7.13 Parameter values

7.13.a Derive the state equations and check their units.

7.13.b Derive output equations for the following variables:

i The angular velocity of the pinion shaft.

ii The angular velocity gear N _{2} .

iii The torque in the compliant shaft.

iv The force acting to accelerate the crosshead.

v The translational velocity of the crosshead.

7.13. c Write the state equations and output equations in vector-matrix form.

7.13.d Calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω_{ n } , the damped natural frequency ω_{d} , and the damping ratio ζ.

The system is de-energized before the input torque pulse shown in Fig. P7.13b was applied to the system.

7.13.e Solve the state equations using Mathcad or MATLAB for the input torque pulse shown in Fig. P7.13b with the parameter values of Table 7.

7.13.f Solve and plot the responses of the output variables using Mathcad or MATLAB.

Problem 7. 14 A fluid-rotational mechanical system is shown in Fig. P7.14 . The fluid power unit is modeled as pressure source p (t ), drawing fluid from a reservoir vented to the atmosphere. The pump’s high-pressure line has fluid resistance R and inertance I . The rotational hydraulic motor produces 600 ft.-lbs. of torque at a hydraulic fluid pressure of 3,000 psi. The fluid discharging from the motor returns to the reservoir. The output shaft of the motor turns a compliant shaft with torsion spring constant K . The compliant shaft drives a flywheel with mass moment of inertia J supported on bearings with damping b .

Table 7.14 Parameter values

Fig. P7.14 Hybrid fluid-rotational mechanical system

7.14.c Write the state equations and output equations in vector-matrix form.

7.14.d Calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω_{ n } , the damped natural frequency ω_{d} , and the damping ratio ζ. If the real components of the eigenvalues vary by two or more orders of magnitude, then the duration of the components of the response vary by like magnitude. If so, solve the state equations twice, once using a short duration to see the fast response.

Fig. P7.14b Pressure input

The system is running in steady state at the input pressure of 1,500 psi before the input torque pulse shown in Fig. P7.13b was applied to the system at time t = 0.

7.14.e Solve the state equations using Mathcad or MATLAB for the input pressure pulse shown in Fig. P7.13b with the parameter values of Table 7.14 .

7.14.f Plot the responses of the output variables using Mathcad or MATLAB.

Problem 7. 15 A rotational system is shown in Fig. 7.15 . An angular velocity source drives the input shaft to a fluid coupling with damping b _{1} . The output shaft of the fluid coupling drives shaft is compliant, with spring constant K _{1} , which drives a flywheel J _{1} with rotational inertia J _{1} . A compliant shaft with spring constant K _{2} , from flywheel J _{1} drives flywheel J _{2} with rotational inertia J _{2} . A drag cup with damping b _{2} is connected between the hub of flywheel J _{2} and the machine frame, which is ground.

Fig. P7.15 A rotational mechanical system

Table 7.15a Parameter values

Table 7.15b Fluid coupling damping b _{1} values

7.15.a Derive the state equations and check their units.

7.15.b Derive output equations for the following variables:

i The angular velocity of the pinion.

ii The torque acting through the compliant shaft.

iii The torque acting to accelerate inertia J _{2} .

iv The torque acting through the fluid coupling b _{1} .

7.15. c Write the state equations and output equations in vector-matrix form.

7.15.d Calculate the system’s eigenvalues for Cases I, II, and III. If the system is underdamped, determine the ideal, undamped natural frequency ω_{ n } , the damped natural frequency ω
_{d} , and the damping ratio ζ.

7.15.e The system was de-energized before a step input of 1,500 RPM was applied. Solve the state equations and plot their responses using Mathcad or MATLAB for cases I, II, and III.

7.15.f Solve and plot the responses of the output variables using Mathcad or MATLAB for cases I, II, and III.

7.15.g Calculate and plot the power dissipated in the fluid coupling b _{1} and the drag cup b _{2} for cases I, II, and III.

Problem 7.16 The schematic of an electric circuit with a voltage source, two inductors, two capacitors and a resistor is shown in Fig. P7.16 .

Fig. P7.16 An RLC circuit

Table 7.16a Parameter values

Table 7.16b Resistor R values

7.16.a Derive the state equations and check their units.

7.16.b Derive output equations for the following variables:

i The current flowing through capacitor C _{1} .

ii The current flowing through capacitor C _{2} .

iii The voltage across inductor L _{1} .

iv The current through resistor R .

7.16. c Write the state equations and output equations in vector-matrix form.

7.16.d Calculate the system’s eigenvalues for Cases I, II, and III. If the system is underdamped, determine the ideal, undamped natural frequency ω
_{ n } , the damped natural frequency ω
_{d} , and the damping ratio ζ.

7.16.e The system was de-energized before a step input of 24 VDC was applied. Solve the state equations and plot their responses using Mathcad or MATLAB for cases I, II, and III.

7.16.f Solve and plot the responses of the output variables using Mathcad or MATLAB for cases I, II, and III.

7.16.g Calculate and plot the power dissipated in resistor R for cases I, II, and III.

Problem 7.17 The schematic of an electric circuit with a voltage source, two inductors, two capacitors and a resistor is shown in Fig. P7.17 .

Fig. P7.17 An RLC circuit

Table 7.17a Parameter values

Table 7.17b Resistor R values

7.17.a Derive the state equations and check their units.

7.17.b Derive output equations for the following variables:

i The current flowing through capacitor C _{1} .

ii The current flowing through capacitor C _{2} .

iii The voltage across inductor L _{1} .

iv The current through resistor R .

7.17. c Write the state equations and output equations in vector-matrix form.

7.17.d Calculate the system’s eigenvalues for Cases I, II, and III. If the system is underdamped, determine the ideal, undamped natural frequency ω
_{ n } , the damped natural frequency ω
_{d} , and the damping ratio ζ.

7.17.e The system was de-energized before a step input of 24 VDC was applied. Solve the state equations and plot their responses using Mathcad or MATLAB for cases I, II, and III.

7.17.f Solve and plot the responses of the output variables using Mathcad or MATLAB for cases I, II, and III.

7.17.g Calculate and plot the power dissipated in resistor R for cases I, II, and III.

Problem 7.18 An electromechanical system is shown in Fig. P7.18a . A voltage source drives a DC motor, which has resistance R and torque constant K _{ T } . The DC motor turns a compliant shaft, modeled as a torsion spring with spring constant K _{1} .

The shaft turns flywheel_{1} with mass-moment of inertia J _{1} . Flywheel_{1} is supported on bearings with damping b _{1} . Flywheel_{2} has mass-moment of inertia J _{2} bearings with damping b _{2} , and is belt driven. The belt which runs between flywheels one and two is long enough that the energy stored in the taut or stretched side of the belt, Fig. P7.18b , must be included in the dynamic system. The compliant belt is represented by a translational spring between two transducers, which interface between the torque on the pulley’s shafts and the force carried by the belt, Fig. P7.18c . The sign reversal in the translational spring represents the taut and slack sides of the belt’s switching positions.

Fig. P7.18a Electromechanical system with a compliant shaft and a compliant belt

Fig. P7.18b The compliant belt’s taut and slack sides switch positions during an oscillation

Fig. P7.18c Linear graph representing a compliant belt as a translational spring between two rotational to translational transducer interfaces

Table 7.18 Parameter values

7.18.a Derive the state equations and check their units.

7.18.b Derive output equations for the following variables:

i The current drawn from the voltage source.

ii The back-EMF of the DC motor.

iii The torque acting to accelerate flywheel J _{1} .

iv The torque acting to accelerate flywheel J _{2} .

7.18. c Write the state equations and output equations in vector-matrix form.

7.18.d Calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω
_{ n } , the damped natural frequency ω
_{d} , and the damping ratio ζ.

7.18.e The system was de-energized before a step input of 48 VDC was applied. Solve the state equations and plot their responses using Mathcad or MATLAB. 7.18.f Solve and plot the responses of the output variables using Mathcad or MATLAB.

Problem 7.19 A hybrid rotational-translational mechanical system is shown schematically in Fig. P7.16 . The system consists of a force source F (t ) acting on a lever, to which are attached a damper b _{1} and two springs, K _{1} and K _{2} . The lever has mass moment of inertia J . Spring K _{2} is attached to mass M , which slides on a rail with a lubricating film, damping b _{2} . The system is de-energized before the input step force of 1,000 N acts on the system.

Fig. P7.19 Hybrid rotational-translational mechanical system

Table P7.19 Parameter Values

7.19.a Derive the state equations for this system and check their units.

7.19.b Derive output equations for

i The force acting to accelerate mass M .

ii The velocity of mass M .

iii The force acting through spring K _{1} .

iv The force acting through spring K _{2} .

v The angular velocity of the lever.

vi The velocity of the point of application of the input force F (t ).

7.19.c Express the state and output equations in matrix form.

7.19.d Calculate the system’s eigenvalues. If the system is underdamped, determine the ideal, undamped natural frequency ω
_{ n } , the damped natural frequency ω
_{d} , and the damping ratio ζ.

7.19.e Solve the state equations and the output equations using Mathcad or MATLAB.

7.19.f Plot the responses of the output variables using Mathcad or MATLAB.

Problem 7.20 The stainless steel and aluminum thermal system of Sect. 5.8.4 is reproduced as Fig. 5.52 . The stainless steel is 0.5 in. thick. The aluminum is 2 in. thick. Both metals are divided into layers 0.25 in. thick, parallel to the external surfaces. The thermal system model consists of thermal resistances in series and thermal capacitances in parallel. The linear graph is Fig. 5.53 . The thermal resistances of the stainless steel and aluminum are calculated for 0.25 in. layer. The thermal capacitances are calculated pairing adjacent layers and using the temperature node between them as the temperature of the 0.5 in. thick capacitance.

Fig. 5.52 Thermal system divides into five regions, each with a thermal capacitance between the center temperature node of the region and ground temperature

Fig. 53 Linear graph of the stainless steel and aluminum thermal system. The linear graph is split between nodes 6 and 7 for this figure

The thermal resistances of the 0.25 in. thick stainless steel and aluminum layers are:

\({R_{SS}} = 8.23 \times {10^{ - 5}}\frac{{{}^{\rm{o}}{\rm{K}}}}{{\rm{W}}}\) and \({R_{Al}} = 6.91 \times {10^{ - 6}}\frac{{{}^{\rm{o}}{\rm{K}}}}{{\rm{W}}}\)

The thermal capacitances of the stainless steel and aluminum are:

\( {C_{{p_{SS}}}} = 204\,\,\frac{{{\rm{kJ}}}}{{^{\rm{o}}{\rm{K}}}} \) and \( {C_{{p_{Al}}}} = 137\,\,\frac{{{\rm{kJ}}}}{{^{\rm{o}}{\rm{K}}}} \)

\( {R_{S{S_1}}} \) , \( {R_{a{l_1}}} \) , \( {R_{a{l_3}}} \) , \( {R_{a{l_5}}} \) , \( {R_{a{l_{\,7}}}} \) , \( {R_{a{l_9}}} \) , and \( {R_{a{l_{10}}}} \) .

The thermal system is at the uniform temperature of 20 °C when the temperature of node 1 is given a step increase to 100 °C. The temperature of node 11 is held at 20 °C.