Reminders

1. Use nodal notation for the across variable drop, and indicate the positive direction of the through variable in the direction of the drop in the across variable. Sources are the exception. Sources raise the across variable in the direction of the through variable “flow.”

2. Check units in terms of the power variables of the system, not fundamental units or conventional units.

3. Although system dynamics calculations can be performed using US customary units, the set of US units can lead to significant errors, due to the mishandling of the gravitational constant. The units of mass moment of inertia are a case in point. US customary units commonly used ounce in^{2} , lb in^{2} , lb ft^{2} , and slug ft^{2} . It is easier to perform the calculations in SI Units and then convert to US customary units to present the results, if so required.

4. A dynamic system has only one characteristic equation, independent of our choice of the output variable. Consequently, there is a time constant of the system, only one . All power variables in the system vary at the same rate, although their step responses will differ. Some variables will grow, while others will decay, but they do so in synchronizativon.

5. Remember to convert from kPa and MPa to Pa for SI units.

6. There are 1,000 L in 1 m^{3} .

7. Engineers practicing in the United States must be able to convert between US customary units and SI units without resorting to a reference or the internet. Before you leave school, commit to memory at least one conversion factor for each unit you will work with. The most common conversion needed is length, i.e., \(1\ \text{in}=2.54\ \text{cm}=25.4\ \text{mm}\) , Table 5.4 .

Table 5.4 U.S Customary to SI Unit Conversions

Problem 5.1 A fluid system consisting of two fluid resistances, R = 100 MPa =· sec/m^{3} and R = 200 MPa · sec/m^{3} , and a fluid accumulator (capacitance), C = 6 × 10^{−8} m^{3} /Pa, is acted upon by a pressure source Fig. P5.1 . Resistance R _{2} discharges to atmospheric pressure in the system’s reservoir.

Fig. P5.1 Fluid system schematic

5.1.a Derive the system equations for:

i The volume flow rate from the pressure source.

ii The volume flow rate into the fluid accumulator.

iii The pressure in the fluid accumulator.

and check their units

The capacitor is de-energized at time, \(t={{0}^{-}}\) . A step change in pressure, \(P\left( t \right)=200\,\text{kPa}\,{{u}_{s}}(t)\) is applied at time, \(t=0\) .

5.1.b Solve the system equations of part a .

5.1.c Plot the responses using Mathcad or MATLAB

Problem 5.2 An RL electric circuit where \({{R}_{1}}=10\,\Omega \) , \({{R}_{2}}=20\,\Omega \) , and \(L=3\times {{10}^{-3}}\,\text{H}\) is shown schematically in Fig. P5.2 .

Fig. P5.2 RL circuit schematic

5.2.a Derive the system equations for:

i The current through the inductor.

ii The voltage drop across the inductor.

iii The current through resistor R _{2} .

and check their units.

The circuit is de-energized at time, \(t={{0}^{-}}\) . At \(t=0\) , a step input voltage \(v\left( t \right)=48\,\text{VDC}\,{{u}_{s}}(t)\) to the circuit.

5.2.b Solve the system equations of part a .

5.2.c Plot the responses using Mathcad or MATLAB.

Problem 5.3 An electric circuit with \({{R}_{1}}=1\,\text{k}\!\!\Omega\!\!\text{ }\) , \({{R}_{2}}=2\,\text{k}\!\!\Omega\!\!\text{ }\) , \(C=1\,\text{ }\!\!\mu\!\!\text{ F}\) and is shown schematically in Fig. P5.3 .

Fig. P5.3 RC circuit schematic

5.3.a Derive the system equations for:

i The current through the capacitor.

ii The voltage across the capacitor.

iii The voltage across resistor R _{1} .

and check their units.

The circuit is de-energized at time, \(t={{0}^{-}}\) . A step input of voltage \(v\left( t \right)=15\,\text{VDC}\,{{u}_{s}}\left( t \right)\) is applied to the circuit at time, \(t=0\) .

5.3.b Solve the system equations of part a .

5.3.c Plot the responses using Mathcad or MATLAB.

Problem 5.4 The fluid system shown schematically in Fig. 5.4a consists of a fluid power unit modeled as a pressure source, two contractions modeled as fluid resistances, \({{R}_{1}}=100\,\text{MPa}\cdot \text{sec/}{{\text{m}}^{\text{3}}}\) and \({{R}_{2}}=200\,\text{MPa}\cdot \text{sec/}{{\text{m}}^{\text{3}}}\) , and a long run of piping modeled as a fluid inertance, \(I=40\times {{10}^{6}}\,\text{kg/}{{\text{m}}^{\text{4}}}\) .

Fig. P5.4a Fluid circuit schematic

5.4.a Derive the system equations for:

i The volume flow rate from the pressure source.

ii The pressure difference across the fluid resistance, R _{1} .

iii The volume flow rate through the fluid inertance.

iv The pressure difference across the fluid inertance.

and check their units.

The system was de-energized before the pressure pulse shown in Fig. 5.4b was applied to the system.

Fig. P5.4b Input pressure pulse

5.4.b Solve the system equations of part a .

5.4.c Plot the responses using Mathcad or MATLAB.

Problem 5.5 An electric circuit with two resistors, R_{1} =10Ω and R_{2} =20Ω, and an inductor, L = 3×10^{-3} H , is shown schematically in Fig. P5.5 .

Fig. P5.5 RL circuit schematic

5.5.a Derive the system equations for:

i The current from the voltage source.

ii The voltage across resistor R_{1} .

iii The current through the inductor.

iv The voltage across the inductor.

v The current through resistor R_{2} .

and check their units.

The circuit is de-energized at time, \({{R}_{1}}=10\,\Omega \) ^{-} . At time, t = 0, a step input voltage \({{R}_{2}}=20\,\Omega \) is applied to the circuit.

5.5.b Solve the system equations of part a.

5.5.c Plot the responses using Mathcad or MATLAB.

Problem 5.6 An electrical system consisting of a voltage source, two resistors, \(L=3\times {{10}^{-3}}\,\text{H}\) and \(t={{0}^{-}}\) , and a capacitor,\(v\left( t \right)=24\,\text{VDC}\,{{u}_{s}}(t)\) is shown in the schematic Fig P5.6a .

Fig. P5.6a RC circuit schematic

5.6.a Derive the system equations for:

i The current from the voltage source.

ii The voltage across resistor R_{1} .

iii The current through the capacitor.

iv The voltage drop across the capacitor.

v The current through resistor R_{2} .

and check their units.

The voltage applied to the system is plotted in Fig 5.6b .

5.6.b Solve the system equations of part a .

5.6.c Plot the responses using Mathcad or MATLAB.

Problem 5.7 An electric circuit with \({{R}_{1}}=10\,\Omega \) , \({{R}_{2}}=20\,\Omega \) , \(C=1\times {{10}^{-6}}\,\text{F}\) is shown schematically in Fig. P5.7 .

Fig. P5.7 RC circuit schematic

5.7.a Derive the system equations for:

i The current through the capacitor.

ii The voltage across the capacitor.

iii The voltage across resistor R_{1} .

and check their units.

The circuit is de-energized at time, \({{R}_{1}}=1\,\,\text{k }\!\!\Omega\!\!\text{ }\) . A step input of voltage \({{R}_{2}}=2\,\,\text{k }\!\!\Omega\!\!\text{ }\) is applied to the circuit at time, \(C=1\,\,\text{ }\!\!\mu\!\!\text{ F}\) .

5.7.b Solve the system equations of part a .

5.7.c Plot the responses using Mathcad or MATLAB.

Problem 5.8 An electric circuit with \(t={{0}^{-}}\) , \(v\left( t \right)=15\,\text{VDC}\,{{u}_{s}}\left( t \right)\) , \(t=0\) and is shown schematically in Fig. P5.8 .

Fig. P5.8 RC circuit schematic

5.8.a Derive the system equations for:

i The current through the capacitors.

ii The voltage across the capacitors.

iii The voltage across the resistors.

iv The current through resistor R_{1} .

v The current through resistor R_{2} .

vi The voltage across capacitor C_{1} .

vii The voltage across capacitor C_{2} .

and check their units.

The circuit is de-energized at time, \({{R}_{1}}=1\,\text{k }\!\!\Omega\!\!\text{ }\) . A step input of voltage \({{R}_{2}}=2\,\text{k }\!\!\Omega\!\!\text{ }\) is applied to the circuit at time, \(C=1\,\text{ }\!\!\mu\!\!\text{ F}\) .

5.8.b Solve the system equations of part a .

5.8.c Plot the responses using Mathcad or MATLAB.

Problem 5.9 The electrical system shown in Fig. P5.9a consists of voltage source, a resistor with resistance, \(t={{0}^{-}}\) , and a coil wound around a ferrite core. The coil has both inductance, \(v\left( t \right)=15\,\text{VDC}\,{{u}_{s}}\left( t \right)\) and resistance, \(t=0\) .

Fig. P5.9a Electrical system

5.9.a Derive the system equations for:

i The current from the voltage source.

ii The voltage across resistance R_{1} .

iii The current through the inductor.

iv The voltage across the inductor.

v The current through resistance R_{2} .

and check their units.

The system was de-energized before the voltage pulse shown in Fig. P5.9b was applied to the system.

Fig. P5.9b Input voltage pulse

5.9.b Solve the system equations of part a .

5.9.c Plot the responses using Mathcad or MATLAB.

Problem 5.10 The fluid system shown schematically in Fig. 5.10a consists of a fluid power unit modeled as a pressure source, two contractions modeled as fluid resistances \({{R}_{1}}=100\,\Omega \) and \(L=3\times {{10}^{-3}}\,\text{H}\) , a long run of piping modeled as a fluid inertance, \({{R}_{2}}=20\,\Omega \) and a fluid accumulator (capacitance), \({{R}_{1}}=100\,\text{MPa}\cdot \text{sec/}{{\text{m}}^{\text{3}}}\) .

Fig. P5.10a Fluid circuit schematic

5.10.a Derive the system equations for:

i The volume flow rate from the pressure source.

ii The pressure in the fluid accumulator.

iii The volume flow rate through the fluid inertance.

iv The pressure difference across the fluid inertance.

and check their units.

The system was de-energized before the pressure pulse plotted in Fig. P5.10b was applied to the system.

Fig. P5.10b Input pressure pulse

5.10.b Solve the system equations of part a .

5.10.c Plot the responses using Mathcad or MATLAB.

Problem 5.11 An electrical system consisting of a voltage source a resistor, \({{R}_{2}}=200\,\text{MPa}\cdot \text{sec/}{{\text{m}}^{\text{3}}}\) , a capacitor\(I=40\times {{10}^{6}}\,\text{kg/}{{\text{m}}^{\text{4}}}\) , and an inductor, \(C=6\times {{10}^{-8}}{{\text{m}}^{\text{3}}}\text{/Pa}\) , is shown in the schematic P5.11a .

Fig. P5.11a RLC circuit schematic

5.11.a Derive the system equations for:

i The current through the inductor.

ii The voltage across the inductor.

iii The current through the capacitor.

iv The voltage across the capacitor.

and check their units.

The system was in steady-state under the previously applied step input of 12 VDC when a step change was made at time, \({{R}_{1}}=1\,\Omega \) , increasing the voltage to 36 VDC, as shown in Fig. 5.11b .

5.11.b Solve the system equations of part a .

5.11.c Plot the responses using Mathcad or MATLAB

Problem 5.12 A firefighting system consists of a pump and a 135 ft. long, horizontal, circular cross-sectioned steel pipe, terminated with a nozzle, as shown in Fig. P5.12 . The pump draws water from a reservoir at atmospheric pressure. The pump is modeled as a pressure source in series with the internal resistance of the pump. The pressure of the source is 150 psi. The maximum steady-state flow rate of the pump is 1250 gallons per minute when discharging directly to the atmosphere. The maximum steady-state flow rate of the pump discharging through the pipe and nozzle is 800 gallons per minute.

Fig. P5.12 Firefighting system

5.12.a What is the internal resistance of the pump?

5.12.b What is the internal diameter of the pipe in inches, if the system reaches 70 % of its steady-state flow in 2 sec after the nozzle valve is opened?

Problem 5.13 An electric circuit consisting of a voltage source, two resistors, \(C=1\times {{10}^{-6}}\,\text{F}\) and \(L=3\times {{10}^{-3}}\,\text{H}\) , a capacitor \(t=0\) , and an inductor, \({{R}_{1}}=10\,\Omega \) , is shown as the schematic Fig. P5.13a .

Fig. P5.13a RLC circuit schematic

5.13.a Derive the system equations for:

i The voltage across the capacitor.

ii The current through resistor R_{1} .

iii The current through the inductor.

iv The voltage drop across the inductor.

v The voltage across resistor R_{2} .

and check their units.

The system is initially de-energized before the voltage pulse shown in Fig. P5.13b was applied.

Fig. P5.13b Input voltage

5.13.b Solve the system equations.

5.13.c Plot the responses using Mathcad or MATLAB.

Problem 5.14 The electric circuit shown in Fig. P5.14 consists of a voltage source, a resistor, \({{R}_{2}}=2\,\Omega \) , a capacitor, \(C=2\times {{10}^{-6}}\,\text{F}\) , and coil with 1.4 millihenrys of inductance and 2 Ω of resistance.

5.14.a Derive the system equations for:

i The voltage across the capacitor.

ii The current through resistor R.

iii The current through the inductor.

iv The voltage drop across the inductor.

and check their units.

The system is initially de-energized before a step input of 24 VDC was applied.

5.14.b Solve the system equations.

5.14.c Plot the responses using Mathcad or MATLAB.

Problem 5.15 The electric circuit shown in the schematic, Fig. P5.15a , consists of a voltage source, a resistor, R = 10Ω, two capacitors, C _{1} = C _{2} = 2 Ù 10^{-6} F, and two inductors, L Ù L Ù 0.015 H,

Fig. P5.15a RLC circuit schematic

5.15.a Derive the system equations for:

i The voltage across capacitor C _{2} .

ii The current through resistor R .

iii. The current through inductor L _{1} .

iv The voltage drop across the inductors.

v The voltage across resistor R.

and check their units.

The system is initially de-energized before the voltage pulse shown in Fig. P5.15b was applied.

Fig. P5.15b Input voltage

5.15.b Solve the system equations.

5.15.c Plot the responses using Mathcad or MATLAB.

Problem 5.16 A hydraulic system consists of a pump modeled as a pressure source which draws fluid from a reservoir vented to the atmosphere, two fluid resistances, \(R=10\,\Omega \) and, \(C=2\times {{10}^{-6}}\,\text{F}\) and a fluid accumulator (capacitor), \(L=0.015\,\text{H}\) . The return line is 3/4 in. I.D. and 40 feet long. The density of the hydraulic oil is approximately 875 kg/m^{3} .

5.16.a Derive the system equations for:

i The pressure in the fluid accumulator.

ii The volume flow rate into the fluid capacitor.

iii The volume flow rate through fluid resistance R _{2} .

iv The volume flow rate from the pump.

and check their units.

5.16.b Calculate the fluid inertance of the return line

The system was running in steady-state with the pump pressure of 1,800 psi when, at time, t = 0 the pressure was given a step increase to 3,000 psi.

5.16.c Solve the system equations of part a and plot the responses using Mathcad or MATLAB.

5.16.d The fluid reservoir of a fluid power unit is sized so that the residence time, the average time that fluid remains in the reservoir during operation, is two minutes, in order to degas the hydraulic fluid.

i Determine the capacity of the pump needed to provide 125 % maximum flow calculated in part c .

ii Design a cubical steel box for the reservoir, such that there is 2 in. of air above the fluid. Round up the dimensions to the nearest inch.

Problem 5.17 The hydraulic system schematic shown in Fig. 5.17a consists of a pump modeled, as a pressure source, which draws fluid from a reservoir vented to the atmosphere, three fluid resistances, \({{R}_{2}}=200\,\text{MPa}\cdot \sec /{{\text{m}}^{3}}\) _{,} \(C=0.0040\,{{\text{m}}^{\text{3}}}\text{/MPa}\) , and \(t=0\) , and two fluid accumulators (capacitors), \({{R}_{1}}=100\,\text{MPa}\cdot \text{sec/}{{\text{m}}^{\text{3}}}\) and \({{R}_{2}}=200\,\text{MPa}\cdot \text{sec/}{{\text{m}}^{\text{3}}}\) .

Fig. P5.17a Fluid system schematic

5.17.a Derive the system equations for:

i The pressure in the fluid accumulator, C _{1} .

ii The volume flow rate into the fluid capacitor, C _{1} .

iii The pressure in the fluid accumulator, C _{2} .

iv The volume flow rate into the fluid capacitor, C _{2} .

v The volume flow rate through fluid resistance R _{2} .

vi The volume flow rate from the pump.

and check their units.

The system was de-energized, when it was given the pressure pulse plotted in Fig. P5.17b .

Fig. P5.17b Input pressure pulse

5.17.b Solve the system equations of part a and plot the responses using Mathcad or MATLAB.

Problem 5.18 The fluid system shown in the Fig. P5.18a consists of a pump, modeled as the pressure source, P (t ), two fluid resistances, R _{1} and R _{2} , and fluid capacitance, C . The pump’s reservoir is vented to the atmosphere. The value of the fluid resistance, \({{R}_{3}}=300\,\text{MPa}\cdot \text{sec/}{{\text{m}}^{\text{3}}}\) is known , \({{C}_{1}}=0.04\,{{\text{m}}^{\text{3}}}\text{/MPa}\) . The values of the parameters \({{C}_{2}}=0.05\,{{\text{m}}^{\text{3}}}\text{/MPa}\) and C are unknown and to be determined from a dynamic test. The pump was running in steady-state with a pressure of 500 psi when, at time \({{R}_{1}}\) sec, the pressure was suddenly increased to 1,000 psi. The pump’s pressure and volume flow rate are plotted, Fig. P5.18b , and tabulated, Table P5.18 , in US customary units.

Determine the unknown fluid resistance, R _{2} and capacitance, C . Report your results in SI units.

Fig. P5.18a Fluid system schematic

Fig. P5.18b Pump test data

Table P5.18 Pump Test Data

Problem 5.19 The RLC circuit shown schematically in Fig. P5.19a consists of a capacitor in series with 2 m of 22 AWG solid copper wire wound around a ferromagnetic core. The resistance 22 AWG solid copper wire is 16.8 ohms per 1,000 ft. Note the coil has two energetic properties, resistance and induction. Induction enhanced by the presence of a ferromagnetic material leads to energy loss, due hysteresis of the magnetic field in the ferromagnetic material. Induction enhanced by ferromagnetic material is also non-linear.

Fig. P5.19a RLC circuit schematic

The circuit was charged to 15 VDC. Then the connection between the voltage source and the circuit and the end of the coil was grounded. The response of the voltage across the capacitor was captured on an oscilloscope. A portion of the response is shown in Fig. P5.19b .

Fig. P5.19b The voltage across the capacitor during the latter portion of the discharge of the RLC circuit

5.19.a Use the value of the resistance of the copper wire above to determine the capacitance and inductance of the circuit.

5.19.b Calculate the value of the apparent resistance, R_{coil} , from the observed response of the circuit. Determine the capacitance and inductance of the circuit.

5.19.c How different are the values calculated in parts a and b ? If the difference is significant, what may account for the discrepancy?