Power Transmission, Transformation, and Conversion

Abstract

A “simple machine,” such as a lever, transforms the magnitudes of the forces and velocities at its points of action but does not change the mechanical power, product of force and velocity. Modern machinery both transforms power and converts power from one type to another. Linear graphs of energetic models of machinery include interfaces which both couple and separate the subsystems. The interfaces consist of two branches, one for each subsystem. There is no direct action or flow of the through variable over the interface into the other subsystem. Hybrid systems are systems which contain subsystems of different types. Electric and hydraulic motors, generators, and pumps are hybrid systems. The linear graph method couples similar and dissimilar subsystems using interfaces, which differ only in the equations that describe their effect.

Appendices

Summary

Transformers and transducers interface subsystems with a larger dynamic system. Transformers interface like subsystems, and transducers interface dissimilar subsystems. Power flows across the interface between the subsystems, but there is no direct connection between the through variables in the two branches of the transducer or transformer interface. The power flowing down one branch of the interface, equal to the product of the through variable and the across variable difference of that branch, is equal to the product of the through and across variables in the other branch.

The sign convention is based on (a) assuming that the power flows on both sides of the interface is positive and (b) orienting the through variables in the two branches of the interface towards ground. Because the positive direction of power flow is into the transformer or transducer interface, and the interface cannot store power, at any instant, one of the two power flows must be negative. Consequently, one of the two equations, relating a power variable in a branch on one side of the interface to a power variable on the other side, will have a negative sign. Although a negative sign can be physically possible, as in the case of a pair of gears, often, the negative sign is nonsense. The negative sign is a result of the linear graph sign convention. When there is a non-sensical negative sign, it should be tolerated, until the derivation of the system equation is completed. Then fix the sign, if necessary, before using the results. Correcting the sign prior to completion of the reduction leads to further errors. Sign conflicts between subsystems are common, because the subdisciplines of mechanical engineering have different histories.

There are three equations associated with a transformer or transducer. The first equation is the summation of power flow into the interface, Fig. 6.162a, which is the sum of the products of the power variables in the two interface branches. The second and third equations each relate a single power variable on one side of the interface with a single power variable on the other side. The latter two equations are useful in the derivation of a system equation, since there is a one-for-one power variable substitution. The power summation equation is valuable but inefficient in a reduction, because the power variable substitution is one-for-three.

Fig. 6.162

Linear graph transformer and transducer sign conventions for positive power flow

It is often possible to derive a relationship between the displacements or velocities on the two sides of the interface, using the geometry of the mechanism of the volume displacement of a pump or hydraulic motor. In other cases, such as with electric motors, generators, or turbines, experimental data is needed to establish a transducer equation. Once a relationship is obtained, then substitution into the power summation yields the second one-to-one equation, for example,

$$ {{\mathit\Omega }_{1g}}=-\frac{{{N}_{2}}}{{{N}_{1}}}{{\mathit\Omega }_{2g}}\quad \to\quad {{\mathsf{\mathit{P}}}_{A}}+{{\mathsf{\mathit{P}}}_{B}}=0\quad \to\quad {{\mathit\Omega }_{1g}}{{T}_{A}}+{{\mathit\Omega }_{2g}}{{T}_{B}}=0 $$

When drawing linear graphs of systems with transformers and transducers:

• 1. Find nodes of distinct values of the across variables for each subsystem, and identify those nodes on the schematic.

• 2. Draw and identify the nodes of the linear graph.

• 3. Draw the two interface branches for each transducer or transformer, and name the through variable of each interface branch.

• 4. Add the remaining elements of the system between their respective nodes.

• 5. There is no branch across a transformer or transducer interface! There is no through variable flow across the interface. Power flows across the interface.

A transformer constant is dimensionless. A transducer constant has two sets of units. Generally, both sets must be used to check the units of a system equation with a transducer.

Problems

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 $b = 6\,{\rm{N}} \cdot {\mathop{\rm s}\nolimits} /{\rm{m}}$.

• 6.10.a Derive the system equations for the following:

  • i. The velocity of the rack.

  • ii. The force acting to accelerate the rack.

  • iii. The force acting to shear the lubricating fluid film.

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 $b = 0.1\,{\rm{N}} \cdot {\rm{m}} \cdot {\mathop{\rm s}\nolimits} /{\rm{rad}}$.

• 6.11.a Derive the system equations for

  • i. The current through resistance R.

  • ii The voltage drop across resistance R.

  • iii. The back EMF.

  • iv. The motor’s torque TM.

  • v. The torque acting through damping b.

  • vi. The torque acting to accelerate mass moment of inertia JM + JL.

  • vii. The angular velocity of mass moment of inertia JL.

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, $b = 4\,{\rm{N}} \cdot {\rm{s/m}}$.

• 6.12.a Derive the system equations for the following:

  • i. The force acting through the spring.

  • ii. The force acting through the damper.

  • iii. The velocity of the point of application of the input force.

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.

• 6.12.b Solve the system equations.

• 6.12.c Plot the responses from time $t = 0$ to $t = 2{\mathop{\rm s}\nolimits} \; + 6\tau$ using Mathcad or MATLAB.

Fig. P6.12

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

Problem 6.13 The rotational mechanical system shown in Fig. P6.13a 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 $J = 25\,{\rm{kg}} \cdot {{\rm{m}}^2}$.

• 6.13.a Derive the system equations for the following:

  • i. The torque from the velocity source.

  • ii. The torque acting to accelerate mass moment of inertia J.

  • iii. The angular velocity of mass moment of inertia J.

  • iv. The torque acting through damper b2.

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.

• 6.13.b Solve the system equations.

• 6.13.c Plot the responses from $t = 0$ to $t = 1 + 6\tau$ sec using Mathcad or MATLAB.

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 ${b_2} = 10\,{\rm{N}} \cdot {\rm{s/m}}$.

• 6.14.a Derive the system equations for the following:

  • i. The force acting to accelerate mass M.

  • ii. The force acting through the damper b1.

  • iii. The force acting through the damper b2.

  • iv. The velocity of mass M.

  • v. The velocity of the point of application of the input force.

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.

• 6.14.b Solve the system equations.

• 6.14.c Plot the responses from time $t = 0$ to $t = 10 + 6\tau$ using Mathcad or MATLAB.

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 ${b_2} = 60\,{\rm{N}} \cdot {\rm{s/m}}$

• 6.15.a Derive the system equations for the following:

  • i. The torque acting to accelerate the motor’s mass moment of inertia J.

  • ii. The angular velocity of the lead screw.

  • iii. The force acting to accelerate the load carriage mass M.

  • iv. The velocity of the load carriage mass M.

Check their units.

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

• 6.15.b Solve the system equations.

• 6.15.c Plot the responses from $t = 0$ to $t = 3 + 6\tau$ using Mathcad or MATLAB.

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 $b = 2\,{\rm{N}} \cdot {\mathop{\rm s}\nolimits} /{\rm{m}}$.

• 6.16.a Derive the system equations for the following:

  • i. The velocity of the rack.

  • ii. The force acting to accelerate the rack.

  • iii. The force acting to shear the lubricating fluid film.

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 $M = 2\,{\rm{kg}}$.

• 6.17.a Derive the system equations for the following:

  • i. The volume flow rate from the pump.

  • ii. The force acting to accelerate the piston.

  • iii. The velocity of the piston.

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.

• 6.18.a Derive the system equations for the following:

  • i. The force acting through spring K ₁.

  • ii. The force acting through spring K ₂.

  • iii. The force acting through the damper b ₁ .

  • iv. The force acting through the damper b ₂.

  • v. The velocity of the point of application of the input force.

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.

• 6.18.b Solve the system equations.

• 6.18.c Plot the responses from time $t = 0$ to $t = 2 + 6\tau$ using Mathcad or MATLAB.

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.

• 6.19.a Derive the system equations for the following:

  • i. The torque acting through spring K.

  • ii. The angular velocity of inertia J.

  • iii. The velocity difference across spring K.

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.

• 6.19.b Solve the system equations.

• 6.19.c Plot the responses from using Mathcad or MATLAB.

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 ${b_2} = 0.1\,{\rm{N}} \cdot {\rm{m}} \cdot {\mathop{\rm s}\nolimits} /{\rm{rad}}$.

• 6.20.a Derive the system equations for the following:

  • i. The current through resistance R.

  • ii. The back EMF.

  • iii. The motor’s torque T _M .

  • iv. The torque acting through the shaft.

  • v. The angular velocity of mass moment of inertia J.

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