Problem 8.1 The rotational system shown schematically in Fig. P8.1 comprises an angular velocity input driving a fluid coupling attached to a compliant shaft modeled as a spring. The compliant shaft drives a flywheel with mass moment of inertia J . Two sets of parameters, Case I and Case II, are tabulated.

Fig. P8.1 A rotational mechanical system

Table P8.1 Parameter values

8.1.a Derive the state equations for this system.

8.1.b 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 .

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

8.1.d 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.

8.1.e The system is de-energized when it is acted on by a step input of 10 rad/s. Write a MATLAB program for the Euler method to solve the state equations, and, using the solution of the state equations, solve the output equations for Cases I and II. Plot the responses of the output variables.

8.1.f Repeat part e using the Runge–Kutta algorithm.

8.1.g Plot the Euler method and Runge–Kutta algorithm results for the angular velocity of flywheel on the same plot. How does the difference between the results of the two methods vary over the solution?

Problem 8. 2 A mass M supported on frictionless roller is shown in Fig. P8.2 a. A dashpot with damping b and a spring with spring constant K are connected between the mass and ground. The mass is acted upon by an applied force F (t ).

The system has a linear and a non-linear configuration. The linear configuration has linear dampers, which are described by the equation F _{ b } = bv _{1g} . The non-linear configuration has non-linear dampers designed so that they exert a damping force when the damper is in compression but not when the damper is in extension, as shown in Fig. P8.2 b.

Fig. P8.2 a Second-order translational mechanical system, b Nonlinear damper, which exerts a damping force only in compression

Table P8.1 Parameter values

8.2.a Derive the state equations for this system.

8.2.b 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 .

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

8.2.d For the linear 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.

8.2.e Write a MATLAB program to solve the linear state equations using the Runge–Kutta algorithm for a step input of 10 N. Assume that the system was de-energized at time t = 0. Include the code to solve the output equations and plot their responses. Run your code using the parameter values for Cases I and II.

8.2.f Save your MATLAB script to a different file name and edit the script to include the non-linear behavior of the dashpot. Run your program for the same input and parameter values as part e .

Problem 8.3 The fluid system shown schematically in Fig. 8.3 a consists of a fluid power unit, modeled as a pressure source, two fluid resistances R _{1} = 100 MPa sec/m^{3} and R _{2} = 200 MPa sec/m^{3} , a long run of piping, modeled as a fluid inertance, I = 40 × 107 kg/m^{4} and a fluid accumulator (capacitance), C = 6 × 10^{−8} m^{3} /Pa. The system will be modeled twice, a linear and a non-linear model. In the linear model, the fluid resistance R _{1} has a constant resistance and allows flow in both directions. In the non-linear model, the fluid resistance R _{1} is a check valve which only allows flow in the direction shown. The fluid resistance in the forward direction is the same as the linear case.

Fig. P8.3a Fluid circuit schematic

and check their units.

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

8.3.d Calculate the system’s eigenvalues configured with the linear fluid resistance R _{1} . 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 was de-energized before the pressure pulse plotted in Fig. P8.3 b was applied to the system.

8.4.e Write a MATLAB program to solve the linear state equations using the Runge–Kutta algorithm for the pressure input shown in Fig. P8.3 b. Assume that the system was de-energized at time t = 0. Include the code to solve the output equations and plot their responses.

Fig. P8.3b Input pressure pulse

8.3.f Save your MATLAB script of part e under a different file name and edit it to solve the non-linear state equations using the Runge–Kutta algorithm for the pressure input shown in Fig. P8.3 b.

8.3.g Compare the pressure across the fluid inertance in the linear and non-linear cases and calculate the maximum change in pressure due to the check valve.

Problem 8.4 The electric circuit shown in the schematic, Fig. 8.4 a, consists of a voltage source, two resistors, \( {R_{\,1}} = 1\,\,\Omega\) and \( {R_{\,2}} = 0.2\,\Omega , \) a capacitor, C = 20 × 10^{−6} F, an inductor, L = 1 × 10^{-6} H, and a diode D .

Diodes conduct current only in the direction that the triangle of their symbol points Fig. 8.4 b. This is a diode’s “forward” direction. Diodes have a threshold voltage, which varies with the material and design of the diode. A typical silicon diode has threshold voltage of 0.63 VDC before it conducts in the forward direction, Fig. 8.4 c. An actual diode’s voltage-current plot is similar to Fig. 8.4 c. The small increase in resistance to the forward conduction with increased current is usually neglected and the diode modeled as having the threshold voltage drop but no additional resistance.

Fig. P8.4a Schematic of a RLC circuit with a diode D

Fig. P8.4b Diode schematic symbol showing forward direction (direction of current flow). c Diode voltage-current plot

The system will be modeled in two configurations, a linear configuration without the diode, and a non-linear configuration with the diode.

8.4.a Derive the state equations for the linear configuration and check their units.

8.4.b Derive output equations for the following variables of the linear system.

i The voltage across capacitor C .

ii The current flowing through inductor L .

iii The voltage across inductor L .

iv The current through resistor R _{ 1 } .

v The current through resistor R _{ 2 } .

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

8.4.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 initially de-energized before the voltage pulse shown in Fig. P8.4 d was applied.

8.4.e Write a MATLAB program to solve the linear state equations using the Runge–Kutta algorithm for the voltage pulse input shown in Fig. P8.4 d. Assume that the system was de-energized at time t = 0. Include the code to solve the output equations and plot their responses.

8.4.f Save your MATLAB script of part e under a different file name and edit the script to solve the state equations for the non-linear configuration, which includes the diode, for the voltage input shown in Fig. P8.4 d.

8.4.g Compare the voltage across the inductance in the linear and non-linear systems and calculate the maximum change in voltage due to the diode.

Problem 8.5 Two rotational mechanical systems are shown schematically in Figs. P8.5 a and P8.5 b. Both systems contain a torque source, a gear with rotational inertia, J _{1} , supported by bearings with damping b_{1} , a shaft with torsional stiffness, K , and a flywheel with rotational inertia, J _{2} , supported by bearings with damping, b _{2} . In System I, the gear is a spur gear, driven by a pinion with negligible inertia. In System II, the gear is a worm gear, driven by a worm with negligible inertia. The gear ratios of Systems I and II are the same: 20 rotations of the pinion, or worm, to one rotation of the gear.

Table P8.5 Parameter values for Systems I and II

Fig. P8.5a System I spur gear drive

Fig. P8.5b System II worm drive

There is one significant difference between the two systems. The spur gears of System I can be driven by either the torque source or the output shaft, K . However, the worm gears of System II can only be driven by the torque source. A worm drive is “self-locking.” It cannot be driven by its output shaft on the worm gear, only the input shaft on the worm.

The angular velocity of the pinion in System I need not have the same sign as the torque applied by the torque source. However, the angular velocity cannot have the opposite sign of the torque from the torque source. The angular velocity of the worm in System II can be of the same sign as the torque applied by the torque source or zero.

The spur gears are counter-rotating. Consequently, the angular velocities of the two shafts have opposite signs. We are free to define the signs for the rotation of the input and output shafts of the worm drive. For convenience, we will define the sense of rotation of the input and output shafts to have opposite signs to correspond to the rotation of the spur gear system.

8.5.a Derive the state equations for Systems I and II.

8.5.b The shaft, K , is carbon steel, solid with a circular cross-section, 2 in. in diameter and 12 ft long. Calculate its torsional spring constant, K , as described in Chap. 4 .

8.5.c The flywheel, J _{2} , shown in the plan and cross-section in Fig. P8.5.c , is carbon steel. Calculate the mass-moment of inertia (rotational inertia), J _{2} , using superposition, as described in Chap. 5 . The spokes are 1 in. wide.

Fig. P8.5c Flywheel J _{2}

8.5.d Draw a flowchart of the logic needed in the state equation solver loop, to prevent “backwards” (positive) rotation of the worm gear, and to lock the worm gear, when the input torque is turned off.

The system is at rest and relaxed for \( t < 0, \) before an input torque pulse with magnitude of \( 10\,\,{\rm{N}} \cdot {\rm{m}} \) and a duration of one second is applied at time, t = 0, as shown in Fig. P8.5 d.

Fig. P8.5d Input torque pulse

8.5.e Write a MATLAB code to solve the state equations of System I with the Runge–Kutta algorithm for the pulse input shown in Fig. P8.3 d. The duration of the simulation should be 2 s. Plot the state variables.

8.5.f Modify the code to solve the state equations of System II, again plotting the state variables.

8.5.g Calculate the maximum stress in the shafts of System I and System II. If the yield strength of the steel is 60 ksi, did either shaft fail?

Problem 8.6 A translational mechanical system contains a spring, two masses, and two dampers acted upon by an applied force is shown in Fig. P8.6 a. The system has linear and non-linear configurations. There are two cases for both configurations. The two cases for the linear configuration of the system use the parameters in Table P8.6 . The two cases of the non-linear configuration use the mass and spring constant values from Table P8.6 , but have non-linear dampers which have half the damping coefficient b in extension as they do in compression, as shown in Fig. P8.6 b. The values of damping coefficient b _{1} and b _{2} in Table P8.6 are the higher compression values for the non-linear dampers.

Table P8.6 Parameter values

Fig. P8.6a Third-order mechanical system

Fig. P8.6b Non-linear damper force, damper b _{2} Case I. The damping coefficient in compression is that in Table P8.6 . The damping coefficient in extension is one half of the tabulated value

8.6.a Derive the state equations for this system.

8.6.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} .

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

8.6.d For the linear configuration, 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 .

8.6.e For Cases I and II of the linear configuration, write a MATLAB program to solve the state equations and the output equations for the force pulse plotted in Fig. P8.6 c. Plot the responses of the output variables.

8.6.f Edit a copy of the MATLAB program of part e to solve state and output equations of the non-linear configuration for Cases I and II. Plot the responses of the output variable to the force pulse of Fig. P8.6 c.

Fig. P8.6c Input force pulse

Problem 8.7 A translational mechanical system contains a mass M sliding on a lubricating fluid film with damping b . The mass is acted upon by an applied force F (t ). Spring K _{1} is attached between the mass and ground. Spring K _{ 2 } is attached to ground but is not attached to the mass. When the system is at rest and relaxed, there is a gap of width d between Spring K _{2} and the mass M . Spring K _{2} and is only in contact with the mass when the position of the mass compresses spring K _{2} .

Fig. P8.7 Second-order translational mechanical system

Two linear and one non-linear configurations of the system will be investigated. The two linear configurations are (1) Spring K _{2} is absent from the system, and (2) Spring K _{2} is attached to mass M . The non-linear configuration is that shown in Fig. P8.7 with gap d between the mass and spring K _{2} when the system is at rest and relaxed. We will investigate two cases. (1) The gap d of the de-energized system equals zero, i.e. the spring is in contact with but not attached to the mass, and (2) the gap equals half the steady-state displacement of the mass under the load of 1,000 N. There are two sets of parameters, Case I and Case II.

Table P8.7 Parameter values

8.7.a Derive the state equations for this system using K _{ equiv } as the spring constant.

8.7.b Derive the output equations for:

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

8.7.d For the linear configuration, calculate the system’s eigenvalues for the parameter values of 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 .

8.7.f Write a MATLAB program to solve the linear state equations using the Runge–Kutta algorithm for a step input force F (t ) = 1,000 N u _{ s } (t ). Assume the system is at rest and relaxed before the input acts on the system. Solve the output equations for the parameter values of Cases I and II and plot their responses.

8.7.g Save your MATLAB script of part f under a different name and edit it to solve the non-linear state equations which include a gap between the mass and spring. Solve the state and output equations for the input and conditions of part f .

Problem 8.8 The schematic of an electric circuit with a voltage source, two inductors, two capacitors, a resistor, and a diode is shown in Fig. P8.8 a.

Fig. P8.8a A non-linear RLC circuit

Table 8.8a Parameter values

Diodes conduct current only in the direction that the triangle of their symbol points Fig. 8.8 b. This is a diode’s “forward” direction. Diodes have a threshold voltage which varies with the material and design of the diode. A typical silicon diode has threshold voltage of 0.63 VDC before it conducts in the forward direction, Fig. 8.8 c. An actual diode’s voltage-current plot is similar to Fig. 8.8 c. The small increase in resistance to the forward conduction with increased current is usually neglected and the diode modeled as having the threshold voltage drop but no additional resistance.

The system will be modeled in two configurations, a linear configuration without the diode, and a non-linear configuration with the diode.

8.8.a Derive the state equations for the linear system and check their units.

8.8.b Derive output equations for the following variables of the linear system.

i The voltage across capacitor C _{1} .

ii The voltage across capacitor C _{2} .

iii The current through inductor L _{1} .

iv The current through inductor L _{2} .

v The voltage across inductor L _{1} .

vi The voltage across inductor L _{2} .

vii The current through resistor R .

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

8.8.d Calculate the linear 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 .

Fig. P8.8b Diode schematic symbol showing forward direction (direction of current flow), c Diode voltage-current plot

The system is initially de-energized before the voltage pulse shown in Fig. P8.8 d was applied.

Fig. P8.8d Input force pulse

8.8.e Write a MATLAB program to solve the linear state equations using the Runge–Kutta algorithm for the voltage pulse input shown in Fig. P8.8 d. Include the code to solve the output equations and plot their responses.

8.8.f Save your MATLAB script of part e under a different file name and edit it to solve the non-linear state equations, which include the diode in the circuit, for the voltage input shown in Fig. P8.8 d.

8.8.g Compare the voltage across the inductances in the linear and non-linear systems and calculate the maximum change in voltage due to the diode.

Problem 8. 9 A cantilevered beam with rectangular cross-section is shown in Fig. P8.9 a. Attached to the beam is a cylinder of mass M . The diameter of the cylinder equals the width of the beam w . The height of the cylinder is twice its diameter.

8.9.a Determine the undamped angular frequency of the beam and mass system if they are carbon steel and L = 50 mm, w = 5 mm, and h = 1 mm.

8.9.b Determine the damping coefficient b needed for the system to have a damping ratio of ζ = 0.04, using the velocity the point on the beam at distance L from the wall as the location of the velocity node. Determine the eigenvalues of the system.

Fig. P8.9a Linear cantilevered beam with a cylindrical mass attached at distance L from the support

8.9c The design is modified adding a support such that there is a gap d = 1 mm below the bottom surface of the beam, when the system is at rest and relaxed, at the distance of x = 30 mm from the support, Fig. P8.9 b. When the beam is in contact with the support during downward deflection, approximate it as a cantilevered beam 20 mm long. The spring constant of the beam is greatly increased. Conversely, the effective mass of the beam is reduced when beam is in contact with the support.

The deflection equation for a point at distance x from the support of a cantilevered beam of length L is

$${{\delta }_{x}}=\frac{F{{x}^{2}}}{6EI}\left( 3L-x \right)$$

The area moment of inertia of a rectangular cross-section about a horizontal line through its centroid is

$$I=\frac{w{{h}^{3}}}{12}$$

Determine the effective mass and stiffness of the beam with the support as a function of the vertical displacement y of node 1, at the distance L from the wall.

Fig. P8.9b Support positioned under the beam at distance x from the wall and a gap d from the bottom of the beam when it is unloaded and at rest

8.9.d Derive the state equations for this system.

8.9.e Derive the output equations for:

i The velocity of node 1.

ii The force acting through the equivalent spring K .

iii The angular velocity difference across spring K .

8.9.f Express the state and output equations in matrix form.

8. 9 g. Write a MATLAB program to solve the linear state equations using the Runge–Kutta algorithm for a step input of F _{0} = 10 N. Assume that the system was de-energized at time t = 0. Solve the output equations and plot the responses of the output variables.

8.9.h Save your MATLAB script with a different file name and edit the script to include the effect of the support positioned under the beam. Run your program for the same input of F _{0 = } 10 N.

Problem 8.10 An electromechanical system is shown in Fig. P8.10a . 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 . 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. A belt runs between flywheels one and two. The belt slips on the 4-in. diameter pulley when the force carried by the belt equals 20 N, limiting the maximum force in the belt to 20 N.

Fig. P8.10a Electromechanical system with a compliant shaft belt slip

Fig. P8.10b The compliant belt’s taut and slack sides switch positions during an oscillation.

Fig. P8.10c Linear graph representing a compliant belt as a translational spring between two rotational to translational transducer interfaces.

Table P8.10 Parameter values

The system model without belt slip is the linear model. The system model with belt slip is the non-linear model.

8.10.a Derive the state equations of the linear model and check their units.

8.10.b Derive output equations for the following variables:

i The current drawn from the voltage source.

ii The torque acting through the compliant shaft K _{1} .

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

iv The angular velocity flywheel J _{2} .

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

8.10.d Calculate the system’s eigenvalues for the linear case with no belt slip. 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 was de-energized before a step input of 48 VDC was applied.

8.10.e Write MATLAB code to solve the state equations and output equations of the linear model with the Runge–Kutta algorithm for the step input of 48 VDC. Plot the responses of the output variables.

8.10.f Save the MATLAB script of part e under a different name and edit the code to solve the state equations and output equations of the non-linear model for the step input of 48 VDC. Plot the responses of the output.