Reminders

1. Write energetic equations with proper notation. The problem statements do not explicitly state that the problems require (a) the energetic equations and (b) proper notation, because those are part of the linear graph method and implied by drawing a linear graph. The energetic equations consist of compatibility, continuity, element, and energy equations. Proper notation refers to using (a) node subscripts to identify the positive direction of the drop in the across variable of an element, and (b) the element parameter as the subscript, which identifies the through variable of an element.

2. Clear fractions and create common denominators as you work the reduction . Improper fractions must be cleared to present the result in standard form. It is generally easier to place a sum of ratios over a common denominator when it is created , rather than carrying improper ratios forward and clearing them at the end of the reduction. One advantage of placing sums of ratios over a common denominator is that the resulting ratio can be cleared from a product by multiplying its fractional inverse.

3. Standard form. (a) Time constant form applies only to the first-order system equations. The terms are ordered, with the derivative first, followed by the zero^{-} order term. (b) Standard form for higher order differential equations requires the coefficient of the highest order derivative of the output variable to be cleared. This is the same standard form used for polynomials. (c) The system equation cannot have improper ratios. Improper ratios are not standard form, because they promote error.

4. Unit checks. Check units of the system equation before solving it, by expressing element parameters in terms of the system’s power variables and time. Do not check units in terms of fundamental units. Although fundamental units are straightforward in a system of single energy type, they become very cumbersome in “hybrid” systems of more than one energy type, for example, electric motors. For example, how would you combine mechanical units of torque with electrical units of voltage?

5. Convert units to SI for calculations . Convert result to US customary units, if required. Remember that metric units may not be SI, for example, centimeter. The SI unit of length is meter. Likewise, remove scaling prefixes and express values in the base unit, for example, \( \text{kN}=1,000\,\,\text{N} \) .

6. Causality. If the system is de-energized for time, \( t<0 \) , then the response of the system for time \( t<0 \) is zero. However, evaluating response functions for negative time will yield a non-zero result, if the response function is not multiplied by a Heaviside step function time Shifted to when the corresponding input is applied to the system. Do not plot negative time unless you use a Heaviside unit step function to zero-out each response, until its corresponding input is applied to the system, and the response has the correct time Shift.

7. Inspect the plots of your results. Look at your results, and check that they are reasonable and complete. Is the response a reasonable shape? Does the plot show a response prior to the application of the input? Is the vertical extent of the trace shown, or must the axis limits be edited? Are the limits of the vertical axis too far from the extent of the response, thereby squashing the trace? Does the plot show the beginning of steady-state but not excessively so?

8. Title plots and label axes . Unidentified plots and unlabeled axes are unacceptable.

9. Mathcad plots . When an x –y plot is inserted in a Mathcad worksheet, Mathcad automatically sets the ranges of both axes. Mathcad always sets the limits of the abscissa (the independent variable) from - 10 to + 10. Click on a limit to edit it. Set the lower limit of time to zero, and set the upper limit, such that the response has reached steady-state. Show the entire transient period of the response but not too much of steady-state. Edit the limits of the ordinate to maximize the proportion of the plot occupied by the trace. Right-click on the plot to bring up the “Format” menu. Add gridlines to the two axes. Select “Traces” in the Format menu to change the thickness, color, and type of line.

10. MATLAB plots . Show the entire transient period of the response but not too much of steady-state. Edit the limits of the ordinate to maximize the proportion of the plot occupied by the trace. Add gridlines and change the trace thickness.

Problem 3.1 For the pulses shown in Fig. P3.1 :

3.1.a Sketch the Time Shifted and scaled Heaviside unit step functions which superpose to form the pulse.

3.1.b Express the pulse as a function consisting of Time Shifted and scaled Heaviside unit step functions.

3.1.c Using Mathcad or MATLAB.

Problem 3.2 A translational mechanical system consisting of a force source, a mass, \( M=5\,\,\text{kg} \) supported on ideal frictionless rollers, and damper, \( b=3\,\,\text{N}\cdot \text{s}/\text{m} \) , is shown schematically in Fig. P3.2a and b .

3.2.a Use the information presented in the schematics and linear graph shown in Fig. P3.2 to derive a complete set of energetic equations for the system.

3.2.b Derive the system equation that relates the force input to these variables:

i The velocity of the mass.

ii The force acting to accelerate the mass.

iii The force acting through the damper.

Check the units of the system equation in terms of power variables and time.

3.2.c The system is at rest at time, t < 0. Determine the unit step responses of the system equations derived in Part b . Use superposition to determine the responses to the input force pulse, F (t ), plotted in Fig. P3.2d .

3.2.d Plot the responses, using Mathcad or MATLAB.

Fig. P3.2 a System with a translational force source, mass, and damper system. b Schematic annotated with nodes of distinct values of velocity. c Linear graph of the system. d Force input pulse which acts on the system

Problem 3.3 A translational mechanical system consisting of a velocity source, a spring with spring constant, \( K=4\,\,\text{N}/\text{m} \) , and a dashpot (damper) with damping coefficient, \( b=2\,\,\text{N}\cdot \text{s}/\text{m} \) , is shown in the schematics, as given in Fig. P3.3a and b.

3.3.a Use the information presented in the schematics and linear graph shown in Fig. P3.3 to derive a complete set of energetic equations for the system.

3.3.b Derive the system equation that relates the applied velocity input to

Check the units of the system equations, in terms of power variables and time.

3.3.c The system is relaxed at time, t < 0. Determine the unit step responses of the system equations derived in Part b. Use superposition to determine the responses to the applied velocity pulse, v (t ), plotted in Fig. P3.3d .

3.3.d Plot the responses, using Mathcad or MATLAB.

Fig. P3.3 a System with a translational spring, damper system acted on by a velocity source. b Schematic annotated with nodes of distinct velocity. c Linear graph of system. d Velocity input pulse which acts on the system

Problem 3.4 A translational mechanical system consisting of a velocity source, a spring with spring constant, \( K=2\,\,\text{N}/\text{m} \) , and a dashpot (damper) with damping coefficient, \( b=4\,\,\text{N}\cdot \text{s}/\text{m} \) , is shown in the schematics, as given in Fig. P3.4a and b. The velocity, \( 10\,\text{cm/s} \) ec, was applied at an unknown time \( t<0 \) . The system reached steady-state by time \( t={{0}^{-}} \) . At time \( t=0 \) , the velocity applied to the system is reversed, v (t ) = -10 cm/sec, for t > 0, as shown in Fig. P3.3d .

3.4.a Use the information, as presented in the schematics and linear graph shown in Fig. P3.4 , to derive a complete set of energetic equations for the system.

3.4.b Derive the system equation that relates the applied velocity input to

i The force in the spring.

ii The velocity drop across the spring.

Check the units of the system equations, in terms of power variables and time.

3.4.c Use the initial condition method to determine the response of the system’s power variables from Part b to the velocity applied at time \( t=0 \) , shown in Fig. P3.4d .

3.4.d Plot the responses using Mathcad or MATLAB.

Fig. P3.4 a System with a translational spring, damper system acted on by a velocity source. b Schematic annotated with nodes of distinct velocity. c Linear graph of system. d Velocity input applied to the system

Problem 3.5 A translational mechanical system which consists of a velocity source, a mass, and a damper is shown in Fig. P3.5 . Its mass is \( M=5\,\,\text{kg} \) , supported on ideal, frictionless rollers. The damping constant is \( b=7\,\,\text{N}\cdot \text{s}/\text{m} \) . The system was at rest before the input shown in Fig. P3.5d was applied.

Check the units of the system equations in terms of power variables and time.

3.5.b The input velocity, v (t ), plotted in Fig. P3.5d is applied to the damper at time \( t=0 \) . Determine the unit step response of the system equations, derived in part a. Use superposition to solve the system equations for the velocity input, as shown in Fig. P3.5d .

3.5.c Plot the responses using Mathcad or MATLAB.

Fig. P3.5 a System with a velocity source, damper, and mass. b Schematic annotated with nodes of distinct values of velocity. c Linear graph of the system. d Velocity input applied to the system

Fig. P3.6 a Translational mechanical system. b Schematic annotated with nodes of distinct values of velocity. c Linear graph of the system. d Force input applied to the system

Fig. P3.7 a System with a force source, mass, spring, and damper. b Schematic annotated with nodes of distinct values of velocity. c Linear graph of the system. d Force input applied to the system

Problem 3.6 A translational mechanical system consisting of a force source, a spring with \( K=100\,\,\text{N}/\text{m} \) , a damper with \( b=300\,\,\text{N}\cdot \text{s}/\text{m} \) , and a massless, rigid bar constrained to translation is shown in Fig. P3.6a . The system has reached steady-state under the application of an input of − 40 N, applied at an unknown time, \( t<0 \) . At time \( t=0 \) , the input force is increased to + 60 N, as shown in Fig. P3.6d .

Check the units of the system equations in terms of power variables and time.

3.6.b Use the method of undetermined coefficients to solve the system equations of part a , for the input shown in Fig. P3.6d .

3.6.c Plot the response using Mathcad or MATLAB.

Fig. P3.8 a System with a force source, mass, spring, and damper. b Schematic annotated with nodes of distinct values of velocity. c Linear graph of the system. d Force input applied to the system. The system was in steady-state under a previous step input at time, \( t=0 \) , when pulse input, F (t ), was applied

Problem 3.7 A translational mechanical system which consists of a force source, a mass, a spring, and a damper is shown in Fig. P3.7 . The system was de-energized, before the input shown in Fig. P3.7d was applied. The parameter values are Case I:\( b=800\,\,\text{N}\cdot \text{s}/\text{m} \) , \({\rm{b}} = 400\,{\rm{N}} \cdot {\rm{sec/m}},\,\,{\rm{M}} = 250\,\,{\rm{kg}},\) , and \({\rm{K}} = 800\,\,{\rm{N/m}}\) \({\rm{b}} = 400\,{\rm{N}} \cdot {\rm{sec/m}},\,\,{\rm{M}} =25\,\,{\rm{kg}},\) , and \({\rm{K}} = 1,200\,\,{\rm{N/m}}\) Case II.

Check the units of the system equations in terms of power variables and time.

3.7.b The input force, F (t ), plotted in Fig. P3.7d is applied to the damper at time \( t=0 \) . Determine the unit step response of the system equations derived in part a. Use superposition to solve the system equations for the velocity input, shown in Fig. P3.7d .

3.7.c Plot the responses using Mathcad or MATLAB.

Fig. P3.9 a System with a velocity source, damper, mass, and spring. b Schematic annotated with nodes of distinct values of velocity. c Linear graph of the system. d Velocity input applied to the system. The system was in steady-state under a previous step input at time, \( t=0 \) , when the pulse input, v (t ), was applied

Problem 3.8 A translational mechanical system which consists of a force source, a mass, a spring, and a damper is shown in Fig. P3.8 . The system was in steady-state under a previous step input, before the force pulse shown in Fig. P3.8d was applied. The parameter values are Case I: \( b=400\,\,\text{N}\cdot \text{s/m} \) , \( M=250\,\,\text{kg} \) , and \( K=5,000\,\,\text{N/m} \) and Case II.

Check the units of the system equations in terms of power variables and time.

3.8.b The input force, F (t ), plotted in Fig. P3.8d is applied to the damper at time \( t=0 \) . Determine the unit step response of the system equations, derived in part a. Use superposition to solve the system equations for the velocity input, shown in Fig. P3.8d .

3.8.c Plot the responses using Mathcad or MATLAB.

Problem 3.9 A translational mechanical system which consists of a velocity source, a damper, a mass, and a spring is shown in Fig. P3.9 . The system was in steady-state under a previous step input, before the velocity pulse shown in Fig. P3.9d was applied. The parameter values are Case I: \({\rm{b}} = 600\,{\rm{N}} \cdot {\rm{sec/m}},\,\,{\rm{M}} = 150\,\,{\rm{kg}},\) , and \({\rm{K}} = 5,000\,\,{\rm{N/m}}\) and Case II:.

Check the units of the system equations in terms of power variables and time.

3.9.b The input velocity, v (t ), plotted in Fig. P3.9d is applied to the damper at time \( t=0 \) . Determine the unit step response of the system equations derived in part a. Use superposition to solve the system equations for the velocity input, shown in Fig. P3.9d .

3.9.c Plot the responses using Mathcad or MATLAB.

Problem 3.10 A translational mechanical system which consists of a velocity source, a damper, a mass, and a spring is shown in Fig. P3.10 . The system was in steady-state under a previous step input, before the velocity pulse shown in Fig. P3.10d was applied. The parameter values are Case I:\({\rm{b}} = 800\,{\rm{N}} \cdot {\rm{sec/m}},\,\,{\rm{M}} = 50\,\,{\rm{kg}},\) , and \({\rm{K}} = 2,000\,\,{\rm{N/m}}\) and Case II:.

Check units of the system equations, in terms of power variables and time.

3.10.b The input velocity, v (t ), plotted in Fig. P3.10d is applied to the damper at time \( t=0 \) . Determine the unit step response of the system equations, derived in part a. Solve the system equations for the velocity input shown in Fig. P3.10d .

3.10.c Plot the responses using Mathcad or MATLAB.

Fig. P3.10 a System with a velocity source, damper, mass, and spring. b Schematic annotated with nodes of distinct values of velocity. c Linear graph of the system. d Velocity input applied to the system. The system was in steady-state under a previous step input at time, \( t=0 \) , when pulse input, v (t ), was applied