Modification of a Sophomore Linear Systems Course to Reflect Modern Computing Strategies

Transcription

1 Session 3220 Modification of a Sophomore Linear Systems Course to Reflect Modern Computing Strategies Raymond G. Jacquot, Jerry C. Hamann, John E. McInroy Electrical Engineering Department, University of Wyoming Abstract This paper reports on an effort currently underway to integrate modern computing strategies into a sophomore course in linear systems. The course material includes Laplace transforms, system modeling and simulation, Fourier series and Fourier transforms. The course has a laboratory which meets biweekly. The effort reported here is one to incorporate the use of two software packages across the lecture and laboratory portions of the course. Introduction Getting Electrical Engineering students started in their curriculum with up-to-date attitudes toward the role of computing in the engineering problem solving process is extremely important. This attitude should not be one that automated computing is a course of last resort for problems that cannot be solved by hand, but rather that computing is something that the practitioner does in practice to increase productivity and to make the workload more bearable. In addition, computing should not be done in a setting where the engineer does not understand the process being implemented. That is, a full understanding of the process is still very important so that the engineer can know when computed results make no sense. Freshman students in the Electrical Engineering Department at the University of Wyoming are first introduced to computing in the Introduction to Engineering Computing course wherein they are exposed to tksolver and Excel in an engineering problem solving atmosphere addressing problems of material balance, engineering economics, electric circuits, and statistics and data presentation. In the past, the introduction of computing tools appropriate to electrical engineering (namely Spice, MATLAB and an additional simulation tool) was done on a hit-and-miss basis and it was never clear what computing skills upper division students had. Reported here is an effort to remedy this situation and provide a unified approach such that all students have a set of known basic modern computational skills. Course Setting In the second semester of the sophomore year electrical engineering students are required to take a first course in linear systems which is preceded by a circuit analysis course the previous semester. Also they should have taken or be taking concurrently an ordinary differential equations course in the mathematics department. The linear systems course is a prerequisite for further study in electrical networks, network synthesis, controls and discrete signals and systems. Page

2 The course meets three times weekly and has a biweekly laboratory. The nominal topic outline of the course is given below in Table 1. Table 1. Topic Outline for the Linear Systems Course Solution of Linear Equations The Unit Step Function The Dirac Delta Function The Laplace Transform Development of Transform Pairs Time and Exponential Multiplication Partial Fraction Expansion Linear Ordinary Differential Equations System Modeling Mechanical Systems Electrical Networks System Concepts The Transfer Function Poles, Zeros and System Stability The Frequency Response Function Bode Plots First- and Second-Order Systems The Convolution Integral Fourier Methods The Fourier Series Mean Square Error and the Parseval Relation Filtering of Periodic Signals Complex Fourier Series The Fourier Spectrum Introduction to the Fourier Transform Appropriate Software In the selection of software there are many possibilities and it is best to introduce students to packages that will have maximum utility in their future education and practice. The software packages chosen for this course were MATLAB [1]and VisSim [2]. MATLAB was chosen because it has become the standard for system analysis, signal processing and many other engineering computing tasks. VisSim was chosen as a typical simulation tool because of its short learning curve and its excellent graphical user interface. This was done with full consciousness that Simulink fits better with MATLAB since it is an adjunct to MATLAB. This was done as a tradeoff because of the ease with which VisSim is learned. After this, the use of Simulink in the future is more readily achievable. Perhaps the novel part of this reported work is the simultaneous and synergistic introduction of both packages. Both of these software packages are available in Student Editions [3, 4] at very reasonable prices. It is important for completeness to note that Spice is introduced in the Electrical Networks course for which this course is a prerequisite. Page

3 The Theme Problem The primary use of the software is associated with the biweekly laboratory and is accomplished in the framework of a problem which provides a recurrent theme for all aspects of the course. In the fall semester of 1996 the problem involved a two degree-of-freedom vehicle suspension system as illustrated in Fig. 1 although it could as well be an electrical network. The system is fourth order and thus hand computations will not suffice for the calculation of various system responses to various road profile inputs. m 1 x 1 b k 1 x 2 m 2 k 2 x 3 Figure 1. Vehicle Suspension System Theme Problem. In this system m 1 and x 1 are respectively the mass and displacement of the chassis while m 2 and x 2 are respectively the mass and displacement of the axle and wheels (the unsprung mass). Coefficient b represents the viscous damping of the shock absorber, k 1 the stiffness of the springs and k 2 the stiffness of the tires. Displacement x 3 represents the road profile which serves as the forcing function for the system. A meaningful set of parameters for this system are: m 1 = 360 kg m 2 = 30 kg k 1 = 15 kn/m k 2 = 65 kn/m b = 0, 600, 1200, 1800 N-s/m For all exercises the students are asked to predict the system response using transform methods and MATLAB, and to verify that prediction by time-domain simulation using VisSim. The Laboratory Exercises In the course of the laboratory six exercises are accomplished and are detailed below by number. Lab Exercise 1 This provides an introduction to MATLAB in the context of polynomial assembly from the roots, finding polynomial roots, polynomial evaluation, function evaluation and plotting and solution of linear equations. Lab Exercise 2 Students gain expertise in the use of MATLAB as an aid in the Laplace transform solution of linear, constant coefficient differential equations and to plot the solution. The functional form of the solution is to be verified by simulation using VisSim and the conventional analog computer paradigm employing integrators, summers, inverters and function generators. Page

4 Lab Exercise 3 In this exercise the students model the system of Fig. 1 and solve the governing differential equations to get responses x 1 (t) and x 2 (t) when x 3 (t) is a negative step function for several values of the shock absorber coefficient. These solutions are to be corroborated by appropriate simulations using VisSim. Lab Exercise 4 Students are asked to derive the transfer function X 1 (s)/x 3 (s) and invert it to obtain the impulse response function. They are asked to examine the effect of the shock absorber coefficient b on the locations of the poles and zeros of the transfer function. Lab Exercise 5 Students are asked to use frequency response concepts to examine the steady-state response of the vehicle to a sinusoidal washboard road for several different forward velocities. Again these results are to be verified by time-domain simulation. Lab Exercise 6 Here the road profile is taken to be a sawtooth function which is to be represented by a Fourier series. Using the frequency response function and the principle of superposition the students are asked to predict the steady-state response to this sawtooth forcing function. Using VisSim they are also asked to synthesize the sawtooth waveform using a sum of sinusoidal signals. After this synthesis, the resulting waveform is to be used as a driving function for the vehicle dynamics. As a specific example consider Lab Exercise 6 in which the vehicle travels over a sawtooth road surface at a single fixed forward velocity of v = 0.12 m/s. The wavelength of the washboard is λ = 0.3 m so the fundamental radian frequency is ω 1 = 2.51 r/s. This waveform is illustrated in Fig. 2 below. The Fourier synthesis of the sawtooth waveform can be accomplished in VisSim employing a series of sinewave generators and summation. This signal can then be used as the forcing function to the system treated as a transfer function. x 3 (t) 0.08m t, sec. Figure 2. Sawtooth Road Waveform. The results for b = 600 N-s/m are illustrated in Fig. 3. It can be seen that in the case of the simulation solution there is a transient period before steady state is reached. Conclusion The authors have reported here an effort to rejuvenate a sophomore level linear systems course by incorporating state-of-the-art computing tools into the laboratory portion of the course. One aspect of this effort which was viewed as very positive was the simultaneous use of MATLAB and VisSim to evaluate responses of a reasonably complex system from both a mathematical and Page

5 MATLAB VisSim Figure 3. Responses Calculated by Fourier Series and by Simulation. a simulation point of view. The tools introduced at this time form a solid foundation for computing tasks in subsequent signals, systems and electronics courses. For interested faculty the m-files and the vsm-files are available for each lab exercise and the lab handouts are available as Word documents. An request to the first author at quot@uwyo.edu will initiate the sending of said materials. References [1] The Math Works, Inc., MATLAB Reference Guide, Natick, MA, [2] Visual Solutions, Inc., VisSim User s Guide--Version 1.2, Westford, MA, [3] The Math Works, Inc., The Student Edition of MATLAB, Prentice-Hall, Englewood Cliffs, NJ, [4] Darnell, K. and A. K. Mulpur, Visual Simulation with Student VisSim, PWS Publishing, Boston, MA, Raymond G. Jacquot Ray Jacquot received his BSME and MSME degrees at the University of Wyoming in 1960 and 1962 respectively. He was an NSF Science Faculty Fellow at Purdue University where he received the Ph.D. in He is a member of ASEE, IEEE and ASME and has been active the Computers in Education Division of ASEE and has educational interests in control of dynamic systems and is currently Professor of Electrical Engineering. Jerry C. Hamann Jerry Hamann received the BS degree in Electrical Engineering/Bioengineering from the University of Wyoming in After employment with Hewlett-Packard he returned to the University of Wyoming as a NSF Graduate Fellow completing the MS degree in He completed the Ph.D. at the University of Wisconsin in 1993 with a focus in robust control. His Page

6 interests are in systems and real-time signal processing. He is currently Assistant Professor of Electrical Engineering. John E. McInroy John McInroy received the BS degree in Electrical Engineering from the University of Wyoming in He received the MS and Ph.D. degrees in Electrical Engineering from Renselaer Polytechnic Institute in 1988 and 1991 respectively. His interests are in robotic systems, sensor data fusion and precision motion control. He is currently Assistant Professor of Electrical Engineering. Page