Enhancement of Student Learning Via Laboratory ...

Session T1B

Enhancement of Student Learning Via Laboratory Demos and Course Project in Vibration Control Subha K. Kumpaty, Rich Phillips and Daniel Haeg Mechanical Engineering Department Milwaukee School of Engineering, Milwaukee, WI 53202 [email protected], [email protected], [email protected]

Abstract - Teaching vibration control, a senior level course at Milwaukee School of Engineering, has been enhanced by the introduction of laboratory demonstrations on both single and two degree-of-freedom (dof) systems. These demonstrations include measurements under both free and forced (harmonic) vibration. Transient vibration of single dof systems is treated using convolution integral, Fourier series and numerical analysis/ MATLAB® programming. Numerical analysis is discussed for multi-dof systems as well. The course structure leads to a terminal project that emphasizes modal analysis and solving multi-dof systems via modal matrix approach. The course is enriched with both theory and experiments working in tandem so that the student learning is enhanced in a powerful way. The course structure, laboratory demonstrations, project details and other critical parts of the course are presented in this paper. Student feedback on the course material and structure is also discussed. Index Terms – Systems, Vibration Measurement/Control, Modal Displacements, Degrees of Freedom INTRODUCTION Milwaukee School of Engineering is dedicated to excellence in undergraduate education. The goal of the undergraduate curriculum is to produce well-rounded engineers, which is achieved through strong emphasis in excellent technical preparation, strong laboratory orientation with faculty teaching labs in small size sections and required Senior Design projects. Accordingly, MSOE graduates are highly sought by industry (over 95% placement). The mechanical engineering students are introduced to MATLAB programming in the freshman year itself and are taught numerical modeling and analysis in the junior year. Bridging the gap is our four-credit class “Dynamics of Systems” in the last quarter of sophomore year for which currently, Close, Frederick and Newell text [1] is employed. The author of this paper, Dr. Kumpaty, is the course coordinator. He is also the course coordinator for a senior level course, “Vibration Control” which he has been teaching for a decade and thus holds a unique privilege of developing the course structure and format that he has found to be most useful to enhance student learning. Having taught the class with his hand-outs and Schaum’s series supplement as well as using several texts such as Thompson and Dahleh [2], Inman [3], Tongue [4] and Rao [5] over this decade-long

experience, the author is joined by co-authors in sharing the current course ingredients that have been effective. The demonstrations employed in this course could not have been successful without excellent cooperation of our senior laboratory technician, Mr. Richard Philips. Mr. Daniel Haeg is currently a senior student in our mechanical engineering program who provided the student thinking to the process of teaching and learning and is a MATLAB Simulink genius. The feedback on the course material and its presentation has been very affirming and this paper intends to highlight the sweet story of successful integration of laboratory demonstrations and the course project that has immensely impacted student learning. COURSE STRUCTURE The 10-week, quarter system, three-credit hour course begins with a quick review of modeling of mechanical systems and solution of ordinary differential equations by classical methods (Week 1). Next, free vibration of various single dof, translational and rotational mechanical systems is addressed (Week 2). Weeks 3-4 are dedicated for the treatment of forced vibration of single dof systems with several applications to harmonic forcing function. They include vibration isolation, support motion, rotational unbalance, whirling of shafts and vibration measuring instruments such as accelerometer and seismometer. In these two weeks, there is ample readiness for students to involve in the laboratory and increase their comprehension of the concepts first-hand. The laboratory provides students an opportunity to recognize and grasp the details of time response and frequency response clearly. They get to write a report on free and forced vibration of a single dof system. The first exam will concentrate on the material thus far learnt. An emphasis is given to testing the modeling of differential equations for given systems. Weeks 5 and 6 are utilized for transient analysis of single dof systems. Fourier series is employed for periodic functions. Once harmonics are determined, the solution due to each harmonic is summed to give the response for linear systems. Convolution integral is taught to solve system response due to various non-periodic forcing functions such as step and ramp. Numerical analysis of ordinary differential equations is revisited allowing MATLAB programming and employing Runge-Kutta methods. A take-home exam is fitting for transient vibration at the end of Week 6. Shock

San Juan, Puerto Rico

July 23–28, 2006 International Conference on Engineering Education T1B-7

Session T1B studies could be easily undertaken by a programmatic assignment. Two dof systems are addressed in weeks 7 and 8, both free and forced vibration. Lagrange equation is stressed so as to derive equations of motion for multi-dof systems. Dynamic vibration absorber is emphasized as a great application of vibration control. Students get to see various types of coupling of differential equation sets and recognize the use of MATLAB functions such as ode45 to solve the equations quickly. That there are two natural frequencies with definite mode shapes for a two dof system is illustrated by the second laboratory. Students appreciate seeing the theory in practice. A second exam is in order addressing the two dof systems. Modal analysis is introduced as a solution approach to multi-dof systems in Weeks 9 and 10. While illustrating the solution technique for a two dof system, a course project is assigned on a three or four dof system. An application could be for a machine tool system or an automobile suspension system. Special topics such as the Tacoma Narrows disaster, an introduction to continuous systems, solving natural frequencies by Holzer’s method could be discussed in Week 10 as time allows. The course culminates in the course project described above and a final examination in Week 11.

The following second order differential equation for this rotational system is obtained by summing the moments about the point where the system is pivoted and applying Newton’s law.

Mp 2θ + bq 2θ + kr 2θ = 0

(1)

The mass M refers to the effective mass of the rotor and the bar at distance p from the pivot. Similarly q and r are the distances from the pivot the damper and spring are located respectively. Equation (3) can be written as the more general form shown in equation (4).

θ + 2 ζ ω nθ + ω n 2θ = 0

(2)

SINGLE DOF SYSTEM LABORATORY I. Free Vibration This laboratory demonstrates how the damping coefficient (b) and spring constant (k) could be found for a second order system from the acceleration data. More importantly, this demonstration helps students learn about the important characteristics of second order differential equations: the natural frequency (ωn) and the damping ratio (ζ). An accelerometer was attached to a bar that had a spring and damper on it. The effective mass and its location were provided by the instructor. The accelerometer was connected to an oscilloscope which gave the acceleration data. The bar was given an initial input and allowed to oscillate. From the readout on the oscilloscope the data was collected: the amplitudes of consecutive cycles and the time between them. Figure I shows the setup for free vibration of a bar; Figure II portrays the data from the accelerator on the oscilloscope.

FIGURE II FREE VIBRATION RESPONSE OF SINGLE DOF MECHANICAL SYSTEM

The damping ratio is calculated from the logarithmic decrement, δ which is the natural logarithm of the ratio of two δ consecutive peaks. The two are related by ζ = . 2 2 4π + δ

Then using the time period of the damped cycle, damped natural frequency is obtained ( ω d = 2t π ) and natural d

frequency is gleaned from ω n =

ωd 1−ζ 2

. Substitution of the

values and comparison of (1) and (2) lets students calculate b and k. Also, the students get to see the experimental data agree with the following analytical solution for free vibration where A and B are constants that can be obtained from initial conditions, θ (0) and θ (0).

θ (t ) = e−ζω t ( A sin ω d t + B sin ω d t ) n

(3)

II. Forced Vibration

FIGURE I SINGLE DOF MECHANICAL SYSTEM WITH THE ACCELERATOR

The purpose of this demonstration is to show how a mechanical system reacts to a sinusoidal input. The springmass-damper system has a frequency response that is related to the ω (ratio of driving frequency and natural frequency,



Session T1B ω/ωn). For this portion of the laboratory, instead of just giving the system an initial input, forced sinusoidal input in the form of a rotating unbalance (m=0.0614 lbm; e=2.25 in.) was provided. A precise, harmonic oscillator was employed to provide varying frequency (Figure III). The frequency and the acceleration of the system were measured on the oscilloscope.

then multiplied by 15/28 to move it from where the accelerometer was located to the position where the mass was located. Table I shows the calculations. The experiment verified the theoretical response quite well as depicted in Figure IV. Frequency Response

9.0000 8.0000

Amplitude (Normalized)

7.0000 6.0000 5.0000 4.0000 3.0000 2.0000 1.0000 0.0000 0.7000

0.8000

0.9000

1.0000

1.1000

1.2000

ω/ωn

Theoretical Data

Experimental Data

FIGURE IV NORMALIZED FREQUENCY RESPONSE OF SINGLE DOF MECHANICAL SYSTEM FIGURE III HARMONIC EXCITATION OF SINGLE DOF MECHANICAL SYSTEM

TWO DOF SYSTEM LABORATORY

TABLE I EXPERIMENTAL DATA AND THEORETICAL CALCULATIONS Theoretical Frequency Hz

Experimental

Input Xth, p-p rad/s w/wn in Normalized mV

Acceleration in/s^2

Xexp Xexp cor., in Normalized

4.63

29.09 0.7405 0.0104

0.5702

76.8

15.14 0.0179

0.0096

0.5271

5.1

32.04 0.8156 0.0169

0.9283 137.5

27.11 0.0264

0.0141

0.7778

5.68

35.69 0.9084 0.0387

2.1282 343.8

67.78 0.0532

0.0285

1.5679

5.747

36.11 0.9191 0.044

2.4221

525

103.5 0.0794

0.0425

2.3388

6.211

39.02 0.9933 0.1395

7.6744 1906

375.75 0.2467

0.1322

7.2697

6.329

39.77 1.0122 0.135

7.424 1438

283.49 0.1793

0.096

5.2821

6.666

41.88 1.0661 0.0647

931

183.54 0.1046

0.056

3.0827

3.5612

Shown in Figure V is a two dof system, fixed at one end to the table with equal masses and equal stiffnesses made of spring steel (with negligible or small amount of structural damping). Two accelerometers, identical to the one employed in single dof system measurement, are attached to each mass and the output is measured on the oscilloscope. Also seen is the rotating unbalance that can be easily attached or detached to any of the masses to perform frequency response.

In this lab, the theoretical frequency response of the system was compared to the experimental data collected. The amplitude is related to the input frequency by the equation

X=

meω 2 M ω n2

(1 − ω 2 ) 2 + (2ζω ) 2

(4)

where the mass of the main system is M and the small rotating mass m, the input frequency ω, and the radius of the rotating mass e. Since the oscilloscope read the acceleration and not the amplitude, several conversions were needed. First, the peak to peak voltage had to be divided by two to get the amplitude, and then it was necessary to divide by the instrument sensitivity 98mV/g and then by accelerometer and oscilloscope gains. Then g s were converted into appropriate units and displacement was calculated by dividing by

ω 2 and

FIGURE V GENERAL SETUP OF TWO DOF MECHANICAL SYSTEM

First, the system was tested experimentally to observe the natural frequencies. The theoretical natural frequencies and corresponding mode shapes for this system are 0.618 k / m {0.618 1.000} and 1.618 k / m {-1.618 1.000}. The experiment matched the theory quite well for the known values of stiffnesses and masses. Figure VI depicts the first



Session T1B natural frequency and the corresponding outputs whose ratios match the theoretical (0.62). Notice the displacements are in phase as expected at the lower natural frequency. An out-ofphase response was noticed, as expected, at the higher natural frequency.

COURSE PROJECT A three dof system is assumed for a heavy machine tool system with the floor, machine tool base and tool head as three masses connected by respective springs and dampers. The model yields the following system of differential equations (5). ⎛ m1 0 ⎜ ⎜ 0 m2 ⎜0 0 ⎝

x1 ⎫ ⎛ (c1 + c2 ) −c2 − k2 0 ⎞ ⎧ x1 ⎫ ⎛ (k1 + k2 ) 0 ⎞ ⎧ x1 ⎫ ⎧0 ⎫ ⎞ ⎧ ⎪ ⎟ ⎪ ⎪ ⎜ ⎟⎪ ⎪ ⎜ ⎟⎪ ⎪ ⎪ (c2 + c3 ) −c3 ⎟ ⎨ x2 ⎬ + ⎜ − k2 (k2 + k3 ) − k3 ⎟ ⎨ x2 ⎬ = ⎨0 ⎬ (5) ⎟ ⎨ x2 ⎬ + ⎜ −c2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟ ⎜ ⎟ ⎜ ⎟ m3 ⎠ ⎩ x3 ⎭ ⎝ 0 c3 ⎠ ⎩ x3 ⎭ ⎝ 0 k3 ⎠ ⎩ x3 ⎭ ⎩ F (t ) ⎭⎪ −c3 − k3 0 0

The goal of the project is to solve the system by modal analysis. The first item of interest is to solve for eigen values (square of natural frequencies) from (6) and eigen vectors (mode shapes) by plugging back each natural frequency and solving for displacement ratio. The 3X3 modal matrix P is assembled by placing the mode shapes together as three column vectors.

K − Mω2 = 0 FIGURE VI FREE VIBRATION RESPONSE OF TWO DOF MECHANICAL SYSTEM (FIRST NATURAL FREQUENCY)

A rotating unbalance was employed and with a built-in circuit to vary the frequency, forced response of the two dof system was attempted. The motion of the two masses was studied at various frequencies. Noteworthy was the nullifying of the motion of the mass on which the harmonic excitation was supplied. This occurred when the excitation frequency was very closely matched with k / m which is the frequency of the decoupled mass-spring systems. Figure VII shows the system under harmonic excitation and its response on the oscilloscope.

(6)

The modal matrix is normalized by finding generalized mass matrix (PTMP) and dividing each column vector of the modal matrix by the square root of the corresponding generalized mass. The modified modal matrix is shown in (7).

P =

⎡ ⎧ X1 ⎫ ⎢ 1 ⎪ ⎪ 1 P=⎢ ⎨X2 ⎬ M ii ⎢ M 11 ⎪ X ⎪ ⎩ 3 ⎭ω1 ⎣

⎧ X1 ⎫ 1 ⎪ ⎪ ⎨X2 ⎬ M 22 ⎪ ⎪ ⎩ X 3 ⎭ω2

1 M 33

⎤ ⎥ (7) ⎥ ⎥ 3⎦

⎧ X1 ⎫ ⎪ ⎪ ⎨X2 ⎬ ⎪ ⎪ ⎩ X 3 ⎭ω

Employing X = P Y and pre-multiplying the equation set by

P T we notice decoupling of the equations into three single dof T systems in modal displacements with force being P F and T initial, modal displacements, being P MX0 and likewise, the

initial velocities. Notice how the decoupling occurs (8).

⎛ ω12 0 0 ⎞ ⎛1 0 0⎞ ⎜ ⎟ T T 2 P KP = ⎜ 0 ω2 0 ⎟ ; P MP = ⎜⎜ 0 1 0 ⎟⎟ ⎜0 0 1⎟ ⎜ 0 0 ω32 ⎟⎠ ⎝ ⎠ ⎝ 0 0 ⎞ ⎛ 2ξ1ω1 ⎜ T P CP = ⎜ 0 (8) 2ξ 2ω2 0 ⎟⎟ ⎜ 0 ⎟ 0 2ξ3ω3 ⎠ ⎝ Modal damping factors can be thus obtained easily from (8). The three decoupled equations take the form (9). FIGURE VII FORCED VIBRATION RESPONSE OF TWO DOF MECHANICAL SYSTEM

yi + 2ξiωi yi + ωi 2 yi = Fo cos ωi t

(9)

The total solution for modal displacements, Y is given by



Session T1B yi ( t ) =

cos(ωt − φi )

Fo

ωi 2 ⎛

2

2 2 ⎛ω ⎞ ⎞ ⎛ ⎛ω ⎞⎞ ⎜ 1 − ⎜ ω ⎟ ⎟ + ⎜ 2ξi ⎜ ω ⎟ ⎟ i⎠ ⎠ i ⎠⎠ ⎝ ⎝ ⎝ ⎝

⎡ ⎤ y o + ξiωi yoi + e −ξiωit ⎢ yoi cos( 1 − ξi 2 ωi t ) + i sin( 1 − ξi 2 ωi t ) ⎥ 2 ωi 1 − ξi ⎢⎣ ⎥⎦ − ξiωi t e Fo ... − 2 2 ωi ⎛ ⎛ ω ⎞2 ⎞ ⎛ ⎛ ω ⎞ ⎞2 ⎜ 1 − ⎜ ω ⎟ ⎟ + ⎜ 2ξi ⎜ ω ⎟ ⎟ i⎠ ⎠ i ⎠⎠ ⎝ ⎝ ⎝ ⎝ ⎡ ⎤ ω sin (φi ) + ξiωi cos (φi ) sin( 1 − ξi 2 ωi t ) ⎥ ⎢ cos (φi ) cos( 1 − ξi 2 ωi t ) + 2 ωi 1 − ξi ⎣⎢ ⎦⎥ ⎛ ⎛ ⎞⎞ ⎜ 2ξi ⎜ ω ω ⎟ ⎟ i⎠ ⎝ where φi = tan ⎜ ⎟ 2 ⎜1− ⎛ω ⎞ ⎟ ⎜ ⎜ ω⎟ ⎟ i⎠ ⎠ ⎝ ⎝ −1

(10)

dof systems. The only limitation is that the damping matrix is proportional to the stiffness matrix. The authors show that the same result could be obtained by solving the coupled equation set by employing ode45. The Simulink diagram (Figure IX) shows how complex the three dof system is. The more the dof, the more complicated the coupled set will be. A graphical user interface was created to monitor various inputs (masses, spring constants, damping coefficients, initial conditions, forcing functions- impulse, step and harmonic) with the resulting solution on display. Figures X and XI depict the solutions under a given set of initial conditions and/or forcing function (sample cases I and II). The modal matrix approach was also found to yield the same solution.

FIGURE IX SIMULINK MODEL FOR THE USER INTERFACE

STUDENT FEEDBACK

FIGURE VIII SIMULINK MODEL TO SOLVE BY MODAL ALALYSIS

Or MATLAB could be employed to solve modal displacements, one at a time as single dof systems. Figure VIII depicts the MATLAB Simulink model to solve modal displacements by ode45 (Runge-Kutta scheme). The only thing that is left is to convert the solution into actual

“Lab sessions were very helpful in understanding the theory very well.” “The instructor’s depth of knowledge gets passed on very well.” “The instructor makes an effort to explain the material with examples and realistic models.” “This course provides material that is most useful to MEs.” “I like the applications of vibrations.” “Great examples, and very interesting final project.” “Demonstration of vibratory systems (applications) was awesome.” “The instructor brings in real world examples.. shows applications.. cares what students think and accommodates to them.” “The project requirements were spelled out clearly in doable steps.” “A very solid course taught by a solid instructor.”

displacements, X, which is achieved by P Y. The solution is easily obtained for given set of initial conditions and the cutting force applied on machine tool head. The modal analysis is indeed a very user friendly tool for solving multi San Juan, Puerto Rico International Conference on Engineering Education T1B-11

July 23–28, 2006

Session T1B I liked learning about how to apply differential equations. Labs provided the depth.” “I liked the labs which reflect what we learned in class and the movie on Tacoma Narrows Bridge disaster.” “The class is tough- hard math and real pain but the instructor makes it lovable and doable- he has an interesting perspective of the material.” “The project really helped me understand the material by application. The course showed how critical vibrations can be to mechanical systems.” “Practical applications.” “Props were very helpful for visualization of higher vibration modes.”

FIGURE X USER INTERFACE TO SOLVE 3DOF SYSTEMS: SAMPLE CASE I

The above statements describe the positive influence of employing student-friendly tools on learning vibration control concepts- laboratory demonstrations, numerical solution techniques and project based learning. The course is quite challenging for some students who may have enrolled in the elective class treating it as a filler and for those who are fully tied up with senior design (prototype construction) and hence unable to give as much time as the course would need. Time issue is the only negative comment from students. Fine-tuning the laboratory demonstrations continues to be a top priority for the authors. CONCLUSION The laboratory demonstrations, programming assignments on transient vibration and the course project employing modal analysis have proved to be effective tools in enhancing student learning of Vibration Control course concepts in the presenter’s classroom at Milwaukee School of Engineering. This type of in-depth integration of concepts and tools in a course offering will go a long way in students’ perception and comprehension. The positive feedback from students in the course evaluations affirms the assessment of the authors. In conclusion, integration of laboratory demonstrations and a comprehensive course project is highly recommended since it will facilitate the student-centered learning become a rewarding experience for all involved- both faculty and students. REFERENCES [1] Close, C.M., Frederick, D.K. and Newell, J.C., Modeling and Analysis of Dynamic Systems, 3rd Edition, John Wiley, 2002.

FIGURE XI USER INTERFACE TO SOLVE 3DOF SYSTEMS: SAMPLE CASE II

[2]

Thompson, W.T. and Dahleh, M.D., Theory of Vibration with Applications, 5th Edition, Prentice Hall, 1998.

[3]

Inman, D.J., Engineering Vibration, 2nd Edition, Prentice Hall, 2000.

[4]

Tongue, B.H., Principles of Vibration, 2nd Edition, Oxford University Press, 2002.

[5] Rao, S.S., Mechanical Vibration, 4th Edition, Pearson Prentice Hall, 2002.

