Improving the ME4840 Vibration System Design Course Final Report

2006 Curriculum Improvement Fund (CIF)

Submitted by P.B. Ravikumar, Professor, Mechanical and Industrial Engineering

Summary: The ME 4840 Vibration System Design course is an important technical elective for Mechanical Engineering majors. Vibration studies have become increasingly important to mechanical engineers due to demands placed on performance of mechanical systems. Vibration often adversely affects performance due to noise and fatigue failures. Vibration studies are also important in positive applications such as vibratory conveyors. The Vibration System Design course at present lacks experimental insight due to lack of equipment and experiments to go with them. One of the objectives of this CIF project was to develop the laboratory part of the course. This objective was achieved by formulating necessary experiments that fit the course, identifying potential vendors who can provide the equipment and other resources to do so those experiments, and submitting a DIN funding request. Another objective of this CIF project was to generate course materials that support the understanding of mathematical development through use of software such as Matlab, Mathlab, and Simulink. This has been accomplished through the development of pertinent examples using Mathcad, Matlab, and Simulink. A third objective was to use the software for specific vibration application problems. This has been realized for applications such as balancing, vibration isolation, accelerometers and automobile suspension systems. This report provides representative material of the project outcomes that were generated to meet these three major objectives. Laboratory Experiments Through careful study of the needs of industry in the field of vibrations and balancing those needs with curriculum requirements (a representative Course of Study is given in Appendix 1), the following experiments were identified to support the ME4840 Vibration System Design Course. It should be realized that not all of the experiments may be implemented right away due to budget and curriculum constraints that might prevent the purchase of all necessary equipment. 1. Simple pendulum 2. Compound pendulum 3. Center of percussion 4. Bifilar suspension 5. Mass-spring systems 6. Torsional oscillations of a single rotor 7. Torsional oscillations of a single rotor with viscous damping

8. Torsional oscillations of a two-rotor system 9. Transverse vibration of a beam with one or more bodies attached 10. Undamped vibration absorber 11. Forced vibration of a rigid body -spring system with negligible damping 12. Free-damped vibrations of a rigid-body spring system 13. Forced-vibrations of a rigid-body spring system 14. Bump shock testing to produce various shock pulses and waveforms in the vertical

direction. 15. Package drop testing to test a package’s ability to withstand impacts from various

heights and precise angles. To conduct experiments 1 through 13, a frame such as the TM16 Universal Vibration Apparatus [1] along with necessary Ancillaries will have to be purchased. A photograph of the Apparatus is shown below. A DIN funding request has been submitted to the College of EMS through the Chair of the Mechanical Engineering Department for the procurement of TM16 or any equivalent system. (See Appendix 2)

Universal Vibration Apparatus TM-16

Bump shock testing (Experiment #14) to produce various shock pulses and waveforms in the vertical direction will require a bump shock tester such as the DS-PS110 Bump Shock Tester [2] shown below. A DIN funding request has been submitted to the College of EMS through the Chair of the Mechanical Engineering Department. (See Appendix 2) Bump Shock Tester DS-PS110

Package drop testing (Experiment #15) will require equipment such as the DS-PD5 [2] shown below and can perhaps be justified if grants can be obtained from industry that have interest in such tests. Some companies will be contacted once the basic experiments 1 through 14 are implemented in the first place. A DIN funding request has been submitted to the College of EMS through the Chair of the Mechanical Engineering Department. (See Appendix 2)

Package Drop Tester DS-PD5

Use of Software such Mathcad [4], Matlab [5], and Simulink[6] In addition to improving the experimental side of the course, current and capable programs such as Mathcad, Matlab, and Simulink will be used to support the learning of many key areas of the course. Developed examples include the application of such software to solving differential equations, performing matrix operations, and using the simulation aspect of Simulink to demonstrate the inter-relationship between vibration and automatic controls. Appendix 3 contains representative examples that have been developed through the CIF project. Example 3-1 demonstrates the use of Mathcad to solve vibration problems using non-differential equations. The example calculates the damping factor, the logarithmic decrement, and the ratio of any two successive amplitudes for a given vibration system defined by its mass, spring constant, and damping coefficient. Example 3-2, an inverted pendulum, demonstrates the use of Mathcad to solve vibration problems that involve differential equations. The example demonstrates the use of differential equations’ command syntax as well as the use of graphing techniques to visualize responses that are so important in vibration studies. The particular example also demonstrates stable conditions in vibration studies. Example 3-3 is also an inverted pendulum except that its parameters force an unstable condition as demonstrated by the response. Example 3-4 addresses the import topic and application of balancing in vibration. The example illustrates the use of computing software such as Mathcad to solve single-plane balancing problems. Example 3-5 is an illustration of the use of Mathcad to solve vibration isolation problems. The example computes the statical deflection that the isolators that isolate engine vibrations from an aircraft radio under certain conditions. Example 3-6 shows the application of Mathcad to finding automobile performance due to the effects of inputs to its suspension. Example 3-7 is an example of application of software such as Mathcad to vibration measuring instruments such as accelerometers. The example demonstrates the robust performance of an accelerometer due to combined harmonic input frequencies. Example 3-8 demonstrates the importance of using computer software to solve matrix problems that are so common in vibration problems. The example generates the eigenvalues (and hence natural frequencies) and eigenvectors (and hence mode shapes) of the body of an automobile whose suspension is modeled as a two-degree of freedom system. Example 3-9 demonstrates the use of Matlab to finding the unit-step response of a higher-

order vibration system. Although Matlab and its ancillary program Simulink suit the needs of an Automatic Controls course very well, it helps to introduce Matlab in the vibration class. Doing so highlights the idea that vibration systems are also part of the many systems that are covered by automatic controls. Of course, Matlab could be used to solve many of the problems that were solved using Mathcad. But Matlab calls for a programming sort of approach to do so. This may not be always convenient. Example 3-10 shows the use of Matlab to find performance characteristics such as maximum overshoot of a vibration system. The example is a system that has a rotational inertia and damper and is subjected to a unit-step input. Example 3-11 is a demonstration of the use of Simulink which can accommodate block diagram representations of transfer functions “intelligently” and thereby enable simulation of performance predictions of vibration systems. Although Simulink is often used in advanced levels of automatic controls classes, an introduction to the topic in vibration through examples such as Example 3-11 are very valuable. Use of Software to Specific Application Problems The examples outlined in the above section include software usage to specific application problems in vibrations such as balancing, vibration isolation, accelerometers, and automobile suspension systems. The in-built differential equation portions of the software are in fact applicable to any vibration system (a door damped for example) that can represented by governing differential equations. Conclusions The CIF funding (although lower in amount funded than requested) was very helpful to the author to synthesize suitable vibration experiments, identify potential vendors of necessary equipment to run those experiments, request DIN funding, and develop course materials in Mathcad, Matlab, and Simulink to support the teaching of the ME4840 Vibration System Design course. It is hoped that DIN funding will be allocated in 2007 to develop further on the outcomes of this CIF project.

Bibliography 1. www.tq.com

TQ Education and Training Ltd Bonsall Street, Long Eaton Nottingham, NG10 2AN Tel: (44) 115 972 2611 Fax: (44) 115 973 1520 email: info@tq.com

2. http://www.dynsolusa.com/products.htm?trackcode=bizcom

Dynamic Solutions P.O. Box 7963 Northridge, CA 91327 Sales and Marketing: Aimmee Hagler aimmee@dynsolusa.com 877-767-7077

3. Thompson, William T., Dahleh, Marie, Dillon, Thoery of Vibration with Applications,

Prentice-Hall Inc., 5th Edition. 4. Mathcad is a product of Mathsoft by PTC. PTC Corporate Headquarters, Needham, MA 02494, USA mathsoft.com 5. Matlab is a product of The Mathworks, Inc. Mathworks, Inc., 3 Apple Hill Drive, Natick, MA 01760 mathworks.com 6. Simulink is a product of The Mathworks, Inc. Mathworks, Inc., 3 Apple Hill Drive, Natick, MA 01760 mathworks.com

APPENDIX 1

Course of Study

CCOOUURRSSEE OOFF SSTTUUDDYY ME 4840 01 VIBRATION SYSTEM DESIGN, SPRING 2006 TIME : Tue, Thu 10:00 am to 11:18 am Room: 44, Ottensman

PROFESSOR : Dr. Ravikumar ; OFFICE : 61, Ottensman.

OFFICE HOURS Mon,Wed,Fri : 10:00 am to 11 am and 2 pm to 5 pm; Tue 9 am to 10 am and 2 pm to 5 pm; Wed 9 am to 11 am

CLASS DATES

Jan 24,26,31 Feb 2,7,9,14,16,21,23,28 Mar 2,7,9,21,23,28,30 Apr 4,6,11,13,18,20,25,27 May 2,4,9,11,16

TEXT Theory of Vibration with Applications, W. T. Thomson & M.D. Dahleh, Prentice-Hall, 1998, Fifth Edition

COURSE DESCRIPTION

Oscillatory Motion Harmonic Motion, Periodic Motion, Vibration Terminology Free Vibration Vibration Model, Equations of Motion, Natural Frequency, Energy Method, Rayleigh Method, Principle of Virtual

Work, Viscously Damped Free Vibration, Logarithmic Decrement, Coulomb Damping Harmonically Excited Forced Harmonic Vibration, Rotating / Rotor Unbalance, Whirling of Rotating Shafts, Support Vibration Motion, Vibration Isolation, Energy Dissipated by Damping, Structural Damping, Sharpness of Resonance, Vibration

Measuring Instruments Transient Vibration Impulse Excitation, Arbitrary Excitation, Laplace Transform Formulation, Pulse Excitation and Rise Time, Shock

Response Spectrum, Shock Isolation, Finite Difference Numerical Method, Runge-Kutta Method Two or More Degrees Normal Mode Analysis, Initial Conditions, Coordinate Coupling, Forced Harmonic Vibration, Digital of Freedom Systems Computation, Vibration, Absorber, Centrifugal Pendulum Vibration Absorber, Vibration Damper Properties of Flexibility Influence Coefficients, Reciprocity Theorem, Stiffness Influence Coefficient, Orthogonality

Vibrating Systems Eigenvectors, Modal Matrix P, Decoupling Forced Vibration Equations, Modal Damping in Forced Vibration, Normal Mode Summation

Lagrange’s Equation Generalized Coordinates, Virtual Work, Lagrange’s Equation Computational Various Computational Methods Methods Brief overview of Continuous Systems and Random Vibrations

EXAMINATIONS - SCHEDULE AND WEIGHAGE

EXAMINATION DUE DATE/ MAIN STUDY WEIGHAGE TYPE EXAM DATE MATERIAL

VIBRATION DESIGN PROJECT PROPOSAL DOCUMENT Thu, Feb 23 5 %

TEST #1 Thu, Mar 2 1/ 24 to 2/ 28 20 % TEST #2 Thu, Apr 27 3/ 2 to 4/ 25 20 %

VIBRATION DESIGN PROJECT FINAL REPORT Thu, May 4 1/ 24 to 4 / 27 10 %

VIBRATION DESIGN PROJECT FINAL PRESENTATION Tue, May 9 1/ 24 to 4/ 27 10 %

HOME WORK AND/OR QUIZ 10 %

FINAL EXAM (Room 44 OTTS) (1pm to 2:52 pm) Tue, May 16 1/ 24 to 5 / 11 25 %

TOTAL 100 %

GRADING POLICY

A: 90 TO 100 %; B: 80 TO 89 %; C: 70 TO 79 %; D: 60 TO 69 %; F: 0 TO 59 %

APPENDIX 2

DIN Funding Request

From: P. B Ravikumar <ravikump@uwplatt.edu> To: rolle@uwplatt.edu kunzd@uwplatt.edu Cc: ravikump@uwplatt.edu Date: 08/12/06 10:55 pm Subject: Vibration Lab Equipment Request in next DIN Proposal for ME4840

Vibration System Design Attachments:

To: Dr. Rolle and Dr. Kunz The following vibration equipment (or equivalent) is hereby requested for inclusion in the next DIN proposal for the ME4840 Vibration System Design course. At the time the DIN proposal will be put together, I will provide accurate cost estimates as well as who can provide the needed equipment at the best price. 1. Universal Vibration Apparatus TM-16 by TQ Education and Training Limited, Nottingham, UK. 2. Bump Shock Tester, DS-PS110 by Dynamic Solutions, Northridge, CA. 3. Package Drop Tester, DS-PD5 by Dynamic Solutions, Northridge, CA. I hope to specify the above models or other equivalent ones with a total budget of around $ 30,000 when the DIN proposal is put together. If you have any specific questions or clarifications, please contact me. P.B. Ravikumar

APPENDIX 3

Examples

EXAMPLE 3-1 Non-Differential Equations for Spring-Mass-Damper System

ANS (a)

ωnkm

:= ωn 20= rad / s

ωd ωn 1 ζ2

−⋅:= ωd 19.9= rad /s ANS (b) or fdωd2 π⋅

:= fd 3.167= Hz ANS (b)

δ2 π⋅ ζ⋅

1 ζ2

−

:= δ 0.6315= ANS (c)

δ ln x1_over_x2( ) or x1_over_x2 exp δ( ):= (that is, e δ )

x1_over_x2 1.88= ANS (d)

Spring - mass - damper system

m 17.5:= kg k 70:= N / cm c 0.7:= N / cm / s

k k 100⋅:= k 7000= N / m c c 100⋅:= c 70= N / m / s or N-s / m

(a) . Damping ratio or factor ζ = ? (b) Natural frequency of damped oscillation ωd = ?(c). Logarithmic decrement = ? (d) Ratio of any two consecutive amplitudes = ?

cc 2 m k⋅:= cc 700=

ζccc

:= ζ 0.1= Damping ratio or factor

EXAMPLE 3-2 Stable Inverted Pendulum: Use of differential equation solver and graphs

For k = 158.46, the response is stable. If you changed k to 55, as an example, you will find unstable response.

Note : y axisis radians.Verify by IC.

0 0.450.91.351.82.252.73.15 3.64.05 4.50.4

0.2

0

0.2

0.4

θ t( )

t

t 0 0.001, 4.5..:=

Right Clicked on Odesolve and chose Adaptive rather thanfixed because output otherwise had numerical errors ofslight damping type response decay for some value trials (not for the ones chosen finally here, so fixed would still be ok) although there is no damping. Adaptive choice fixed it !

θ Odesolve t 4.5,( ):=

θ' 0( ) 0θ 0( ) 0.3

13

m1⋅ m2+⎛⎜⎝

⎞⎠

l2⋅ θ'' t( )⋅ k l2⋅ m1 g⋅l2

⋅ m2 g⋅ l⋅+⎛⎜⎝

⎞⎠

−⎡⎢⎣

⎤⎥⎦

θ t( )⋅+ 0

Givenτ 2.3433=τ

1f

:=

f 0.4267=fωn2 π⋅

:=ωn 2.6813=ωn

k l2⋅ m1 g⋅l2

⋅ m2 g⋅ l⋅+⎛⎜⎝

⎞⎠

−

13

m1⋅ m2+⎛⎜⎝

⎞⎠

l2⋅:=

k 158.46:=l 0.5:=m2 5:=m1 2:=g 9.81:=

Inverted Pendulum (Stable)

Example 3-3 Unstable Inverted Pendulum: Use of differential equation solver and graphs

Note : If the scaling everywhere was changed from 4.5 to 2 above, you can clearly see theinitial condition of 0.3 radians at t=0 followed by the unstable response zoomed in up to 2.

Note : y axisis radians.Verify by IC.

0 0.450.91.351.82.252.73.153.64.054.50

2 .105

4 .105

6 .105

θ t( )

t

t 0 0.001, 4.5..:=

θ Odesolve t 4.5,( ):=

θ' 0( ) 0θ 0( ) 0.3

13

m1⋅ m2+⎛⎜⎝

⎞⎠

l2⋅ θ'' t( )⋅ k l2⋅ m1 g⋅l2

⋅ m2 g⋅ l⋅+⎛⎜⎝

⎞⎠

−⎡⎢⎣

⎤⎥⎦

θ t( )⋅+ 0

Givenτ 1.8886i−=τ

1f

:=

f 0.5295i=fωn2 π⋅

:=ωn 3.3269i=ωn

k l2⋅ m1 g⋅l2

⋅ m2 g⋅ l⋅+⎛⎜⎝

⎞⎠

−

13

m1⋅ m2+⎛⎜⎝

⎞⎠

l2⋅:=

k 55:=l 0.5:=m2 5:=m1 2:=g 9.81:=

Inverted Pendulum (Unstable)

Example 3-4 Single-Plane Balancing

x11 x11 x1−:= x11 1.197−=

y11 y11 y1−:= y11 5.221= Alone x112 y112+:= Alone 5.356=

r1x1− x11⋅ y1 y11⋅−( )2 x1 y11⋅ y1 x11⋅−( )2

+⎡⎣ ⎤⎦

x112 y112+

:= r1 0.597=

BWphi1 atan2 x1− x11⋅ y1 y11⋅− x1 y11⋅ y1 x11⋅−,( ):= BWphi1 1.869= BWphi1 107.09deg=

BALANCING SOLUTION:

BWphi1 107.09deg= BW_New_phi1 BWphi1 TW1_Angle+:= BW_New_phi1 250.09deg=

BWmag1 r1 TW1⋅:= BWmag1 1.494=

ENTER INITIAL VIBRATION AMP AND PHASE IN THE PLANE BELOW

z1 3.2:= ph1 30 deg⋅:=

x1 z1 cos ph1( )⋅:= x1 2.771=

y1 z1 sin ph1( )⋅:= y1 1.6=

ENTER TRIAL WEIGHT AT THE REFERENCE LINE ON THE PLANE: TW1 2.5:=

ENTER TRIAL WEIGHT ANGLE FROM REFERENCE LINE: TW1_Angle 143 deg⋅:=

ENTER VIBRATION AMP AND PHASE IN THE PLANE DUE TO TW1 BELOW:

z11 7:= ph11 77 deg⋅:=

x11 z11 cos ph11( )⋅:= x11 1.575=

y11 z11 sin ph11( )⋅:= y11 6.821=

Example 3-5 Vibration Isolation

By the way, knowing the weight W, we can select the right spring with stiffness to give ∆.

static deflection is the answer.∆ 2.679mm=Thus ,

This is lower than 0.15, so ok.TR2 0.074=TR2

1

2 π⋅ f2( )2 ∆

g⋅ 1−⎡⎢

⎣⎤⎥⎦

:=

For the above found ∆ , at 2200 cpm, the transmissibility TR2 is found below:

∆ 2.679mm=∆

14

g⋅TR 1+( )

TR π2

f12⋅⋅

⋅:=TR11

2 π⋅ f1( )2 ∆

g⋅ 1−⎡⎢

⎣⎤⎥⎦

g 9.81m

s2⋅:=

f2 22001

min⋅:=f1 1600

1min

⋅:=Not needed here.W 106.75 N⋅:=

TR 0.15=TR 1 Isolation−:=Isolation 0.85:=

Example 3-6 Unfavorable Speed And Amplitude Vs Speed of a Vehicle with a Spring-Model Suspension

L 14.63m:= ∆ 10.16cm:= g 9.81m

s2:=

k_over_mg∆

:= k_over_m 96.5551

s2=

VcritL

2 π⋅k_over_m⋅:= Vcrit 22.88

ms

=

Y 7.62cm:= V 64.4kmhr

:=

ωn k_over_m:= ωn 9.826rads

=

ω2 π⋅ V⋅

L:= ω 7.683

rads

=

XY

1ω

ωn

⎛⎜⎝

⎞

⎠

2−

:= X 0.196m= X 19.604cm=

Example 3-7 Accelerometer Performance

As a different example, change f2 to 12.5. Input period will be 0.4 s. Again,calibrated output z(t) will be same as input a(t) except for phase shift.

Plot of a(t)and calibratedz(t) vs t

0 0.2 0.4 0.6 0.8 1400

200

0

200

400

a t( )

z t( )

t

Note above:Instead of 1/ωn2 first term on RHS, we have 1/1 to include calibration so

that numerical input acceleration a(t) and output acceleration z(t) are same except forphase shift as seen by their plots below..

z t( )1−

1ω1

2Y1⋅ sin ω1 t⋅ φ1−( )⋅ ω2

2Y2⋅ sin ω2 t⋅ φ2−( )⋅+⎛

⎝⎞⎠⋅:=

a t( ) ω12

− Y1⋅ sin ω1 t⋅( )⋅ ω22

Y2⋅ sin ω2 t⋅( )⋅−:=

y t( ) Y1 sin ω1 t⋅( )⋅ Y2 sin ω2 t⋅( )⋅+:=

t 0 0.001, 50..:=

φ2 0.157=φ2π

2

ω2ωn

⋅:=φ1 0.079=φ1π

2

ω1ωn

⋅:=

Input period is 0.2 s.ωn 628.319=ωn 2 π⋅ fn⋅:=

τ2 0.1=τ2

1f2

:=ω2 62.832=ω2 2 π⋅ f2⋅:=

τ1 0.2=τ1

1f1

:=ω1 31.416=ω1 2 π⋅ f1⋅:=

fn 100:=Y2 0.05:=Y1 0.1:=f2 10:=f1 5:=ζ 0.707:=

Example 3-8

Matrix Operations: Eigenvalues and Eigenvectors

EVA eigenvals D( ):= EVA47.602

81.929⎛⎜⎝

⎞⎠

= N_fEVA1

EVA2

⎛⎜⎜⎝

⎞

⎠:= N_f

6.899

9.051⎛⎜⎝

⎞⎠

=

EVE eigenvecs D( ):= EVE0.998

0.068−

0.739

0.674⎛⎜⎝

⎞⎠

=

P EVE:= P0.998

0.068−

0.739

0.674⎛⎜⎝

⎞⎠

= PT 0.998

0.739

0.068−

0.674⎛⎜⎝

⎞⎠

=

GMM PT M⋅ P⋅:= GMM107.008

3.123 10 15−×

8.653− 10 15−×

781.321

⎛⎜⎜⎝

⎞

⎠=

ONP

P1 1,

GMM1 1,

P2 1,

GMM1 1,

P1 2,

GMM2 2,

P2 2,

GMM2 2,

⎛⎜⎜⎜⎜⎜⎝

⎞

⎟⎟⎟

⎠

:= ONP0.096

6.607− 10 3−×

0.026

0.024

⎛⎜⎝

⎞

⎠=

CheckIdntyMtx ONPT M⋅ ONP⋅:= CheckIdntyMtx1

0

0

1⎛⎜⎝

⎞⎠

= Checks !!

CheckEigenVal ONPT K⋅ ONP⋅:= CheckEigenVal47.602

0

0

81.929⎛⎜⎝

⎞⎠

= Checks !!

Set Math - Options - Array ORIGIN to 1.

W 3220:= l1 4.5:= l2 5.5:= k1 2400:= k2 2600:=

r 4:= g 32.2:= JcWg

r2⋅:= Jc 1600= l 10:=

mWg

:= m 100= Mm

0

0

Jc

⎛⎜⎝

⎞

⎠:= M

100

0

0

1600⎛⎜⎝

⎞⎠

=

Kk1 k2+

k2 l2⋅ k1 l1⋅−

k2 l2⋅ k1 l1⋅−

k1 l12

⋅ k2 l22

⋅+

⎛⎜⎜⎝

⎞

⎠:= K

5000

3500

3500

127250⎛⎜⎝

⎞⎠

=

D M 1− K⋅:= D50

2.188

35

79.531⎛⎜⎝

⎞⎠

=

Example 3-9

Unit-Step Response of a Higher Order Vibration System

%Unit-Step Response of C(s)/R(s) and Partial-Fraction Expansion of C(s) num=[0 3 25 72 80] den=[1 8 40 96 80] step(num,den) v=[0 3 0 1.2],axis(v),grid title('Problem A-5-9 Pages 302 to 304') num1=[0 0 3 25 72 80] den1=[1 8 40 96 80 0] [r,p,k]=residue(num1,den1)

0 0.5 1 1.5 2 2.5 3 0

0.2

0.4

0.6

0.8

1

Time (sec)

Amplitude

Example 3-10

Maximum Overshoot and other Transient Response Characteristics of a Second-Order Inertia-Damper System

ts5 6.579=ts53σ

:=

ts2 8.772=ts24σ

:=

Mp 0.254=Mp e

σ

ωd

⎛⎜⎝

⎞

⎠− π⋅

:=

tp 3.007=tpπ

ωd:=

tr 1.897=trπ β−

ωd:=

β deg 66.422deg=β deg β:=β 1.159=

β atanωdσ

⎛⎜⎝

⎞

⎠:=

σ 0.456=σ ζ ωn⋅:=

ωd 1.045=ωd ωn 1 ζ2

−⋅:=

ωn 1.14:=ζ 0.4:=

Example 3-11

Use of Simulink to Demonstrate Block Diagram Concepts in Vibrations

Simulink enables representation of transfer function blocks “intelligently” and thereby simulate the system performance. The Simulink version of Example 3-10 is shown below. Due to graphic limitations, the text is not visible clearly. They are as follows: The left block diagram is a unit-step input icon. The summer is the one to its right. The system transfer function is 1.42 / (1.09 s^2 + s). The block on the right is the scope to view the response and the system has a unit feedback.

1.42

1.09s +s2

Transfer FcnStep Scope