: ANALYSIS AND DESIGN OF VEHICLE SUSPENSION SYSTEM USING MATLAB AND SIMULINK

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1 : ANALYSIS AND DESIGN OF VEHICLE SUSPENSION SYSTEM USING MATLAB AND SIMULINK Ai Mohammadzadeh, Grand Vae State Univerit ALI R. MOHAMMADZADEH i urrent aitant profeor of Engineering at Shoo of Engineering at Grand Vae State Univerit. He reeived hi B.S. in Mehania Engineering from Sharif Univerit of Tehnoog And hi M.S. and Ph.D. both in Mehania Engineering from the Univerit of Mihigan at Ann Arbor. Hi reearh area of interet i fuid-truture interation. Saim Haidar, Grand Vae State Univerit SALIM M.HAIDAR i urrent aoiate profeor of Mathemati at Grand Vae State Univerit. He reeived hi B.S. in Mathemati with a Minor in Phi from St. Vinent Coege, and hi M.S. and Ph.D. in Appied Mathemati from Carnegie-Meon Univerit. Hi reearh tudie are in appied noninear anai: partia differentia equation, optimization, numeria anai and ontinuum mehani Amerian Soiet for Engineering Eduation, 006 Page.3.

2 Anai and Deign of Vehie Supenion Stem Uing MATLAB and SIMULINK Aireza Mohammadzadeh, Saim Haidar Grand Vae State Univerit Overview Athough tetboo,, 3, 4, 5, 6 in the area of vibration empo oftware too, uh a MATLAB, Mathad, Mape, in their treatment of vibration prinipe and onept; however mot of their overage of the ever important roe of tehnoog in teahing vibration i imited to ioated uage of thee too in ome end of the hapter omputer probem. Seond, their treatment appear to fou primari on the preentation of the programming apet of the iue without muh anai and deign of vibration tem. In vibration, the impet mode repreenting a tem i a inear, umped parameter, direte tem mode, whih require oniderabe anatia and omputationa effort for tem with more than two degree of freedom. In uh irumtane, the ue of oftware program, uh a MATLAB and Mathad are eentia in obtaining numeria reut in order to undertand and predit tem phia behavior. For eampe, the natura frequenie and mode hape of a four degree of freedom mode of an automobie upenion tem are, in genera, pair of ompe onjugate for whih hand auation and etration i a formidabe ta, if not impoibe. Suh tudie an be eai done in MATLAB or a Mathad environment. Eampe ie thi, mae it more and more evident to the teaher of vibration that the bet approah to teahing vibration onept and prinipe i to arefu integrate omputationa method avaiabe in mot oftware program with the theor. Athough the treatment of automobie upenion tem i a tandard appiation of vibration theor, the appiation of MATLAB and SIMULINK to it i an origina frame wor. A a frequent intrutor of vibration oure, one of the author reguar reeive ompimentar opie of tetboo on the ubjet of vibration eah and ever ear from a number of pubiher. In neither the graduate eve tetboo, uh a the one b, Weaver and Timoheno 7, Meirovith 8, Ginberg 9, de Siva 0, Benaroa, or the undergraduate eve tet, uh a the one b Thomon, Tongue, Inman, Rao, Beahandron, Ke have we een or notied a ompete treatment of upenion probem. For intane, Thomon over the free vibration mode of upenion tem with no damping eement invoved. Inman onider damping in the mode but regard on free vibration and avoid the ompe onjugate eigenvaue invoved. On the other hand, Meirovith preent a fored vibration formuation of the upenion mode, however, avoid the oution part a together. None of thee tetboo mentioned above, preent derivation and formuation for bae eitation of the upenion tem a it i preented in our paper. That i a -degree of freedom mode. Beide, in deriving Page.3.

3 the equation of motion for their probem, a tetboo author mentioned above ue Newtonian mehani, wherea in our aroom projet we introdued our tudent to anatia approah of Lagrange equation. We haven t een uh treatment done in an tetboo, in onnetion with the upenion probem. A we mentioned before, no oution to atua repone of the upenion tem eept for impe ae of free Vibration i avaiabe in an of the above tetboo. In our ae of bae eitation, one i atua deaing with two input one in the front, and one in the rear tire to the tem. A it i hown in the paper, the tranfer funtion due to eah input ha a 3 rd degree ponomia in the nominator and a 4 th degree ponomia in the denominator Let u not forget that there are 4 of uh tranfer funtion in the paper. It i a formidabe ta to find the repone of the tem b hand auation. It i in here and in thi apait where we introdued our tudent to thee eiting feature and too in MATLAB and SIMULINK to provide inight about the tem repone, and, at the ame time, guide our tudent to better undertanding of vibration onept b tring to engage them in deign of a better tem. The main objetive of the foowing projet, given to our tudent in vibration a, i to hep tudent undertand and appreiate prinipe and onept of vibration through an effetive integration of oftware program, MATLAB and SIMULINK, with theor. Thi further highight the need for integration between mathematia anai and engineering tem deign. After the aignment of the foowing projet it beame inreaing evident to the author of thi artie that the ombination of MATLAB and SIMULINK i a powerfu too whih add a new dimenion to reearh in vibration tem ontro and to the intrution of vibration oure ine it ha the promie of aiding tudent to undertand muh better the vibration prinipe. Our tudent howed deep undertanding of uh prinipe, a a reut. SIMULINK i an interative environment for tem imuation and embedded tem deign. A a patform for muti-domain modeing and imuation, SIMULINK et tudent preie deribe and epore a tem behavior. In addition, SIMULINK, provide a graphia uer interfae that i often muh eaier to ue than traditiona ommand-ine program. Integration of SIMULINK into the vibration intrution wi therefore be of great pedagogia vaue. To meet thee objetive and to atif the ABET requirement for enhaning the deign ontent of engineering urriuum, the foowing projet wa aigned to tudent in vibration a. Student ued both MATLAB and SIMULINK in thi projet to both anaze and deign automobie upenion tem. Negeting the ma of tire and the roing motion of the vehie, and ombining the tiffne and damping effet of tire and upenion tem into an equivaent damping and tiffne tem, a preiminar mode baed on the boune and pith motion of the vehie i onidered. Student were then aed to ue Lagrange equation to derive the governing differentia equation of motion, for the boune and pith motion of the vehie. MATLAB wa then ued to arrive at the natura frequenie and mode hape of the tem. SIMULINK wa empoed to verif the reut obtained in MATLAB b potting the Power Spetra Denit of the repone due to initia ondition proportiona to one of the eigenvetor of the tem. Student further utiized SIMULINK to invetigate the repone to an arbitrar initia ondition, and the reaized whih of the two motion of boune and pith wa the dominant one in the enuing motion. Page.3.3

4 Frequen Repone Funtion, FRF, for both motion wa obtained uing MATLAB. MATLAB and SIMULINK were then utiized to arrive at tem repone to the rough terrain. To ower the intenit of the annoing pith motion of the vehie SIMULINK, a a deign too thi time, wa ued to find a proper damping for upenion tem to ahieve thi goa. Student feedba with repet to the projet wa ver poitive. The a enjoed woring with SIMULINK epeia due to the reative eae in buiding the tem mode in omparion with the orreponding MATLAB mode. In hort, tudent indiated that SIMULINK heped them a ot in ahieving a deeper, hoiti undertanding of the oure materia and it objetive b promoting a virtua aborator for vibration onept. Probem Statement Figure Center of ma Figure Page.3.4

5 An automobie on a rough terrain, uh a the one hown in the Figure, ehibit boune, pith, and ro on top of it rigid bod motion. In thi anai, we aume that the roing motion ompared to the two other tpe of oiator motion i negigibe. Negeting the roing motion and ma of tire, and ombining the tiffne and damping effet of tire and upenion tem into an equivaent damping and tiffne tem, a preiminar mode for automobie upenion tem i preented in the Figure. Initia vaue for the repetive inertia, damping oeffiient, and pring rate are a foow: m = 000 g J = 500 g.m = = N/m = = 3000 N./m = m and =.5 m Where m i the auto bod ma, J i it moment of inertia about the enter of ma, inde refer to front upenion tem wherea inde refer to rear upenion tem, and and are the ditane between the enter of ma and front and rear upenion repetive. The ar i aumed to be traveing at 50 m/hr and the road i approimated a inuoida in ro etion with ampitude of 0 mm and the waveength = 5 m. Formuation a Uing Lagrange equation derive the governing differentia equation of motion, deribing the boune and pith motion. b Uing MATLAB, obtain the natura frequenie of the tem and the orreponding mode hape. Verif the reut in part b b buiding a SIMULINK mode of the tem. Simuate eah mode and how that the tem oiate at the repetive natura frequenie. d Auming free vibration of the tem under the initia ondition 0 = 4 mm and 0 = 0.05 radian with the initia veoitie aumed zero, whih mode ontribute the mot to the enuing motion of the tem? Subtantiate our anwer uing SIMULINK. e With the hep of MATLAB obtain the Tranfer Funtion for both the boune and pith motion. f Uing SIMULINK, obtain the tem repone to the road eitation a i deribed above. g It i we etabihed that the pith motion i the mot annoing motion for the ar paenger. Deign our upenion tem to ower the boune motion from it urrent vaue. Jutif our anwer b imuating the reut in SIMULINK. Doe our deigned upenion ower the boune magnitude ao? The governing tem of differentia equation whih deribe the boune and pith motion of the tem hown in Figure i found uing Lagrange Equation. The generaized oordinate t and t are ued to deribe the boune and pith motion of the auto bod. The ineti energ i deribed in Equation a: Page.3.5

6 J m T The potentia energ i deribed in Equation a: U Raeigh diipation funtion deribing viou diipation in the damper i: Q 3 The Lagrangian evauated from and, and together with 3 ubtituted in 4 and 5 one obtain equation of motion. U T L Q L L dt d 4 Q L L dt d 5 The appiation of Equation 4 and 5 ied: m J The equation of motion an ao be hown in matri form a: 0 0 t t t t J m 6 t t Soution Part b Our firt attempt i to find the damped natura frequenie and the mode hape of the damped tem. To thi end we et the right ide of equation 6 to zero. Auming a harmoni repone, Page.3.6

7 the harateriti equation for the tem i found b etting the determinant of the harateriti matri to zero. m det J 0 7 We an now ue MATLAB to do the agebra and find the harateriti root. The foowing MATLAB eion wa performed to get the ompe onjugate pair of root. We tae advantage of MATLAB funtion onv, and root to obtain the harateriti root. MATLAB Code to Obtain Damped Natura Frequenie and the Mode Shape >> m = 000; J = 500; = 30000; =30000; = 3000; =3000; >> =; =.5; >> a = [m + +]; >> b = [J *^+*^ *^+*^]; >> C = onva,b C =.0e+009 * >>d = [*-* *-*]; >> e = onvd,d e = >> f=.0e+009*[ ] - [ ] f =.0e+009 * >> r = rootf r = i i i Page.3.7

8 i The reut above indiate that the firt and eond damped natura frequenie are: / rad. / rad. n d n d The negative ign in front of the rea part of the ompe root indiate the deaing nature of the oiation / rad. / rad. n n Equation 9 and 0 render; n rad/ and n 6.55 rad/ = The mode hape an be found b: i m B A i m B A So the firt mode hape i: B A And the eond mode hape i: B A The mode hape indiate that there i no phaing in the mode a epeted in the proportiona damping ae. Let u ee if we an get the natura frequenie and the mode hape of the tem b etting the damping matri and the right ide of equation 6 equa to zero. We empo MATLAB funtion Page.3.8

9 eig to etrat natura frequenie and mode hape b running the ript fie AeeEigen.m. Thi fie i: % Cauating Eigenvaue and Eigenvetor m = 000; J = 500 ; = 30000; =30000; =; =.5; %Etabihing Ma Matri and Stiffne Matri m=[m 0; 0 J]; = [+ *-*; *-* *^+*^]; % Caing Funtion "eig" to Obtain Natura Frequenie and Mode Shape [u,amda]=eig, m; fprintf'\n' dip'natura Frequenie are:' % Print Natura Frequenie w = qrtamda fprintf'\n' % Print the Mode Shape dip'mode hape are:' fprintf'\n' dip'u=' fprintf'\n' dipu Running AeeEigen in MATLAB provide undamped natura frequenie and it orreponding mode hape. >> AeeEigen Natura Frequenie are: w = Mode hape are: u= Page.3.9

10 Thi indiate natura frequenie of n rad/ and n 6.55 rad/, whih are amot eat the ame frequen vaue obtained b equation and. From the MATLAB fie above, the firt mode i: Whie the eond mode i: Thee are amot eat the ame a mode hape found in 3 and 4, for the damped ae. Part SIMULINK i ued in thi part to verif the reut obtained above in part b. Foowing i the mode buit for thi purpoe. Figure 3 Page.3.0

11 We provide the firt eement of the mode vetor, a the initia ondition for integrator and the eond eement of the mode vetor,, a the initia ondition for the integrator 3 in the Figure 3. Upon running the imuation, the power petra denit bo in the Figure 3 wi provide the frequen ontent of the repone for both boune and pith motion. Sope in the diagram ao wi provide the damped tem repone for both boune and pith motion, with ope in the diagram rendering the pith motion and ope bo in Figure 3 ieding the boune motion. Let u eamine the reut of uh a imuation. Figure 4 in the net page how the output from the power petra denit for the boune motion of the auto bod. Notie the time hitor of the repone whih ear depit the initia ondition, Figure 4 Page.3.

12 A it i een from Figure 4 the power petra denit indiate that the repone of the boune motion i taing pae at a damped frequen of 5 rad/. Thi agree ver we with our previou auation for the damped natura frequenie in part a, whih rendered a firt damped natura frequen of rad/ See equation 9. For the ae of pae, and brevit we wi not how the reut of the power petra ope for the boune motion for the firt mode. However the reut from uh ope reonfirm that the boune motion i ao ha a frequen of 5 rad/. Intead we wi provide the ope reut for the boune motion for the eond mode of vibration. To thi end we wi et the initia ondition in the appropriate integrator bo aording to the eond moda vetor obtained in part b. That i: A B Firt et u oo at the ope reut whih wi provide the time domain repone for both boune and pith motion. Thee are: Time hitor for the boune free repone Figure 5 A it i ear een from Figure 5 the deaing oiator motion of the repone i evident from the ope reut. B zooming on the repone tudent an obtain the damped natura frequen of the repone and b empoing the onept of the Logarithmi Derement the an arrive at the damping ratio of thi mode. We wi not provide the detai of uh a proedure, but uh meaurement and auation wi reonfirm the earier reut obtained in part b above and render the ame damped natura frequen and damping ratio for the eond mode. That i: d n 6. 68rad/.363 Page.3.

13 The reut from the pith ope and it power petra denit are: Time hitor and power petra denit of the pith motion for the eond mode Figure 6 Page.3.3

14 Obviou thee reonfirm our previou obervation. Part d In Figure 3, the initia ondition 0.04 m and 0.05 rad wi reut in the foowing: Power petra denit ope reut for Pith motion for arbitrar initia ondition Figure 7 Due to pae imitation and for the ae of brevit we wi not how here the Power Spetra Sope reut for the boune motion. However, the reut from the dipa of that ope indiate, a Figure 7 doe, that the eond mode pa the dominant roe for thee partiuar initia ondition. That i the ampitude of the ontribution of the firt mode i muh e than the eond mode ampitude for the tem free repone. Page.3.4

15 Part e Upon taing the Lapae tranform from equation 6 and auming zero initia ondition, we wi arrive at the domain equation for the tead repone of the tem a foow: Y Y X m 8 Y Y J X 9 Uing Cramer method and apping the prinipe of uperpoition, we an obtain the tranfer funtion for both boune and pith motion for eah input Y and Y. Due to pae imitation we on provide the reut for input Y a foow: J m J Y X And Y J m m A we mentioned we wi not provide the reut for input Y here, however, the MATLAB ode provided beow, and the SIMULINK imuation in part f, wi ear provide thoe tranfer funtion aoiated with thi input. The MATLAB ode for arriving at thee tranfer funtion and it i reut for running the MATLAB ript TraferFn.m i: MATLAM m Fie for Obtaining Tranfer Funtion % Input Data*********************************************** m = 000; J = 500; = 30000; =30000; = 3000; =3000; =; =.5; % Etabihing the Ponomia***************************** a = [m + +]; b = [J *^+*^ *^+*^]; C = onva,b; d = [*-* *-*]; e = onvd,d; e = [0 0 e]; % Finding Charateriti Ponomia' Coeffiient************ f= C-e g= [ ]; h=onvb,c; i=[* *]; p=[*-* *-*]; q=onvb,g+[0 onvi,p]; % Tranfer Funtion for Boune Motion Due to Y Input******** Page.3.5

16 = tfq,f o= onv-a,i ; = onvg,p; =o-[0 ]; % Tranfer Funtion for Pith Motion Due to Y Input******** = tf,f gg=[ ]; ii=[* *]; qq=onvb,gg- [0 onvii,p]; % Tranfer Funtion for Boune Motion Due to Y Input********** =tfqq,f oo=onva,ii; =oo-[0 onvgg,p]; % Tranfer Funtion for Pith Motion Due to Y Input********** =tf,f Running the above.m fie give: >> TranferFn Tranfer funtion: X Y 7.5e006 ^ e008 ^ e e e006 ^ e007 ^ e008 ^ +.5e e009 Tranfer funtion: Y -6e006 ^3-8.5e007 ^ - 4.5e e e006 ^ e007 ^ e008 ^ +.5e e009 Tranfer funtion: X Y 7.5e006 ^ e007 ^ + 4.5e e e006 ^ e007 ^ e008 ^ +.5e e009 Tranfer funtion: Y 9e006 ^3 +.5e008 ^ + 4.5e e e006 ^ e007 ^ e008 ^ +.5e e009 Page.3.6

17 Thi i obviou what i aed in part e. Part f We wi now imuate the motion of the ar on the road. We have aumed that the road i approimated a inuoida in ro etion with ampitude of 0 mm and the waveength = 5 m. The ar i traveing at 50 m/hr. thee ondition provide the input Y and Y for the imuation. The period and e frequen for the harmoni input and the phae dea due to input Y are a foow: Then: 5 T 0.36 V rad / T.5 rad 5 Y Ain t Y Ain t Whih upon ubtitution of,, and ampitude of the motion in the above equation ied: Y Y 0.0in 7.453t 0.0in7.453t Thee are the input to the SIMULINK mode in part f, whih i depited in Figure 8. Notie that thee harmoni funtion are hown in Figure 8 a ine wave and ine wave bo in Figure 8. The Tranfer Funtion whih were derived in the above MATLAB ode i ao een for eah input and tem repone. The bo propert for the ine wave bo wi enabe u to furnih the frequen, ampitude and phae for the eitation. Notie that the ummation bo impement the prinipe of the uperpoition and the ope wi provide u with the time repone of both the boune and pith motion of the repone. Figure 9 and 0 provide u with the time repone for the boune and the pith motion. Notie that the ope indiate the ear tranient ontribution to the repone with the tead ampitude for the boune motion being about 4mm and the pith ampitude of.60-3 rad. obviou the urrent vaue for the upenion tem effetive redue the vibration tranmitted to the automobie bod. A it wa tated before the ampitude of the road wave i 0 mm, whih on 4 mm i tranmitted to the bod of the automobie and even e than that to the paenger due to ioation tem for paenger eat. In the net and fina part of thi projet tudent reate a mode to further redue the tranmitted vibration a it i deribed in part g of thi doument. Page.3.7

18 Figure 8 Boune repone of the tem Figure 9 Page.3.8

19 Pith repone of the tem Figure 0 Part g In thi part we wi utiize the power of SIMULINK to teah tudent about the deign method in pratia vibration probem. Obviou optimization probem an be handed through MATLAB optimization paage; however we wi how here that SIMULINK mode preented i an eeent too to optimize the repone of the tem. Figure, i the mode whih wa buit to arrive at damping oeffiient vaue that further redue the ampitude of the pith repone of the tem. The bue oored bo in Figure wi depit a ider gain, whih wi hange the vaue of the damping oeffiient of the tem. B running the mode for evera vaue of damping oeffiient we wi obtain a trend for the tem repone. B tabuating the pi repone for different vaue of damping oeffiient one an zero in at the optimum vaue of the damping oeffiient. In thi paper we wi firt verif the tem repone for the initia vaue of the damping oeffiient, a wa done in part f in the above, and then how the effet of oupe of damping oeffiient vaue in the output repone. A it i evident from the output hitorie in Figure and 3, the reut for part f i eat repeated in the imuation of the mode hown in Figure. After mode verifiation, the net ta i to ower the unwanted pith motion, due to the road eitation, whih i diued ater. Page.3.9

20 Figure Figure Page.3.0

21 Figure 3 Now et u tr different vaue for damping oeffiient b hanging the vaue in the ider gain. We wi hooe a = 6000 N./m in the mode, depited beow, in Figure 4. Figure 4 Page.3.

22 The repone for boune and pith motion are hown in Figure 5 and 6 repetive. Figure 5 Figure 6 B further zooming at the pea of the tead repone we obtain that: Boune Ampitude = 7 mm Pith Ampitude =.50-3 rad Thi i higher than previou vaue obtained when the damping oeffiient were 3000 N./m. We now et the damping oeffiient to 000 N./m. and obtain the repone of the tem a hown for boune and pith motion in Figure 7 and 8 repetive. Page.3.

23 Figure 7 Figure 8 Further zooming on the pea of the tead repone in Figure 7 and 8 revea that: Boune Ampitude =.6 mm Pith Ampitude = 0-3 rad The reut indiate that b owering the damping oeffiient from initia vaue of the upenion tem, we wi redue the tranmitted oiation. Page.3.3

24 Aement Pedagogia tudent earn an ubjet matter in engineering, the bet, b atua apping it prinipe and onept via hand-on eperiment or in an appiation-oriented projet. Thi projet, whih too tudent 3 wee to ompete, gave them ampe opportunit to ue what the eaned in the aroom to anaze and deign an indutria mode of an automobie upenion tem. The projet wa onidered a 0% of tudent fina grade and gave an appiation oriented dimenion to mathematia nature of the topi diued in the oure. It made tudent go through man topi overed in the eture, uh a the ignifiane of ompe eigenvaue and eigenvetor and the roe the pa in tabiit of the tem, omparion of free undamped repone and damped repone, ogarithmi derement, tranfer funtion ignifiane, uperpoition onept in inear tem, -domain and time-domain oution tehnique, Lagrange equation, bo diagram and feedba onept. The ao earned new tehnoogia too to arr out thee tehnique in a omewhat reaiti etting b woring with MATLAB and SIMULINK. In partiuar, tudent wrote their own ode in MATLAB and buit their own mode uing SIMULINK. Upon ompetion of the projet, tudent feedba indiated that the projet wa ver intrumenta in undertanding the onept of the oure b requiring them to empo their aquired nowedge in the proe of anai and deign of the upenion tem. The overwheming preferred SIMULINK over MATLAB, due to it graphia and viua apabiitie and reative eae in buiding and modifing the appropriate mode. In ight of the imuation mode buit in thi projet and eperiene gained, the author beieve that there are ertain advantage in uing SIMULINK in a vibration and differentia equation oure. The diuion to foow i in agreement with the tudent onenu that SIMULINK mode are ver uefu in verifiation of man oure topi, both diret and indiret. The main benefit in uing SIMULINK in vibration oure i that it provide it uer with what might be aed a a virtua vibration aborator! That i tudent an imuate a tem and tud the nature of the tem repone, due to different input and initia ondition, b heing the output of the ope bo in their mode. Student an ee the effet of hanging tem parameter on the tem repone, b eai tweaing thee parameter in their graphia mode and oberving the outome on the mode ope. For eampe, tudent ued the ope output of part of the projet Figure 4 to arrive at damped natura frequen of the mode b imp meauring the time between the ubequent pea, and obtained tem damping ratio b meauring the ubequent ampitude ratio and empoing the ogarithmi derement formua. B etting the damping oeffiient to zero in their mode Figure 3 tudent oberved the hange in tem repone b notiing a ontant ampitude oiator repone. The ao oberved, in the proe, the oene of the numeria vaue of tem natura frequen and it damped natura frequen; omething that the ame aro before whie doing part b of thi projet. The were ao intruted to ue negative damping ratio and oberve the untabe repone of their tem. The onept of eigenvetor were peifia iutrated in the SIMULINK mode Figure 3 b mutiping the eigenvetor of the tem b an ontant vaue and notiing that thi reuted in no hange in the repone of the tem. A part d of the Page.3.4

25 projet indiate tudent were ao abe to ee the ontribution of eah mode to the tem repone a a reut of arbitrar initia ondition. One of tudent diffiutie, notied b the author through teahing vibration and differentia equation oure, i the idea behind the appiation of the Lapae tranform method to inear differentia equation. It wa intereting to oberve that how tudent appreiated the notion of onverting the ouped tem of imutaneou differentia equation to tem of agebrai inear equation in part e of thi projet, uing the dreaded Lapae tranform approah. SIMULINK mode buit in part f of the projet ha the advantage of howing the tranfer funtion due to eah input and output ear. Obviou, no ommand-ine programming oftware an how thee four tranfer funtion a ear and a effetive a SIMULINK doe See Figure 8, for eampe. Beide the SIMULINK mode impement the Invere Lapae tranform to obtain the time domain repone of the tem to the input eitation in a graphia approah. Something that i not done a eai a it i hown here b mean of an ommandine programming. The prinipe of uper poition i ao ear depited in the SIMULINK mode of part f Figure 8-0 b the ummation bo in the mode. The big advantage of uing SIMULINK in thi projet i it abiit to engage it uer in improving the deign of the deired tem with reative eae. The at part of the projet ertain mae ue of thi trength in the oftware b requiring tudent to deign for a maer tranmiibiit ratio. Obviou, we are taing about optimization tehnique, whih theoretia i beond the ope of a vibration oure in the junior eve ear. In addition, thi abiit i not provided in uh an effetive manner in ommand-ine oftware. Yet, thi i done eai in SIMULINK b having iterari a virtua aborator at our dipoa in term of a mode utiizing variabe gain e.g., variabe damping oeffiient to he for the improved repone of the tem. Moreover, a it i evident from the time-domain mode Figure, SIMULINK unie the ommand-ine oftware anguage doe not onvert the eond order differentia equation to tate form to obtain the oution. We beieve SIMULINK i a great too in eduation and/or indutr, due to it GUI feature and imuation apabiitie! Thi approah ontitute a new frame wor in vibration eduation. We woud ie to emphaize again, that the oution part of thee governing equation for the upenion tem, uing SIMULINK and MATLAB i ao new. Conuion Thi projet ear how how hepfu MATLAB and SIMULINK are to epoe vibration tudent to pratia probem with indutria impiation. Cear our aia method are not uffiient enough to ove appiation probem uh a thi one. In their feedba, tudent indiated a great ene of appreiation for thee oftware too, epeia SIMULINK, in heping them ahieve hoiti undertanding of phia onept of vibration oure. Out of thi eperiene, the author great beieve that integrating thee oftware too in vibration oure great improve tudent abiit to fae haenging appiation probem, find an appropriate oution uefu, and gain a trong ene ritia thining that hep them unite nowedge with human eperiene. Page.3.5

26 Bibiograph. Inman, Danie J., Engineering Vibration, /E, Prentie Ha, 00.. Rao, Singireu S., Mehania Vibration, 4/E, Prentie Ha, Thomon, Wiiam T., Daheh, Marie Dion, Theor of Vibration with Appiation, 5/E Prentie Ha, Tongue Benon, Prinipe of Vibration, /E Oford, Ke, S. Graham, Fundamenta of Mehania Vibration, /E, MGraw Hi, Baahandran, Baaumar, Magrab, Edward B., Vibration, Thomon, Weaver, W. JR., Timoheno, S.P., and Young, D.H., Vibration Probem in Engineering, 5 th. Edition Wie, Meirovith, Leonard, Fundamenta of Vibration, MGraw Hi, Ginberg, J. H., Mehania and Strutura Vibration: Theor and Appiation, Wie, de Siva, C.W., Vibration, Fundamenta and Pratie CRC Pre, Benaroa, H., Mehania Vibration, Anai, Unertaintie and Contro, nd. Edition, Mare Deer, Math Wor In., MATLAB, verion 7, Nati, MA: Math Wor, In., 005. Page.3.6