Numerical Impulsive Response Analysis of a Frame Structure A MATLAB Computational Project-Based Learning

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1 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú. umercal Impulsve Respose Aalyss of a rame Srucure A ATLAB Compuaoal Projec-Based Learg Hosse Rahem, Ph.D. Professor ad Charma, Deparme of Egeerg ad Techology, Vaugh College of Aeroaucs ad Techology,86-0 rd Aveue, lushg, Y, 69-USA, Emal:hosse.rahem@vaugh.edu Shazm Baksh echacal Egeerg Techology Program, Vaugh College of Aeroaucs ad Techology,86-0 rd Aveue, lushg, Y, 69-USA, Emal: shazm.baksh@vaugh.edu ABSTRACT The am s o mpleme a compuaoal projec-based learg mehodology egeerg courses ha ca corbue o he success ad mproveme of sudes egeerg ad egeerg echology programs. Ths learg mehodology s based o boh compuaoal ad programmg process ha ca ehace sude s mahemacal udersadg, crcal hkg, ad problem solvg sklls hrough he core egeerg courses. A compuaoal projec based o boh closed-form ad umercal echques wh ATLAB programmg wll be mplemeed o sudy a mpulsve respose of a frame srucure. The closed-form respose of such sysem ca be evaluaed hrough covoluo egral. or he case ha a srucure s subjeced o a rregular mpulsve forcg fuco such as sesmc vbrao, he umercal egrao ca be mplemeed wh ATLAB programmg o deerme he mpulsve respose of such sysem. I hs vesgao we wll aalyze he sysem respose hrough a fe dfferece umercal echque. The echque ca be mplemeed o vesgae he mpulsve respose for boh udamped ad damped sysem wh much less complex mahemacal volveme. A ATLAB scrp based o fe dfferece echque ca be developed o geerae umercal resuls of such excao durg ad afer acg mpulsve force. Keywords: Impulsve respose, xed fe dfferece, umercal-based ATLAB Scrp. ITRODUCTIO The excao of a sysem due o a suddely o-perodc appled force ha ca oly say wh he sysem for a very shor perod of me s kow as rase mpulsve respose. A closed-form soluo hrough covoluo egral ca be mplemeed o deerme he mpulsve excao uder a acg rase mpulsve force. or a rregular forcg fuco he umercal egrao aro Paz, 980; Sgresu S. Rao, 00 of covoluo egral s mplemeed o deerme he mpulsve excao. A ATLAB umercal-based scrp s developed o sudy ad deerme respose of oe-sory frame srucure durg ad afer acg mpulsve force. I hs vesgao a mxed forward-backward fe dfferece echques H. Rahem, 009 s mplemeed o develop umercal form of he goverg equao of moo due o a suddely appled mpulsve force. A ATLAB scrp s developed based o a fe dfferece umercal formulao o deerme respose of such sysem for boh damped ad udamped cases wh a rregular acg mpulsve force. 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE- Jue -4, 00

2 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú. However ay umercal soluo volved wh error, hrough Taylor s seres, usg boh forward ad backward fe dfferece, he rucao error bee esmaed o be order of sep sze. To crease he accuracy of he umercal resuls, oe should cosder he sep sze as small as possble. I hs vesgao we wll sudy wo cases of mpulsve vbrao; Udamped Sysem, Damped Sysem. or he case of udamped sysem he resuls by fe dfferece echque wll be compared by he umercal egrao of covoluo egral.. COPUTATIOAL PROJECT-BASED LEARIG ODEL The am s o mpleme a mehodology based o compuaoal projec-based learg model o ehace sude s learg objecves aalycal, umercal, crcal hkg, ad problem solvg sklls hrough he egeerg core courses H. Rahem, 009. gure s a graphcal model of compuaoal projec-based learg. gure : Compuaoal Projec-Based Learg odel The behavor of may egeerg sysems deformao, vbrao, hea rasfer, ad ohers ca be defed by he soluo of he goverg equao of he body H. Rahem, 007. The focus of hs sudy s gve o boh closed-form ad umercal soluo usg ATLAB programmg. I hs vesgao, we wll aalyze he mpulsve respose of oe-sory frame srucure by boh closed-form umercal egrao ad a mxed forwardbackward fe dfferece umercal echque.. IPULSIVE RESPOSE AALYSIS O OE-STORY RAE STRUCTURE I hs vesgao we wll sudy a problem relaed o he mpulsve vbrao aalyss of a oe-sory srucure wh four beam colums W8x4 supporg srucure. or hs problem boh closed-form umercal egrao ad umercal fe dfferece mehods ca be employed o deerme he respose of he srucure. A ATLAB scrp ca be developed o geerae resuls ad defy he vbrao behavor of srucure uder a acg mpulsve force as show g.. 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE- Jue -4, 00

3 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú. Kowg ha colums are rgd o he flour slab g. C, he oal sprg rae K T ca be expressed as follow 48EI K T 4K L Le s cosder W8x4 seel beam for he colum Wh I ad E 0e6 ps, he sprg Rae for four colums ca be evaluaed as 40e68.8 k T 0,000lb / 9 x The wegh of flour slab wh all oher dead loads ca be cosdered as 8.7 kps, hece, he mass ca be calculaed as m W g 8,700lb lb s / s. UERICAL ITEGRATIO The mpulsve respose based o covoluo egral aro Paz, 980 for a udamped sysem ca be expressed as x s d m 0 Above ca also be expressed as x m { s cos d cos s d } 0 0 The umercal egrao of a mpulsve respose based o a pecewse lear forcg fuco has bee suded by may researchers aro Paz, 980; Sgresu S. Rao, 00. Here, we wll develop a ATLAB scrp based o such umercal egrao o vesgae he mpulsve respose behavor of oe-sory fame srucure. g., s a represeao of a pecewse ler forcg relaoshp bewee me ad, ha s 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE- Jue -4, 00

4 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú. 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE-4 Jue -4, 00 4 Hece, he umercal form of respose aro Paz, 980 ca be expressed as { } cos s m x 5 where cos d [ ] { } s s cos cos ] s [s s d ad [ ] { } cos cos s s ] cos [cos A ATLAB scrp based o above umercal egrao s wre o geerae he umercal respose of a sysem uder mpulsve force. I hs sudy, he respose behavor of a oe-sory frame srucure based o a mpulsve forcg fuco as show fgure 4 has bee vesgaed. 0 0 for for for o >

5 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú. ATLAB Program - umercal Iegrao % umercal Aalyss of Covoluo Iegral % Impulsve Vbrao Aalyss of a Udamped Sysem % or a acg recagular force up o me % ad a Tragular decreasg force up o me % h mus be /0 o geerae smooh respose graph pu'ial me '; f pu'al me f'; pu'tme Durao of Impulsve force Row arx '; h pu'tme sep sze h'; m pu'ass, m '; k pu 'Sprg Cos. k '; o pu 'orce, o '; w k/m^.5; f-/h; zeros,; xzeros,; ; 0; 0; j; whle jjh; f j<,; jo; jj; elsef j<, & j>, j o/,-,*,-j; j,-j/,-j*j; else j 0; jj; ed jjj-j*j-j/h*/w*sw*j- sw*jj-j/w^*h*cosw*j- cosw*jw*j*sw*j-j*sw*j; jjj-j*j-j/h*/w*cosw*j- cosw*jj-j/w^*h*sw*j- sw*j-w*j*cosw*j-j*cosw*j; xj j*sw*j-j*cosw*j/m*w; f j > f, break, ed jj; ed dsp ' Tme Respose ' dsp [ x ] plo,x Upo execuo he resuls ca be geeraed as show Ial me 0 al me f Tme sep sze h0.0 Tme Durao of Impulsve force Row arx [ ] ass, m 00 Sprg Cos. k orce, o The codoal loop ca be adjused o cosder he effec of a specfc appled mpulsve force. 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE-5 Jue -4, 00

6 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú.. IITE DIERECE UERICAL AALAYSIS The goverg equao of a sysem uder a exeral force ca be expressed as d x c dx k x 6 d m d m m or a acg mpulsve force wh varous force fucos, s o feasble o deerme he aalycal soluo. However, a soluo based o fe dfferece echques ca be mplemeed wh ATLAB o deerme respose of he sysem durg ad afer acg mpulsve force. Above equao ca also be expressed as dv d c K V x 7 m m m Usg forward dvded dfferece, frs dervave ca be approxmaed as dv d V V ± O h h The frs forward dfferece of he goverg equao wll provde velocy a me, ha s V m c V m k x m h 9 The frs backward dfferece of above equao wll provde respose a me, ha s c k x V x h x m m m 0 A geeral ATLAB program based o mxed dfferece forward-backward approxmao of above fe dfferece umercal formulaos s developed o defy he vbrao behavor of a damped ad udamped srucure uder a mpulsve acg force. 8 ATLAB Program xed orward-backward e Dfferece Techque % Impulsve Vbrao, x e Dfferece % h mus be very small comparso o h«/0 pu'ial :'; f pu'fal f:'; h pu'sep sze h:'; x pu'ial x:'; m pu'mass, m '; c pu 'Dampg Coef., c '; k pu 'Sprg Cos. k '; o pu 'orce ampl., o '; v pu'ial v:'; pu'tme Durao of Impulsve force Row arx '; f-/h; zeros,; xzeros,; vzeros,; ; x x; v v; j; 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE-6 Jue -4, 00

7 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú. whle f j<,; jo; elsef j<, & j>,; j,-j/,-,*o; else j0; ed xj /m*j- k/m*xj-c/m*vj*h^ vj*h xj; vj /m*j-k/m*xj-c/m*vj*h vj; j jh; f j > f, break, ed jj; ed dsp ' Tme Respose velocy ' dsp [ x v ] plo,x Upo execuo he resuls ca be geeraed as show al :0 fal f: sep sze h:0.00 Ial x:0 mass, m 00 Dampg Coef., c 0 Sprg Cos. k orce ampl, o Ial v:0 Tme Durao of Impulsve force Row arx [ ] or a me sep sze, h /00, he resuls by fe dfferece echque Table comparso o he resuls of closed-form umercal egrao Table produce a error of ± % error order of sep sze. By cosderg he ar dampg effec, C 50 lb-sec/, upo execuo of ATLAB program wh he eree of he followg pu daa hrough commad wdow, he mpulsve vbrao behavor ca be defed for a damped oe-sory frame srucure, ha s 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE-7 Jue -4, 00

8 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú. Ial :0 fal f:4 sep sze h:0.00 Ial x:0 mass, m 00 Dampg Coef., c 50 Sprg Cos. k orce ampl, o Ial v:0 Tme Durao of Impulsve force Row arx [ ] I hs case due o dampg effec, he mpulsve respose eveually des ou afer several vbrag cycles. 4. COCLUSIO As ca be observed, he umercal mx fe dfferece echque wh a me sep sze, h /00, provde resuls ha are closely compable wh closed-form umercal egrao approach. Developme of fe dfferece umercal echque for mpulsve respose aalyss s much less me cosumg whou ay complex mahemacal mplcao. The ATLAB programmg o oly faclaes he compuaoal process bu also eable us o udersad he behavor of a egeerg sysem based o a gve mahemacal model. I hs vesgao he ATLAB programmg eabled us o udersad graphcally he behavor of a mpulsve respose based o boh he closed-form umercal egrao ad fe dfferece umercal approach. The resuls showed ha he accuracy fe dfferece umercal aalyss ca be acheved by selecg a proper sep sze, whch s h me durao of acg force/00. The compuaoal projec-based learg model wh ATLAB programmg delvered learg objecves ha are requred o movae sude s mahemacal, crcal hkg, ad problem solvg sklls. I ca be observed ha ATLAB programmg eable boh researchers ad sudes wh a ool o udersad dffere mahemacal echques aalyzg ad sudyg behavor of a egeerg sysem. 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE-8 Jue -4, 00

9 Eghh LACCEI La Amerca ad Carbbea Coferece for Egeerg ad Techology LACCEI 00 Iovao ad Developme for he Amercas, Jue -4, 00, Arequpa, Perú. The preseed aalyss s based o a acg mpulsve force o he srucure. However, due o sesmc groud excao, a fuure sudy wll cosder a mpulsve behavor of oe-sory frame srucure based o earhquake groud excao. REERECES aro Paz, Srucural dyamcs, heory ad compuao, Va osrad Rehold Compay Ic, 980 Sgresu S. Rao, echacal Vbrao, 4h edo, Pearso prece Hall,.J., 00. Hosse Rahem, A e Dfferece ad e Eleme umercal Approach wh ATLAB d draf, Vaugh College Book Sore, ew York, 009 Hosse Rahem, Deermsc Thermal racure Lfe Aalyss of a Vsco-elasc Cylder Paper preseed a ffh LACCEI Ieraoal La Amerca ad Carbbea Coferece for Egeerg ad Techology, ay 9 - Jue 0 007, Tampco, éxco Hosse Rahem ad Paul LaVerge, Egeerg Aalyss wh ATLAB programmg-a approach o ehace eachg ad learg effecveess Paper preseed a seveh LACCEI Ieraoal La Amerca ad Carbbea Coferece for Egeerg ad Techology, Jue 0-05, 009, Sa Crsobal, Veezuela AUTHORIZATIO AD DISCLAIER Auhors auhorze LACCEI o publsh he paper he coferece proceedgs. eher LACCEI or he edors are resposble eher for he coe or for he mplcaos of wha s expressed he paper. 8 h La Amerca ad Carbbea Coferece for Egeerg ad Techology Arequpa, Perú WE-9 Jue -4, 00