Bullet of te Trslv Uversty of Brşov Vol 5 (54-1 Seres 1: Specl Issue No 1 MATLAB PROGRAM FOR THE NUMERICAL SOLUTION OF DUHAMEL CONVOLUTION INTEGRAL M BOTIŞ 1 Astrct: I te ler lyss of structures troug modl superposto fter decouplg te moto equtos, tere must e solved te moto equtos for ec degree of dymc freedom, troug umercl tegrto, y usg te Dumel covoluto tegrl To fd structurl respose to ertquke cto t s lso eeded to determe te mxmum spectrl vlues dsplcemet, speed d ccelerto for y ccelerogrm of te ste; tese vlues re determed y umercl tegrto wt Dumel tegrl I ts pper t s preseted Mtl progrm for tegrtg umercl of Dumel covoluto tegrl wt trpezods metod tt c e used ot for modl lyss d spectrl lyss of structures Key words: Dumel tegrl covoluto, dymc lyss y modl superposto, spectrl lyss ertquke egeerg, dymc respose of te structures for rdom forces 1 Itroducto Dymc lyss of cvl structures s ecessry we ctos re tme-vryg If dymc lyss s ler, modl superposto prcple c e used To use modl superposto prcple we must frst ceve modl decomposto d te mke modl superposto After decomposto modl vrto modes re oted, wc glgt modl deformed spes ssocted wt ec mode of vrto d ertl mss correspodg to ec mode of vrto Troug modl decomposto two mportt results re oted, oe s qulttve d s represeted y te form of ow modes of vrto d oter s qutttvely d dctes level of prtcpto of te ertl msses te movemet of dymc system Te lyss of te vrto mode s very mportt ecuse ec vrto mode s cotruto for te moto of te dymc system After spectrl decomposto, modl superposto c e mde, wc volves composg sgfct vrto modes moto dymc system I cse of system wt degrees of freedom (fg1 dymc equto of moto s: [ M ]{ x} [ C]{ x} [ K]{ x} { F} (1 were: -[M ] ertl mtrx of dymc system; -[C] dmpg mtrx of dymc system; -[K] stffess mtrx of dymc system; 1 Dept of Cvl Egeerg, Trslv Uversty of Brşov
46 Bullet of te Trslv Uversty of Brşov Vol 5 (54-1 Seres 1 ẋ -{ } ccelertos vector for te degrees of freedom dymc; -{ẋ} velocty vector for te degrees of freedom dymc; -{x} dsplcemet vector for te degrees of freedom dymc; -{F} force vector for te degrees of freedom dymc Istteous dymc respose o te degree of freedom k s oted y ler superposto of stteous dymc resposes correspodg to te modes of vrto fg: k x ( x ( t ( 1 k, To dscoect equtos of moto te followg ler trsformto s used: k 1 k, ( k, ( 1 x ( x (3 Reltosp etwee turl coordtes dymc system x k (, k 1 d prcpl coordtes or modl (, 1 c e expressed y te modl mtrx s follows: Fg 1 Te system wt degrees of freedom dymc x x x x 1 1 1,1,1 1,1,1 1, 1,,, 1, 1 1, 1, 1, 1 1, 1,,, 1 1 Dsplcemet vector velocty d ccelerto o te dymc degrees of freedom k 1 ecome: Fg Vrtos mode d egevlues for system wt degrees of freedom dymc { } [ ]{ ( }; { } [ ]{ ( }; (4 { } [ ]{ ( } were: -[] modl mtrx -{ ( } dsplcemets vector turl coordtes -{ ( } velocty vector turl coordtes
M BOTIŞ: Mtl progrm for te umercl soluto of Dumel covoluto tegrl 47 -{ ( } ccelertos vector turl coordtes Replcg te system equtos of moto (1 rel coordtes wt modl coordtes, te equtos of moto s relesed Due to ortogolty of vectors ssocted vrto modes system of equtos (1 s reduced to decoupled equtos of moto for ec degree of freedom dymc: As te system of dfferetl equtos of moto s decoupled, we c wrte te correspodg equto of moto for ec degree, s follows: F { ( } { ( } { ( } (5 M 1 were: T - F { F( } correspodg equvlet force mode ; T - M M equvlet mss mode ; - correspodg dmpg mode -[ C] [ M ] [ K] Ryleg dmpg mtrx To solve decoupled equtos of moto, Dumel covoluto tegrl c e used Dsplcemet respose oted usg Te covoluto tegrl Dumel tegrl o ec dymc degrees of freedom s: t 1 ( t ( F ( e s ( t d M 1 (6 were, - 1 dmped pulsto If you kow te swer dsplcemets o ec dymc degree of freedom te we c clculte veloctes ccelertos d ertl forces for dymc system: { } [ ]{ ( }; { } [ ]{ ( }; { } [ ]{ ( } Te lyss of te dymcs of system wt degrees of freedom dymcs c e see s solvg te system of equtos s reduced to clculto covoluto tegrl Dumel For ts reso ts pper presets Mtl progrm for umercl tegrto of covoluto tegrl Dumel If te t pply H mpulse of te system, ts s removed from equlrum (7 posto wt velocty x H / m, (fg3 d performs free dmped vrtos descred y te equto : H e m H ( t e m ( t s ( t s ( t for t (8 Fg 3 Free respose dsplcemet due to mpulse fte H If te dymc system rdom dsturg force cts,(fg4 equto
48 Bullet of te Trslv Uversty of Brşov Vol 5 (54-1 Seres 1 of moto of te dymc system s: F( m Acto of dsturg force ppled dymc system t rest c e represeted s cotuous elemetry mpulses, o te tme tervl from t t After pplyg elemetry mpulse dsplcemet respose of te dymc system dymc o degree of freedom s: dh ( t d e s ( t d m (9 F( ( t e s ( t d m Dymc dsplcemet respose of te system s ceved y superposto of effects s tegrl mout of moto geerted y elemetry pulse successo o te legt of tme tt force cts [1]: F( ( t e s ( t d m f ( ( t F e s ( t d m Fg 4 Dsplcemet respose oted y superposto of te effects of ftesml mpulses dh If dsturg force s ppled to dymc system wle t s moto due to tl dsplcemet d velocty codtos, te dsplcemet respose s expressed: e [( x x cost s st] f ( ( t F e s ( t d m were, - x tl dsplcemet for t= - x tl velocty for t= It c e see tt f te dymc system wt sgle degree of freedom dymc system to determe dymc respose Dumel covoluto tegrl must e clculted Mtemtcl spects of umercl soluto of te Dumel tegrl To tegrte umerclly covoluto tegrl Dumel, furter s preseted trpezods metod wc s te smplest metod used prctce Trpezods metod s sed o pproxmtg te re delmted y te grp of fucto f :[, ] R wt te re of trpezod If te tervl [, ] s dvded to two sutervls [, x1] d [ x 1, ] te tegrl of fucto f(x c e pproxmted y te relto: dx x1 dx dx [ f ( f ( x1] [ f ( f ( x1] [ f ( f ( x1 f ( ] [ f ( f ( x1 f ( ] 4 To reduce te error of tegrto, te tervl [,] my e dvded sutervls wt dvso ( /, x1
M BOTIŞ: Mtl progrm for te umercl soluto of Dumel covoluto tegrl 49 tegrl of fucto f(x c e pproxmted y te relto: dx x1 dx x11 dx [ f ( f ( x1] [ f ( f ( x 1] [ f ( f ( x1 f ( x 1 f ( ] [ f ( f ( x1 f ( x 1 f ( ] Error clcultg troduced y te metod of trpezods depeds o te tegrto step d s te followg expresso o tervl [ x x+]:: 3,, E ( f ( 1 Full error s: 1 3 1,, ET ( E ( f ( 1 I 1 I 1 3 Implemetto of Mtl progrm for clculto Dumel tegrl For umercl clculto of te covoluto tegrl Dumel strts from respose dsplcemet for system wt oe degree of freedom: F( ( t e s( t d m F( t e cos d e st m F( t e s d e cot m t t A( e st B( e cost were, F( A( e cos d m F( B( e s d m To solve te Dumel tegrl trpezods metod wll e used to clculte tegrls A( d B( wt Mtl progrm [] % Solvg covoluto tegrl Dumel %y umercl tegrto metod of %trpezods %Aswer dplcemet, velocty d %ccelerto for dymc sstem oe degree %of freedom % force rmp cler;clf; % Tme step used =; % Mss system wt 1GLD m=175; % Rgdty of te system wt 1GLD k=7; % Frcto of crtcl dmpg u=5/1; % Pulsto system 1GLD wtout %dmpg omeg=sqrt(k/m; % Pulsto system 1GLD wt %dmpg omegd=omegsqrt(1-u^; % Tme tervl tt s computed %respose system wt 1 DOF for %dsplcemet, velocty d ccelerto tmx=14; % Force rmp fucto vrto cure=[5 14 5 1 1 ]; % Numer of pots of fucto curves pt=msze(cure; t=; f=terp1([cure(1,:p]', [cure(,:p]',; y=f1exp(uomeg cos(omegd(1momegd^-1; y=f1exp(uomeg s(omegd(1momegd^-1; % Numer of steps for umercl tegrto ps=tmx/; suplot(,1,1 for =1:ps t=t+; % Clculte te force t tme t fort=terp1([cure(1,:p]', [cure(,:p]', f1=fort5;
5 Bullet of te Trslv Uversty of Brşov Vol 5 (54-1 Seres 1 f(=f1; tmp(=t; % Vrto represetto force %fucto tme p=plot(t,f1,'--rs','lewdt',, 'MrkerEdgeColor','', 'MrkerFceColor','', 'MrkerSze',; old o; grd o; % Clculto of vlues y d y %from Dumel covoluto tegrl y(=f11exp(uomeg cos(omegd(1momegd^-1; y(=f11exp(uomeg s(omegd(1momegd^-1; ed % Fl vlues y=[y y(1,:' ]; y=[y y(1,:']; % Metod trpezods t= suplot(,1, for =:ps+1 t=t+; A(1=; B(1=; A(=A(-1+(y(+y(-1/; B(=B(-1+(y(+y(-1/; % Respose moto for rmp sgl %cse 1GLD dymc system d(=(a(exp(-uomeg s(omegd- B(exp(-uomeg cos(omegd(k/5; % Grpcl represetto of te rto of %dymc dsplcemet d sttc %dsplcemet plot(t,d(,'--rs','lewdt',, 'MrkerEdgeColor','r', 'MrkerFceColor','r', 'MrkerSze',; old o; grd o; ed 4 Results d cocluso Te progrm proposed te pper s ee tested te cse of dymcl system wt oe degree of freedom dymc Dymc lyss ws performed for two steps of tegrto = d =1 Te results fter umercl tegrto re preseted fg5 d fg6 Fg 5 Te respose dsplcemet for dymc system wt 1 degree of freedom, tegrto step, Fg 6 Te respose dsplcemet for dymc system wt 1 degree of freedom, tegrto step,1 Te result oted y umercl tegrto ws compred wt te exct soluto, for te tegrto step =, ot error 13,7 % d f tegrto step s = error ws,3% Altoug te progrm preseted s smple, t c e successfully used f te tegrto step s lower t were T/1, were T s ow perod for dymc system Refereces 1 Ifrm, M: Dmc structurlor s gere sesmc Edtur ddctc s pedgogc, 1973 Adelw Kr, Rold B Gueter,: Itroducto to Numercl Metods: A MATLAB Approc CRC Press Tylor &Frcs Group, 1