AN APPLICATION OF SUBSTRUCTURE METHOD

Transcription

1 U.P.B. Sci. Bull., Series A, Vol. 68, No., 6 AN APPLCATON OF SUBSTRUCTURE ETHOD Daiela DOBRE Lucrarea descrie uele aspecte matematice privid metoda substructurii aplicate uui sistem elastic cu mase cocetrate, supus uei miscari armoice, i domeiul timp si frecveta. Pri itroducerea metodei substructurii devie posibila studierea uui sistem discretizat i doua subsisteme, petru care sut scrise ecuatiile diamice de echilibru. Sistemul de ecuatii este rezolvat utilizad schema explicita Newmark de iterare i timp, obtiadu-se astfel cotributiile celor doua subsisteme. The paper deals with the mathematical descriptio of the substructure method applied to a elastic lumped mass system subjected to a harmoic motio, i time ad frequecy domai. By itroduci the substructure method, it becomes possible to study a system separately i two subsystems (substructures), with a iterface betwee them, ad for which the dyamic equilibrium equatios are writte. The system of equatios is solved usi explicit Newmark time iteratio scheme, distiuishi betwee the quatities comi from the substructure ad substructure. Keywords: substructure method, equilibrium equatios, umerical iteratio, Fourier trasform, impedaces fuctios troductio case of omoe systems it is ot ecessary to make distictios betwee some compoet parts, but i other cases, i which these compoets have differet properties, the aalyse o compoets it is useful ad the the subsequet assembli of the results (kiematics or topoloical partitio). The substructures are disjuctive parts, with commo boudary poits but, for which iteral or exteral (o boudaries) derees-of-freedom are cosidered. Throuh substructure method, eerally the reductio of dyamic problem dimesios is followed, by static or dyamic codesatio, related to exteral derees-of-freedom of substructures [, ]. Researcher, Natioal stitute for Buildi Research- NCERC Bucharest, ROANA

2 8 Daiela Dobre A elastic lumped mass system i time ad frequecy domai The aim of this work is to idetify the ifluece of each substructure i the modelli of iteractio betwee those two substructures (subsystems) as matrices from the split system of equatios matrix. The appropriate way to do this is to cosider a -deree-of-freedom liiar oscillator, supported o a riid layer (substructure ) ad resti o aother layer modelled with elastic isotropic homoeous halfspace (substructure ), Fi.. The substructure has a stiffess matrix [K], mass matrix [] ad dampi matrix [C], satisfyi the coditio [ ] [ K][ ] [ C] = [ ] [ C][ ] [ K], a ecessary ad sufficiet coditio for the substructure to admit decompositio ito classical real modes []. The system has siificat derees of freedom, amely, horizotal traslatio of each floor, horizotal traslatio of the base mass ad rotatio of the system i the plae of motio. Substructure Substructure Table Characteristics of the substructures Riid layer mass m, momet of iertia, Structure mass [], (momet of iertia ), stiffess [K], dampi [C], heiht H Displacemet: vt () hθ () t v() t Elastic halfspace Poisso s ratio υ, mass desity ρ, shear wave velocity c s Riid massless plate o the its displacemet compatibility with the surface of the halfspace lower surface of the riid basemat Displacemet: v () t v () t The equilibrium equatios of motios will be developed for the eeral case of masses, i terms of the parameters of the overall system ad the ukow displacemets: v () t - deformatio at free-field surface, vt ()- deformatio of the substructure relative to the base, v () t - base displacemet caused by substructure -substructure iteractio, ad θ () t - base rotatio caused by substructure -substructure iteractio.

3 A applicatio of substructure method 9 Fi.. A lumped elastic system with -deree-of-freedom (Substructure ) Fi.. - Physical models to represet dyamic stiffess for Substructure (traslatio motio/rotatioal motio/coupli of horizotal ad rocki motios)substructure, i the time domai

4 4 Daiela Dobre Substructure, i the time domai - all masses are isolated i order to et the horizotal force equilibrium equatio: - m[ v () t θ h v ()] t cv () t kv () t c [ v () t v ()] t k [ v () t v ()] t = mv [ v ( t) θ h v ( t) ] c v ( t) v ( t) v () t v () t k v t v t = m v [ ] k[ v ( t) v ( t) ] [ ] [ () ()]... m c [ ( ) ( )] [ ( ) ( )] [ ( ) m v t θ h v t c v t v t k v t v ( t)] = m v - the etire structure is isolated from the elastic halfspace i order to et the horizotal force ad momets about the cetroidal x-axis of the basemat equilibrium equatios: - [ ( ) m ( )] v t θ h v t [ ( ) m ( )] v t θ h v t m [ v ( t) θ h v ( t)] mv () t mv () t mv () t [ ( ) m ( )] v t v t = V () t ( V () t is the base iteractio shear force) θ () t m h v t θ h v t [ ( ) ( )] m h v t θ h v t [ ( ) ( )] m h [ v ( t) θ h v ( t)] m h v () t m h v ( t) m h v () t = () t ( () t is the base iteractio momet) the matrix form oe ca obtai:

5 A applicatio of substructure method 4 m... m hm m... m hm m... m hm m m hm m m m... m m m... m m hm hm... hm h h h... h hm hm... hm hm 4 hm... hm J v () t v v () t... v () t v θ c c c... v () t c c c c... v c c c4... v () t c v () t... v... θ () t k k k... v () t k k k k... v k k k4... v () t =... k v () t... v... θ () t = - m m m... v... m m m... m m V hm hm... hm () t

6 4 Daiela Dobre The system of equatios is iterated with respect to time usi explicit Newmark time iteratio scheme with β =, γ = ad Δt vi = vi Δt v i v i Δt v i = v i ( v i v i ) Thus: v Cv Kv = P i i i i v = P. Δt Δt Δt C v i = Pi vi K v i ( C K Δt) v i C K ( ) v = f v, v, v ad the discrete system of liear equatios at time t is i i i i i i Distiuishi betwee the quatities comi from the substructure ad substructure, the system v i = Pi may be split ito: substr substr substr substr v substr P = v substr substr substr substr substr P substr substr, where = C Δt, m... m hm m... m hm = m..., substr substr = m hm, m m hm =, T substr substr substr substr m m... m m hm hm... hm = substr substr hm hm... hm h m 4 h m... h m

7 A applicatio of substructure method 4 aother simplified form, the mass matrix ca be expressed as: = ;... H h h = h ;... h substr substr substr substr = substr substr substr substr or H T T T = { } m H T T T T { H} { H} J H H Substructure, i the frequecy domai, usi the Fourier trasformi of equatios ad the impedaces fuctios (because the stiffess ad dampi properties of the substructure are frequecy depedet) The Fourier trasform is a mathematical techique for coverti time domai data ( vt ()) to frequecy domai data ( V ( ω )), ad reversely, iωt V( ω) = v( t) e dt ad is applied to each equilibrium equatio for substructure. ore over, the equatios of motio ivolve oly the two substructure - substructure iteractio derees-of-freedom, v () t ad θ ( t), ad each impedaces fuctios are expressed i terms of the halfspace: ( ) ( ) ( ) mω ciω k c iω k V iω mω hθ ( iω) mω V ( iω) c iω k V ( iω) = m V ( iω)

8 44 Daiela Dobre ( ) ( ) ( ) ( ) mω c iω k c iω k V iω c iω k V iω mω hθ ( iω) mω V ( iω) ( ) ciω k V ( iω) mv ( iω) = m ω c iω k V iω m ω hθ ( iω) m ω V ( iω) c iω k V ( iω) ( ) ( ) ( ) m V ( iω) = ( ) ( ) ( ) mω V ( iω) mω V ( iω)... m ω V ( iω) mω h mω h... m ω h Θ ( iω) mω mω... m ω mω V ( iω) m m... m m V ( iω) = V ( iω) m hω V ( iω) m hω V ( iω)... m hω V ( iω) mh ω mω 4h... mω h ) Θ ( iω) ( m hω m hω... m hω ) V ( iω) ( m h m h... m h) V ( iω) = ( iω) Usi the complex impedace fuctios (frequecy depedet, havi the R form ia ( ) = ( a) i( a) ), the iteractio forces acti o substructure are ive i the frequecy domai by: V ( iω ) ( iω) ( iω) V ( iω ), i.e. vv vθ = ( iω) ( iω) ( iω) θ v ( i ) θ θ Θ ω...,, V ( iω )...,, V ( iω ) =...,, V ( iω ),,...,,, V ( iω ),,...,,, Θ ( iω ) m m... =- V... ( iω ), where m m m... m m V ( iω ) hm hm... hm ( iω )

9 A applicatio of substructure method 45 = mω ciω k c iω k = ( c iω k ) = =... = = = mω,, = mω h = mω c iω k c iω k ( ω ) = c i k 4 =... = = = mω, m h, = ω =... = = ( ω ) = c i k, = m ω c iω k = mω,, = mω h = mω,, = mω = m ω, ( ) ω = m... m m ( iω ), vv ( ) ω... ( ), = mh m h m h iω v θ, mω h =, = mω h, = mω h ( ) ω =, mh... m h ( iω ) θ v ( ), = mh m h... m h ω ( iω ) θθ

10 46 Daiela Dobre For substructure, ωr dyamics, the dimesioless frequecy a is itroduced, a =, cs with r represeti a characteristic leth ad c s the shear-wave velocity from the motio, cs =, shear modulus. Usi the static-stiffess coefficiet K, is ρ formulated the dyamic stiffess coeficiet, S(a ): Sa ( ) = Kka ( ) iaca ( ) [ ] The spri with the stiffess ka ( ) overs the force, which is i phase with the displacemet, ad the dampi coefficiet ca ( ) describes the force which is 9 out of phase. The dyamic-stiffess coefficiet Sa ( ) ca be iterpreted as a spri with the frequecy-depedet coefficiet Kk( a) ad a r dashpot i parallel with the frequecy-depedet coefficiet Kca ( ), Fi.. c s Fi.. - terpretatio of dyamic-stiffess coefficiet for harmoic excitatio as spri ad as dashpot i parallel with frequecy-depedet coefficiets Usi coes to model the halfspace, there are the followi expressios for stiffess ad dampi (Fi. ) []: z cs koriz ( a ) = a ; z c ( s coriz a ) =. r c π r c

11 A applicatio of substructure method 47 (for the horizotal motio, c = c s, = for all υ, for the vertical motio, c =c p, = for υ ad c = c s ad for υ ) 4 z c a krot ( a ) a θ s = π r c rc a zc s ; c rot z c s = r c rc a zc s ( a ) a (for the torsioal motio, c = c s, = for all υ, for the rocki motio, c =c p, = for υ ad c = c s ad for υ ) Also, for stiffess ad dampi there are the followi formulae: koriz 8 = r ; c υ oriz 4.6 = ρcsr ; υ k rot 8r = ; c ( υ) rot.4 = ρc r υ 4 s Coclusios The theoretical aalysis of a lumped elastic system with deree-offreedom supported by a riid layer resti o a elastic halfspace is preseted. The quatities comi from the substructure ad the substructure are put ito evidece, bei determied the matrices for ad at the iterface from the both structures. This aspect is importat from mathematical poit of view, but importat too from practical poit of view for the possibility to combie the umerical simulatio of the aalytical part of the system with the effective laboratory testi of the remaii part of the system. The theoretical poit of view is oi to be cotiued with some umerical studies related to the substructure -substructure iteractio, usi the computer modeli of differet systems.

12 48 Daiela Dobre R E F E R E N C E S. Ray W. Clouh, Joseph Pezie, Dyamics of structures, craw-hill c, 99.. Horea Sadi, Elemete de diamica structurilor, Ed. Tehica, 98.. T. K. Cauhey, Classical Normal odes i Damped Liear Dyamic Systems, Applied echaics, Divisio of Eieeri, Califoria stitute of Techoloy, Pasadea, 965.