{ FlexPDE REL. 5.0.4, VERIF. St2LH_1.pde } { ***************************************************** Structural analysis, theory of elasticity (St) Two-dimensional problem (2) Linear problem (L) Harmonic analysis (H) Plane stress Compound Double-T Region (steel and copper) Bottom boundary is fixed ***************************************************** } title "Harmonic Vibrations of a Double-T Region " variables Ur { Ur,Ui are the real and imaginary parts of the displacement along axis X } Ui Vr { Vr,Vi are the real and imaginary parts of the displacement along axis Y } Vi definitions Um = sqrt(Ur^2+Ui^2) { Amplitude of the displacement along axis X } Vm = sqrt(Vr^2+Vi^2) { Amplitude of the displacement along axis Y } Scal=0.2*globalmax(magnitude(x,y))/globalmax(magnitude(Um,Vm)) {geometrical parameters in meters} h=2e-2 {thickness} b=16e-2 {half-heigth} l=5e-2 {half-width of cleat} nu = 0.29 { Poisson's ratio for steel } E = 2e11 {Youngs modulus in Pascals for steel} rho = 7.86e3 {density in kg/m^3 for steel } mu = E/2/(1+nu) {shear module} lambdaL = 2*mu*nu/(1-2*nu) {Lame's parameter} {Lame's parameters for plane stress } lambdaz = 2*mu*lambdaL/(lambdaL+2*mu) force=1e3 Freq1=98.17 Freq2=735 Q=1000 {Q-factor } Fsum=Freq1+Freq2 {damping coefficients} alpha=pi*2*Freq1*Freq2/(Fsum*Q) beta = 1/(pi*2*Fsum*Q) omega = 2*pi*freq1 {calculates results for frequency freq1, for another frequency set another value} {real and imaginary parts of the strains} Srxx = dx(Ur) Sryy = dy(Vr) Srxy = (dy(Ur)+dx(Vr))/2 Sixx = dx(Ui) Siyy = dy(Vi) Sixy = (dy(Ui)+dx(Vi))/2 {real and imaginary parts of the stresses} Trxx= lambdaz*(Srxx+Sryy)+2*mu*Srxx-omega*beta*(lambdaz*(Sixx+Siyy)+2*mu*Sixx) Tryy= lambdaz*(Srxx+Sryy)+2*mu*Sryy-omega*beta*(lambdaz*(Sixx+Siyy)+2*mu*Siyy) Trxy=2*mu*(Srxy-omega*beta*Sixy) Tixx= lambdaz*(Sixx+Siyy)+2*mu*Sixx+omega*beta*(lambdaz*(Srxx+Sryy)+2*mu*Srxx) Tiyy= lambdaz*(Sixx+Siyy)+2*mu*Siyy+omega*beta*( lambdaz*(Srxx+Sryy)+2*mu*Sryy) Tixy=2*mu*(Sixy+omega*beta*Srxy) equations { дифференциальные уравнения гармонических колебаний для плоской задачи теории упругости при учете затухания по Релею} Ur: dx(Trxx) + dy(Trxy) + rho*omega^2*Ur - omega*alpha*rho*Ui = 0 Ui: dx(Tixx) + dy(Tixy) + rho*omega^2*Ui + omega*alpha*rho*Ur = 0 Vr: dx(Trxy) + dy(Tryy) + rho*omega^2*Vr - omega*alpha*rho*Vi = 0 Vi: dx(Tixy) + dy(Tiyy) + rho*omega^2*Vi + omega*alpha*rho*Vr = 0 boundaries Region 1 {upper region 1 from steel} start (h/2,0) natural(Ur)=0 natural(Vr)=0 natural(Ui)=0 natural(Vi)=0 {free boundary} line to (h/2,b-h) to (l,b-h) to (l,b) point load(Ur)=force point load(Vr)=force line to (-l,b) point load(Ur)=-force point load(Vr)=-force line to (-l,b-h) natural(Ur)=0 natural(Vr)=0 line to (-h/2,b-h) line to (-h/2,0) to close Region 2 {lower region 2 with another material properties} {material properties for copper} nu=0.33 E=1.2e11 rho=8.9e3 start (h/2,0) natural(Ur)=0 natural(Vr)=0 natural(Ui)=0 natural(Vi)=0 {free boundary} line to (h/2,-b+h) to (l,-b+h) to (l,-b) value(Ur)=0 value(Vr)=0 value(Ui)=0 value(Vi)=0 {fixed displacements} line to (-l,-b) natural(Ur)=0 natural(Vr)=0 natural(Ui)=0 natural(Vi)=0 line to (-l,-b+h) to (-h/2,-b+h) to (-h/2,0) to close monitors elevation(Vr,Vi) from(-l,b) to (l,b) report(omega) plots grid(x+Scal*Ur,y+Scal*Vr) as "Real displacement" report(freq1) grid(x+Scal*Ui,y+Scal*Vi) as "Imag displacement" report(freq1) end