Dynamic loading(DMA) Laplace-plane shear operator >G[L]: =G[R]+(G[d]*s)/(s+1/tau[sigma])i d s Applied strain in time plane unprotect(gamma)igamma(t):=gamma[0]*cos(omega*t)i (1)=Yc0(01) Applied strain in laplace plane with (inttrans): gamma(s):=laplace(gamma(t),t, s)i Ms) Dynamic modulus in laplace plane >G bar: =G[L]*gamma(s)/gamma[0]; S g bar Invert for time-plane modulus >G t: =invlaplace(G bar, s, t)i ot Gasin(ot) Gr O to cos(ot) Gr coS(ot) o to G, cos(ot) Simplifying 'G(t)'=factor(collect((G t), cos(omega(t))))i G,e ot, Ga sin(ot)+Gr o t, cos(o t)+Gr coS(o0+o, G, cos(on) Simplifying further and rearranging manually 1+2r2 1+aDynamic loading (DMA) Laplace-plane shear operator > G[L]:=G[R]+ (G[d]*s)/(s+1/tau[sigma]); Gd s GL := GR + 1 s + τσ Applied strain in time plane: > unprotect(gamma);gamma(t):=gamma[0]*cos(omega*t); γ ( )t := γ0 cos( ω t) Applied strain in laplace plane: > with(inttrans):gamma(s):=laplace(gamma(t),t,s); γ0 s γ ( )s := s 2 + ω2 Dynamic modulus in laplace plane: > G_bar:=G[L]*gamma(s)/gamma[0]; Gd s GR + 1 s + τσ               s G_bar := s 2 + ω2 Invert for time-plane modulus: > G_t:=invlaplace(G_bar,s,t);          t − τ σ ω 2 τσ 2 GR ω 2 τσ 2 cos( ω t) GR cos( ω t)  Gd e ω τσ Gd sin( ω t) − + Gd cos( ω t) G_t := + + ω 2 τσ 2 ω 2 τσ 2 ω 2 τσ 2 + 1 ω 2 τσ 2 2 1 ω 2 τσ + 1 + 1 + + Simplifying: > 'G(t)'=factor(collect((G_t),cos(omega(t))));          t − τ σ ω 2 τσ 2 GR ω 2 τσ 2 cos( ω t) GR cos( ω t)  Gd e − ω τσ Gd sin( ω t) + Gd + + cos( ω t) G ( )t = ω 2 τσ 2 + 1 Simplifying further and rearranging manually: −t  22 σ G* = Gd eτ σ + GR + Gd ω τ σ   cos ( ) −   Gd ωτ  ωt 22 2 2 2 2  1+ω τ σ  1+ω τ σ  1+ω σ  Page 1 1