Coupled oscillators: 2 masses m1, m2, 3 springs with constants k1, k2, k3. All springs relaxed = equilibrium

x1[eta1_]:=eta1*Sqrt[m1]

x2[eta2_]:=eta2*Sqrt[m2]

eta[ac_,omeg_,t_,C_]:=C*ac*Exp[-i*omeg*t]

tmat={{1,0},{0,1}}

vmat={{(k1+k2)/m1,-k2/Sqrt[m1*m2]},{-k2/Sqrt[m1*m2],(k2+k3)/m2}}

MatrixForm[vmat]

det[lambda_]:=Det[vmat-lambda*tmat]

lams=Simplify[Solve[det[lam]==0,lam]]

m1=m

m2=m

lamda2[k1_,k2_,k3_]:=((k1+2 k2+k3-(Sqrt[(k1^2)+4 (k2^2)-2 k1 k3+(k3^2)]))/(2 m))

lamda1[k1_,k2_,k3_]:=((k1+2 k2+k3+(Sqrt[(k1^2)+4 (k2^2)-2 k1 k3+(k3^2)]))/(2 m))

k1=10

k3=10

m=1

Plot[{lamda1[k1,k2,k3],lamda2[k1,k2,k3]},{k2,0,k1},PlotRange-> {0,50}]

k3=k1

Clear[lamda1]

a1={1/Sqrt[2],-1/Sqrt[2]}

MatrixForm[a1]

a2={1/Sqrt[2],1/Sqrt[2]}

aa={a1,a2}

{{(1/(Sqrt[2])),-(1/(Sqrt[2]))},{(1/(Sqrt[2])),(1/(Sqrt[2]))}}

MatrixForm[Transpose[aa].vmat.aa]

k2=10

lamb1=lamda1[k1,k2,k3]

lamb2=lamda2[k1,k2,k3]

eta1[t_]:=c1*a1[[1]]*Cos[Sqrt[lamb1]*t]+c2*a2[[1]]*Cos[Sqrt[lamb2]*t+phi2]

eta2[t_]:=c1*a1[[2]]*Cos[Sqrt[lamb1]*t]+c2*a2[[2]]*Cos[Sqrt[lamb2]*t+phi2]

{Slider[Dynamic[c1],{0,1,0.1}],Style[NumberForm[Dynamic[c1],{3,2}]]}

{Slider[Dynamic[c2],{0,1,0.1}],Style[NumberForm[Dynamic[c2],{3,2}]]}

{Slider[Dynamic[phi2],{0,Pi,Pi/20}],Style[NumberForm[Dynamic[phi2],{3,2}]]}

N[eta1[0]]

Dynamic[Plot[{eta1[tt],eta2[tt]},{tt,0,50}]]

3