MATH528

The following model describes response of a system with inertia, dissipation, storage under excitation

In[6]:= rhs = m y”[t] + c y'[t] + k y[t]; r[t_] = F Cos[omega t]; hODE = rhs == 0; hsol[t_,m_,c_,k_] = y[t] /. DSolve[hODE,y[t],t][[1,1]]

$c_{1} e^{\frac{1}{2} t (- \frac{\sqrt{c^{2} - 4 k m}}{m} - \frac{c}{m})} + c_{2} e^{\frac{1}{2} t (\frac{\sqrt{c^{2} - 4 k m}}{m} - \frac{c}{m})}$

In[2]:= psol[t_] = a Cos[omega t] + b Sin[omega t]

$a \cos (ω t) + b \sin (ω t)$

In[3]:= uCoef = Simplify[Evaluate[rhs /. y->psol] - r[t]]

$\cos (ω t) (a k - a m ω^{2} + b c ω - F) + \sin (ω t) (b (k - m ω^{2}) - a c ω)$

In[4]:= sys = {Coefficient[uCoef,Cos[omega t],1],Coefficient[uCoef,Sin[omega t],1]}=={0,0}

${a k - a m ω^{2} + b c ω - F, b (k - m ω^{2}) - a c ω} = {0, 0}$

In[5]:= yp[t_,m_,c_,k_] = psol[t] /. Solve[sys,{a,b}][[1]]

$\frac{c F ω \sin (ω t)}{c^{2} ω^{2} + k^{2} - 2 k m ω^{2} + m^{2} ω^{4}} + \frac{F (k - m ω^{2}) \cos (ω t)}{c^{2} ω^{2} + k^{2} - 2 k m ω^{2} + m^{2} ω^{4}}$