Exercise:
A linearly damped oscillator can be described by the following system of differential s: fracddtleftmatrixy v_ymatrixright leftmatrix & -omega_^ & -deltamatrixright leftmatrixy v_ymatrixright abcliste abc Find the eigenvalues and eigenvectors. Show that the system corresponds to a stable spiral for deltaomega_ underdamped. abc Derive the solutions for the initial conditions yy_ v_y start from rest and y v_yv_ initial push. abcliste

Solution:
abcliste abc The trace tau and the determinant Delta for the matrix describing the system are tau -delta -delta Delta -delta--omega_^omega_^ Using the general expression for eigenvalues of a times matrix we find lambda_ fractaupmsqrttau^-Delta frac-deltapmsqrtdelta^-omega_^ result-deltapmsqrtdelta^-omega_^ For the underdamped case we have delta^-omega_^ and therefore the square root is an imaginary number: lambda_ -deltapm isqrtomega_^-delta^ -deltapm iomega where omegasqrtomega_^-delta^ is a real value. The eigenvectors are found to be bf v_ resultleftmatrix-delta-iomega omega_^ matrixright bf v_ resultleftmatrix-delta+iomega omega_^ matrixright Both eigenvalues are complex and have a negative real part so the system corresponds to a stable spiral. abc The fundamental solutions are bf xi_t bf v_ e^lambda_ t bf xi_t bf v_ e^lambda_ t The general solutions are superpositions of the fundamental solutions: leftmatrixyt v_ytmatrixright a_ bf v_ e^lambda_ t+a_ bf v_ e^lambda_ t The coefficients a_ and a_ are given by the initial conditions. In the first case we have leftmatrixy_ matrixright a_ leftmatrix-delta-iomega omega_^ matrixright + a_ leftmatrix-delta+iomega omega_^ matrixright The second tells us that a_-a_ so the first can be written as y_ a_left-delta-iomega+delta-iomegaright -a_ i omega with the solution a_ fracy_-iomegafraci y_omega-a_ The displacement for the case where the pulum is started from rest with an initial displacement y_ is then yt fraciy_omegaleft-delta-iomega e^-delta te^iomega t--delta+iomega e^-delta te^-iomega tright fraci y_omega e^-delta tleftdelta lefte^-iomega t-e^iomega tright-iomegalefte^-iomega t+e^iomega trightright The terms with the complex exponentials can be simplified as follows: e^-iomega t-e^iomega t cos-omega t+isin-omega t-cosomega t-isinomega t cosomega t-cosomega t-isinomega t-isinomega t -isinomega t e^-iomega t+e^iomega t cos-omega t+isin-omega t+cosomega t+isinomega t cosomega t+cosomega t-isinomega t+isinomega t cosomega t It follows for the displacement yt y_ e^-delta tleftfrac-delta i^omegasinomega t-fracomega i^omegacosomega tright resulty_ e^-delta tleftcosomega t+fracdeltaomegasinomega tright For the velocity we find v_yt fraci y_omega omega_^ e^-delta tlefte^iomega t-e^-iomega tright fraci y_omega omega_^ e^-delta t isinomega t result-y_ fracomega_^omega e^-delta tsinomega t For the pulum starting from the equilibrium position with an initial velocity v_ the coefficients are defined as follows: leftmatrix v_matrixright a_ leftmatrix-delta-iomega omega_^ matrixright + a_ leftmatrix-delta+iomega omega_^ matrixright From the first we get a_ -delta-iomega+a_-delta + iomega Longrightarrow a_ -a_fracdelta+iomegadelta-iomega The second is then v_ omega_^ a_+a_ omega_^ a_ left-fracdelta + iomegadelta-iomegaright omega_^ a_frac-iomegadelta-iomega Longrightarrow a_ -v_fracdelta-iomegaiomegaomega_^ i v_fracdelta-iomegaomegaomega_^ a_ -a_fracdelta+iomegadelta-iomega -i v_fracdelta-iomegaomegaomega_^fracdelta+iomegadelta-iomega -iv_fracdelta+iomegaomegaomega_^ The displacement is given by yt i v_fracdelta-iomegaomegaomega_^-delta-iomegae^-delta te^iomega t &quad - iv_fracdelta+iomegaomegaomega_^-delta+iomegae^-delta te^-iomega t fraci v_omegaomega_^e^-delta tleft-delta-iomegadelta+iomegae^iomega t & quadquadquadquad-delta+iomega-delta+iomegae^-iomega tright fraci v_omegaomega_^e^-delta tleft-omega^+delta^ e^iomega t+omega^+delta^e^-iomega tright fraci v_omegaomega_^e^-delta tleftomega_^ e^-iomega t-omega_^ e^iomega tright fraci v_omega e^-delta t-isinomega t resultfracv_omegae^-delta tsinomega t The velocity is as follows: v_yt i v_fracdelta-iomegaomegaomega_^omega_^ e^-delta te^iomega t &quad - iv_fracdelta+iomegaomegaomega_^omega_^ e^-delta te^-iomega t fraci v_omega e^-delta tleftdelta-iomegae^iomega t-delta+iomegae^-iomega tright fraci v_omega e^-delta tleftdelta lefte^iomega t-e^-iomega tright-iomegalefte^iomega t+e^-iomega trightright fraci v_omega e^-delta tleftdelta i sinomega t-iomega cosomega tright resultfracv_omega e^-delta tleftomegacosomega t-deltasinomega tright abcliste