Three-Mass System
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Exercise:
Derive the system of differential s for the system of three coupled masses in the figure below. Find the eigenvalues and eigenvectors for K k and describe the normal modes in words. center includegraphicswidthtextwidth#image_path:thremass-system-# center
Solution:
The differential s are mddot x_ -K x_+kx_-x_ -K+k x_+k x_ mddot x_ kx_-x_+kx_-x_ k x_-k x_ +k x_ mddot x_ kx_-x_ -K x_k x_-K+kx_ The coefficient matrix is thus A pmatrix -fracK+km & frackm & frackm & -frackm & frackm & frackm & -fracK+km pmatrix The characteristic polynomial is given by detleftbf A-lambda bf Iright detpmatrix -fracK+km-lambda & frackm & frackm & -frackm-lambda & frackm & frackm & -fracK+km-lambda pmatrix -leftfracK+km+lambdaright^leftfrackm+lambdaright+leftfracK+km+lambdarightleftfrackmright^ leftfracK+km+lambdarightleftleftfrackmright^-leftfracK+km+lambdarightleftfrackm+lambdarightright leftfracK+km+lambdarightleftfrack^m^-frack^m^-fracKkm^-fracK+kmlambda-frackmlambda-lambda^right -leftfracK+km+lambdarightleftlambda^+fracK+kmlambda+fracKkm^right with zeros for lambda_ -fracK+km lambda_ fracleft-fracK+kmpmsqrtleftfracK+kmright^-fracKkm^right fracmleft-K+kpmsqrtK^+Kk+k^-Kkright frac-K+kpmsqrtK^-Kk+k^m For Kk this can be simplified: lambda_ -frackm -omega_^ lambda_ frac-k-sqrtk^m-+sqrtomega_^ lambda_ frac-k+sqrtk^m--sqrtomega_^ with omega_^ frackm The three normal modes have angular frequencies omega_ sqrtomega_ omega_ sqrt+sqrtomega_ omega_ sqrt-sqrtomega_ with omega_omega_omega_. vspacemm It can easily be verified that the corresponding eigenvectors can be written as hat x_ pmatrix - pmatrix hat x_ pmatrix -sqrt pmatrix hat x_ pmatrix sqrt pmatrix The three normal modes can be characterised as follows: itemize item All three masses moving in sync with the amplitude of the middle mass being sqrt greater than that of the masses on the sides lowest frequency omega_. item Left and right masses moving in opposite directions with the middle mass at rest ermediate frequency omega_. item Left and right mass moving in sync with the middle mass moving in opposite direction with amplitude sqrt greater than that of the other masses highest frequency omega_. itemize
Derive the system of differential s for the system of three coupled masses in the figure below. Find the eigenvalues and eigenvectors for K k and describe the normal modes in words. center includegraphicswidthtextwidth#image_path:thremass-system-# center
Solution:
The differential s are mddot x_ -K x_+kx_-x_ -K+k x_+k x_ mddot x_ kx_-x_+kx_-x_ k x_-k x_ +k x_ mddot x_ kx_-x_ -K x_k x_-K+kx_ The coefficient matrix is thus A pmatrix -fracK+km & frackm & frackm & -frackm & frackm & frackm & -fracK+km pmatrix The characteristic polynomial is given by detleftbf A-lambda bf Iright detpmatrix -fracK+km-lambda & frackm & frackm & -frackm-lambda & frackm & frackm & -fracK+km-lambda pmatrix -leftfracK+km+lambdaright^leftfrackm+lambdaright+leftfracK+km+lambdarightleftfrackmright^ leftfracK+km+lambdarightleftleftfrackmright^-leftfracK+km+lambdarightleftfrackm+lambdarightright leftfracK+km+lambdarightleftfrack^m^-frack^m^-fracKkm^-fracK+kmlambda-frackmlambda-lambda^right -leftfracK+km+lambdarightleftlambda^+fracK+kmlambda+fracKkm^right with zeros for lambda_ -fracK+km lambda_ fracleft-fracK+kmpmsqrtleftfracK+kmright^-fracKkm^right fracmleft-K+kpmsqrtK^+Kk+k^-Kkright frac-K+kpmsqrtK^-Kk+k^m For Kk this can be simplified: lambda_ -frackm -omega_^ lambda_ frac-k-sqrtk^m-+sqrtomega_^ lambda_ frac-k+sqrtk^m--sqrtomega_^ with omega_^ frackm The three normal modes have angular frequencies omega_ sqrtomega_ omega_ sqrt+sqrtomega_ omega_ sqrt-sqrtomega_ with omega_omega_omega_. vspacemm It can easily be verified that the corresponding eigenvectors can be written as hat x_ pmatrix - pmatrix hat x_ pmatrix -sqrt pmatrix hat x_ pmatrix sqrt pmatrix The three normal modes can be characterised as follows: itemize item All three masses moving in sync with the amplitude of the middle mass being sqrt greater than that of the masses on the sides lowest frequency omega_. item Left and right masses moving in opposite directions with the middle mass at rest ermediate frequency omega_. item Left and right mass moving in sync with the middle mass moving in opposite direction with amplitude sqrt greater than that of the other masses highest frequency omega_. itemize