MATLAB Programming –Eigenvalue Problems and Mechanical Vibration0)(=⋅−=⋅xIAxxAλλCite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].A Coupled Mass Vibration ProblemEOM:0)(0)(122212122111=−++=−−+xxkxkxmxxkxkxmCite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Vibration Solutions – harmonic response)cos(2121ϕω+⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡tccxx0/)(///)(2121222212=⎥⎦⎤⎢⎣⎡⋅⎥⎦⎤⎢⎣⎡++−−−++−ccmkkmkmkmkkωωTrial solution:Matrix representation of EOM:Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Vibrational Frequencies and Mode Shapes⎥⎦⎤⎢⎣⎡−=⎥⎦⎤⎢⎣⎡+=⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=11211222121221121121AccmkkAccmkωωCharacteristic Equation (Determinant = 0 ):2221kmkk ±=−+ωCite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Vibrations as a general class of “Eigenvalue Problems”0/)(///)(2121222212=⎥⎦⎤⎢⎣⎡⋅⎥⎦⎤⎢⎣⎡++−−−++−ccmkkmkmkmkkωωRecast EOM:As:0)(1001/)(///)(00/)(///)(21221212221212221212221=⋅−⋅=⋅⎥⎦⎤⎢⎣⎡⋅⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡⋅⎥⎦⎤⎢⎣⎡+−−+⎥⎦⎤⎢⎣⎡⋅⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡⋅⎥⎦⎤⎢⎣⎡+−−+xIAxIxAccccmkkmkmkmkkccccmkkmkmkmkkλλωωωCite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Eigenvalue equation, Eigenvalues, Eigenvectors0)( =⋅−=⋅xIAxxAλλEigenvalue equation:Eigenvalues (angular frequencies of the vibration):2ωλ=Eigenvectors (mode shape of the vibration):⎥⎦⎤⎢⎣⎡=21ccxCite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Solving Eigenvalue Problem in MATLABLook at the problem numerically:mNkmNkkgm /2/1121===m=1;k1=1;k2=2;A=[(k1+k2)/m -k2/m; -k2/m (k1+k2)/m][X,L]=eig(A);XLSimple m-file:Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].MATLAB output of simple vibration problemX =-0.7071 -0.7071-0.7071 0.7071L =1.0000 00 5.0000eigenvector 1eigenvector 2eigenvalue 1eigenvalue 2Ok, we get the same results as solving the characteristics equation… so what is the big deal?Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].A more complex vibrations problemCite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].EOM for a more complex problemEOM can be gotten from free body diagrams of each mass 0)(0)(0)(0)(3426464444423153543334633122632223522152111=−−++=−−−+++=−−−+++=−−+++xkxkxkkxmxkxkxkxkkkxmxkxkxkxkkkxmxkxkxkkkxmCite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Characteristic equation of more complex problem:0004321464244463435422333526232632222151215212=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⋅⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡+−−++−−++−−++−−ccccmkkmkmkmkmkkkmkmkmkmkmkkkmkmkmkmkkkωωωωSolving this and find roots is no longer so simple!It is now an eighth order polynomial….Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Look at more complex vibration as eigenvalue problem⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⋅⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡⋅⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡+−−−++−−−−++−−−++43212432146444463435423335262326322215121521100001000010000100ccccccccmkkmkmkmkmkkkmkmkmkmkmkkkmkmkmkmkkkωCite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Solving more complex problem by MATLABPick some numerical values:mNkmNkmNkmNkmNkmNkkgmkgmkgmkgm/6/5/4/3/2/143216543214321==========Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Solving complex vibration problem by MATLABm1=1;m2=2;m3=3;m4=4;k1=1;k2=2;k3=3;k4=4;k5=5;k6=6;A=[(k1+k2+k5)/m1 -k2/m1 -k5/m1 0; -k2/m2 (k2+k3+k6)/m2 -k3/m2 -k6/m2; -k5/m3 -k3/m3 (k3+k4+k5)/m3 -k4/m3; 0 -k6/m4 -k4/m4 (k4+k6)/m4][X,L]=eig(A);XLCreate another m-file:Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Characteristic matrix of the eigenvalue problemA =8.0000 -2.0000 -5.0000 0-1.0000 5.5000 -1.5000 -3.0000-1.6667 -1.0000 4.0000 -1.33330 -1.5000 -1.0000 2.5000Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Frequencies and mode shapes of complex problemX =0.4483 0.9456 0.6229 -0.16500.5156 -0.1826 -0.0411 -0.90060.5031 -0.2591 0.5788