The tool SD-06 computes the natural frequencies of a uniform bar/shaft/string carrying a lumped mass in free vibration, and plots and animates the associate mode shapes. The tool also gives analytical expressions of modes shapes. The determined natural frequencies and mode shapes are solutions of an eigenvalue problem; see the subject review below.
Example: Free vibration of a longitudinal bar with a lumped mass.
The input in the tool is
After clicking "Run", the results will be shown in another page. The first four modes of the system are
The eigenvalues and the analytical expression of eigenvectors are also shown as follows.
w1 = 13.76280869 rad/s; f1 = 2.19041903 Hz
w2 = 27.47183244 rad/s; f2 = 4.37227793 Hz
w3 = 46.42640582 rad/s; f3 = 7.38899198 Hz
w4 = 58.35241499 rad/s; f4 = 9.28707529 Hz
For 0 ≤ x ≤ 0.7
Flexible Modes: u(x) = a1*cos(beta*x)+a2*sin(beta*x)
Mode #1: a1=0, a2=0.36329794 ;
Mode #2: a1=0, a2=0.1820046 ;
Mode #3: a1=0, a2=0.10769733 ;
Mode #4: a1=0, a2=0.08568626 ;
For 0.7 < x ≤ 1.0
Flexible Modes: u(x) = a1*cos(beta*(x-xm))+a2*sin(beta*(x-xm))
Mode #1: a1=0.34051898, a2=-0.31407829 ;
Mode #2: a1=-0.11787081, a2=-0.00915494 ;
Mode #3: a1=0.02313596, a2=0.06221812 ;
Mode #4: a1=0.08146071, a2=-0.21671323 ;
where beta = (wk2*ρ*A/(E*A))1/2, xm = 0.7