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.

## Tutorial

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.

EigenValues (wn)

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

EigenVectors

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

## Subject Review

Under Development