String Vibration Analysis
This program preforms forced response analysis of spring in tension for multiple harmonic excitation through range of frequencies.
Download URL: String Vibration Analysis
Some screenshots below:
Picture: 1 Mode shape Display
Picture: 2 Nodal Displacement Response
Picture: 3 FRF Amplitude
Picture: 4 Individual Nodes Displacement Response Plot (Bode Diagram)
Forced Response Analysis (Frequency Domain) of String in Tension
The theoretical background for the forced response analysis for String Vibration analysis program is below
Element mass matrix
Element Stiffness matrix
Equation of Motion after forming the Global stiffness matrices
Natural frequency and mode shape
Characteristics matrix
Where,
Ω2 – Spectral matrix
φ – Modal matrix
General Eigen value problem to Standard Eigen value problem
M is diagonal matrix, therefore can be easily inverted
Where, [A] = [M-1][K]
Which is a standard Eigen value problem
The eigen value solution is of the form,
Modal expansion
The equation of motion becomes
Pre-multiply by some [φi]T
Now due to orthogonality
Which implies for each mode i = 1 to n the equation of motion becomes
Using Rayleigh proportional damping
The above equation becomes
where
and
The particular solution of the above equation is of the form (Note that the transient response decays with damping leaving only the steady state part)
where
and
Now using the modal expansion, the nodal displacement response can be found