String Vibration Analysis

This program preforms forced response analysis of spring in tension for multiple harmonic excitation through range of frequencies.

Its a Special case of the repository: https://github.com/Samson-Mano/Dynamics_of_Springs_and_Rigids

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