The equation of motion for beam subjected to moving load can be represented as
EI (∂^4 y(x,t))/(∂x^4 )+m.(∂^2 y(x,t))/(∂x^2 )+c.(∂y(x,t))/∂x=F(x,t)
where m is the constant mass per unit length, EI is the flexural rigidity, c the damping of the beam, y(x, t) the displacement of the beam, and F(x, t) is the external force from moving oscillators.
I want to write a Matlab code to predict the natural frequency and damping coefficient for different mode.
The first step is to write this equation in a discrete format. You can use many vibration book that they show how to do this method. In specific you can use "Principles and Techniques of Vibrations". I also briefly have an introduction of this method in my paper called "Specified Motion of Piezoelectrically Actuated Structures".
When you write your equation of motion as M\ddot x+ C \dot x+ K x= B u
then it is easy to find the eigen values (natural frequency of the system).
In matlab it is wn=sqrt(eig(inv(M)*K))
Hello Prakash Kumar,
The best method to solve your problem is the finite element method (FEM) hence you need to get very much acquainted with this procedure. Assuming harmonic motion and forcing function by taking y(x,t) = Y(x)exp(iwt) (where i =sqrt(-1) and F(x,t) = Fo(x)exp(iwt) the PDE of motion can be converted into the 4th order ODE, namely
EIYiv(x) + (icw - mw2)Y = Fo ______ (1) w = circular frequency of vibration.
When this equation is discretized via the FEM the following expression is obtained
KedDe = Pe ___________(2) where Ked is the element dynamic stiffness matrix, Pe is the element nodal force vector and De = the element nodal displacements.
Ked = Kes + (iwc/m - w2)Mc _________ (3) where Kes = static element stiffness matrix and Mc is the consistent mass matrix. Thus, the dynamic stiffness matrix for a damped beam is a matrix with complex entries .Kes is the well known symmetrical 4 by 4 matrix
Kes = (EI/L3)[12 6L -12 6L ; 6L 4L2 -6L 2L2 ; -12 -6L 12 -6L; 6L 2L2 -6L 4L2] __(4)
Semicolon is used to separate the matrix rows. Mc is the symmetrical 4 by 4 matrix given by
Mc =(mL/420)*[156 22L 54 -13L; 22L 4L2 13L -3L2; 54 13L 156 -22L; -13L -3L2 -22L 4L2] ______ (5)
It is quite difficult to determine the damped natural frequency of a beam though it can be done. In the alternative, the undamped natural frequency of the beam is determined from the Matlab expression
w = sqrt(eig(Mgc-1*Kgs))_______(6) where Kgs = the global static stiffness matrix of the beam and Mgc-1 is the global consistent mass matrix. However, in forced vibration problems damping has to be included in order to determine a realistic beam deflection. An equation similar to Eq. 2 needs to be used with the element matrices replaced by their global equivalents as per Eq.3 Thus, Prakash Kumar, the ball is now in your court to implement the analysis of the damped beam. The only problem you may likely encounter is the method for synthesizing the global matrices. This is explained in the attached pdf file. Good luck.
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