Partial Differential Equation Course (ins 214E) Term Homework Free Vibrations of Simply Supported Beam

Consider the following linear fourth-order partial differential equation

with the boundary conditions

and initial conditions





where _ and _ are given functions and v is vertical displacement (or deflection) of the beam and c is constant and called as the wave speed. This equation corresponds to free vibrations of simply supported beam with L length. The equation for wave speed is

where EI is flexural rigidity of the section of the beam with length L, _ is density and A is area of cross-section of the beam. The load is suddenly removed the beam will vibrate freely.

1. 
   Obtain the expression for the free undamped transverse vibration of the simply supported beam and natural frequency expression _ of the beam by using the Method of Separation of Variables.
   
2. 
   Obtain expression of free vibration for simply supported beam if P (w, Q) is removed suddenly at time t = 0.
   
3. 
   Draw the modes of deflection along the beam and modes of deflection with time along the beam (for three different _ values).
   
4. 
   Obtain and draw the solution by using the Finite Difference Method as a numerical solution approach (based on Excel).
   



G
_
roup II:

B

G
A
roup III:

B

In order to get the deflection due to the static load, also assume that the deflected shape is represented by half range Fourier series:

and the differential equation relating the deflection (displacement) and the load is

Use the following values to draw the mode shapes of the deflection: