Natural Frequencies and Mode shapes of a Bridge with a Sprin

\"Natural Frequencies and Mode shapes of a Bridge with a Spring-Mass System Attached\" In order to change the natural frequencies and decrease the vibration amplitudes of bridges, springs-damper-mass systems are commonly used. Consider a bridge which can be modeled as a simply supported uniform beam (EI, m, L). Neglecting the damping, a bridge with a spring-mass system attached at location x = L_1 is given in the below figure. Obtain the first three exact natural frequencies and mode shapes of the bridge for the representative parameters given in the following table. (m = 1kg/m, EI = 1Nm^2, L = 1m)

Solution

The approach used is to idealise as acoupled two mass two spring system.

The stiffness of the spring represnting the bridge as a spring mass sytem is to be found:

if a concentrated load P is acting at a distance a from theleft end of a simply supported beam of length l,. the defln is

P a^b(l-a)^{^2}/(3EIL)

if a=L/2, K =48EI/L^3

if a =L/3 K= 243EI/(4L^3)

The matrix eqns correspond to a coupled two mass two spring system

The eigenvalues satisfy, for symmetric spring position :

|49- X^2 -48 |

|-48 48-10X^2| =0

where EI=1.,L=1, M=1, m=10

The freq corresponding to this are .298 and 2.88, and mode shapes can be found accordingly from the eigenavlue eqn.

Similalry when L/L = 1/3, the eigenvalue matrix canbe comuted.

It is tedious to do this by hand, so it needs to be programmed as per the stiffness es given above.

Thank you.