# Vibration of a cantilevered beam with a Tip mass

1. Apr 10, 2012

### sbro238

Hey guys,

I was wondering if anyone could point me in the next, correct direction for this problem.
I understand how to determine the mode shapes and the natural frequencies of a cantilevered beam without a tip-mass, but adding the tip-mass baffles me a little bit.

The boundary conditions are as follows:
Assume that the beam is clamped and rigid at the wall, therefore the deflection and the slope are zero at the wall.
I also understand that the bending moment at the tip would be zero (this assumes that the tip mass can be modelled as a point-load with no dimensions).
The shear force at the tip baffles me a little bit. But from what I can gather it should be the negative product mass of the tip mass and the acceleration of the beam at the tip.

So, I take the general equation for the motion of a continuous system such as this

y(x) = Asin(beta*x)+Bcos(beta*x)+Csinh(beta*x)+Dcosh(beta*x)

I integrate through three times giving -
First Integration
y'(x) = A*beta*cos(beta*x)- B*beta*sin(beta*x)+C*beta*cosh(beta*x)
+D*beta*sinh(beta*x)

Second Integration
y''(x) = -A*beta^2*sin(beta*x)- B*beta^2*cos(beta*x)+C*beta^2*sinh(beta*x)
+D*beta^2*cosh(beta*x)

Third Integration
y'''(x) = -A*beta^3*cos(beta*x)+ B*beta^3*sin(beta*x)+C*beta^3*cosh(beta*x)
+D*beta^3*sinh(beta*x)

The first two boundary conditions at the wall give
A+C = 0 & B+D = 0

The last two equations can be rearranged to:
y''(x) = -A*beta^2(sin(beta*x)+sinh(beta*x)) - B*beta^2(cos(beta*x)+cosh(beta*x))

y'''(x) = -A*beta^3(cos(beta*x)+cosh(beta*x))+ B*beta^3(sin(beta*x)-sinh(beta*x))

The first boundary condition at the tip gives:

0= -A*beta^2(sin(beta*x)+sinh(beta*x)) - B*beta^2(cos(beta*x)+cosh(beta*x))

and the second boundary condition at the tip gives:

-mδ^2y/δt^2 = -A*beta^3(cos(beta*x)+cosh(beta*x))+ B*beta^3(sin(beta*x)-sinh(beta*x))

This is the part which confuses me slightly.

The equation for y(x,t) is given by:

y(x,t) = Y(x)*e^iwt

Does this mean, in order to determine δ^2y/δt^2, I should differential this twice with respect to t?
This would yeild:

δ^2y/δt^2 = -w^2*Y(x)*e^iwt

Am I doing this correctly? If so, what would I do AFTER this point? If I recall correctly, the solution for an un-loaded cantilever allows you to eventually determine points at which cos(bL) and cosh(bL) cross, giving the natural frequencies. How would this proceed for a cantilever with a tip mass?

Any help would be greatly appreciated.

Regards,

Seth.