Vibration of a cantilevered beam with a Tip mass

[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 modeled 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.

 

[eljose79]

Dear Seth,

Thank you for reaching out for help with your problem. Adding a tip-mass to a cantilevered beam does introduce some complexities, but it is definitely solvable. Your understanding of the boundary conditions is correct, and your approach to solving the problem is also on the right track.

To determine the mode shapes and natural frequencies of the cantilevered beam with a tip-mass, you will need to solve the differential equation you have set up, taking into account the additional boundary conditions at the tip. This will involve solving for the coefficients A, B, C, and D in your general equation for the motion of a continuous system.

Once you have the general equation, you can use the boundary conditions at the tip to solve for the unknown coefficients. This will involve setting up a system of equations and solving for the coefficients. Once you have the coefficients, you can then solve for the natural frequencies and mode shapes by finding the values of beta that satisfy the equations.

The equation you have for y(x,t) is correct, and you are also correct in your understanding of how to determine δ^2y/δt^2. After this point, you will need to substitute your general equation for y(x) into the differential equation and solve for the coefficients. This will give you the values of A, B, C, and D, which you can then use to solve for the natural frequencies and mode shapes.

I hope this helps guide you in the right direction. If you have any further questions or need clarification, please don't hesitate to ask.