Beam boundary condition problem

[Motorbiker]

Homework Statement

derive the boundary conditions of a beam whichcarries masses and is supported by springs and dampers

Relevant Equations

equations are shown in my working

In my working, I have drawn a free body diagram of the LHS of the beam, I obtained some equations but after that not sure what I need to do. Usuaully in vibration problems you are given BC's then you plug them into the general solution to obtain the frequency equation.

Please can you help me understand how to do this problem? I'm really struggling to understand how to do it.

 

[Chestermiller]

Please use LaTex to render equations. A guide is in the lower left hand corner of your response window.

 

[Motorbiker]

Please use LaTex to render equations. A guide is in the lower left hand corner of your response window.

I think I showed the equations in my working?

 

[Chestermiller]

I think I showed the equations in my working?

It's unreadable.

 

[Motorbiker]

It's unreadable.

I’ve attached another picture of my working, please let me know if you can read this one.

 

[Chestermiller]

It should be $$-ku-C\frac{du}{dt}-mg+Q=\frac{d^2u}{dt^2}$$where Q is the upward shear force exerted by the bar on the mass.

 

[Motorbiker]

It should be $$-ku-C\frac{du}{dt}-mg+Q=\frac{d^2u}{dt^2}$$where Q is the upward shear force exerted by the bar on the mass.

Thanks for this. Could you please explain how you got this equation? I want to understand this better.

 

[Chestermiller]

Thanks for this. Could you please explain how you got this equation? I want to understand this better.

it is from the free body diagram on the mass. It is the force balance equation on the mass.

 

[Motorbiker]

it is from the free body diagram on the mass. It is the force balance equation on the mass.

I'm guessing this equation applies to both sides?
Also how do we know that there is shear force acting on the beam?

 

[Motorbiker]

Please can I get someone help with this? I don't really understand the solution fully, where does the shear force come from?

 

[Chestermiller]

Please can I get someone help with this? I don't really understand the solution fully, where does the shear force come from?

Are the masses attached to the end of the bar or not?

 

[Motorbiker]

Are the masses attached to the end of the bar or not?

Yes they are.

 

[Chestermiller]

Yes they are.

The shear force makes sure that the mass moves with the bar.