# 2-DOF problem with unknown stiffness and velocity

• NTibbs
Your Name]In summary, the problem involves a 2-DOF system with two bodies, A and B, connected by a spring and a viscous damper. The velocity and amplitude of body A are known, as well as the stiffness value for body A. The mass and damping coefficient of body B are also known. The goal is to solve for either the velocity or stiffness of body B, but one of these values must be known in order to solve for the other since the equation of motion for body B has two unknowns. More information or assumptions may be needed to solve for both values.

## Homework Statement

I have a 2-DOF system, whereby I have one body that is grounded by a spring (body A), and a second body (body B) attached to the first by a spring and a viscous damper. For body A, I know the velocity and amplitude (before body B is added). I think I also have the stiffness value for body A (I'm looking at some old notes and I think this was given as part of the original problem). For body B, I know the mass and the damping coefficient.

I would like to solve the velocity or stiffness of body B (by knowing one I can solve the other). With the information I have, is this possible, or must one of these values be known to begin with?

## Homework Equations

equation of motion for body A:
MAx"A+KAxA+KB(xA-xB)+c(x'A-x'B)=P0sinωt

body B:
MB+x"B+KB(xB-xA)+c(x'B-x'A)=0

## The Attempt at a Solution

I've attempted to rearrange the equation above, but ultimately I think I have 2 unknowns here, the velocity and stiffness of B, and I'm uncertain if it is possible to calculate these values without firstly knowing the other. I have Den Hartog's Mechanical Vibrations book which has an example of such a system, but it assumes that these values are already known.

I have all the above information in my notes, so I have been able to check my attempts against the real values, but I am certain that in this problem I did not have stiffness/velocity for B to begin with.

Any assistance would be greatly appreciated, I am quite new to this area of study so please don't hold back if I have made any glaring errors or I'm missing anything obvious!

Thanks,

Nigel

Dear Nigel,

Thank you for sharing your problem with us. Based on the information you have provided, it is indeed possible to solve for either the velocity or stiffness of body B. However, one of these values must be known in order to solve for the other. This is because the equation of motion for body B has two unknowns, x'B and KB, and you would need two equations to solve for them.

If you know the velocity of body B, you can use the equation of motion for body B to solve for the stiffness KB. On the other hand, if you know the stiffness KB, you can use the equation of motion for body B to solve for the velocity x'B.

I would suggest going back to your notes and see if there is any additional information that can help you solve for one of these values. If not, you may need to gather more data or assumptions in order to solve for both the velocity and stiffness of body B.

I hope this helps and good luck with your problem!

Best regards,