Vibration Problem: Equations of Motion using Hamilton's Principle

In summary, the conversation is about a frame consisting of three rigid bars connected by rotational springs and a viscous damper. The frame is subjected to base lateral motion and the two vertical bars rotate. The goal is to derive the equation of motion using Hamilton's Principle, which requires the calculation of kinetic and potential energy as well as virtual work. The issue at hand is how to handle the diagonal damper in the virtual work equation, and the solution is to consider both the vertical and lateral components of the damper force.
  • #1
terpman1
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1. Frame made up of 3 rigid bars linked to each other and the ground by rotational springs, k, a viscous damper, c which connects opposite corners of the frame. Subject to base lateral motion (s(t)), and the two vertical bars rotate by an amount theta(t). Assuming small displacements, derive EOM. Diagram is attached.



2. Using Hamiltons Principle, need T (kinetic energy), V(Potential Energy), and dW (virtual Work).



3. I have the main framework for the problem laid out, the only issue I am having is how to handle the diagonal damper (c) in the virtual Work equation, any tips/hints?
 

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  • #2
The diagonal damper can be handled using the same approach as for the rotational springs. You will need to account for both the vertical and lateral components of the damper force. The total force is the sum of the two components, and the virtual work equation should include both components.
 

1. What is Hamilton's Principle and how is it used in solving vibration problems?

Hamilton's Principle is a mathematical principle used in mechanics to describe the motion of a system. In the context of vibration problems, Hamilton's Principle is used to find the equations of motion for a vibrating system by minimizing the total potential and kinetic energy of the system.

2. How do you determine the boundary conditions for a vibration problem using Hamilton's Principle?

The boundary conditions for a vibration problem can be determined by applying the principle of virtual work to the system. This involves considering small virtual displacements of the system and setting the virtual work done by external forces and virtual work done by internal forces equal to zero.

3. Can Hamilton's Principle be used for any type of vibration problem?

Yes, Hamilton's Principle can be applied to any type of vibration problem, whether it involves a single degree of freedom or multiple degrees of freedom. The principle provides a systematic approach for finding the equations of motion, regardless of the complexity of the problem.

4. How does Hamilton's Principle differ from other methods used to solve vibration problems?

Unlike other methods such as Lagrange's equations or Newton's laws, Hamilton's Principle does not require the explicit knowledge of the forces acting on the system. Instead, it relies on the total potential and kinetic energy of the system, making it a more general and elegant approach.

5. Are there any limitations to using Hamilton's Principle for vibration problems?

While Hamilton's Principle is a powerful tool for solving vibration problems, it does have some limitations. It may not be suitable for problems involving non-conservative forces or problems with complex geometries. Additionally, the application of the principle can become more challenging for systems with a large number of degrees of freedom.

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