Vibration of a cantilever beam problem

In summary, the conversation discusses the setup of a rectangular cantilever beam attached to a wall, with a mass and motor attached to one end. The motor applies a force described by an equation, and the task is to derive an expression for the vertical vibration of the mass. The conversation also mentions the need to find a stable solution, which may imply simple harmonic motion without damping.
  • #1
Feodalherren
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Homework Statement


A rectangular cantilever beam of length L and side lengths b with negligible mass is attached horizontally to a wall. A mass m is attached to the end of the beam where a motor is also located. The motor applies a force on the end of the beam described by the equation P*sin(w*t).

Derive an expression for the vertical vibration of the mass m, that is, find x(t) for a stable solution.

Homework Equations


k = 3EI / L^3
I = b^4/12

The Attempt at a Solution


I'm at a loss here. I am unaware of any equation that describes the damping in a cantilever beam. I want to set up the problem as

x'' + c/m x' + k/m x = Psin(wt)/m

which is how I usually solve these types of problems. But I'm at a loss when it comes to finding out how to express c. Furthermore, when it says to find a stable solution, is it hinting towards harmonic motion?
 
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  • #2
Perhaps they mean that it is simple harmonic motion without damping?
 
  • #3
Feodalherren said:
Perhaps they mean that it is simple harmonic motion without damping?
Yes, it is forced, undamped.
 
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1. What is a cantilever beam?

A cantilever beam is a type of structural element that is supported at one end and free to move at the other end. It is commonly used in engineering and construction as a support for bridges, buildings, and other structures.

2. What is the vibration of a cantilever beam?

The vibration of a cantilever beam refers to the movement or oscillation of the beam in response to an external force or disturbance. This vibration can occur in different modes, such as transverse, torsional, or longitudinal, depending on the direction of the force and the properties of the beam.

3. Why is the vibration of a cantilever beam important?

The vibration of a cantilever beam is important because it can affect the structural integrity and stability of a building or bridge. Excessive vibration can lead to fatigue and failure of the beam, causing damage or collapse of the structure. It is also important in the design and analysis of structures, as it helps engineers understand and predict the behavior of the beam under different loading conditions.

4. What factors affect the vibration of a cantilever beam?

The vibration of a cantilever beam is influenced by several factors, including the material and geometry of the beam, the magnitude and direction of the external force, and the boundary conditions at the support. Other factors such as damping, temperature, and environmental conditions can also affect the vibration of the beam.

5. How is the vibration of a cantilever beam analyzed?

The vibration of a cantilever beam can be analyzed using mathematical models and equations, such as the Euler-Bernoulli beam theory or the Rayleigh-Ritz method. These methods take into account the properties of the beam and the external loading to determine the natural frequencies and mode shapes of the vibration. Advanced techniques such as finite element analysis (FEA) can also be used for more complex and accurate analysis of the beam's vibration.

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