Vibrations problem - Deriving the natural frequencies

In summary, the problem involves a thin beam with a length L, connected to linear springs at each end, and is subject to transverse vibrations. The governing equation for the natural frequencies can be derived using the beam equation and boundary conditions, using concepts such as flexural stiffness, Hooke's law, and the fourth derivative of deflection. Further information can be found on the Wikipedia page for the beam equation.
  • #1
alexisonsmith
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Homework Statement



A thin beam of length L (flexural Stufness EI, cross-sectional area A, density p) is connected to a linear spring of stiffness K_s at each end. Derive the governing equation for the natural frequencies of transverse vibrations from the beam equation and boundary conditions

Homework Equations



Not sure

The Attempt at a Solution



I am really not sure how to start this one, can someone help please
 

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  • #2
Try to use
Flexural stiffness definition (EI=E*I/L),
Hooke's law (F=-kx)
Beam equation
[tex]\frac{\partial^2}{\partial x^2}\left(EI \frac{\partial^2 u}{\partial x^2}\right) = w[/tex]
, in the simple case
[tex]EI \frac{d^4 u}{d x^4} = w(x)[/tex]

See also http://en.wikipedia.org/wiki/Beam_equation
 

FAQ: Vibrations problem - Deriving the natural frequencies

1. What is a vibration problem?

A vibration problem is a physical phenomenon in which a system or structure is subjected to oscillatory motion or movement, causing it to vibrate at a certain frequency. This can be caused by external forces or internal disturbances.

2. How are natural frequencies derived?

Natural frequencies are derived by solving the equations of motion for a system or structure that is subjected to external forces or internal disturbances. This involves determining the equations of motion, applying boundary conditions, and solving for the eigenvalues of the resulting matrix.

3. Why are natural frequencies important?

Natural frequencies are important because they determine the dynamic behavior of a system or structure. Knowing the natural frequencies allows for the prediction of the system's response to external forces or disturbances, and can help identify potential issues or areas of concern.

4. How do natural frequencies affect the stability of a system?

Natural frequencies can affect the stability of a system by determining its response to external forces. If the natural frequency of a system matches the frequency of an external force, resonance can occur, causing the system to vibrate at a larger amplitude. This can lead to instability and potential failure of the system.

5. What factors can affect the natural frequencies of a system?

The natural frequencies of a system can be affected by various factors such as the material properties, geometry, and boundary conditions of the system. Changes in any of these factors can alter the stiffness and mass distribution of the system, thereby changing its natural frequencies.

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