Designing a Cantilever Beam for Resonance Frequency

[engineer23]

Homework Statement

The problem is to design a cantilever beam with a certain resonant frequency.

Homework Equations

w = sqrt (k/m)

The Attempt at a Solution

I know k for a beam and m can be written as density * volume.

My question deals with the w. If I am given a frequency in Hz, can I plug it in for omega directly? Or do I need to multiply by 2*pi first?

 

[PhanthomJay]

Homework Statement

The problem is to design a cantilever beam with a certain resonant frequency.

Homework Equations

w = sqrt (k/m)

The Attempt at a Solution

I know k for a beam and m can be written as density * volume.

My question deals with the w. If I am given a frequency in Hz, can I plug it in for omega directly?

No.

Or do I need to multiply by 2*pi first?

Yes. The angular frequency w, in radians/sec , is the natural frequency, f , in Hz, multiplied by 2*pi. The equation you note is for the angular frequency.

 

[Mene]

I would approach this problem by first understanding the underlying principles of resonance and how it relates to the design of a cantilever beam. Resonance occurs when an object is subjected to a periodic force that matches its natural frequency, causing it to vibrate with a larger amplitude. In the case of a cantilever beam, its natural frequency is dependent on its stiffness (k) and mass (m), as shown in the equation w = sqrt (k/m).

To design a cantilever beam with a specific resonant frequency, we need to manipulate the stiffness and mass of the beam. This can be done by adjusting the material properties and dimensions of the beam. For example, increasing the stiffness by using a stiffer material or increasing the beam's cross-sectional area can result in a higher resonant frequency. Similarly, decreasing the mass by using a lighter material or reducing the beam's length can also increase the resonant frequency.

In regards to the question about the use of frequency in Hz and angular frequency (w), it is important to note that the equation for resonant frequency uses angular frequency (w) rather than frequency (f) in Hz. This is because angular frequency is a more fundamental quantity and is directly related to the natural frequency of an object. Therefore, when given a frequency in Hz, it is necessary to convert it to angular frequency by multiplying it by 2*pi before plugging it into the equation w = sqrt (k/m).

In conclusion, designing a cantilever beam for a specific resonant frequency requires an understanding of the underlying principles of resonance and the manipulation of stiffness and mass. It also involves converting frequency in Hz to angular frequency before using it in the equation for resonant frequency.