Is there a func' for ideal x-section area in a resonant beam?

[silverslith]

Obviously it has to taper from the fixed end to the most mobile one. Or from the centre (node) of a symmetrical one to the ends (antinodes).

Talking about longatudinal harmonic resonance only here.

Ideally the static instants at full compression or tension would give a constant percentage length change from unloaded along the rod.

Does it follow that at any giventime, a point X distance from the node, has Velocity and acceleration: both proportional to X or am I confused?

I'm pretty sure this is an integration of shm mass/spring systems or somesuch.

Is there a formula that emerges for the rate of taper of the crossectional area for a given material elastic modulus and frequency?

 

[Almsco]

Yes, there is a formula that can be used to calculate the rate of taper of the cross-sectional area based on the given material elastic modulus and frequency. The formula is F = k/x, where F is the frequency, k is the spring constant, and x is the distance from the node. This formula can be used to calculate the rate of taper of the cross-sectional area.

 

[charng]

I can provide some insights into the concept of ideal cross-section area in a resonant beam. Firstly, it is important to understand that the ideal cross-section area of a resonant beam is dependent on several factors such as the material properties, frequency of resonance, and the mode of vibration (longitudinal in this case). Therefore, there is no single formula that can be used to calculate the ideal cross-section area for all resonant beams.

However, there are some general principles that can guide the design of the ideal cross-section area for a resonant beam. One of these principles is that the cross-section area should taper from the fixed end to the most mobile end. This is because the fixed end experiences the least displacement during resonance, while the most mobile end experiences the highest displacement. Therefore, the cross-section area should be larger at the fixed end and gradually decrease towards the most mobile end to ensure that the beam can withstand the stress and strain caused by the resonance.

Another important principle is that the cross-section area should be proportional to the distance from the node (centre) of the beam. This means that the cross-section area should decrease in a linear or exponential manner from the centre to the ends (antinodes). This is because the displacement, velocity, and acceleration of a point on the beam are all proportional to its distance from the node. Therefore, the cross-section area should also decrease in a similar manner to ensure that the beam can withstand the varying levels of stress and strain.

To answer the question about whether velocity and acceleration are both proportional to distance from the node, the answer is yes. This is a fundamental principle of resonance in which the velocity and acceleration of a point on the beam are both proportional to its distance from the node. This can be mathematically represented using the equations of motion for a simple harmonic oscillator.

In conclusion, the ideal cross-section area for a resonant beam is dependent on several factors and there is no single formula that can be used to calculate it. However, there are some general principles that can guide the design of the ideal cross-section area, such as tapering from fixed end to mobile end and being proportional to the distance from the node. These principles can be further refined and calculated using mathematical models and equations of motion.