- #1
In summary, we are discussing the effect of an increase in mass on the response amplitude of a forced single-degree-of-freedom oscillator. We are looking for a mathematical proof for the reduction in amplitude. By taking the derivative with respect to mass, we can find the change in amplitude corresponding to a small variation in mass. The only variable that contains the mass is the damping ratio, and we should step through the process to ensure accuracy. There may be different methods for finding this change in amplitude, as seen in the article referenced.
- #2
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- 796
\(\displaystyle X = \dfrac{F}{k} \dfrac{1}{\sqrt{ \dfrac{ 4 \zeta ^2 \omega ^2}{ \omega _n ^2 } + \left ( 1 - \dfrac{ \omega ^2 }{ \omega _n ^2 } \right ) ^2}}\)
So given a small variation in m, dm, we get a corresponding change in X by dX:
\(\displaystyle dX = \dfrac{d}{dm} \left \{ \dfrac{F}{k} \dfrac{1}{\sqrt{ \dfrac{ 4 \zeta ^2 \omega ^2}{ \omega _n ^2 } + \left ( 1 - \dfrac{ \omega ^2 }{ \omega _n ^2 } \right ) ^2}} \right \} ~ dm\)
It looks pretty bad but the only variable that contains the mass is \(\displaystyle \zeta\). Do it step by step.
-Dan
So given a small variation in m, dm, we get a corresponding change in X by dX:
\(\displaystyle dX = \dfrac{d}{dm} \left \{ \dfrac{F}{k} \dfrac{1}{\sqrt{ \dfrac{ 4 \zeta ^2 \omega ^2}{ \omega _n ^2 } + \left ( 1 - \dfrac{ \omega ^2 }{ \omega _n ^2 } \right ) ^2}} \right \} ~ dm\)
It looks pretty bad but the only variable that contains the mass is \(\displaystyle \zeta\). Do it step by step.
-Dan
- #3
anum
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- 0
$\omega_n=\sqrt(k/m)$ is also mass-dependent. Thank you for your reply.
- #4
- #5
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I'll presume that Maple got it right. Though taking that derivative by hand is good practice!
-Dan
-Dan
- #6
- #7
anum
- 6
- 0
its different from what i got
- #8
- 2,021
- 796
Yes, it's different. I'm not sure what method your paper is using.
-Dan
-Dan
1. How does mass affect the amplitude of a wave?
The mass of an object does not directly affect the amplitude of a wave. The amplitude of a wave is determined by the energy or force that creates the wave, not the mass of the object. However, the mass of an object can indirectly affect the amplitude by influencing the energy or force that creates the wave.
2. Is there a relationship between mass and amplitude in a wave?
There is no direct relationship between mass and amplitude in a wave. The amplitude is determined by the energy or force that creates the wave, while the mass of an object only affects this indirectly. However, in some cases, a larger mass may result in a larger amplitude due to the increased energy or force required to create the wave.
3. How does changing the mass of an object affect the amplitude of a standing wave?
Changing the mass of an object in a standing wave will not directly affect the amplitude. The amplitude of a standing wave is determined by the distance between the nodes and antinodes, not the mass of the object. However, changing the mass may affect the frequency of the wave, which can indirectly impact the amplitude.
4. Does increasing the mass of a vibrating object increase the amplitude of the sound wave it produces?
No, increasing the mass of a vibrating object will not directly affect the amplitude of the sound wave it produces. The amplitude of a sound wave is determined by the energy or force that creates the wave, not the mass of the object. However, a larger mass may require more energy or force to create the wave, resulting in a larger amplitude.
5. How does mass affect the amplitude of a water wave?
The mass of an object does not directly affect the amplitude of a water wave. The amplitude of a water wave is determined by the energy or force that creates the wave, not the mass of the object. However, the mass of an object can indirectly affect the amplitude by influencing the energy or force that creates the wave.
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