- #1
Does Increasing Mass Reduce the Amplitude of a Forced SDOF Oscillator?
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Discussion Overview
The discussion revolves around the effects of increasing mass on the response amplitude of a forced single-degree-of-freedom (SDOF) oscillator. Participants are seeking mathematical proof and derivations related to this relationship.
Discussion Character
- Technical explanation, Mathematical reasoning, Debate/contested
Main Points Raised
- One participant is investigating how an increase in mass affects the response amplitude and is looking for mathematical proof of this effect.
- Another participant provides a formula for amplitude that includes mass as a variable and suggests deriving the change in amplitude with respect to mass.
- A participant notes that the natural frequency, which is mass-dependent, also plays a role in the amplitude response.
- There are attempts to use software (Maple) to compute derivatives related to the problem, indicating a reliance on computational tools for verification.
- Discrepancies in results are noted, with participants expressing uncertainty about the methods used in different articles.
Areas of Agreement / Disagreement
Participants express differing views on the methods and results related to the effect of mass on amplitude, indicating that multiple competing perspectives remain unresolved.
Contextual Notes
Participants have not reached consensus on the mathematical derivations or the implications of their findings, and there are indications of missing assumptions or differing methodologies in the literature referenced.
- #2
- 2,020
- 843
[math]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}}[/math]
So given a small variation in m, dm, we get a corresponding change in X by dX:
[math]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[/math]
It looks pretty bad but the only variable that contains the mass is [math]\zeta[/math]. Do it step by step.
-Dan
So given a small variation in m, dm, we get a corresponding change in X by dX:
[math]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[/math]
It looks pretty bad but the only variable that contains the mass is [math]\zeta[/math]. Do it step by step.
-Dan
- #3
anum
- 6
- 0
$\omega_n=\sqrt(k/m)$ is also mass-dependent. Thank you for your reply.
- #4
- #5
- 2,020
- 843
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
- 2,020
- 843
Yes, it's different. I'm not sure what method your paper is using.
-Dan
-Dan
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