- #1
Does Increasing Mass Reduce the Amplitude of a Forced SDOF Oscillator?
Click For Summary
SUMMARY
The discussion focuses on the impact of increasing mass on the response amplitude of a forced single-degree-of-freedom (SDOF) oscillator. The mathematical relationship derived is \(\delta x X = \dfrac{F}{k} \dfrac{1}{\sqrt{ \dfrac{ 4 \zeta ^2 \omega ^2}{ \omega _n ^2 } + \left ( 1 - \dfrac{ \omega ^2 }{ \omega _n ^2 } \right ) ^2}}\). Participants explore the derivative of the amplitude concerning mass changes, emphasizing that the damping ratio \(\zeta\) is the only variable dependent on mass. The discussion also references the use of Maple for calculations, confirming its reliability for deriving these equations.
PREREQUISITES- Understanding of forced single-degree-of-freedom (SDOF) oscillators
- Familiarity with damping ratios and their impact on oscillatory systems
- Knowledge of mathematical derivatives and their application in physics
- Experience with computational tools like Maple for mathematical modeling
- Study the derivation of the damping ratio \(\zeta\) in forced oscillators
- Learn about the effects of mass on natural frequency \(\omega_n\) in SDOF systems
- Explore the use of Maple for solving differential equations in mechanical systems
- Investigate the relationship between amplitude and frequency in forced oscillations
Mechanical engineers, physicists, and students studying dynamics and oscillatory systems will benefit from this discussion, particularly those interested in the mathematical modeling of forced oscillators.
- #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
Similar threads
- · Replies 3 ·
- · Replies 1 ·
Undergrad
Oscillation with constrained amplitude
- · Replies 2 ·
- · Replies 11 ·
- · Replies 10 ·
- · Replies 8 ·
- · Replies 20 ·