Differential Equations: Is there Damping?

In summary, the conversation discusses a spring mass system with an equation of motion involving a sine function and negative exponential terms. The question is whether there is damping and resonance in the system. The participant suggests that there is damping due to the negative exponents, but no resonance because the solution does not increase for large t. They ask for clarification and help on their answer.
  • #1
undrcvrbro
132
0

Homework Statement



A spring mass system has the equation of motion:

c1e^(-t)*sin(t) + c2e^(-t)*cos(t) + 3*sin(t)

Is there damping in the system? Is there resonance in the system?


The Attempt at a Solution


If I had to guess I would say that the 3*sin(t) at the end of the equation would give it resonance because of the periodic motion of the sin function.

Also--this is also a guess-- there could be damping that's caused by the negative exponents on both of the exponentials. The graph for e^(-t) decreases to zero, so maybe that signifies how the movement of the spring is constantly being counteracted by the force of friction until eventually it comes to rest.

Then again, I could be completely wrong. I'm only typing this so that you know I'm giving it a shot. Can someone please help me out here. Obviously, my answers are wrong!

Thanks in advance for any help!
 
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  • #2
Bump.

Anyone?
 
  • #3
undrcvrbro said:

Homework Statement



A spring mass system has the equation of motion:

c1e^(-t)*sin(t) + c2e^(-t)*cos(t) + 3*sin(t)

Is there damping in the system? Is there resonance in the system?


The Attempt at a Solution


If I had to guess I would say that the 3*sin(t) at the end of the equation would give it resonance because of the periodic motion of the sin function.

Also--this is also a guess-- there could be damping that's caused by the negative exponents on both of the exponentials. The graph for e^(-t) decreases to zero, so maybe that signifies how the movement of the spring is constantly being counteracted by the force of friction until eventually it comes to rest.

Then again, I could be completely wrong. I'm only typing this so that you know I'm giving it a shot. Can someone please help me out here. Obviously, my answers are wrong!

Thanks in advance for any help!
Yes, there is damping because, as you say, there are negative exponents. There is no resonance because no part of the solution gets large for large t.
 

FAQ: Differential Equations: Is there Damping?

What is damping in differential equations?

Damping in differential equations refers to the effect of a resistive force that acts against the motion of a system. This force is usually proportional to the velocity of the system and is represented by the damping coefficient.

How does damping affect the behavior of a system?

Damping affects the behavior of a system by reducing the amplitude of its oscillations over time. This means that the system will eventually come to rest instead of continuing to oscillate indefinitely.

What types of damping are there in differential equations?

There are three types of damping: underdamping, critical damping, and overdamping. Underdamping occurs when the damping coefficient is less than a critical value, resulting in oscillatory behavior. Critical damping occurs when the damping coefficient is equal to this critical value, resulting in the fastest return to equilibrium without oscillation. Overdamping occurs when the damping coefficient is greater than the critical value, resulting in a slower return to equilibrium without oscillation.

How is damping represented in differential equations?

Damping is represented in differential equations using the damping coefficient, which is usually denoted by the letter "c." This coefficient is multiplied by the velocity term in the equation, representing the resistive force acting against the motion of the system.

Can damping be beneficial in certain systems?

Yes, damping can be beneficial in certain systems. For example, in mechanical systems, damping can reduce the effects of vibrations and prevent damage. In electrical circuits, damping can prevent oscillations and improve the stability of the system. However, in other systems such as musical instruments, too much damping can negatively affect the sound quality.

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