Spring-Mass-Damper by Recurrence Relations

In summary, the conversation is about solving the Mass-Spring-Damper Differential equation, which involves finding the coefficients of a power series on the left by comparing them to the coefficients on the right. The confusion lies in the fact that the right side is in terms of "t" while the left side is in terms of "x" which is a function of "t". The solution involves expanding both sides into series solutions and grouping like terms.
  • #1
tanky322
43
0

Homework Statement



Solve the Mass-Spring-Damper Differential equation

mx''+bx'+kx=exp(-t)cos(t) (Where x'' is d2x/dt2 etc, don't know how to do the dots above :confused:)




I understand how to solve this problem, but the thing that confuses me is that the right is in terms of "t" and left is in terms of "x". I understand that x is a function of t such that x''=d2x/dt2, but I am confused about how to solve the equation by recurrence relations. Once I expand out each side by its series solutions and group like terms, how do I compare terms of x to terms of t?

For example, if this works out to be:

(a1+3a3)+(a2+4a2)x+(a3+6a5)x^2+...=1+t+t^2/2!+...

Can I say that a1+3a3=1; a2+4a2=1; and a3+6a5=1/2!?


Thanks!
 
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  • #2
There are no x2 terms on the left, and it certainly doesn't make any sense to expand x as a power series in x, because that's just x.

The x's are functions of t. x=x(t). So to get a power series on the left, you would write out x as a power series of t, and you want to find the coefficients of the power series by comparing them to the coefficients on the right.
 
  • #3
This slapped me in the face about 10 minutes after I posted, made me feel rather dumb...

Thanks for your reply!
 

1. What is the Spring-Mass-Damper system?

The Spring-Mass-Damper system is a mechanical system that consists of a mass connected to a fixed point by a spring and a damper. It is a commonly used model in physics and engineering to describe the behavior of objects that oscillate or vibrate.

2. How are recurrence relations used in the Spring-Mass-Damper system?

Recurrence relations are used to describe the motion of the Spring-Mass-Damper system by relating the position, velocity, and acceleration of the mass at different points in time. These relations are derived from the equations of motion and can be used to analyze the behavior of the system over time.

3. What are the key parameters in the Spring-Mass-Damper system?

The key parameters in the Spring-Mass-Damper system are the mass of the object, the spring constant of the spring, and the damping coefficient of the damper. These parameters determine the behavior of the system and can be adjusted to study different scenarios.

4. How does the Spring-Mass-Damper system behave under different initial conditions?

The behavior of the Spring-Mass-Damper system is highly sensitive to its initial conditions, such as the initial position, velocity, and acceleration of the mass. These conditions can affect the amplitude, frequency, and stability of the system's oscillations.

5. What are some real-world applications of the Spring-Mass-Damper system?

The Spring-Mass-Damper system has various applications in engineering and physics, including earthquake-resistant building design, shock absorbers in vehicles, and modeling the motion of a pendulum or a swinging door. It is also used in vibration testing and analysis to study the effects of vibrations on structures and machines.

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