# Overdamped oscillator solution as hyperbolic function?

• Vitani11
In summary, the conversation discusses the equation for the general solution of an overdamped harmonic oscillator, which is x(t) = e-βt(C1eωt+C2e-ωt). The equation involves several constants, including β, C1, C2, ω, and t. The goal is to rewrite the equation in terms of the hyperbolic functions cosh(ωt) and sinh(ωt). The conversation also mentions that the solution involves solving for the constants, which can lead to a complex expression involving x(o) and v(0). However, the individual was able to successfully rewrite the equation using the hyperbolic functions.
Vitani11

## Homework Statement

Here is the equation for the general solution of an overdamped harmonic oscillator:
x(t) = e-βt(C1eωt+C2e-ωt)

β decay constant
C1, C2 constants
ω frequency
t time

## The Attempt at a Solution

I know (eωt+e-ωt)/2 = coshωt and (eωt-e-ωt)/2 = sinhωt but how do I implement this if there are constants? I tried to solve for the constants, but I just get a nasty expression in terms of x(o) and v(0) for each C (where v is the velocity) and this doesn't help with rewriting the functions.

Vitani11 said:

## Homework Statement

Here is the equation for the general solution of an overdamped harmonic oscillator:
x(t) = e-βt(C1eωt+C2e-ωt)

β decay constant
C1, C2 constants
ω frequency
t time

## The Attempt at a Solution

I know (eωt+e-ωt)/2 = coshωt and (eωt-e-ωt)/2 = sinhωt but how do I implement this if there are constants? I tried to solve for the constants, but I just get a nasty expression in terms of x(o) and v(0) for each C (where v is the velocity) and this doesn't help with rewriting the functions.
Write eωt and e-ωt in terms of the hyperbolic functions cosh(ωt) and sinh(ωt).

Vitani11
Yes I know that is the goal but how do I do it if there are two constants that are different in front of each e term?

Nevermind I did it

Thank you

## 1. What is an overdamped oscillator?

An overdamped oscillator is a type of system in which the damping force is strong enough to prevent the oscillations from continuing indefinitely. This results in a slower decay of the oscillations until eventually, the system reaches equilibrium.

## 2. How is the solution of an overdamped oscillator represented?

The solution of an overdamped oscillator is represented by a hyperbolic function, specifically a decaying exponential function. This is because the system no longer has oscillations, but rather a smooth and continuous decay towards equilibrium.

## 3. What factors influence the behavior of an overdamped oscillator?

The behavior of an overdamped oscillator is influenced by the damping coefficient, the mass of the object, and the initial conditions. A higher damping coefficient will result in a faster decay towards equilibrium, while a lower damping coefficient will result in a slower decay. A heavier mass will also result in a slower decay, while lighter masses will decay faster. The initial conditions, such as the initial position and velocity, also play a role in the behavior of the oscillator.

## 4. How does the solution of an overdamped oscillator differ from a critically damped or underdamped oscillator?

The solution of an overdamped oscillator is different from a critically damped or underdamped oscillator in that it does not exhibit oscillatory behavior. A critically damped oscillator will reach equilibrium in the shortest amount of time without oscillations, while an underdamped oscillator will exhibit oscillatory behavior before reaching equilibrium.

## 5. Can the solution of an overdamped oscillator be used to model real-world systems?

Yes, the solution of an overdamped oscillator can be used to model real-world systems. Many physical systems, such as shock absorbers, exhibit overdamped behavior. By understanding the solution of an overdamped oscillator, scientists and engineers can design and optimize these systems for various applications.

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