# Problems with Critical Damping and Underdamping

In summary, the conversation discusses the concept of damping and two problems related to it. The first problem involves finding the displacement of an automobile suspension at t=1s, given that it is critically damped and has a period of free oscillation of 1s. The second problem asks to show that the ratio of two successive maxima in the displacement of an underdamped harmonic oscillator is constant. The equations provided include the ones for critical and underdamping, but the student is unsure if they are the correct ones to use. They attempt to solve the first problem by finding the value of γ, but it does not seem to work. For the second problem, they are not sure where to begin.

## Homework Statement

The concept of damping is new to me and the problems I have seen have had different known values than I see in the equations I have. Here's two I am working on.

1) An automobile suspension is critically damped, and its period of free oscillation with no
damping is 1s. If the system is initially displaced to a distance x0 and released from rest, find the displacement at t = 1 s.

2) Show that the ratio of two successive maxima in the displacement of an underdamped
harmonic oscillator is constant.

## Homework Equations

Critical: x(t) = A e^ (-γ t ) + B e^ (-γ t) where A and B are constants and γ= (2c)/m

Underdamping: x(t) = A0 e ^ (-γt) cos (ω't + θ0)

I honestly don't know if these are the equations I need to find the answers.

## The Attempt at a Solution

1. γ=ω0
= ω0 = √k/m
I found this Using the equation for the frequency of a spring (without damping)
∴ γ = 2π
If I plug this into the equation I have, I am still left with those constants... so I must have the wrong approach.

2. I can't think of where to begin on this one... it seems like the variables wouldn't be constant if I set up a ratio like x2/x1

1. The equation you have for the solution of a critically damped system in incorrect. What is the correct equation?

2. Just do it: set up the ratio for two successive maxima and see where that gets you.

## What is critical damping?

Critical damping is a type of damping that occurs when a system is able to return to its equilibrium position without oscillating or overshooting. It is the ideal level of damping for a system to reach equilibrium quickly and without any residual motion.

## What are the consequences of underdamping?

Underdamping occurs when the damping force in a system is less than the critical damping level. This results in the system oscillating with decreasing amplitude over time, and it can take longer for the system to reach equilibrium. Underdamping can also cause overshooting of the equilibrium position.

## How do problems with critical damping and underdamping arise?

Problems with critical damping and underdamping can arise due to various factors such as errors in the system design, incorrect selection of damping coefficient, or changes in system parameters such as mass or stiffness. They can also occur due to external disturbances or improper maintenance of the system.

## What are some common techniques for dealing with problems with critical damping and underdamping?

Some common techniques for dealing with these problems include adjusting the damping coefficient, changing the system parameters, or implementing control systems to regulate the damping force. Additionally, proper maintenance and monitoring of the system can help prevent these issues from arising.

## What are the implications of not addressing problems with critical damping and underdamping in a system?

If problems with critical damping and underdamping are not addressed, it can lead to decreased system performance, increased energy consumption, and potential damage to the system. It can also affect the stability and safety of the system, especially in critical applications such as in the aerospace or medical industries.

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