# Oscillation with friction - Analytical mechanics

• NODARman
NODARman
Homework Statement
.
Relevant Equations
.
Hi, I had those exercises and want to know if they're correct. Also, feedback/tips would be great from you, professionals.

$$A$$

1. Let's consider the oscillator with a friction parameter...

m \ddot{x}+\alpha \dot{x}=-\kappa x

but with

\alpha^2=4 m \kappa

and after inserting, show that the general solution will be like this:

x(t)=\mathrm{e}^{-\alpha t / 2 m}[\mathcal{A} t+\mathcal{B}]

Express A and B constants with the initial coordinates and velocity and analyze.

My solution:\begin{aligned}
&m\ddot{x} + \dot{x}\sqrt{4mk} +kx=0\\
&x(t)=e^{-\frac{\alpha t}{2m}}[\mathcal {A}t+\mathcal{B}]\\
&\\
&\dot{x}(t)=-\frac{\alpha}{2m} e^{-\frac{\alpha t}{2m}} [\mathcal {A}t+\mathcal{B}]+e^{-\frac{\alpha t}{2m}}\mathcal{A}\\
&\ddot{x}(t)=-\frac{4\mathcal{A}\alpha m-\mathcal{A}{\alpha^{2}}t-\mathcal{B}{\alpha^{2}}}{4m^2}e^{-\frac{\alpha t}{2m}}\\
&\\
&x(0)=\mathcal{B}\equiv x_0 \\
&\dot{x}(0)=-\frac{\alpha}{2m} \mathcal{B}+\mathcal{A} = -\frac{\alpha}{2m} x_0 +\mathcal{A} \\
&\\
&\mathcal{A} = \dot{x}_0 + \frac{\alpha}{2m} x_0\\
&\\
&x(t)=e^{-\frac{\alpha t}{2m}}[\dot{x}_0 + \frac{\alpha}{2m} x_0 t+x_0] = e^{-\frac{\alpha t}{2m}}\left[\dot{x}_0 + \left(\frac{\alpha}{2m} t+1\right)x_0\right]\\
\end{aligned}

Last edited:
NODARman said:
Homework Statement: .
Relevant Equations: .

##\dots## and want to know if they're correct.
How about substituting your solution back in the original equation? That's the first thing I would do to verify my solution. Needless to say this doesn't look right because you have a damped harmonic oscillator with no oscillatory term(s) in the equation.

kuruman said:
Needless to say this doesn't look right because you have a damped harmonic oscillator with no oscillatory term(s) in the equation.
It should turn out to be the critical damping case.

hutchphd

## What is oscillation with friction in the context of analytical mechanics?

Oscillation with friction refers to the study of systems that exhibit periodic motion where frictional forces, such as damping, are present. In analytical mechanics, this involves analyzing how these frictional forces affect the system's motion, typically leading to a gradual decrease in amplitude over time.

## How does friction affect the amplitude of an oscillating system?

Friction causes the amplitude of an oscillating system to decrease over time. This phenomenon is known as damping. The energy lost to frictional forces results in a gradual reduction of the maximum displacement from the equilibrium position, leading to a decay in the oscillatory motion.

## What are the different types of damping in oscillatory systems?

There are three primary types of damping in oscillatory systems: underdamping, critical damping, and overdamping. Underdamping occurs when the system oscillates with a gradually decreasing amplitude. Critical damping is the condition where the system returns to equilibrium as quickly as possible without oscillating. Overdamping is when the system returns to equilibrium without oscillating, but more slowly than in the critically damped case.

## How is the equation of motion for a damped harmonic oscillator derived?

The equation of motion for a damped harmonic oscillator is derived by considering the forces acting on the system. For a mass-spring-damper system, Newton's second law gives $$m\ddot{x} + b\dot{x} + kx = 0$$, where $$m$$ is the mass, $$b$$ is the damping coefficient, $$k$$ is the spring constant, $$x$$ is the displacement, $$\dot{x}$$ is the velocity, and $$\ddot{x}$$ is the acceleration. This second-order differential equation describes the motion of the damped oscillator.

## What is the significance of the damping ratio in an oscillating system?

The damping ratio, often denoted by $$\zeta$$, is a dimensionless measure of damping in a system. It is defined as the ratio of the actual damping coefficient to the critical damping coefficient. The damping ratio determines the nature of the system's response: if $$\zeta < 1$$, the system is underdamped and oscillates; if $$\zeta = 1$$, the system is critically damped and returns to equilibrium without oscillating; if $$\zeta > 1$$, the system is overdamped and returns to equilibrium without oscillating, but more slowly.

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