# Solving ode of forced oscillator with dumping

• Cypeq
In summary, the conversation discusses the process of solving a second-order ODE that describes the motion of a forced oscillator with dumping and constant friction. The equation is rearranged and divided into two steps: finding the homogenous solution and the particular solution. The homogenous solution is solved using the characteristic polynomial and the particular solution is found using "guessing". The general solution is then composed of the superposition of the homogenous solutions and the sum of the responses to the external force. The conversation ends with a thank you to the expert for their help.
Cypeq
Hi i have to solve this ODE which descirbes motion of forced oscillator with dumping and constant friction :p

I'm already solving it numerically with Runge-Kutta 4 yet I'm totaly puzzeled how to do it analytically.

equation:

$$mx'' + kx' + w^2_0x + F_f = A cos(\delta t)$$ Ff delta k and w are constant

moving acceleration x'' to one side we get

$$x'' = \frac 1 m (- kx' - w^2_0x - F_f + A cos(\delta t))$$

i need to solve this equation twice to get velocity x' than position x. Yet i have no clue i know only how to solve x' = f(x) first order ODE :/

first of all rearrange the equation

$$mx''+kx'+w^{2}_{0}x=Acos(\delta t)-F_{f}$$

Divide the solution into two steps.

1) Homogenous solution
This is the behaviour of the system without the external force acting on it

$$x''+\frac{k}{m}x'+w^{2}_{0}x=0$$

This is a constant coefficient equation, and solved with the characteristic polynomial:

$$r^{2}+\frac{k}{m}r+\frac{w^{2}_{0}}{m}=0$$

The roots are given by:

$$r_{1,2}=\frac{-k\pm \sqrt{k^{2}-4mw^{2}_{0}}}{2m}$$

Now I'm going to assume $$4mw^{2}_{0}>k^{2}$$ and define:

$$\alpha = \frac{k}{2m}; \omega=\sqrt{-\frac{k^{2}}{4m}+w^{2}_{0}}$$

so
$$r_{1,2}=\alpha\pm i\omega$$

Therefore the set of homogenous solutions is
$$x_{1}(t)=e^{-\alpha t}cos(\omega t); x_{2}(t)=e^{-\alpha t}sin(\omega t)$$

The general homogenous solution is given by superposition.

2) Particular Solution
The response of the system to the external force

I'm going to use "guessing", but first we will divide the response into two:

2.1) Response to the constant force $$F_{f}$$
The easiest guess is that x itself will be constant. So if we choose x=c, we'll have

$$x_{p,1}=c=-\frac{F_{f}}{w^{2}_{0}}$$

2.2) Response to the harmonic force
Now we going to choose some $$x=Ccos(\delta t)+Dsin(\delta t)$$

Substituting this into the equation gives:

$$-mC\delta^{2} cos(\delta t) -mD\delta^{2}sin(\delta t)-kC\delta sin(\deltat) + kD\delta cos(\delta t)+w^{2}_{0}Ccos(\delta t)+w^{2}_{0}Dsin(\delta t)=Acos(\delta t)$$

Comparing coefficient of cosines and sines gives:

$$-mC\delta^{2}+kD\delta+w^{2}_{0}C=A$$
$$-mD\delta^{2}-kC\delta+w^{2}_{0}D=0$$

which gives

$$C=\frac{A(w^{2}_{0}-m\delta^{2})}{(w^{2}_{0}-m\delta^{2})^{2}+k^2\delta^{2}}; D=\frac{Ak\delta}{(w^{2}_{0}-m\delta^{2})^{2}+k^2\delta^{2}}$$

So $$x_{p,2}=\frac{A(w^{2}_{0}-m\delta^{2})}{(w^{2}_{0}-m\delta^{2})^{2}+k^2\delta^{2}}cos(\delta t)+\frac{Ak\delta}{(w^{2}_{0}-m\delta^{2})^{2}+k^2\delta^{2}}sin(\delta t)$$Now finally we compose all of the results together to get the general solution of the ODE:

$$x(t)=c_{1}e^{-\alpha t}cos(\omega t)+c_{2}e^{-\alpha t}sin(\omega t)-\frac{F_{f}}{w^{2}_{0}}+\frac{A(w^{2}_{0}-m\delta^{2})}{(w^{2}_{0}-m\delta^{2})^{2}+k^2\delta^{2}}cos(\delta t)+\frac{Ak\delta}{(w^{2}_{0}-m\delta^{2})^{2}+k^2\delta^{2}}sin(\delta t)$$

The first & second term are superposition of the homogoneous solutions.
The third & fourth term are sum of the responses to the external force.

The coefficients c1 & c2 are arbitrary but given initial conditions you can find them.

Nicely done.

big thanks it took me some time how to get this results but u helped me a lot thanks mate :)

## 1. What is an "ode of forced oscillator with dumping"?

An ode (ordinary differential equation) of forced oscillator with dumping is a mathematical model that describes the motion of a damped harmonic oscillator under the influence of an external force. It is represented by a second-order differential equation.

## 2. How do you solve an ode of forced oscillator with dumping?

To solve an ode of forced oscillator with dumping, you can use various mathematical techniques such as the substitution method, Laplace transform, or numerical methods. The specific method used may depend on the complexity of the equation and the desired level of accuracy.

## 3. What is the significance of the damping factor in an ode of forced oscillator with dumping?

The damping factor in an ode of forced oscillator with dumping represents the rate at which the amplitude of the oscillator decreases over time. It is a measure of the system's energy dissipation and can affect the stability and behavior of the oscillator.

## 4. How does the external force affect the motion of an oscillator in an ode of forced oscillator with dumping?

The external force in an ode of forced oscillator with dumping can alter the natural frequency and amplitude of the oscillator's motion. Depending on the frequency and magnitude of the force, the system may exhibit resonance or undergo chaotic behavior.

## 5. Can an ode of forced oscillator with dumping be applied to real-world systems?

Yes, the ode of forced oscillator with dumping has many practical applications in various fields such as engineering, physics, and biology. It can be used to model the behavior of systems such as electrical circuits, mechanical systems, and biochemical reactions.

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