# Spring Damping

#### DaveMan

so, f = ma, and f = -kx -bv.
rearrange to the form of a differential equation.
i am stuck when the next line simply states the general solution.
i have done this for first differential only, not second derivatives.

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#### jamesrc

Gold Member
For the homogeneous solution to ma = -kx -bv, it is standard practice to find the characteristic equation:

First, rewrite into a standard form:

$\ddot{x} + \frac{b}{m}\dot{x} + \frac{k}{m}x$

Set
$\frac{k}{m} = \omega_n^2$

$\frac{b}{m} = 2\zeta\omega_n$

(the reason why should be clear by the end of the problem; natural frequency and damping ration are useful, meaningful quantities in the study of oscillations)

characteristic equation:

$s^2 + 2\zeta\omega_n s + \omega_n^2 = 0$

find the roots of the characteristic equation (it's just a quadratic in s), s 1,2 , so that the solution to the differential equation is written:

$x(t) = C_1 e^{s_1 t} + C_2 e^{s_2 t}$

using the Euler identity and some algebra, you end up with the solution:

$x(t) = A e^{-\zeta\omega_n t}\cos\left(\omega_d t + \phi \right)$

where the damped frequency $\omega_d = \omega_n \sqrt{1-\zeta^2}$ and the constants A and &phi; (magnitude and phase) are determined by the initial conditions. (You can solve it with a sine or cosine, you'll just end up with a different phase.) That's just the basics off the top of my head, but I hope that helps.

#### DaveMan

I dont understand how you get the charateristic equation.

#### HallsofIvy

Homework Helper
That's usually one of the first things you learn in an introductory differential equation course.

The linear homogeneous differential equation a y"+ by'+ cy= 0 has
"characteristic equation" ar2+ br+ c= 0.

More generally, you replace the nth derivative with rn.

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