Classification of Harmonic Oscillators with Damping

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m \frac{d^{2} y}{d t^{2}} + b \frac{d y}{d t} + k y = 0

$m frac(d^(2)y)(d t^(2))+ b frac(dy)(dt) + ky = 0$

(m) Mass

$(m) "Mass"$

(b) Damping coefficient

$(b) "Damping coefficient"$

(k) Spring constant

$(k) "Spring constant"$

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Solutions to the second-order equation
$m \frac{d^{2} y}{d t^{2}} + b \frac{d y}{d t} + k y = 0$ $m frac(d^(2) y)(d t^(2)) + b frac(dy)(dt) + ky =0$

typically serve as models for harmonic oscillators. Variables $m$ $m$ and $k$ $k$ are always greater than zero, representing the mass and spring constant, whereas $b$ $b$ is the damping coefficient and is greater than or equal to zero.

You may also see the form above written as:
$\frac{d^{2} y}{d t^{2}} + p \frac{d y}{d t} + q y = 0$ $frac(d^(2) y)(d t^2) + p frac(dy)(dt) + qy =0$

because $m \neq 0$ $m!= 0$ , in which case $p = b / m$ $p=b"/"m$ and $q = k / m$ $q=k"/"m$ are non-negative constants. A linear system representing this second-order equation is written as follows:
$frac( d bb(Y))(dt)= ((0, 1), (-p, -q))bb(Y)$

The Linearity Principle allows us to produce new solutions from known ones by adding solutions to each other and multiplying solutions by constants. This means that second-order equations of the form $a frac(d^(2) y)(d t^(2)) + b frac(dy)(dt) + cy =0$ where $a$ , $b$ , $c$ are arbitrary constants are said to be linear. Blanchard, Devaney, and Hall say, "More precisely these equations are homogeneous, constant-coefficient, linear, second-order equations." This is because of the constants $a$ , $b$ , $c$ and the fact that the right side is equal to zero, making it homogeneous (325).

Depending on the variables, $m$ , $b$ , and $k$ , we can tell a lot about how it relates to harmonic oscillators. Harmonic oscillators that are undamped, when $b=0$ , are spring systems that are unaffected by friction. The motion created by an undamped harmonic oscillator is referred to as simple harmonic motion. When the damping coefficient effects the motion of the spring, the harmonic oscillator is referred to as being either underdamped, overdamped, or critically damped.

Whether the harmonic oscillator is underdamped, overdamped, or critically damped the constants $m$ , $b$ , and $k$ lead us to the characteristic equation
$m s^2 + bs + k = 0$

with roots (hence eigenvalues) given by the quadratic formula
$frac(-b pm sqrt(b^2 -4mk))(2m)$ .

If the discriminant $b^2 - 4mk$ <0, the harmonic oscillator is said to be underdamped.
If the discriminant $b^2 -4mk$ >0, the harmonic oscillator is said to be overdamped.
If the discriminant $b^2-4mk$ =0, the harmonic oscillator is said to be critically damped.

If you think about it, by using the quadratic formula we are simply using a short cut to get to the answer that we got by converting the second-order equation into a first-order linear, homogeneous system of equations; either strategy is effective given an equation of the form:
$a frac(d^(2) y)(d t^(2)) + b frac(dy)(dt) + cy =0$

However, it is important to know how to produce an answer using each method because by thinking about the solution in terms of eigenvalues and eigenvectors you gain a deeper understanding of the subject matter; it relates the algebraic calculations to the geometric interpretation.

You may also wish to view the page for the General Solution of Second-Order, Linear, Homogeneous Equations .

Sources

Blanchard, Paul, Robert L. Devaney, and Glen R. Hall. Differential Equations. 3rd ed. Belmont, CA: Thomson Brooks/Cole, 2006. Print.

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Classification of Harmonic Oscillators with Damping

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