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Modes and the Characteristic Equation

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Introduction of the Characteristic Equation

Several presumptions are made in order to solve

{$$ my^{\prime\prime} + by^{\prime} + ky = 0 $$}.

This form emphasizes the application to spring-damper problems in which m is mass, k is the spring coefficient from Hooke's law, and b is a coefficient of damping. These are taken to be constant over the applicable conditions. Usually tacit is that the independent variable is t which is time in most physical problems. Dividing through by m results in the standard form:

{$$ y^{\prime\prime} + Ay^{\prime} + By = 0 $$}.

Some results depend on A and B being real numbers, which in physical modeling they should be.

From this we find the characteristic equation by substituting {$ e^{rt} $} for y.

{$$ \begin{align} y^{\prime\prime} + Ay^{\prime} + By &= 0 \cr ( e^{rt})^{\prime\prime} + A( e^{rt})^{\prime} + B( ) &= 0 \cr r^2e^{rt} + Are^{rt} + Be^{rt} &= 0 \cr \left(r^2 + Ar + B\right)e^{rt} &= 0 \cr r^2 + Ar + B &= 0 \cr \end{align}$$}.

This is a quadratic equation, because the differential equation is second order. It has two roots, and the possibilities are:

RootsCaseRootsSolution
Two real and unequal rootsoverdamped{$ r_1,r_2 $}{$ y = c_1e^{r_1 t} + c_2e^{r_2 t} $}
Two real and equal rootsunderdamped{$ r_1 = r_2 $}{$ y = c_1e^{r_1 t} + c_2te^{r_1 t} $}
Two complex rootscritically damped{$ r=\alpha \pm \beta i $}{$ y = e^{\alpha t}[ c_1\cos(\beta t) + c_2\sin(\beta t)] $}

<< Unit II: Second Order Constant Coefficient Linear Equations | Trail MIT 18.03SC Differential Equations | Damped Harmonic Oscillators >>

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