Critical Damping

If the system is then called critically damped. Note that both the roots of the characteristic equation become equal in this case:



The theory of ordinary differential equations (2^nd order linear homogeneous equations with constant coefficients) tells that the solution in this case is of the form:



The constants A₁ and A₂ may be written in terms of initial conditions; let at t = 0, q = q₀ and dq/dt = v₀. Hence, we get A₁ = q₀ and from





The above solution is also non-oscillatory. The displacement exponentially falls to zero as time increases.

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