� � � � 60 13. Natural frequency and damping ratio There is a standard, and useful, normalization of the second order homogeneous linear constant coefficient ODE mx¨ + bx˙ + kx = 0 under the assumption that both the “mass” m and the “spring con-stant” k are positive. It is illustrated in the Mathlet Damping Ratio. In the absence of a damping term, the ratio k/m would be the square of the circular frequency of a solution, so we will write k/m = �n 2 with �n > 0, and call �n the natural circular frequency of the system. Divide the equation through by m: x¨ + (b/m) ˙x + �n2 x = 0. Critical damping occurs when the coefficient of ˙x is 2�n. The damping ratio α is the ratio of b/m to the critical damping constant: α = (b/m)/(2�n). The ODE then has the form (1) x¨ + 2α�nx˙ + �n2 x = 0 Note that if x has dimensions of cm and t of sec, then �n had di-mensions sec−1, and the damping ratio α is “dimensionless,” a number which is the same no matter what units of distance or time are chosen. Critical damping occurs precisely when α = 1: then the characteristic polynomial has a repeated root: p(s) = (s + �n)2 . In general the characteristic polynomial is s2 + 2α�ns + �n2, and it has as roots −α�n ± α2�n 2 − �2 = �n(−α ± α2 − 1).n These are real when |α| ∗ 1, equal when α = ±1, and nonreal when |α| < 1. When |α| → 1, the roots are −α�n ± i�d where (2) �d = 1 − α2 �n is the damped circular frequency of the system. These are com-plex numbers of magnitude �n and argument ±ζ, where −α = cos ζ. Note that the presence of a damping term decreases the frequency of a solution to the undamped equation—the natural frequency �n—by the factor 1 − α2 . The general solution is (3) x = Ae−λ�nt cos(�dt − π)� 61 Suppose we have such a system, but don’t know the values of �n or α. At least when the system is underdamped, we can discover them by a couple of simple measurements of the system response. Let’s displace the mass and watch it vibrate freely. If the mass oscillates, we are in the underdamped case. We can find �d by measuring the times at which x achieves its maxima. These occur when the derivative vanishes, and x˙ = Ae−λ�nt (−α�n cos(�dt − π) − �d sin(�dt − π)) . The factor in parentheses is sinusoidal with circular frequency �d, so successive zeros are separated from each other by a time lapse of ν/�d. If t1 and t2 are the times of neighboring maxima of x (which occur at every other extremum) then t2 − t1 = 2ν/�d, so we have discovered the damped natural frequency: 2ν (4) �d = . t2 − t1 We can also measure the ratio of the value of x at two successive maxima. Write x1 = x(t1) and x2 = x(t2). The difference of their natural logarithms is the logarithmic decrement: � ⎨ x1� = ln x1 − ln x2 = ln . x2 Then −� x2 = e x1. The logarithmic decrement turns out to depend only on the damping ratio, and to determine the damping ratio. To see this, note that the values of cos(�dt−π) at two points of time differing by 2ν/�d are equal. Using (3) we find −λ�nt1x1 eλ�n(t2−t1)= = e . x2 e−λ�nt2 Thus, using (4) and (2), � ⎨ x1 2ν 2να � = ln x2 = α�n(t2 − t1) = α�n �d = � 1 − α2 . From the quantities �d and �, which are directly measurable charac-teristics of the unforced system response, we can calculate the system parameters �n and α: � ⎨2�/2ν �d � (5) α = � , �n = � = 1 + �d . 1 + (�/2ν)2 1 − α2 2νMIT OpenCourseWare http://ocw.mit.edu 18.03 Differential Equations���� Spring 2010 For information about citing these materials or our Terms of Use, visit:
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