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9.5: Calculation of Viscous Damping Ratio

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    7681
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    Consider an underdamped 2nd order system in a state of free vibration, i.e., with zero input, \(u(t)=0\). This free vibration can be an initial-condition response or the residual response after input excitation has ceased, e.g., for \(t>t_{d}\) if the input is a pulse. Then the free-decay will have the form of Figure \(\PageIndex{1}\). On the figure, a reference local extreme value \(x\left(t_{0}\right) \equiv x_{0}\) is annotated (at a crest on Figure \(\PageIndex{1}\), but it could just as well be at a trough), and subsequent local extreme values (both crests and troughs) also are annotated. We wish to calculate viscous damping ratio \(\zeta\) from the graph of response. Note also that the exponential envelope is indicated on Figure \(\PageIndex{1}\) with dashed lines.

    clipboard_e2479c46269667936cbbe782110ed1399.png
    Figure \(\PageIndex{1}\): Free-vibration response of a damped 2nd order system

    We use Equation \(\ref{eqn:9.21b}\) for free-vibration response:

    \[x(t)=x_{\max } e^{-\zeta \omega_{n} t} \cos \left(\omega_{d} t+\phi\right)\label{eqn:9.21b} \]

    From this equation, we form the ratio \(x_{0} / x_{r}\) of the reference extreme absolute value \(\left|x\left(t_{0}\right)\right|\) divided by the \(r\)th crest or trough absolute value, where \(r\) as shown on Figure \(\PageIndex{1}\) can have either integer or half-integer values. Note that \(\left|\cos \left(\omega_{d} t+\phi\right)\right|=1\) at each of the local extreme values, and that the time of the \(r\)th extreme value is \(t_{r}=t_{0}+r T_{d}\), where \(T_d = 2 \pi / \omega_{d}\) is the damped natural period. So the required ratio is

    \[\frac{x_{0}}{x_{r}}=\frac{e^{-\zeta \omega_{n} t_{0}}}{e^{-\zeta \omega_{n}\left(t_{0}+r T_{d}\right)}}=e^{\zeta \omega_{n} r T_{d}}=\exp \left(\zeta \omega_{n} r \frac{2 \pi}{\omega_{n} \sqrt{1-\zeta^{2}}}\right)=\exp \left(2 \pi r \frac{\zeta}{\sqrt{1-\zeta^{2}}}\right)\label{eqn:9.25} \]

    Taking the natural logarithm of Equation \(\ref{eqn:9.25}\) gives the so-called logarithmic decrement:

    \[\ln \left(\frac{x_{0}}{x_{r}}\right)=2 \pi r \frac{\zeta}{\sqrt{1-\zeta^{2}}} \Rightarrow \frac{\zeta}{\sqrt{1-\zeta^{2}}}=\frac{\ln \left(x_{0} / x_{r}\right)}{2 \pi r} \equiv \zeta_{s}\label{eqn:9.26} \]

    In the last term of Equation \(\ref{eqn:9.26}\), we define \(\zeta_s\) as being the approximate for "small" damping ratio \(zeta\). It is very common for a system to have positive, but small damping. We define damping to be small if \(\sqrt{1-\zeta^{2}} \approx 1\), which simplifies considerably equations such as Equation \(\ref{eqn:9.26}\). For \(\zeta=0.2\), \(\sqrt{1-\zeta^{2}}=0.980\), so this is a reasonable upper limit for "smallness." We can find the exact equation for \(\zeta\) by squaring Equation \(\ref{eqn:9.26}\) and then proceeding algebraically to derive

    \[\zeta=\frac{\zeta_{s}}{\sqrt{1+\zeta_{s}^{2}}}, \text { for } 0 \leq \zeta<1\label{eqn:9.27} \]

    It might appear that the preceding derivation requires the values of \(x_0\) and \(x_r\) to be at crests and troughs of the response plot, and that these should be zero-to-peak values; but neither of these restrictions is necessary. The values of \(x_0\) and \(x_r\) can be at any convenient instants along the time history (zeros of the response, as well as extremes), provided that we interpret \(x_0\) and \(x_r\) as being the magnitudes of the exponential envelope at the chosen instants. Normally, only the free-vibration response plot is available (from a storage oscilloscope, strip-chart recorder, etc.), so we should sketch in the exponential envelope to aid in measuring \(x_0\), \(x_r\) and \(r\). Moreover, rather than measuring zero-to-peak values, it is more accurate and easier to measure \(x_0\) and \(x_r\) as peak-to-peak values, from the lower exponential boundary to the upper exponential boundary.

    Let us summarize the procedure for measuring/calculating \(\zeta\) from a plot of free-vibration response. First, sketch in the exponential envelope. Next, choose time instants along the graph at which you can measure with reasonable accuracy the number of periods \(r\) (usually an integer, half-integer, or quarter-integer) and the magnitudes \(x_0\) and \(x_r\) between the exponential boundaries. Next, substitute the measured values of \(r\), \(x_0\), and \(x_r\) into Equation \(\ref{eqn:9.26}\) and calculate \(\zeta_s\). If this \(\zeta_{s}\) is \(\leq 0.2\), then \(\zeta \approx \zeta_{s}\) with sufficient engineering accuracy. However, if \(0.2<\zeta_{s}<1\), then calculate \(\zeta\) more accurately from Equation \(\ref{eqn:9.27}\).

    There is a simplified version of Equation \(\ref{eqn:9.26}\) that is often used for quick calculation of small \(\zeta\). If possible, we find the reference magnitude and the \(r\)th magnitude such that \(x_{0} / x_{r}=2\), and we label the number of periods as \(r_{1 / 2}\). Hence, \(\ln \left(x_{0} / x_{r}\right) / 2 \pi \equiv 0.110\), which leads us from Equation \(\ref{eqn:9.26}\) to the half-amplitude formula for small \(\zeta\):

    \[\zeta_{s}=\frac{0.110}{r_{1 / 2}} \approx \zeta\label{eqn:9.28} \]

    Finally, the preceding derivation was based upon Equation \(\ref{eqn:9.21b}\) for “displacement” response \(x(t)\). However, the formulas derived for \(\zeta\) are equally valid if the measurements are made from graphs of “velocity” \(\dot{x}(t)\) or “acceleration” \(\ddot{x}(t)\). This is so because, from Equations 9.4.5 and 9.4.6, the derivatives of \(x(t)\) have the same exponentially-bounded sinusoidal form as \(x(t)\) itself.


    This page titled 9.5: Calculation of Viscous Damping Ratio is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by William L. Hallauer Jr. (Virginia Tech Libraries' Open Education Initiative) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.