9.9: Identification of a Mass-Damper-Spring System

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Suppose we have a mechanical system that is known to be an \(m\)-\(c\)-\(k\) system (or a close enough approximation thereto, for engineering purposes), such as that of Figure 9.1.1, and suppose we need to estimate from experimental measurements the system parameters: mass \(m\), effective viscous damping constant \(c\), and stiffness constant \(k\). This is a form of the process known generally as system identification (ID). There are many methods of system ID using both transient response and frequency response. In this section, we illustrate one common method based upon pulse response.

The theoretical basis for system ID by pulse testing of a mechanical \(m\)-\(c\)-\(k\) system is the ideal impulse response given by Equation 9.7.2, with use of Equation 8.4.6, \(I_{U}=I_{F} / k\), and the equation for natural frequency, \(\omega_{n}^{2}=k / m\):

\[x(t)=\frac{I_{F}}{k} \frac{k / m}{\omega_{d}} e^{-\zeta \omega_{n} t} \sin \omega_{d} t=\frac{I_{F}}{m \omega_{d}} e^{-\zeta \omega_{n} t} \sin \omega_{d} t\label{eqn:9.41} \]

Similarly, the values at \(t = 0\) of the upper and lower boundaries of the exponential envelope (for guidance, see Figure 9.7.1) are

\[\pm I_{U} \frac{\omega_{n}^{2}}{\omega_{d}}=\pm \frac{I_{F}}{m \omega_{d}}\label{eqn:9.42} \]

The practical implementation of these equations is based upon using them to approximate the real response from a short force pulse, as is discussed extensively in Section 8.6.

The experimental procedure is as follows. An engineer or technician strikes the mass lightly but sharply with a specially designed hammer. A force sensor mounted in the head of the hammer measures the pulse. A displacement sensor measures motion of the mass due to the force pulse. For accuracy of the system ID, it is essential that the pulse duration \(t_d\) of the hammer strike be very short relative to a quarter of the system natural period: \(t_{d}<<1 / 4 T_{n}\); achieving this might require some experimental iteration, testing hammer contact tips of different degrees of hardness. The time history \(t_f(x)\) of force input to the mass is recorded; this force pulse might be somewhat irregular, such as Figure 8.5, or it might appear to have a more regular form, such as a half-sine. The time history of displacement response, \(x(t)\), also is recorded; with a properly short pulse duration, response \(x(t)\) should look very much like the damped sinusoid of Fig. 9.7.1, ideal impulse response. The steps of the subsequent calculation algorithm are:

Calculate from measurements on the \(x(t)\) graph the damped natural frequency \(f_d\) (Hz) and the viscous damping ratio \(\zeta\). In order to obtain reasonably accurate values of \(f_d\) and \(\zeta\), be sure to average over as many cycles as possible of the damped sinusoid. To aid in the calculation of \(\zeta\), first sketch in the exponential envelope. Next, use the values of \(f_d\) and \(\zeta\) to calculate the two circular natural frequencies in rad/s: \(\omega_{d}=2 \pi f_{d}\) and \(\omega_{n}=\omega_{d} / \sqrt{1-\zeta^{2}}\). For small \(\zeta\), these two frequencies will be essentially identical.
From the \(f_{x}(t)\) graph, use graphical or approximate theoretical integration to calculate the actual force impulse \(I_F\). From the \(x(t)\) graph, find the value at \(t = 0\) of the upper and lower boundaries of the exponential envelope, which are approximately \(\pm I_{F} / m \omega_{d}\) from Equation \(\ref{eqn:9.42}\). It is important in this step to be very careful with the units of these quantities measured from experimental data. Now, you have the data required to calculate the mass from the identity\[m=\left[\omega_{d} \times \frac{1}{I_{F}} \times \frac{I_{F}}{m \omega_{d}}\right]^{-1}\label{eqn:9.43} \]
Finally, calculate the stiffness constant using \(k=m \omega_{n}^{2}\), and calculate the effective viscous damping constant from Equation 9.1.6, \(c=2 \zeta m \omega_{n}=2 \zeta \sqrt{m k}\).

It is always essential in engineering practice to check your calculations as much as is practical. After you have calculated \(m\)-\(c\)-\(k\) from the procedure described above, you can check the validity of your values by using MATLAB (or some similar calculation software) to graph the ideal impulse response of your calculated system, Equation \(\ref{eqn:9.41}\); then compare the calculated graph with the recorded experimental response. If the two graphs are very similar, then your system ID is probably correct, provided that you calculated \(I_F\) correctly. However, suppose that you use an incorrect value of \(I_F\), then calculate wrong values of \(m\)-\(c\)-\(k\) based upon this wrong \(I_F\); if you then graph the ideal impulse response, using these wrong values in Equation \(\ref{eqn:9.41}\), the result will look very much like the recorded experimental response, giving you a false indication that your system ID is correct. Therefore, be sure to calculate correctly the force impulse \(I_F\).