10.3: Frequency Response of Mass-Damper-Spring Systems

Reviewing the basic 2^nd order mechanical system from Figure 9.1.1 and Section 9.2, we have the \(m\)-\(c\)-\(k\) and standard 2^nd order ODEs:

\[m \ddot{x}+c \dot{x}+k x=f_{x}(t) \Rightarrow \ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=\omega_{n}^{2} u(t)\label{eqn:10.15} \]

with the definitions

\[\omega_{n}=\sqrt{\frac{k}{m}}, \quad \zeta \equiv \frac{c}{2 m \omega_{n}}=\frac{c}{2 \sqrt{m k}} \equiv \frac{c}{c_{c}}, \quad u(t) \equiv \frac{1}{k} f_{x}(t)\label{eqn:10.16} \]

So we can use the correspondence \(U=F / k\) to adapt FRF (10-10) directly for \(m\)-\(c\)-\(k\) systems:

\[\frac{X(\omega)}{F / k}=\frac{1}{\sqrt{\left(1-\beta^{2}\right)^{2}+(2 \zeta \beta)^{2}}}, \quad \phi(\omega)=\tan ^{-1}\left(\frac{-2 \zeta \beta}{1-\beta^{2}}\right), \quad \beta \equiv \frac{\omega}{\sqrt{k / m}}\label{eqn:10.17} \]

In the conceptually simplest form of forced-vibration testing of a 2^nd order, linear mechanical system, a force-generating shaker (an electromagnetic or hydraulic translational motor) imposes upon the system’s mass a sinusoidally varying force at cyclic frequency \(f\), \(f_{x}(t)=F \cos (2 \pi f t)\). The resulting steady-state sinusoidal translation of the mass is \(x(t)=X \cos (2 \pi f t+\phi)\). Calibrated sensors detect and \(x(t)\), and then \(F\), \(X\), \(f\) and \(\phi\) are measured from the electrical signals of the sensors. In principle, the testing involves a stepped-sine sweep: measurements are made first at a lower-bound frequency in a steady-state dwell, then the frequency is stepped upward by some small increment and steady-state measurements are made again; this frequency stepping is repeated again and again until the desired frequency band has been covered and smooth plots of \(X / F\) and \(\phi\) versus frequency \(f\) can be drawn.

For system identification (ID) of 2^nd order, linear mechanical systems, it is common to write the frequency-response magnitude ratio of Equation \(\ref{eqn:10.17}\) in the form of a dimensional magnitude of dynamic flexibility¹:

\[\frac{X(\omega)}{F}=\frac{1}{k} \frac{1}{\sqrt{\left(1-\beta^{2}\right)^{2}+(2 \zeta \beta)^{2}}}=\frac{1}{\sqrt{\left(k-m \omega^{2}\right)^{2}+c^{2} \omega^{2}}}\label{eqn:10.18} \]

Also, in terms of the basic \(m\)-\(c\)-\(k\) parameters, the phase angle of Equation \(\ref{eqn:10.17}\) is

\[\phi(\omega)=\tan ^{-1}\left(\frac{-c \omega}{k-m \omega^{2}}\right)\label{eqn:10.19} \]

Note that if \(\omega \rightarrow 0\), dynamic flexibility Equation \(\ref{eqn:10.18}\) reduces just to the static flexibility (the inverse of the stiffness constant), \(X(0) / F=1 / k\), which makes sense physically. In fact, the first step in the system ID process is to determine the stiffness constant. Electromagnetic shakers are not very effective as static loading machines, so a static test independent of the vibration testing might be required. In principle, static force \(F\) imposed on the mass by a loading machine causes the mass to translate an amount \(X(0)\), and the stiffness constant is computed from

\[k=\frac{F}{X(0)}\label{eqn:10.20} \]

However, suppose that it is more convenient to shake the mass at a relatively low frequency (that is compatible with the shaker’s capabilities) than to conduct an independent static test. Note from Figure 10.2.1 that if the excitation frequency is less than about 25% of natural frequency \(\omega_n\), then the magnitude of dynamic flexibility is essentially the same as the static flexibility, so a good approximation to the stiffness constant is

\[k \approx\left(\frac{X\left(\omega \leq 0.25 \omega_{n}\right)}{F}\right)^{-1}\label{eqn:10.21} \]

Even if it is possible to generate frequency response data at frequencies only as low as 60-70% of \(\omega_n\), one can still knowledgeably extrapolate the dynamic flexibility curve down to very low frequency and apply Equation \(\ref{eqn:10.21}\) to obtain an estimate of \(k\) that is probably sufficiently accurate for most engineering purposes.

Assuming that all necessary experimental data have been collected, and assuming that the system can be modeled reasonably as an LTI, SISO, \(m\)-\(c\)-\(k\) system with viscous damping, then the steps of the subsequent system ID calculation algorithm are:

1. Calculate \(k\) from Equation \(\ref{eqn:10.20}\) and/or Equation \(\ref{eqn:10.21}\), preferably both, in order to check that both static and dynamic testing lead to the same result.
2. Determine natural frequency \(\omega_{n}\) from the frequency response curves. The frequency at which the phase angle is −90° is the natural frequency, regardless of the level of damping. Also, if viscous damping ratio \(\zeta\) is small, less than about 0.2, then the frequency at which the dynamic flexibility peaks is essentially the natural frequency. With \(\omega_{n}\) and \(k\) known, calculate the mass: \(m=k / \omega_{n}^{2}\).
3. Measure the resonance (peak) dynamic flexibility, \(X_{r} / F\). Then the maximum dynamic amplification equation Equation 10.2.9 gives the following equation from which any viscous damping ratio \(\zeta \leq 1 / \sqrt{2}\) can be calculated:\[\frac{X_{r}}{F / k}=k\left(\frac{X_{r}}{F}\right)=\frac{1}{2 \zeta \sqrt{1-\zeta^{2}}}\label{eqn:10.22} \]The calculation is simplified considerably if \(\zeta\) is small, less than about 0.2:\[\frac{1}{2 \zeta} \approx k\left(\frac{X_{r}}{F}\right) \Rightarrow \zeta \approx \frac{1}{2}\left[k\left(\frac{X_{r}}{F}\right)\right]^{-1}\label{eqn:10.23} \]Finally, the viscous damping constant is calculated from Equation \(\ref{eqn:10.16}\):\[c=c_{c} \times \zeta=2 \zeta \sqrt{m k}=2 \zeta m \omega_{n}\label{eqn:10.24} \]

¹However, see homework Problem 10.16 for the practical reasons why it might often be better to measure dynamic stiffness, Eq. (10-31), rather than dynamic flexibility.