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10.5: Common Frequency-Response Functions

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    The term frequency-response function (FRF) is general, meaning physically the magnitude and phase in steady-state sinusoidal variation with time of some subject variable, relative to the magnitude and phase of some other reference variable. Often, but not always, the subject variable is clearly a physical response (output), and the reference variable is clearly a physical excitation (input). During over a century of development, engineers have found uses for many different types of FRF subject and reference variables, and names for these types are widely recognized. Many of the most common of these FRF definitions, their names, and examples are presented in this section.

    Diagrams such as Figure 4.3.1 and Figures 10.1.1 and 10.2.1 display FRFs graphically, and there are many different forms of these diagrams. One of the most well-known and useful forms is the Bode diagram, on which relative magnitude and phase are plotted separately against the sinusoidal frequency, which is on a logarithmic scale (as on Figure 10.2.1). Relative magnitude is usually plotted on the decibel scale [\(\mathrm{dB}=20 \times \log _{10}\text{(relative magnitude)}\)], and relative phase is usually expressed in degrees. Another useful graphical format for FRFs is the Nyquist diagram or plot (after Harry Nyquist, Swedish-born American electrical engineer and physicist, 1889-1976), which is described at length in Section 17.2. Application of Bode diagrams and Nyquist diagrams to analysis of control systems is discussed in Chapter 17.

    The use of FRFs apparently began during the 1880s with the definitions of admittance and impedance in the context of electrical engineering. Electrical admittance is traditionally assigned the symbol \(Y(\omega)\). With \(\Delta e(t)\) being a specified voltage difference and \(i(t)\) being a specified current, admittance is defined by the equation

    \[Y(\omega)=\left\{\frac{L[i(t)]}{L[\Delta e(t)]}\right\}_{s=j \omega}\label{eqn:10.29} \]

    Keep in mind that the Laplace transforms in Equation \(\ref{eqn:10.29}\) and all other definitions of FRFs are for zero initial conditions. Electrical impedance is traditionally assigned the symbol \(Z(\omega)\), and it is defined to be the inverse of admittance: \(Z(\omega)=1 / Y(\omega)\). In general, admittance is the degree to which an electrical system admits (or permits, if you prefer) the flow of current, and impedance is the degree to which the system impedes the flow of current. The simplest examples of electrical admittance and impedance are for an ideal linear resistor of resistance \(R\), with voltage difference \(\Delta e(t)\) across the resistor and current \(i(t)\) flowing through it. Ohm’s law, Equation 5.2.1, is \(\Delta e(t)=R \times i(t)\), so \(Y(\omega)=1 / R\) and \(Z(\omega) = R\), the same for all frequencies1.

    The electrical FRFs for an ideal linear capacitor with capacitance \(C\) are more complicated. The relationship between voltage difference \(\Delta e(t)\) across a capacitor and current \(i(t)\) flowing through it is Equation 5.2.4: \(i(t)=C \times d[\Delta e(t)] / d t\). Taking the Laplace transform of this equation gives \(L[i(t)]=C \times s L[\Delta e(t)]\), so, from Equation \(\ref{eqn:10.29}\), a capacitor’s admittance is \(Y(\omega)=j \omega C\), and its impedance is \(Z(\omega)=1 /(j \omega C)=-j \times 1 /(\omega C)\). The magnitude of this impedance is called capacitive reactance (Hammond, 1961, pp. 97-100), and is usually written as \(X_{C}=1 /(\omega C)=1 /(2 \pi f C)\). Physically, for a given magnitude of voltage difference and a given capacitance \(C\), the magnitude of current permitted by a capacitor is directly proportional to frequency \(\omega\) or \(f\), and the current leads the voltage difference by 90°.

    The relationship between voltage difference \(\Delta e(t)\) across an ideal linear inductor and current \(i(t)\) flowing through it is Equation 5.2.9: \(\Delta e(t)=\mathcal{L} \times d i(t) / d t\) denoting inductance \(\equiv \mathcal{L}\) in this paragraph only, to distinguish it from Laplace transform operator \(L\)). You can show easily that the associated admittance and impedance are, respectively, \(Y(\omega)=1 /(j \omega \mathcal{L})=-j \times 1 /(\omega \mathcal{L})\) and \(Z(\omega)=j \omega \mathcal{L}\). The magnitude of this impedance is called inductive reactance, and is usually written as \(X_{\ell}=\omega \mathcal{L}=2 \pi f \mathcal{L}\). Physically, for a given magnitude of voltage difference and a given inductance \(\mathcal{L}\), the magnitude of current permitted by an inductor is inversely proportional to frequency \(\omega\) or \(f\), and the current lags the voltage difference by 90°.

    The subject and reference variables of FRFs commonly used for mechanical and structural systems are, of course, different than those used for electrical systems. During much of the 20th century, conflicting and ambiguous names for these FRFs appeared in the engineering literature, but the names in English have been more uniformly standardized in recent years, as described by Ewins, 1984, pp. 26-27, and by Maia and Silva, 1997, pp. 38-39. In the following, we will derive theoretical FRF equations for an \(m\)-\(c\)-\(k\) system; however, in engineering practice, these FRF definitions are also applied both theoretically and experimentally for much more general and more complicated mechanical and structural systems.

    For mechanical admittance, the subject variable is a physical displacement (translation or rotation) at some point and in some direction on the mechanical or structural system, and the reference variable is a physical action (force or moment) imposed at some point and in some direction onto the system (Bisplinghoff et al., 1955, pp. 663- 665). Thus, for an \(m\)-\(c\)-\(k\) system, from Laplace transformation of the ODE \(m \ddot{x}+c \dot{x}+k x=f_{x}(t)\), and with use of notation defined in Equations 10.2.5 and 10.3.2, the equation for complex mechanical admittance is

    \[\left\{\frac{L[x(t)]}{L\left[f_{x}(t)\right]}\right\}_{s=j \omega}=\frac{1}{\left(k-\omega^{2} m\right)+j \omega c}=\frac{1}{k}\left[\frac{1}{\left(1-\beta^{2}\right)+j 2 \zeta \beta}\right]\label{eqn:10.30} \]

    The magnitude ratio and phase angle associated with Equation \(\ref{eqn:10.30}\) were written earlier in Equations 10.3.4 and 10.3.5. Some synonyms in English for mechanical admittance are dynamic flexibility, dynamic compliance, and receptance.

    The inverse of mechanical admittance is known as dynamic stiffness (occasionally also mechanical impedance, but this name usually denotes a different FRF, as described below); thus, the dynamic stiffness of an \(m\)-\(c\)-\(k\) system is

    \[\left\{\frac{L\left[f_{x}(t)\right]}{L[x(t)]}\right\}_{s=j \omega}=\left(k-\omega^{2} m\right)+j \omega c=k\left[\left(1-\beta^{2}\right)+j 2 \zeta \beta\right]\label{eqn:10.31} \]

    The definition of Equation \(\ref{eqn:10.31}\), with the Laplace transform of action in the numerator and that of displacement in the denominator, might seem unorthodox, since an action (force or moment) is usually the physical input and a displacement is usually the physical output. However, this type of FRF is useful in practice for experimental measurements, as is described in homework Problem 10.16. Dynamic stiffness is also the basis of an advanced method for analyzing theoretically the dynamics of distributed-parameter structures (Clough and Penzien, 1975, Chapter 20; Fergusson and Pilkey, 1991).

    For mechanical and structural systems, the first and second time derivatives of displacement are also of interest, primarily because velocity sensors and, especially, accelerometers are often used to measure motion. Accordingly, appropriate FRFs have been defined. For mobility, the subject variable is a velocity, and the reference variable is an action. Since \(L[\dot{x}(t)]=s \times L[x(t)]\), the mobility of an \(m\)-\(c\)-\(k\) system, from Equation \(\ref{eqn:10.30}\), is

    \[\left\{\frac{L[\dot{x}(t)]}{L\left[f_{x}(t)\right]}\right\}_{s=j \omega}=\frac{j \omega}{\left(k-\omega^{2} m\right)+j \omega c}=\frac{1}{c}\left[\frac{j 2 \zeta \beta}{\left(1-\beta^{2}\right)+j 2 \zeta \beta}\right]\label{eqn:10.32} \]

    The inverse of mobility is usually given the name mechanical impedance. Accordingly, the mechanical impedance of an \(m\)-\(c\)-\(k\) system is

    \[\left\{\frac{L\left[f_{x}(t)\right]}{L[\dot{x}(t)]}\right\}_{s=j \omega}=\frac{\left(k-\omega^{2} m\right)+j \omega c}{j \omega}=c\left[1-j\left(\frac{1-\beta^{2}}{2 \zeta \beta}\right)\right]\label{eqn:10.33} \]

    For accelerance (also known as inertance), the subject variable is an acceleration, and the reference variable is an action. Since \(L[\ddot{x}(t)]=s^{2} \times L[x(t)]\), the accelerance of an \(m\)-\(c\)-\(k\) system, from Equation \(\ref{eqn:10.30}\), is

    \[\left\{\frac{L[\ddot{x}(t)]}{L\left[f_{x}(t)\right]}\right\}_{s=j \omega}=\frac{(j \omega)^{2}}{\left(k-\omega^{2} m\right)+j \omega c}=\frac{1}{m}\left[\frac{-\beta^{2}}{\left(1-\beta^{2}\right)+j 2 \zeta \beta}\right]\label{eqn:10.32} \]

    It is worth noting that for static loading (\(\omega=0\)), mechanical admittance equals a nonzero static flexibility, but both mobility and accelerance are zero. The inverse of accelerance is called apparent mass, as motivated by the following expression for an \(m\)-\(c\)-\(k\) system:

    \[\left\{\frac{L\left[f_{x}(t)\right]}{L[\ddot{x}(t)]}\right\}_{s=j \omega}=\frac{\left(k-\omega^{2} m\right)+j \omega c}{(j \omega)^{2}}=m\left[\left(1-\frac{1}{\beta^{2}}\right)-j \frac{2 \zeta}{\beta}\right]\label{eqn:10.35} \]

    All of the mechanical and structural FRFs described in this section have dimensional units. For example, common units of accelerance used in the dynamic testing of structures are G’s or meters/s2 (of acceleration) per pound or newton (of force).

    1Clearly, the SI unit for impedance is the ohm (\(\Omega\)), that of resistance; apparently since admittance is the inverse of impedance, the SI unit for admittance was given the whimsical name mho (\(\Omega^{-1}\)).


    This page titled 10.5: Common Frequency-Response Functions 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.