Modal Analysis

Synonyms

Generalized coordinate analysis; Modal expansion; Mode acceleration method; Mode displacement method; Mode superposition method; Multi-degree-of-freedom (MDOF) system decoupling; Order reduction; Response series truncation

Introduction

The current entry attempts to synoptically guide the reader through the merits and critical details of modal analysis having a particular focus on earthquake engineering, which is the main subject of this encyclopedia. Tracking the history of modal analysis, the conception of vibration modes dates back to the eighteenth century and the pioneering studies and debates of Daniel and John Bernoulli, Euler, and d’Alembert (see Kline 1990) who while studying the problem of the vibration of a taut string came up with the notion of the modal shape contributions that build up the total observed oscillations. Such revolutionary for the time vibration knowledge along with a systematic treatment and practical extensions of modal analysis topics, more focused on continuous systems, appeared first in a very concerted way in the historical “Theory of Sound” by Lord Rayleigh (1877). Much later, today, the term modal analysis is used with different context in engineering language depending on the engineering stream and application ranging from the simple eigenvalue analysis, which recovers the normal mode frequencies and shapes, to the complex experimental dynamic properties identification. Its typical and closest to earthquake engineering definition refers to the ensemble of theoretical tools and operations, which are employed in simplifying the dynamic response calculation of a very large extended structural system under generic time-dependent loading. A much wider picture of modal analysis encompassing apart from the previous side a vast experimental background that has an invaluable contribution in understanding, enhancing, and controlling structures pertains primarily in mechanical engineering. This latter broad view was captured for the first time in a very holistic manner by a seminal book publication by Ewins (1984), which was later succeeded by similar influential titles like the ones by his students Maia et al. (1997) and He and Fu (2001).

The description that follows does not intend to be an alike broad exhaustive review of the so-called experimental or operational modal analysis, and the reader is suggested to refer to the quoted titles for this purpose. The definition assumed herein is closer to the first solely analytical approach, and as such it was previously very effectively included in the fundamental dynamics handbooks by Biggs (1964), Clough and Penzien (1993), and Chopra (1995) or in latest educational dynamics literature like Rajasekaran (2009). There the main attributes, limitations, and benefits of expressing a discretized structure’s response to some random or deterministic dynamic excitation (earthquakes being an ideal generic example) as the superposition of modal responses are discussed and exemplified through practical exercises. The influence of damping in the modal decomposition process along with the crucial question of how many modes are needed in accurately reproducing the full dynamic information hidden in response metrics are meticulously addressed for becoming a tool in the hands of practicing structural engineers. The presentation that follows collects and reproduces a synopsis of all this essential digested knowledge, which is of high priority for seismic analysis and design. This information directly links to other entries of the encyclopedia and allows their better understanding and connection. Although it aims to be a self-contained entry, it is founded on some previous knowledge of structural modeling and the dynamics of single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) systems, which are presented assuming the reader was previously exposed to basic vibration textbooks.

The entry begins with the study of a typical linear undamped n-degree-of-freedom system (with the application of multistory buildings always in mind) under a common form of dynamic loading. This system, in a straightforward manner, is transformed to n independent/uncoupled generalized SDOF systems acted upon by certain fractions of the initially defined loading, and two different approaches are introduced for performing any further practical analysis. Subsequently, findings are extrapolated to a variously damped viscous system (the proportional or classical and the generic or nonclassical), which is a very useful and effective approximation of the energy dissipation mechanism even in complex structures consisting of many interconnecting nonuniform parts. The alternatives of working either in the time or frequency domain are both discussed in brief. For the subsequent main topic of modal truncation that emerges, where one has to decide the minimum number of modes r ($r<n$ = number of degrees of freedom) that suffice for a sufficiently accurate reduced representation of the dynamic solutions (which may be displacements, rotations, cumulative shear forces, moments, etc.), all main different approaches and practices are presented. Miscellaneous details on various alterations of the analysis (e.g., mode acceleration or mode displacement form) practiced for increasing its numerical efficacy together with connections to aseismic structural design and brief essential reference to valuable extensions (e.g., nonlinear systems, health monitoring, and control) are provided throughout.

Modal Analysis Basics

The Undamped/Conservative System

The topic of dynamic-degree-of-freedom (DOF) selection, which is typical when transforming continuous systems to discrete models, along with reference to nonholonomic systems (i.e., where independent motion coordinates are essentially accompanied by additional constraints for fully determining motion) is not pursued herein. For the presentation in hand, the simple form of the already discrete engineering models assumed yield such discussions unnecessary. Thus, for this point, imagine a simplified n-degree-of-freedom discrete undamped structural model relieved by any gyroscopic effects. Such can be the translational toy mechanical system in Fig. 1a or the sway multistory shear frames in Fig. 1b, c.

Fig. 1

MDOF structural models (a) toy mechanical discrete system (b, c) multistorey sway shear frames

In all cases depicted in Fig. 1, the envisaged finite degrees of freedom (DOFs) necessary are clearly designated. For all systems additional assumptions (e.g., rigid beams, massless columns, and springs) were employed for reducing the required DOFs. The set of governing equations of motion may be put, by either following the force equilibrium approach on n free-body diagrams or the energy approach, in the matrix form

$$ \mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{K}\mathbf{u}(t)=\mathbf{F}(t), $$

(1)

where M and K are the constant symmetric mass and stiffness matrices ($n \times n$) ü(t) and u(t) are the column matrices or vectors ($n \times 1$) of accelerations and displacements expressed in physical coordinates varying with time t and F(t) the vector ($n \times 1$) of the externally applied (not motion-dependent) dynamic loads. As a convention, all matrices are herein denoted by bold letters, while common letters are used to denote scalar magnitudes. Regarding the stiffness matrix K, we must notice that it encompasses both the elastic and geometric stiffness information of the structural model, e.g., forces causing buckling are assigned to geometric stiffness. Also the constant nature of K translates to a linear elastic operational regime. As it can be easily verified through all Fig. 1 examples, for M it is straightforward to make it diagonal using the so-called lumped mass form, where translational DOFs are defined on each discrete mass. Yet this, which is not always possible or practical, will generally result in a non-diagonal K and, thus, in statically coupled Eq. 1, with the term statically coupled meaning that each of the constituting linear equations of motion includes components of the displacement vector u(t) corresponding to various DOFs and not only the DOF referred to the existing component of the acceleration vector ü(t). This coupling is in fact a coordinate coupling that perplexes the solution of Eq. 1 especially in the common case of large n. Alternatively, a different formulation practiced in finite element (FE) derivations for either M or K, entitled consistent form, leads in general to non-diagonal forms for both these matrices, thereby leading to statically and dynamically coupled equations of motion. This means that each one of these equations includes components of the vectors u(t) and ü(t) corresponding to various DOFs and not only to one DOF. And exactly, modal analysis aims at decoupling these equations. If Eq. 1 gets uncoupled, this will yield n independent equations of motion, i.e., each one of which refers to only one DOF, with the result that the MDOF system described by Eq. 1 can be treated as n equivalent uncoupled SDOF systems. Each of them corresponds to a so-called natural mode of vibration, which is characterized by a particular frequency and shape (modal/eigen frequency ω _i and modal/eigen vector φ _i , $\left(n \times 1\right)$). Such an advantageous simplification can be established by transforming the vector u(t) into a new vector q(t), ($n \times 1$), whose components are called normal or principal coordinates, as well as generalized displacements. This transformation is equivalent to the separation of variables technique, exercised in partial differential equations (PDEs), and is described as below

$$ \mathbf{u}(t)=\boldsymbol{\Phi}\;\mathbf{q}(t)={\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i{q}_i(t)}, $$

(2)

or alternatively for the displacement vector u _i (t) corresponding to each individual mode i

$$ {\mathbf{u}}_i(t)={\boldsymbol{\varphi}}_i{q}_i(t). $$

(3)

In Eqs. 2 and 3, Φ is the modal matrix ($n \times n$) consisting of all n mode shapes φ _i arranged as its columns, which includes the entire modal shape information. And q(t) is a vector (i.e., column matrix) of n components, with the arbitrarily chosen ith component q _i (t) representing the unknown time-functioning generalized displacement for the ith mode.

The modal parameters of frequency ω _i and shape φ _i are deduced by solving the so-called eigenvalue problem. This comes up when replacing Eq. 3 inside the free/unforced counterpart of Eq. 1 resulting from Eq. 1 after setting $\mathbf{F}(t)=0$ and recognizing the pure harmonic nature of the associated q(t). Specifically speaking, considering the free/unforced counterpart of Eq. 1, i.e., the free-vibration equations

$$ \mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{K}\mathbf{u}(t)=0, $$

(4)

and assuming that each modal response ${\mathbf{u}}_i(t)={\boldsymbol{\varphi}}_i{q}_i(t)$ is a solution to Eq. 4, it follows

$$ \mathbf{M}\;{\boldsymbol{\varphi}}_i{\ddot{q}}_i(t)+\mathbf{K}\;{\boldsymbol{\varphi}}_i{q}_i(t)=0, $$

(5)

which after premultiplying with the transposed mode shape vector φ ^T _i , i.e., a row matrix with the same elements as the column matrix φ _i , gives

$$ {\boldsymbol{\varphi}}_i^T\mathbf{M}\;{\boldsymbol{\varphi}}_i{\ddot{q}}_i(t)+{\boldsymbol{\varphi}}_i^T\mathbf{K}\;{\boldsymbol{\varphi}}_i{q}_i(t)=0. $$

(6)

Now, defining the frequency ω _i as equal to

$$ {\omega}_i^2=\frac{{\boldsymbol{\varphi}}_i^T\mathbf{K}\;{\boldsymbol{\varphi}}_i}{{\boldsymbol{\varphi}}_i^T\mathbf{M}\;{\boldsymbol{\varphi}}_i}, $$

(7)

Equation 6 becomes equal to the harmonic vibration in q _i (t)

$$ {\ddot{q}}_i(t)+{\omega}_i^2{q}_i(t)=0, $$

(8)

and combining with Eq. 5 results in

$$ \left(\mathbf{M}-{\omega}_i^2\mathbf{K}\right)\;{\boldsymbol{\varphi}}_i=0, $$

(9)

which has a nonzero solution φ _i only for the zero determinant

$$ \left|\mathbf{M}-{\omega}_i^2\mathbf{K}\right|=0. $$

(10)

Equation 10 is also termed the characteristic or frequency equation of the system. As expected due to the problem’s indeterminate nature, the resulting vectors φ _i are not unique solutions but represent families of solutions of the form α φ _i , with α being an arbitrary scalar constant. The computational effort/time required for a solution is proportional to n ³. It is noticed that deriving such sets of ω _i , φ _i solutions for large structural systems have introduced a whole new field for devising efficient solvers of Eqs. 9 and 10 (i.e., eigensolvers); the curious reader may consult §8 (Rades 2010). One of the most renown equivalent numerical techniques of great application belongs to Rayleigh and Ritz (for some historical notes and disputes on its origins, one can read Leissa (2005)). Further, it is vital to refer to the important orthogonality properties, viz.,

$$ {\boldsymbol{\varphi}}_i^T\mathbf{K}{\boldsymbol{\varphi}}_j={\boldsymbol{\varphi}}_i^T\mathbf{M}{\boldsymbol{\varphi}}_j=0\ \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\ i\ne j, $$

(11)

which result from the symmetrical form of the mass and stiffness matrices M and K, respectively, and allow Eq. 1 to be uncoupled into n SDOF equations. Indeed, premultiplying Eq. 1 with φ ^T _i and replacing u(t) with Φ q(t) according to Eq. 2, one gets

$$ {\boldsymbol{\varphi}}_i^T\mathbf{M}\boldsymbol{\Phi } \ddot{\mathbf{q}}(t)+{\boldsymbol{\varphi}}_i^T\mathbf{K}\boldsymbol{\Phi } \mathbf{q}(t)={\boldsymbol{\varphi}}_i^T\mathbf{F}(t). $$

(12)

Owning to the orthogonality conditions, Eq. 12 becomes

$$ {\boldsymbol{\varphi}}_i^T\mathbf{M}{\boldsymbol{\varphi}}_i{\ddot{q}}_i(t)+{\boldsymbol{\varphi}}_i^T\mathbf{K}{\boldsymbol{\varphi}}_i{q}_i(t)={\boldsymbol{\varphi}}_i^T\mathbf{F}(t), $$

(12a)

or equally

$$ {M}_i\;{\ddot{q}}_i(t)+{K}_i\;{q}_i(t)={P}_i(t), $$

(12b)

which after division with M _i becomes

$$ {\ddot{q}}_i(t)+{\omega}_i\;{q}_i(t)=\frac{P_i(t)}{M_i}, $$

(12c)

where

$$ {K}_i={\boldsymbol{\varphi}}_i^T\mathbf{K}{\boldsymbol{\varphi}}_i,\ {M}_i={\boldsymbol{\varphi}}_i^T\mathbf{M}{\boldsymbol{\varphi}}_i,\ {\omega}_i={K}_i/{M}_i\ \mathrm{and}\ {P}_i(t)={\boldsymbol{\varphi}}_i^T\mathbf{F}(t) $$

(13)

with K _i , M _i , ω _i and P _i (t) being the so-called scalar modal stiffness, mass, frequency, and force, respectively. Note that combining orthogonality with Eq. 9 the typical SDOF relation, ${K}_i={\omega}_i^2{M}_i$ comes up. The values of K _i and M _i are subject to the scaling opted for the modal vectors φ _i and as such on their own are not very informative of the whole system. A common scaling, or else referred to as normalization, used for φ _i is the one that results in unit modal masses. Namely, if each φ _i is divided by $1/\sqrt{M_i}$, then evidently the second of Eq. 13 yields

$$ \frac{{\boldsymbol{\varphi}}_i^T}{\sqrt{M_i}}\mathbf{M}\frac{{\boldsymbol{\varphi}}_i}{\sqrt{M_i}}= 1. $$

(14)

With all the previous data in hand, one can now proceed with solving the initial problem of determining the response of the undamped system to some generic forcing F(t). The complete solution to Eq. 12b, which is a generalized SDOF equation, provides the part of the total response that owes exclusively to the ith mode and can be solved independently of all other modes. As is well known from the solution to an SDOF equation, the modal solution q _i (t) of Eq. 12c consists of a free-vibration solution q ^c _i (t) due to exclusively the initial conditions q _i (0) and ${\dot{q}}_i(0)$ and a particular forced-vibration solution q ^p _i (t) due to exclusively the imposed modal force P _i (t). Namely, the resultant solution q _i (t) in the time domain equals

$$ {q}_i(t)={q}_i^c(t)+{q}_i^p(t), $$

(15)

with

$$ {q}_i^c(t)={q}_i(0) cos{\omega}_it+\frac{{\dot{q}}_i(0)}{\omega_i} \sin {\omega}_it, $$

(16)

and

$$ {q}_i^p(t)=\frac{1}{\omega_i}{\displaystyle \underset{0}{\overset{t}{\int }}\frac{P_i\left(\tau \right)}{M_i}} \sin {\omega}_i\left(t-\tau \right)d\tau . $$

(17)

The forced-vibration solution q ^p _i (t) described in the time domain by Eq. 17 is the Duhamel integral expression, for the calculation of which a number of numerical techniques (e.g., Newton-Cotes formulas, Simpson’s rule, Romberg integration, etc.) may be employed. It is worth pointing out that recovering the modal initial conditions q _i (0) and ${\dot{q}}_i(0)$ from the initial conditions u(0) and $\dot{\mathbf{u}}(0)$ can be a straightforward result of Eq. 2 and the orthogonality properties (11). Namely, by multiplying Eq. 2 with φ ^T _i Μ and taking into account the orthogonality properties (11), it follows that

$$ {\boldsymbol{\varphi}}_i^T\mathbf{M}\mathbf{u}(0)={\boldsymbol{\varphi}}_i^T\mathbf{M} \boldsymbol{\Phi}\;\mathbf{q}(0)={\boldsymbol{\varphi}}_i^T\mathbf{M} {\boldsymbol{\varphi}}_i\;{q}_i(0)={M}_i{q}_i(0), $$

(18a)

$$ {\boldsymbol{\varphi}}_i^T\mathbf{M}\dot{\mathbf{u}}(0)={\boldsymbol{\varphi}}_i^T\mathbf{M} \boldsymbol{\Phi}\;\dot{\mathbf{q}}(0)={\boldsymbol{\varphi}}_i^T\mathbf{M} {\boldsymbol{\varphi}}_i\;\dot{\mathbf{q}}(0)={M}_i{\dot{q}}_i(0), $$

(18b)

from which the magnitudes ${q}_i(0)={\boldsymbol{\varphi}}_i^T\mathbf{M}\mathbf{u}(0)/{M}_i$ and ${\dot{q}}_i(0)={\boldsymbol{\varphi}}_i^T\mathbf{M}\dot{\mathbf{u}}(0)/{M}_i$ are derived.

Finally, after solving Eqs. 16 and 17 for the unknowns q ^c _i (t) and q ^p _i (t), and then Eq. 15 for the unknown q _i (t), Equation 2 will be used for synthesizing the total response time histories u(t) from the individual synchronous modal contributions u _i (t), which have been determined as the products φ _i q _i (t) of the evaluated modal shapes φ _i and generalized displacements q _i (t). This writes as

$$ \mathbf{u}(t)={\displaystyle \sum_{i=1}^n{\mathbf{u}}_i(t)=}{\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i{q}_i(t)}. $$

(19)

The most valuable feature of Eq. 19 is that not all n terms of the sum are required for accurately representing the total response. A large-scale structure like a tall building has hundreds of DOFs, which will produce hundreds of natural modes. Yet, it is common that for typical operational loading conditions, the lower (i.e., of lower frequencies) few modes, whose number r is far smaller than the number n of all the DOFs of the structure, are enough to compute the vector u(t) by using Eq. 19, for the higher (i.e., of higher frequencies) modes of a structure are not excited considerably. The value of r, as well as to what extent it can be reduced with respect to n, is a function not only of the initial loading F(t)’s characteristics (i.e., shape distribution and frequency content) and their match to specific natural modes but also of the response metric that needs to be considered. For instance, representation of dynamic moment, shear, and stress variables in general requires more natural modes than representing the displacement solutions. This finding that was meticulously exemplified in Chopra (1995, 1996) and Clough and Penzien (1993) owes primarily to the fact that higher-order response attributes need additional information for their description (analogous to the added information embedded in derivatives of a function compared to the actual function).

The Classical/Proportional Damping Addition

The conservative (i.e., undamped, i.e., without energy losses) system treated earlier is fundamental for any dynamic study. Yet, it is not completely accurate in representing reality. All structures have the inherent ability of damping with which they can dissipate energy inputs. As a matter of fact, apart from being a structural property, effective damping can also surface from the interaction of the structure with environmental or other types of loads. Distinctive examples of such cases, also described as self-excitation, are the Tacoma Narrows Bridge collapse (with the associated pure torsional flutter) and the large lateral vibrations of London Millennium Bridge (with pedestrians acting as negative dampers). The damping property is almost impossible to faithfully model ab initio. This is a result of a multitude of reasons. There is little information on inherent material damping, and further energy dissipation can originate from sources like connections, crack openings, or inter-element friction that are really hard to capture entirely. The practical and plausible solution is to model damping as a linear viscous term $\mathbf{C}\dot{\mathbf{u}}(t)$, i.e., function of velocity, which efficiently provides a route toward changing the energy balance in the equations of motion. This will lead to modifying the set of Eq. 1 as follows:

$$ \mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{C}\dot{\mathbf{u}}(t)+\mathbf{K}\mathbf{u}(t)=\mathbf{F}(t), $$

(20)

where C stands for the damping matrix ($n \times n$). This is in general non-diagonal. A work-around toward approximating the many unknown elements of C is to connect it to the straightforwardly derived structural matrices K and M. The approximation of stiffness-proportional damping, which is actually appealing to intuition, can easily be shown to result in damping increasing with frequency. This is contrary to many experimental findings, which in general refute such a pattern. Mass-proportional damping on the other hand, which is simplistically linked to air/medium damping, also contradicts experience. However, combining the two proportionality approaches yields the so-called Rayleigh damping, which seems to constitute a good and much-used idealization for the energy dissipation mechanism of evenly distributed structures. A typical example of the latter is a multistory building of similar material and structural system throughout its height. According to the Rayleigh approach, C is given the form

$$ \mathbf{C}={\mathrm{a}}_0\mathbf{M}+{\mathrm{a}}_1\mathbf{K}, $$

(21)

where a₀ and a₁ are unknown scalar constants. Employing Rayleigh damping allows the classical modal decomposition (uncoupling) process earlier performed for the undamped case to apply to the damped case as well. Accordingly, one can firstly insert Eq. 2 in Eq. 20 and then multiply with the transposed mode shape vector φ ^T _i derived from the undamped free-vibration Eq. 4 to get

$$ {\boldsymbol{\varphi}}_i^T\mathbf{M}\boldsymbol{\Phi}\;\ddot{\mathbf{q}}(t)+{\boldsymbol{\varphi}}_i^T\left({\mathrm{a}}_0\mathbf{M}+{\mathrm{a}}_1\mathbf{K}\right)\boldsymbol{\Phi}\;\dot{\mathbf{q}}(t)+{\boldsymbol{\varphi}}_i^T\mathbf{K}\boldsymbol{\Phi}\;\mathbf{q}(t)={\boldsymbol{\varphi}}_i^T\mathbf{F}(t). $$

(22)

Equation 22 due to Eqs. 11 and 13 becomes alike to Eq. 12

$$ {\boldsymbol{\varphi}}_i^T\mathbf{M}{\boldsymbol{\varphi}}_i\;{\ddot{q}}_i(t)+{\boldsymbol{\varphi}}_i^T\left({\mathrm{a}}_0\mathbf{M}+{\mathrm{a}}_1\mathbf{K}\right){\boldsymbol{\varphi}}_i\;{\dot{q}}_i(t)+{\boldsymbol{\varphi}}_i^T\mathbf{K}{\boldsymbol{\varphi}}_i\;{q}_i(t)={\boldsymbol{\varphi}}_i^T\mathbf{F}(t). $$

(22a)

Or alternatively

$$ {M}_i\;{\ddot{q}}_i(t)+\left({\mathrm{a}}_0{M}_i+{\mathrm{a}}_1{K}_i\right){\dot{q}}_i(t)+{K}_i\;{q}_i(t)={P}_i(t). $$

(22b)

In the last pair of equations, it should be noted that exclusively due to the fact that C was assumed to be a weighted sum of M and K, orthogonality still applies and results in uncoupled modal damping expressions C _i for each mode i. Namely,

$$ {\boldsymbol{\varphi}}_i^T\mathbf{C}{\boldsymbol{\varphi}}_j=0\ \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\ i\ne j\ \mathrm{and}\ {C}_i={\mathrm{a}}_0{M}_i+{\mathrm{a}}_1{K}_i={\boldsymbol{\varphi}}_i^T\mathbf{C}{\boldsymbol{\varphi}}_i. $$

(23)

Equation 23 allows Eq. 22b to be rewritten as

$$ {M}_i\;{\ddot{q}}_i(t)+{C}_i{\dot{q}}_i(t)+{K}_i\;{q}_i(t)={P}_i(t), $$

(22c)

which after division with M _i becomes equal to

$$ {\ddot{q}}_i(t)+ 2{\xi}_i\;{\omega}_i\;{\dot{q}}_i(t)+{\omega}_i\;{q}_i(t)=\frac{P_i(t)}{M_i}, $$

(22d)

with the modal mass and stiffness M _i and K _i , the modal frequency ω _i , and the modal force P _i (t) defined by Eq. 13. As for the so-called modal damping ratio (or fraction of the critical damping) ξ _i , it stands for the ratio C _i /2ω _i M _i , i.e.,

$$ {\xi}_i={C}_i/ 2{\omega}_i{M}_i. $$

(24)

It is worth noting that Eq. 23 characterizes not only the Rayleigh damping model but also any damping model that offers the possibility for the classical modes of the undamped free vibration to be used in order to uncouple the equations of damped motion of the structure under consideration. Any such damping model is grouped as classical damping and substantially simplifies all subsequent dynamic calculations. Obviously, the subcases of mass-proportional and stiffness-proportional damping brought up earlier must also be categorized as classical damping.

Subsequently the process of calculating modal responses is identical to the previous description, where someone solves independently each modal equation and then synthesizes the total response by summing up the few lower modal contributions up to a certain order. The modal solution q _i (t) of the Eq. 22 is now also affected by the modal damping as expressed by the modal damping ratio ξ _i . In analogy with Eqs. 15 up to 17, now the modal solution q _i (t) generalizes to

$$ \begin{array}{l}{q}_i(t)={q}_i^c(t)+{q}_i^p(t)=\\ {}={e}^{-{\xi}_i{\omega}_it}\left({q}_i(0) cos{\tilde{\omega}}_it+\frac{{\dot{q}}_i(0)+{\xi}_i{\omega}_i{q}_i(0)}{{\tilde{\omega}}_i} \sin {\tilde{\omega}}_it\right)+{\displaystyle \underset{0}{\overset{t}{\int }}\frac{P_i\left(\tau \right)}{M_i}}{h}_i\left(t-\tau \right)d\tau, \end{array} $$

(25)

where ${\tilde{\omega}}_i$ is the so-called damped modal frequency, which stands for the multiple ${\omega}_i\sqrt{1-{\xi}_i^2}$ of the undamped modal frequency ω _i , i.e.,

$$ {\tilde{\omega}}_i={\omega}_i\sqrt{1-{\xi}_i^2}, $$

(26)

and ${h}_i\left(t-\tau \right)$ is the modal function known as unit-impulse response function given by

$$ {h}_i\left(t-\tau \right)=\frac{1}{{\tilde{\omega}}_i}{e}^{-{\xi}_i{\omega}_i\left(t-\tau \right)} \sin {\tilde{\omega}}_i\left(t-\tau \right). $$

(27)

In the above analysis, one step was tacitly skipped. When adopting the Rayleigh damping expression of Eq. 21, the unknown constants a₀ and a₁ need to be determined for the damping matrix to be fully described. Indicatively this can be done as follows:

The modal damping can be determined experimentally through analyzing response measurements. This is the exact inverse problem of what is dealt in this entry. In short an answer is enabled through different techniques, e.g., the half-power bandwidth method, the free decays of isolated modes, or some other more elaborate modal identification technique. In the absence of such experimental data, figures that have developed from long structural experience can be used. Such information is given in Table 1 noting the stress amplitude dependence.

Table 1 Recommended damping values from Chopra (1995)

With values for the modal damping ratios in hand, one can write Eq. 23 for two modes, customarily the first and the last one considered (i.e., rth), viz.,

$$ \left.\begin{array}{l}{C}_1={\mathrm{a}}_0{M}_1+{\mathrm{a}}_1{K}_1= 2{\omega}_1{M}_1{\xi}_1\\ {}{C}_r={\mathrm{a}}_0{M}_r+{\mathrm{a}}_1{K}_r= 2{\omega}_r{M}_r{\xi}_r\end{array}\right\}\kern0.5em . $$

(28)

The latter system of two algebraic equations is then to be solved simultaneously for a₀ and a₁. A further generalization of classical damping can also be considered when more than two modal damping values are taken into account in capturing the damping distribution. Then we have the Caughey damping approach according to which C is expressed as

$$ \mathbf{C}=\mathbf{M}{\displaystyle \sum_{j=0}^{l-1}{\mathrm{a}}_j{\left({\mathbf{M}}^{-1}\mathbf{K}\right)}^j}, $$

(29)

where l is the number of modal damping ratios to be used. Evidently for $l=2$ this reduces to Rayleigh damping, while for $l=3$ Eq. 29 results in

$$ \mathbf{C}={\mathrm{a}}_0\mathbf{M}+{\mathrm{a}}_1\mathbf{K}+{\mathrm{a}}_2\mathbf{K}{\mathbf{M}}^{-1}\mathbf{K}. $$

(30)

The l unknown constants are found again in a straightforward manner by solving a $l \times l$ system of algebraic equations.

The Nonclassical Damping Case

The classical damping approach although not physically realizable has found extensive use in engineering practice and particularly in standard earthquake engineering design. However, for structures that are heavily damped or unevenly distributed or even when they possess closely spaced modes, the couplings induced by the damping matrix may produce significant shortcomings and complications in the earlier presented modal analysis. An example of such a case, where nonclassical damping is to be expected, would appear in a nuclear reactor containment vessel founded on a soft soil. The quite different damping properties between the stiff structure and the foundation will induce a highly damped, very complex dynamic soil-structure interaction that establishes itself through a very generically shaped damping matrix (Clough and Mojtahedi 1976). For making things clear, in these instances the solution again starts from writing the damped equation of motion

$$ \mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{C}\dot{\mathbf{u}}(t)+\mathbf{K}\mathbf{u}(t)=\mathbf{F}(t). $$

(20)

Still, what changes is that after substituting in the latter the modal decomposition indicated in Eq. 2 and premultiplying with the undamped mode vector φ ^T _i , one gets

$$ {\boldsymbol{\varphi}}_i^T\mathbf{M}{\boldsymbol{\varphi}}_i{\ddot{q}}_i(t)+{\boldsymbol{\varphi}}_i^T\mathbf{C}\boldsymbol{\Phi}\;\dot{\mathbf{q}}(t)+{\boldsymbol{\varphi}}_i^T\mathbf{K}{\boldsymbol{\varphi}}_i{q}_i(t)={\boldsymbol{\varphi}}_i^T\mathbf{F}(t) $$

(31a)

or alternatively

$$ {M}_i{\ddot{q}}_i(t)+{\boldsymbol{\varphi}}_i^T\mathbf{C}\boldsymbol{\Phi}\;\dot{\mathbf{q}}(t)+{K}_i{q}_i(t)={P}_i(t). $$

(31b)

Equation 31b obviously is a system of n-coupled equations, since the row vector $\tilde{\mathbf{C}}={\boldsymbol{\varphi}}_i^T\mathbf{C}\boldsymbol{\Phi }$, ($1 \times n$), has in general nonzero elements that introduce in Eq. 31b apart from ${\dot{q}}_i(t)$ also terms ${\dot{q}}_j(t)$ with $j\ne i$. Thus, contrary to the principle and target of modal analysis, it is not possible to deal with n-independent generalized SDOF modal systems. It actually seems that there is no benefit in transforming the initial coupled equation of motion to its modal form using the undamped modes. There are a number of approaches to tackle this problem:

The first more simplistic approach disregards all the coupled contributions of Eq. 31b by essentially zeroing in the row vector $\tilde{\mathbf{C}}$ all elements apart from C _i , i.e., $\tilde{\mathbf{C}}=\left[0\dots {C}_i\ldots 0\right]$. Subsequently, the remaining analysis becomes identical to the above described for the classically damped case.

A different also approximate approach would be for one to keep all terms inside Eq. 31b and attempt to solve the system of coupled equations using numerical techniques, that is, direct integration (e.g., Newmark’s method). More on the latter will appear in the entry of nonlinear systems. The relative benefit of employing Eq. 31b against applying direct integration to the initial n equations of damped motion described by the matrix Eq. 20 lies in the fact that due to generally few modes dominating response, a much reduced number of r equations can be considered. This is of great importance in large structural systems with thousands of DOFs.

Yet, both of the above analytical approaches may introduce errors of undefined magnitude. The most rigorous treatment of the problem is to uncouple the equations of motion using damped modes instead of the undamped ones. Their derivation, indicatively provided herein, proceeds by first writing the free-vibration (unforced) counterpart of the forced-vibration matrix Eq. 20

$$ \mathbf{M}\ddot{\mathbf{u}}(t)+\mathbf{C}\dot{\mathbf{u}}(t)+\mathbf{K}\mathbf{u}(t)=\mathbf{0}. $$

(32)

And then, assuming a solution of the form

$$ {\mathbf{u}}_i(t)={\boldsymbol{\phi}}_i\;{\mathrm{e}}^{\lambda_it}, $$

(33)

Equation 32 leads to

$$ \left({\lambda}_{\;i}^2\mathbf{M}+{\lambda}_{\;i}\mathbf{C}+\mathbf{K}\right){\boldsymbol{\phi}}_i=\mathbf{0}, $$

(34)

where λ _i and ϕ _i are the damped eigenvalues and eigenvectors, respectively.

The methodology to solve Eq. 34 relies on the so-called state-space formulation. According to it, Eq. 34 may be rewritten as

$$ \left[\begin{array}{cc}\hfill -\mathbf{K}\hfill & \hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill & \hfill \mathbf{M}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\lambda}_i{\boldsymbol{\phi}}_i\hfill \\ {}\hfill {\lambda}_i^2{\boldsymbol{\phi}}_i\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill \mathbf{0}\hfill & \hfill \mathbf{K}\hfill \\ {}\hfill \mathbf{K}\hfill & \hfill \mathbf{C}\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\boldsymbol{\phi}}_i\hfill \\ {}\hfill {\lambda}_i{\boldsymbol{\phi}}_i\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill \end{array}\right], $$

(35)

and introducing the block matrices A and B, ($2n \times 2n$), and column block vector v _i , ($2n \times 1$), i.e.,

$$ \mathbf{A}=\left[\begin{array}{cc}\hfill -\mathbf{K}\hfill & \hfill \mathbf{0}\hfill \\ {}\hfill \mathbf{0}\hfill & \hfill \mathbf{M}\hfill \end{array}\right],\ \mathbf{B}=\left[\begin{array}{cc}\hfill \mathbf{0}\hfill & \hfill \mathbf{K}\hfill \\ {}\hfill \mathbf{K}\hfill & \hfill \mathbf{C}\hfill \end{array}\right]\ \mathrm{and}\ {\mathbf{v}}_i=\left[\begin{array}{c}\hfill {\boldsymbol{\phi}}_i\hfill \\ {}\hfill {\lambda}_i{\boldsymbol{\phi}}_i\hfill \end{array}\right], $$

(36)

reshapes Eq. 35 into

$$ \left({\lambda}_{\;i}\mathbf{A}+\mathbf{B}\right){\mathbf{v}}_i=\mathbf{0}, $$

(37)

which has a nonzero solution v _i only for the zero determinant

$$ \left|{\lambda}_{\;i}\mathbf{A}+\mathbf{B}\right|=0. $$

(38)

The latter is an eigenvalue problem of order 2n, which is known as the complex eigenvalue problem owing to the fact that both λ _i and ϕ _i are in general complex valued. Before proceeding any further, it is worth noticing that the increased order 2n of the complex eigenvalue problem in comparison with the order n of the classical eigenvalue problem requires an increased computational effort/time proportional to ${(2n)}^3= 8{n}^3$ for a solution. This is the reason for this approach to be outside the main pursuits of applied structural dynamics. Now, returning to Eq. 38, if λ _i is a solution, then its conjugate is also one. Typically, λ _i is expressed as

$$ {\lambda}_{\;i}={\omega}_i\left(-{\xi}_i+\mathrm{j}\sqrt{1-{\xi}_i^2}\right), $$

(39)

where $\mathrm{j}=\sqrt{-1}$ is the imaginary unit and ω _i and ξ _i are the modal frequency and modal damping ratio, respectively. The complex nature of the eigenvectors ϕ _i accounts for phase differences other than 0° and 180° between their components. Orthogonality relations now become

$$ {\mathbf{v}}_i^T\mathbf{A}{\mathbf{v}}_j={\mathbf{v}}_i^T\mathbf{B}\;{\mathbf{v}}_j=0\ \mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{all}\ i\ne j, $$

(40)

which allow the uncoupling of Eq. 35 or of its forced counterpart. Having laid the ground for subsequent analysis (e.g., time response solutions), no additional details are presented herein, and the reader is suggested to consult relevant fundamental literature on this special topic for additional resources and worked examples, e.g., (Crandall and McCalley 2002; Itoh 1973; Hansen et al. 2012).

Member Forces

The procedures described above yielded the displacement time response, both modal and total, of a linear elastic structure. Yet, it is of great relevance for design purposes to derive also the force and stress developing on individual members/elements. There are two distinct ways to achieve this:

• 1. Having in hand the modal displacements u _i (t) along the structure, one can use the member stiffness properties in order to derive the modal force R _i (t) on the member and subsequently the required stresses. Considering all the modal contributions, one can write

  $$ R(t)={\displaystyle \sum_{i=1}^r{R}_i(t)}, $$

  (41)

  which apart from forces also applies to the resultant stresses.

• 2. Alternatively the modal displacements u _i (t) can be used to define equivalent modal static forces

  $$ {\mathbf{F}}_i^s(t)=\mathbf{K}\;{u}_i(t)={\omega}_i^2\mathbf{M}\;{u}_i(t). $$

  (42)

Solving the structure statically for the obtained F ^s _i (t), which are now applied as external forces at each time instant, will give again the modal member force R _i (t). All individual modal contributions again have to be added together in accordance with Eq. 41.

The Earthquake Problem

The general modal analysis background presented above will now shift the focus to the earthquake-loading scenario of a classically damped n-DOF structure. Such a structure could be a typical multistory frame, e.g., Fig. 1c, which is customarily used as a benchmark in similar studies. Earthquake loading is the result of seismic action of the ground on the supports of the structure, that is, the result of a vibration motion of the supports of the structure caused by an earthquake. Specifically speaking, the seismic action translates to a ground acceleration input ü _g (t) at the foundation level (this is in most instances horizontal but can also be vertical), whose transmission to the upper structure creates an equivalent dynamic force distribution that constitutes what is called earthquake loading. Disregarding the special topic of multiple support excitation (for such an extension, consult Clough and Penzien (1993) or Chopra (1995)), the earthquake (or seismic) load vector that enters the matrix equation of motion, Eq. 20, can be expressed as

$$ \mathbf{F}(t)=-\mathbf{M}\;\boldsymbol{\upiota}\;{\ddot{u}}_g(t). $$

(43)

The latter implies a synchronous though spatially varying loading of the different lumped masses. The vector ι, ($n \times 1$), is an influence vector of the ground motion on the structure and represents the displacements that will result for each DOF when a static unit ground displacement ${u}_g(t)=1$, of the same direction as ü _g (t), is applied to an undeformed (absolutely rigid) model of the structure. For example, in the frame of Fig. 1b, considering a horizontal earthquake input, ι becomes ${\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]}^T$, while for the frame of Fig. 1c, it turns to ${\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]}^T$. For proof purposes, Eq. 43 and the definition of ι can naturally be derived when expressing the equilibrium at each mass of the MDOF structure in terms of the total displacement u _t (t), which evidently is the vector sum of the relative-to-base displacement u(t) and the ground/base displacement ι u _g (t), i.e.,

$$ {\mathbf{u}}_t(t)=\mathbf{u}(t)+\boldsymbol{\upiota}\;{u}_g(t). $$

(44)

Indeed, the equilibrium of the masses of a seismically excited MDOF structure is expressed by

$$ \mathbf{M}\;{\ddot{\mathbf{u}}}_t(t)+\mathbf{C}\;\dot{\mathbf{u}}(t)+\mathbf{K}\;\mathbf{u}(t)=0, $$

(45)

and substituting ü _t (t) in accordance with Eq. 24, it follows

$$ \mathbf{M}\;\ddot{\mathbf{u}}(t)+\mathbf{C}\;\dot{\mathbf{u}}(t)+\mathbf{K}\;\mathbf{u}(t)=-\mathbf{M}\;\boldsymbol{\upiota}\;{\ddot{u}}_g(t), $$

(46)

which proves that the seismic load vector F(t) equals $-\mathbf{M}\;\boldsymbol{\upiota}\;{\ddot{u}}_g(t)$, thereby proving Eq. 32.

In a broader perspective, the expression for the seismic load can be generalized to describe a wider family of loading cases for which

$$ \mathbf{F}(t)=\mathbf{s}\;f(t), $$

(47)

with the vector s, ($n \times 1$), embodying all spatial variations and the scalar function f(t) embodying all synchronous time variations. Trivially for $\mathbf{s}=-\mathbf{M}\boldsymbol{\upiota }$ and $f(t)={\ddot{u}}_g(t)$ one reverts to the earthquake-related problem.

This is actually the starting point for any subsequent analysis intending to obtain response histories. Yet, it must be noted that for most usual practices in seismic calculations, a different characterization for the earthquake motion is opted. This is the instrumental approach of the so-called response spectra and attempts to remove the tedious numerical burdens that are related to obtaining all the history of the response of a specific structure to the long time sequence of the ground acceleration of a specific earthquake excitation in order that the maximum value of the response is evaluated. Namely, with response spectra, one only has to consider the maximum response output (this may be in terms of displacement, S _d , or velocity, S _v , or acceleration, S _a ) that an SDOF system produces under a specific earthquake excitation. Thus, an earthquake is no more an explicit time function ü _g (t). It is described through its maximum effect on SDOF systems with specific natural frequency and damping. This approach best serves the engineering need for evaluating the strength of a structure against the maximum value of its response to a given seismic excitation.

Whatever the earthquake definition assumed and the analysis opted (time histories or response maxima), there are substantial benefits associated with the use of modal analysis. Yet, the inherent in response spectra need for a decomposition of an initial MDOF structure to equivalent SDOF ones makes modal analysis in that instance more than a beneficial feature an irreplaceable prerequisite.

Excitation and Participation Factors

In what follows, some essential notions relevant to the earthquake analysis of structures are developed through exemplifying the modal analysis application. Combining Eqs. 13 and 47 yields

$$ {P}_i(t)={\boldsymbol{\varphi}}_i^T\mathbf{F}(t)={\boldsymbol{\varphi}}_i^T\mathbf{s}\;f(t)={L}_i\;f(t)\kern1em with\ {L}_i={\boldsymbol{\varphi}}_i^T\mathbf{s}, $$

(48)

as well as

$$ \frac{P_i(t)}{M_i}=\frac{{\boldsymbol{\varphi}}_i^T\mathbf{s}\;}{M_i}f(t)=\frac{L_i\;}{M_i}f(t)={\Gamma}_i\;f(t)\kern1em with\ {\Gamma}_i=\frac{L_i\;}{M_i}, $$

(49)

and, hence,

$$ {\Gamma}_i=\frac{L_i\;}{M_i}=\frac{{\boldsymbol{\varphi}}_i^T\mathbf{s}\;}{M_i}. $$

(50)

The modal factors L _i and Γ _i are termed the modal excitation factor and the modal participation factor, respectively. It is easy to verify that the values of both the factors Γ _i and L _i depend on the normalization used for φ _i (since the ratio L _i /Γ _i equals the modal mass ${M}_i={\boldsymbol{\varphi}}_i^T\mathbf{M}{\boldsymbol{\varphi}}_i$).

In view of Eq. 50 and the orthogonality conditions, it holds true

$$ {\boldsymbol{\varphi}}_i^T\mathbf{s}={\Gamma}_i\;{M}_i={\Gamma}_i\;{\boldsymbol{\varphi}}_i^T\mathbf{M}{\boldsymbol{\varphi}}_i={\boldsymbol{\varphi}}_i^T{\displaystyle \sum_{j=1}^n{\Gamma}_j\;\mathbf{M}{\boldsymbol{\varphi}}_j}, $$

(51)

which allows expanding of the vector s in terms of the modal contributions s _i as follows:

$$ \mathbf{s}={\displaystyle \sum_{i=1}^n{\mathbf{s}}_i=}{\displaystyle \sum_{i=1}^n{\Gamma}_i\;\mathbf{M}{\boldsymbol{\varphi}}_i}. $$

(52)

Evidently, the vector s and its modal components s _i are independent of the absolute magnitudes of φ _i . The modal factors L _i and Γ _i naturally emerge when performing the modal analysis steps indicated in Eqs. 22c and 22d. Namely, when substituting for the load form adopted in Eq. 47 and writing the i modal matrix equation of motion, one gets

$$ {M}_i\;{\ddot{q}}_i(t)+{C}_i\;{\dot{q}}_i(t)+{K}_i\;{q}_i(t)={L}_i\;f(t), $$

(53a)

or equally

$$ {\ddot{q}}_i(t)+2{\omega}_i{\xi}_i{\dot{q}}_i(t)+{\omega}_i^2{q}_i(t)={\Gamma}_if(t). $$

(53b)

The rationale behind the modal decomposition of the load distribution s lies in introducing inertial modal load contributions. To explain this, the inertial force in the ith mode is given as Mü _i or, taking into account Eq. 3, as $\mathbf{M} {\boldsymbol{\varphi}}_i{\ddot{q}}_i(t)$. The Μ φ _i pre-factor, thus, makes apparent the underlying inertial concept linking to Eq. 52. Γ _i evidently from Eq. 53b is a weighting factor that scales the load fraction to be assigned to each specific mode. The obscurity of the normalization dependence for Γ _i hinders it, becoming a direct clear indicator of the relative significance of the ith mode.

Relevant to this, the most important and useful feature of the expansion employed in Eq. 52 is that the load distribution s _i produces response exclusive to mode i or else that the response at each mode i owes solely to s _i . This can be easily derived by observation of Eq. 52. When multiplying it with φ ^T _j and f(t) to create what is defined as the modal load P _j (t) one gets

$$ {P}_j(t)={\boldsymbol{\varphi}}_j^T\mathbf{s}\;f(t)={\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_j^T{s}_if(t)=}{\displaystyle \sum_{i=1}^n{\varGamma}_i\;{\boldsymbol{\varphi}}_j^T\mathbf{M}{\boldsymbol{\varphi}}_i}\;f(t). $$

(54)

This expression due to orthogonality yields a nonzero value only for $j=i$.

Contribution Factors (Chopra’s Physical Interpretation)

Subsequently, if one follows the above procedures, the ensuing earthquake response problem can be solved. Yet, as first developed by Chopra (1995), a transformation is put forward that can further assist in a physical interpretation of modal analysis relevant to earthquake engineering. Namely, due to the linearity of the structural system, if one assumes

$$ {q}_i(t)={\Gamma}_i{D}_i(t), $$

(55)

then the modal Eq. 53b can be replaced by the equivalent

$$ {\ddot{D}}_i(t)+2{\omega}_i{\xi}_i{\dot{D}}_i(t)+{\omega}_i^2{D}_i(t)=f(t), $$

(56)

which describes a unit mass SDOF system, with frequency ω _i and damping ξ _i loaded by f(t); the latter can be either a force or a ground acceleration without any distinction. Accordingly, D _i (t) is but the ith modal response of the system to the loading f(t). The contribution of the ith mode to response can then be given by Eq. 3 as

$$ {\mathbf{u}}_i(t)={\boldsymbol{\varphi}}_i\;{q}_i(t)={\Gamma}_i\;{\boldsymbol{\varphi}}_i\;{D}_i(t). $$

(57)

Defining equivalent modal static forces F ^s _i (t) as in Eq. 42

$$ {\mathbf{F}}_i^s(t)=\mathbf{K}{\mathbf{u}}_i(t)={\omega}_i^2\mathbf{M}{\mathbf{u}}_i(t)={\omega}_i^2\left(\mathbf{M}{\varGamma}_i\;{\boldsymbol{\varphi}}_i\right){D}_i(t)={\mathbf{s}}_i\;{\omega}_i^2{D}_i(t), $$

(58)

one can write for any dynamic modal response quantity R _i (t)

$$ {R}_i(t)={R}_i^s{\omega}_i^2{D}_i(t). $$

(59)

In the latter R ^s _i is the modal response quantity that results from static analysis of the structure under the influence of the vector s _i of static external forces. This clearly shows that within the realm of modal decomposition, any complicated dynamic analysis of a n-DOF structure reduces to the product of two simple steps. In the first one performs static calculations for deriving all the n different R ^s _i and then multiplies with the response solutions D _i (t) of the n different SDOF systems loaded by f(t). The total response is evidently given by the modal summation illustrated in Eq. 41.

The above modal analysis allows the introduction of another set of quantities, named the contribution factors ${\tilde{R}}_i$. These similarly to the participation factors indicate the significance of each mode in the total response. Still, their added merit lies in the fact that they are dimensionless and relieved by the abstractness of normalization. To produce them, one needs to write Eq. 59 as

$$ {R}_i(t)={\tilde{R}}_i\;{R}^s\;{\omega}_i^2{D}_i(t), $$

(60)

in which R ^s is the total response to the vector s of static external forces, and ${\tilde{R}}_i$ is the contribution factor of the ith mode defined as

$$ {\tilde{R}}_i=\frac{R_i^s}{R^s}. $$

(61)

It can easily be proved that

$$ {\displaystyle \sum_{i=1}^n{\tilde{R}}_i}=1, $$

(62)

and, hence, if all modes beyond the r first modes are truncated, the corresponding error e _r is

$$ {e}_r=1-{\displaystyle \sum_{j=1}^r{\tilde{R}}_j}, $$

(63)

which is usually very small for the truncation beyond the first three or four modes.

Extensions and Miscellanea

There are a number of additional topics that were intentionally excluded from the above described backbone of modal analysis. A handful of them is further elaborated in what follows. The particular intention is not to fully capture the extensive available knowledge but rather motivate the reader to further pursue the vast capabilities and multifaceted merits associated with this unique dynamics tool.

Mode Acceleration Method

The entirety of the above approach is based on estimating modal displacements, which subsequently will combine to form the total response. This displacement-based approach that naturally acquires the characterization mode displacement method was physically shown to extend into the calculation of any response variable.

Although the mode displacement method was developed typically first, it is not the only option available. One of the most useful alternatives is the so-called mode acceleration method along with its implementation variations. In identical form, though founded on a different basis, this may also be found under the name of static correction method, and the origin of this naming will below become clear. It was first devised for an undamped system, and its main attribute is that instead of modal displacements, one can seek solutions in terms of only modal accelerations for an undamped system or in terms of modal accelerations and velocities for a damped system. A brief presentation is herein put forward following the derivations and critique proposed by Cornwell et al. (1983).

Starting from the classically damped modal equation Eq. 22d, one can rewrite it as

$$ {q}_i(t)=\frac{1}{\omega_i^2\;}\left(\frac{P_i(t)}{M_i}-{\ddot{q}}_i(t)- 2{\xi}_i\;{\omega}_i\;{\dot{q}}_i(t)\right)=\frac{P_i(t)}{K_i}-\frac{1}{\omega_i^2}{\ddot{q}}_i(t)-\frac{2{\xi}_i}{\omega_i}{\dot{q}}_i(t), $$

(64)

which substituting in the modal superposition Eq. 19 yields

$$ \mathbf{u}(t)={\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i\left(\frac{P_i(t)}{K_i}-\frac{1}{\omega_i^2}{\ddot{q}}_i(t)-\frac{2{\xi}_i}{\omega_i}{\dot{q}}_i(t)\right)} $$

(65a)

or equally

$$ \mathbf{u}(t)={\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i\frac{{\boldsymbol{\varphi}}_i^TF(t)}{K_i}}-{\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i\left(\frac{1}{\omega_i^2}{\ddot{q}}_i(t)+\frac{2{\xi}_i}{\omega_i}{\dot{q}}_i(t)\right)}. $$

(65b)

A plausible replacement is now sought for the first term in Eq. 65b. Namely, taking into account the definition of the equivalent static displacement of the structure u ^s(t) as ${\mathbf{u}}^s(t)={\mathbf{K}}^{-1}\mathbf{F}(t)$, as well as the definition of the equivalent static generalized displacements q ^s _i (t) as ${q}_i^s(t)={P}_i(t)/{K}_i$, and applying Eq. 19, one gets

$$ {\mathbf{K}}^{-1}\mathbf{F}(t)={\mathbf{u}}^s(t)={\displaystyle \sum_{i=1}^n{\mathbf{u}}_i^s(t)}={\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i\frac{P_i(t)}{K_i}}={\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i\frac{{\boldsymbol{\varphi}}_i^T\mathbf{F}(t)}{K_i}}\;. $$

(66)

Thus, the first term of the right-hand member of Eq. 65b only accounts for the total equivalent static displacement, as shown by Eq. 66. This fact evidently transforms the total displacement solution expressed by Eq. 65b into

$$ \mathbf{u}(t)={\mathbf{K}}^{-1}\mathbf{F}(t)-{\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i\left(\frac{1}{\omega_i^2}{\ddot{q}}_i(t)+\frac{2{\xi}_i}{\omega_i}{\dot{q}}_i(t)\right)}. $$

(67)

The latter relation gives this whole alternative solution process a unique physical meaning. To obtain the dynamic response of a structure, one needs to consider first the easy to derive equivalent static (or pseudostatic) displacement and, subsequently, fine-tune it by adding the dynamic effects as introduced by the modal acceleration and velocity responses of the n individual modes. The higher mode, which trivially translates to higher frequency, would impose a relatively faster quadratic convergence for the second term in the right-hand side of Eq. 67 (see ω ² _i denominator). Thus, this would allow less modes to be considered when truncating for economy and practicality the full series expansion. The method actually even when modes higher than r ($\ll n$) are discarded clearly considers partly their influence through their static function.

Evidently, for an undamped system, Eq. 67 reduces to

$$ \mathbf{u}(t)={\mathbf{K}}^{-1}\mathbf{F}(t)-{\displaystyle \sum_{i=1}^n{\boldsymbol{\varphi}}_i\frac{1}{\omega_i^2}{\ddot{q}}_i(t)}\ \mathrm{with}\ {\xi}_i=0, $$

(67a)

where the dynamic contribution of the modal velocities has been canceled out and only remains the dynamic effect of modal accelerations, which justifies the name mode acceleration method.

It is worth noting that even for a damped system, what matters in structural analysis is the maximum displacement max u, which causes the maximum strain and, hence, the maximum stress, required for proportioning the structural members. As a rule of thumb, this max u is taken for equal to a combination of the maximum displacements max u _i of the three or four first modes of the structure, which necessitates that the corresponding synchronous modal velocities become zero, ${\dot{\mathbf{u}}}_i=0$. Thus, the dynamic contribution of the modal accelerations to account for the maximum value of the dynamic response of the damped system only remains.

The method is greatly superior in terms of numerical efficacy outperforming the conventional mode displacement method in all cases by consistently requiring reduced number of modes for accurate, similar minimal error, solutions. Further the mode acceleration solution was found to be more sensitive to damping by means that increasing uniformly the damping of all modes would lead to faster convergence rates of associated dynamic solutions on reduced modal information.

Nonlinearity

The most critical assumption for applying modal analysis lies in the validity of the superposition of modes adopted in Eq. 2. This in simple terms translates to linearity of the structural system. With nonlinear behavior, developing the pursuit for a response-independent stationary modal reference system is not straightforward. Thus, the economy in the analysis of linear structures achieved through the modal order reduction cannot be realized in the analysis of nonlinear structures. Interestingly, in most real-life cases, structures move substantially, yield, or just interact with their loading environment. In all such instances, nonlinear descriptions are needed to capture the underlying phenomena waiving the validity of most of the above analysis.

A generalized manifold-type similar concept of nonlinear normal modes (NNM) has recently developed within the nonlinear dynamics field. Yet, this should be seen by engineers as more of a mathematical intricacy rather than a practical tool credible to be used in ordinary analysis of typical high-order systems. In generic nonlinear structures (this is to distinguish from cases of linear subsystems connecting through nonlinear links), the strict approach consists of numerically integrating the n-coupled equations of motion concurrently. To this purpose the time-continuous matrix Eq. 20 is written in a variational form to enable a time-stepping approach to be used for calculating purposes. Namely, Eq. 20 changes to

$$ \mathbf{M}\Delta {\ddot{\mathbf{u}}}_i+{\mathbf{C}}_i\Delta {\dot{\mathbf{u}}}_i+{\mathbf{K}}_i\Delta {\mathbf{u}}_i=\Delta {\mathbf{F}}_i, $$

(68)

where

$$ \Delta {\ddot{\mathbf{u}}}_i={\ddot{\mathbf{u}}}_{i+1}-{\ddot{\mathbf{u}}}_i,\ \Delta {\dot{\mathbf{u}}}_i={\dot{\mathbf{u}}}_{i+1}-{\dot{\mathbf{u}}}_i,\ \Delta {\mathbf{u}}_i={\mathbf{u}}_{i+1}-{\mathbf{u}}_i,\ \Delta {\mathbf{F}}_i={\mathbf{F}}_{i+1}-{\mathbf{F}}_i, $$

(69)

with i representing the time-step index contrary to its earlier modal meaning, while C _i and K _i are nonlinear damping and stiffness matrices, respectively, whose elements are functions of the displacement u(t) and the velocity $\dot{\mathbf{u}}(t)$ and take on values depending on the considered time step. The index i would translate in terms of absolute time to iΔt, where Δt denotes the time step. Alternatively, the time index $i+1$ can be used (i.e., this is the implicit vs the explicit method when i is used). The overall response calculation problem includes three unknowns ü _i , ${\dot{\mathbf{u}}}_i$ and u _i for which apart from Eq. 68 two more equations could be derived from different approximations for ü _i and ${\dot{\mathbf{u}}}_i$. As a matter of fact, depending on these approximations, which link to an assumed variance of the ü _i and/or ${\dot{\mathbf{u}}}_i$ variables during the time-step duration, a number of different methods have developed. The most broadly used for earthquake-related purposes is Newmark’s β-method.

It is noteworthy that although methods like the latter would easily produce bounded solutions (i.e., stable, meaning results will not “blow up”), whilst the time-step Δt is accurately prescribed. As a matter of fact, this is the most important parameter for any numerical integration scheme and must be sufficiently small. A first approximation for the Δt value can come from the associated modal analysis pertaining in the linear operation regime of the structure. Namely, the highest linear mode would correspond to the lowest natural period T _n . Δt should be sufficiently smaller than this T _n value. It should be reminded that this imposes large numerical limitations and costs and that the inherent softening associated with yielding would increase T _n (on the other hand, hardening would have an adverse effect by further reducing T _n ).

Frequency Domain Solutions

After bringing the structural system to its modal description equivalent, the solutions pursued whether in terms of modal displacements or in terms of modal accelerations and velocities were always expressed in the time domain. Considering the case of the classically damped system with periodic loading and focusing on the probably most significant part of the response, the forced or else for this case steady state, one may suggest some alternatives to Eqs. 17 and 25. The reason is that the Duhamel’s integral that provide the steady-state time response involves the convolution operation between the applied load and the unit-impulse response function. This term tends to perplex calculations.

Without detailing proofs, the alternative representation is in the frequency domain and can easily surface when one assigns inside the SDOF a periodic loading of the form ${\mathrm{P}}_i(t)={\mathrm{P}}_i^o{\mathrm{e}}^{\mathrm{j}\omega t}$, where P ^o _i is a scalar amplitude and again $\mathrm{j}=\sqrt{-1}$. Then by rearranging the generalized SDOF response becomes

$$ {q}_i(t)=\frac{{\mathrm{P}}_i^o{\mathrm{e}}^{\mathrm{j}{\omega}_it}}{\left({K}_i-{\omega}_i^2{M}_i\right)+\mathrm{j}{\omega}_i{C}_i}\;. $$

(70)

Dividing Eq. 70 with P _i (t) in order to create an output over input ratio, one gets

$$ \mathrm{H}\left(\omega \right)=\frac{q_i(t)}{{\mathrm{P}}_i(t)}=\frac{{\tilde{q}}_i\left(\omega \right)}{{\tilde{\mathrm{P}}}_i\left(\omega \right)}=\frac{1}{\left({K}_i-{\omega}^2{M}_i\right)+\mathrm{j}\omega {C}_i}, $$

(71)

where over dash stands for the Fourier transform. The H(ω) function depending only on ω is called the receptance frequency response function. If, instead of displacement response, acceleration or velocity were used for the numerator, one would have the accelerance and mobility frequency response function, respectively. H(ω) is the Fourier transform of the unit-impulse response function given in Eq. 27. Generalizing the concept for an MDOF system, this instead of a scalar becomes a matrix H(ω) with components h _kl (ω), denoting the ratio of displacement at position k and force at position l

$$ {\mathrm{h}}_{kl}\left(\omega \right)={\displaystyle \sum_{i=1}^n\frac{{\boldsymbol{\varphi}}_i(k){\boldsymbol{\varphi}}_i(j)}{M_i\;\left({\omega}_i^2-{\omega}^2\right)+\mathrm{j}\omega {C}_i}}\;. $$

(72)

Subsequently the convolution in the time domain is replaced in the frequency domain from a simple product of the frequency response function with the input force.

Operational Modal Analysis

To this point the presentation was focused on the so-called direct problem, which is concerned with obtaining the dynamic response of a fully known system to a given loading. By fully known it is meant that all the necessary properties to describe its behavior through a model, mainly an FE model, are in hand. Such can be the modal properties instead of the full stiffness, damping, and mass matrices. This in fact can bring a vast simplification and data economy having merits even more exaggerated than the ones modal truncation introduced before. Indicatively, for an n-DOF system, one, instead of determining 3n ² matrix terms, could obtain similar results by employing only a reduced number of modal frequency, damping, and shape sets. Still, this a priori assumed knowledge is not always the case. There are many uncertainties assigned to modeling (e.g., links’ behavior, foundations, damping, etc.) which can bring up substantial errors in any associated model parameters and subsequently response predictions, used either for design, assessment, or control of the structure. To deal with this significant engineering issue, a major part of modal analysis nowadays has been devoted to the so-called inverse analysis. This translates to a mapping of the measured response to the system modal characteristics, and it first appeared with the development of the space program of the USA back in the 1950s. Interestingly, although the direct problem is a single-valued one, the inverse is not. The latter brings up the need for optimizing any of the prediction/analysis routines that will eventually produce modal parameter outputs from measured response data.

The equation-free description that follows leaves aside the very interesting and demanding experimental (including signal preprocessing) techniques devised through the years to obtain the measured response from a structure under different forms of loading. The main scope of this entry is to only refer briefly to some of the most typical available tools able to obtain modal estimates once measured data are available. The mathematical background necessary to fully explain these methods are superseding the space limitations of this entry. Thus, the reader is motivated to also direct to the very insightful and broad §4 of Maia et al. (1997) as well as to a true encyclopedia on the topic by Ljung (1999) for further studying. Excellent software can be found to practice inverse analysis, some even in freeware form, which encompasses the full information provided herein and beyond. Such availability is a true late year’s addition.

Practically all useful inverse analysis methods refer to MDOF systems. For them one may have information on their structural excitation or not. Their difficulty ranges from the simple peak picking method, where one observes peaks in the frequency response functions, to much more demanding alternatives employing complex algebra (Hankel matrices, Markov parameters, etc.) together with the earlier introduced state-space formulation. Most of the MDOF methods break down the structure to a superposition of independent-uncoupled SDOF systems, making even more profound the linearity and modal decomposition demands. In general, the strict target is to fit an analytical function of the form introduced in Eq. 72 to the measured data, which are customarily transformed to frequency response functions. This fitting operation inherently involved in the process lends the name curve fitting to the concept. Depending on the specific algorithm opted for the fitting, one may get the circle fitting, rational function polynomial, or many other refined approaches along the same lines.

Alternatively, one, instead of working in the frequency domain, may work employing the time counterpart of frequency response functions, the impulse response functions. Typically the complex exponential method is used for such fittings. A similar very widespread approach fitting free decays instead of impulse response functions is named after Ibrahim, i.e., Ibrahim time domain method. Alternatively time-based methods like the option of the eigen-realization algorithm which will recover families of systems with identical eigenvalues to the measured one or generic time series tools like the auto-regressive moving average can further be employed.

In all cases there is a major parameter affecting results. This is the order of the assumed model in the identification analysis. Namely, earlier treating the direct problem, one had to decide the order to which the modal solutions were to be truncated. The order that should be decided for the number of modes to be identified here is the direct equivalent.

Summary

Although every effort was made to synoptically present all the critical points that could not be missed of any educational work addressing modal analysis, the reader may still find omissions. To this reason, the wealth of references provided throughout is an excellent resource to complement the current entry as a further reading that could cover practically every aspect relating to the subject.

To recapitulate the main attributes presented herein, a short mind map is put forward that can illustrate the simple steps involved in modal analysis when employed in practical earthquake engineering purposes. This consists of:

• Determining the structural matrices K, M, and C

• Determining the modal natural frequencies ω _i and shapes φ _i

• Computing each modal-generalized response q _i (or ${\ddot{q}}_i$ and ${\dot{q}}_i$) and turning it to modal displacement u _i

• Calculating the total displacement vector u after deciding the minimum number of modes r needed for an accurate solution