.} = c. and WifiCMj} = 0 fori^y where { } T represents the transpose of a vector and the parameters m,, kt and c, are called the modal mass, stiffness and damping coefficients associated with the ith mode of the structure. For a non-classically damped structure the third set of properties, given in Equation 2.6, do not hold. A classically damped structure can therefore be defined as one that satisfies this property. This idealized and desirable situation results from assuming that the damping present in the struc-ture is mass or stiffness proportional (see Humar, 1990). The displacements of the force-excited system can be determined from the equation: where y,(r) are the normalized coordinates which are determined from the solution of the uncou-pled, viscously damped, single degree of freedom system represented by equation 2.8: (2.7) 11 Chapter 2 Background ^ \u2022 ( 0 +^,.(0 + ^ . ( 0 = Pi(t) (2.8) where pt(t) is the modal excitation given by: Pi(t) = Wifpit). (2.9) Solution of equation 2.8 involves the solution of an uncoupled, 2nd order ordinary differential equation similar to that given by equation 2.1. Therefore, a n-degree of freedom system can be broken down into a series of n, single degree of freedom systems, each with modal mass ra,, modal stiffness kt and modal damping c, . These single degree of freedom systems can be solved by a number of methods and then related back to the original structure using its associated mode shape {()),} and equation 2.7. Methods for the solution of the equation of motion are pre-sented in Chopra (1995). 2.1.3 Comment on Natural Frequencies A major focus of this thesis is the application of methods that would help determine if natural fre-quencies of a structural system change in time. However, for truly linear elastic systems, the nat-ural frequencies, by definition, do not change and thus a structure's natural frequencies are independent of the loading conditions and time invariant. A natural frequency which changes in time should be thought of as an instantaneous or temporary natural frequency and the observed change is relative to the actual natural frequency of the linear system. 12 Chapter 2 Background The term \"natural\" frequency is used throughout this thesis to describe the dominant frequencies of all systems, whether they are time invariant or not. 2.2 Frequency Domain Analysis When examining the dynamic response of structures it is convenient to talk in terms of frequency. As part of a typical analysis of structural vibration data, one is concerned with the frequency of the excitation and\/or the natural frequency of the structure. As was seen in the proceeding sec-tion, the vibration of a structure at any point in time can be decomposed into a combination of its mode shapes oscillating at their associated natural frequencies. If a structure is excited by a forc-ing function with a frequency equal to a natural frequency of the structure, resonance occurs. At resonance, the energy present in the input force is added, in phase, to the energy of the structure and thus the response can grow to levels much larger than normal operating levels. The introduction of the Fourier Series and Fourier Transform have greatly facilitated frequency or spectral analysis. The basic mathematical representation of periodic signals is the Fourier Series, which is a weighted sum of harmonically related sinusoids or complex exponentials. Jean Baptist Joseph Fourier (1768-1830), a French mathematician, originally used such a series to describe the phenomenon of heat conduction and temperature distribution through bodies (Proakis and Mano-laksi, 1996). For an aperiodic signal of finite energy, analysis in the frequency domain is made possible by the Fourier Transform. The Fourier Transform is a special case of the Fourier series as the signal is periodic with infinite period. The frequency analysis of aperiodic, continuous time signals involves the following transform pair: 13 Chapter 2 Background Direct Fourier Transform: X((o) = \\ x(t)e'mdt (2.10) \u2014 o o o o Synthesis Equation: x(t) = -z~- [ X(m)e~'m'd(a (2.11) IK J \u2014 o o Equation 2.10 is the Direct Fourier Transform and Equation 2.11 is the synthesis, or inverse, Fou-rier Transform. Most time signals which are analyzed today are discrete, or digitized, time sig-nals. This representation of the signal requires the use of a slightly different transform pair. The discrete value of x(t), at time t = r A , is written {xr} and the series {xr} is called a discrete time series. If T is the length of the signal in seconds and N is the number of discrete points then the time step A is given by A = T\/N. The range of Fourier components Xk is limited to k = 0 to k = N - 1, which correspond to harmonics of frequency co* = 2nk\/T = 2nk\/NA. The Dis-crete Fourier Transform (DFT) pair is given by (Newland, 1993): N- 1 DFT: Xk = ^ \u00a3 x r e i ( 2 n k r \/ N ) k = 0,1,2,..., (N-l) (2.12) r = 0 N- 1 Inverse DFT: xr = J Xke~K2nkr\/N) r = 0,1,2,..., (N-l) (2.13) k = 0 14 Chapter 2 Background Methods for analysis in the frequency domain have been around since the early 19th century but the computational effort required for complex, aperiodic signals made the use of this method practically impossible. In the 1960's an algorithm was developed which enabled the calculation of the Discrete Fourier Transform and Inverse Discrete Fourier Transform quickly and was hence named the Fast Fourier Transform or FFT (Cooley and Tukey, 1965). This algorithm makes use of the harmonic properties of the transform to greatly reduce the computation required. The intro-duction of the FFT lead to an explosion of research in, and application of, analysis in the fre-quency domain. Refer to Ramirez (1985) for a derivation of the Fast Fourier Transform. Analysis which makes use of the Fourier Transform is said to be a frequency domain analysis. Analysis in the frequency domain encompasses many different techniques depending on the nature of the experimental data. There are a few basic representations of a signal, or signals, in the frequency domain which are common to many applications. If Xa(co) is the Fourier Transform of a signal, then the auto power spectrum Gou(co) is given by: Gaa(o\u00bb = X f l(co)<(co) (2.14) where * denotes the complex conjugate. The auto power spectrum is a real valued expression which is proportional to the energy, or power, in the signal. 15 Chapter 2 For 2 time signals xa(t) and xh(t), the cross power spectrum is given by: Background G a b ^ =Xa(C0)XbW. (2.15) The coherence function \"\/^(co), is a measure of the likeness of two signals. It can be used to esti-mate the amount of noise present in the input-output signals and the presence of non-linearities. The coherence function between two signals \"a\" and \"b\" is given by: where Gah((a) is the cross spectrum and G a a(co) and Ghh((\u00a3>) are the auto spectrums of signals \"a\" and \"b\" respectively. The Frequency Response Function (FRF), is used extensively in frequency domain analysis of structures. It is defined as the response of a structure to an harmonic forcing function of unit amplitude. The dynamic equation of equilibrium with a unit amplitude harmonic forcing function is given by: mx + cx + kx = eim. (2.17) If the response x(t) is assumed to be given by x(t) = H(\u00ae)e\"\u00b0', then Equation (2.17) can be rewritten as: yabW = (2.16) G a a ^ G b b ^ 16 Chapter 2 Background (- co2m + \/cue + k)H(a)eiat = eim. (2.18) Now the Frequency Response Function, H(co) can be written as the ratio of output motion to input force: \/\/(co) = - 1 , . (2.19) k - a r m + zcoc Using the frequency ratio P=co\/con Equation (2.19) can be rewritten to yield: H(Q) = (2.20) The FRF H((3), is a complex valued expression and more commonly expressed in terms of its magnitude IH(P)I and phase angle