13 Output-Only Modal Parameter Estimation Using a Continuously Scanning. . . 115 B#∂ 2 z(x, t) ∂2t $+C#∂z(x, t) ∂t $+L[z(x, t)] =0, x ∈D, t ≥0 (13.1) where B(· ), C(· ) and L(· ) are a mass operator, a damping operator and a stiffness operator, respectively, z is the displacement of the structure at the spatial positionx at time t, and Dis its spatial domain. Boundary and initial conditions of the structure are known. Note that the initial conditions can be induced by an external force that the structure is subject to whent <0. A solution to Eq. (13.1) can be obtained using the expansion theorem [18]: z(x, t) = ∞ i=1 φi (x)ui (t) (13.2) where φi is the i-th mass-normalized eigenfunction of the associated undamped structure, which is assumed to be selfadjoint, and ui is the corresponding unknown time function. Orthonormality between φi and φj (j =1, 2, . . . , ∞) with respect toBis expressed by D φj (p)B/φi (p)0dp=δij (13.3) where δij denotes Kronecker delta function, which satisfies δij =1 if i =j andδij =0 if i =j. Assuming that damping of the structure can be modeled by Kelvin-Voigt viscoelastic model, which leads to a classically damped system [18, 19], one canobtainui in Eq. (13.2) by solving an ordinary differential equation: ¨ui (t) +2ζiωi ˙ui (t) +ω 2 i ui (t) =0 (13.4) where ωi is the correspondingi-th undamped natural frequency of the structure, ζi is the i-th modal damping ratio, which is smaller than 1 for an underdamped structure, and an overdot denotes differentiation with respect to t. The initial conditions ui (0) and ˙ui (0) can be determined from the initial conditions of Eq. (13.1). The solution to Eq. (13.4) can be expressed by [20] ui (t) =e−ωiζit ui (0)cos ωi,dt + ˙ ui(0)+ωiζiui(0) ωi,d sin ωi,dt =Aie−ωiζit cos ωi,dt −γi (13.5) where ωi,d =ωi 1−ζ 2 i (13.6) is the i-th damped natural frequency of the structure, Ai = 1 [ui (0)] 2 +#˙ ui (0) +ωiζiui (0) ωi,d $ 2 (13.7) is an amplitude constant, and γi =arctan2 ˙ui (0) +ωiζiui (0) ωi,d ,ui (0) (13.8) is a phase angle; ωiζi in Eq. (13.5) is referred to as the decaying rate of ui. Based on Eqs. (13.2) and (13.5), Eq. (13.2) becomes z(x, t) = ∞ i=1 Aiφi (x)e−ωiζit cos ωi,dt −γi (13.9)
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