18 How Linear Is a Linear System? 187 18.3 Theory This section provides an overview for the theory used in this work. First, the dynamic substructuring theory of the transmission simulator (TS) method is presented. Second, a nonlinear analysis tool, the Hilbert Transform, is presented as a means to verify the linear modal parameters extracted from measurements. 18.3.1 Transmission Simulator Method The Transmission Simulator method was first introduced by Allen and Mayes in [6]. This method provides a quality experimental model and best simulates the boundary conditions between subcomponents by mass-loading the interface between subcomponents. In this study, experimental subcomponent A and B are connected by a central cylinder which acts as a mass-loading of the jointed interface. The theory for the TS method is briefly discussed here for convenience. First, each subcomponent is written as a set of uncoupled modal equations of motion. Modal parameters ω, ζ, and φ are the linear natural frequency, damping ratio, and mode shapes of a subcomponent model. The modal acceleration, velocity and displacement and external force are represented by ¨q, ˙q, q, andF, respectivly. ⎡ ⎣ IA 0 0 0 IB 0 0 0 −ITS ⎤ ⎦ ⎧⎨ ⎩ ¨qA ¨qB ¨qTS ⎫⎬ ⎭ + ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ . . .2ζAωA. . . 0 0 0 . . .2ζBωB . . . 0 0 0 − . . .2ζTSωTS . . . ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ ⎧⎨ ⎩ ˙qA ˙qB ˙qTS ⎫⎬ ⎭ + ⎡ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎣ . . . ω 2 A. . . 0 0 0 . . . ω 2 B. . . 0 0 0 − . . . ω 2 TS. . . ⎤ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎦ ⎧⎨ ⎩ qA qB qTS ⎫⎬ ⎭ =⎧⎨ ⎩ φT A FA φT B FB φT TS FTS ⎫⎬ ⎭ (18.1) To couple the subcomponents, constraints must be enforced on these equations of motion. The constrains are first written in the form of Eq. (18.2). Here, x represents physical displacements of each subcomponent, and ∼ Bis a Boolean matrix such that the motion of shared DOFs between multiple subcomponents are equal. ∼ B⎧⎨ ⎩ xA xB xTS ⎫⎬ ⎭ =0 (18.2) Using a modal approximation, this constraint equation can be transformed into modal coordinates as shown in Eq. (18.3). The constraints have been softened using the pseudo-inverse of the TS shapes as it would be difficult to enforce them strictly when using measured data [6]. φ † TS 0 0 φ † TS φA 0 −φTS 0 φB −φTS ⎧⎨ ⎩ qA qB qTS ⎫⎬ ⎭ = ∼ B⎧⎨ ⎩ qA qB qTS ⎫⎬ ⎭ =0 (18.3) A coordinate transformationq=Lη is used to enforce these constraints on the uncoupled equations of motion. Rewriting the constraints in terms of the new generalized coordinate is shown in Eq. (18.4). ∼ BLη =0 (18.4)
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