278 J.-T. Jiolat et al. 32.2 Model U-K The U-K formulation was originally obtained from the Gauss principle of least action. Then, in the papers by Arabyan and Wu [2] and Laulusa and Bauchau [4], an original algebraic approach was found for deriving the U-K formulation for constrained systems from the classical formulation with Lagrange multipliers [1]. Let us consider a mechanical system with mass matrix Mwhich is subjected to an external force vector Fe(t), which includes all constraint-independent internal and external forces. This system is also subjected to a set of P holonomic and non-holonomic constraints which depend on the system displacement x(t) and velocity v(t). Denoting the dynamical solution xu(t) of the unconstrained system and the one x(t) of the constrained system, which depends on the constraining forces Fc(t), and following [2], one obtains the motion equations of the constrained system proposed by Udwadia and Kalaba [1, 2]: ¨x = ¨xu +M−1/2B+(b−A¨xu). (32.1) ¨xu =M−1F e(t) (32.2) where Ais the constraint matrix, bis a known constrained vector, B+is the Moore-Penrose inversion of matrixB=AM1/2. The original character of this approach is that it can be used for conservative or dissipative, linear or non-linear systems. Moreover, the generalized inverse B+ can be rendered numerically robust, even when the constraint matrix is singular. For a particular excitation Fe(t), we can solve these equations using a suitable time-step integration scheme. Next, we adapt the U-K formulation in order to deal with continuous flexible systems whose dynamics will be described in terms of modal coordinates. We assume a set of S vibrating subsystems, each one defined in terms of its unconstrained modal basis and being coupled through P kinematic constraints. Then, using the usual modal equations that govern the physical motion of the subsystems, we end up with similar equations of motion, which are described now in terms of modal parameters [1]. ¨q =W˜M−1 (− ˜C˙q− ˜Kq+Fext) (32.3) where q represent the vector of modal displacements, ˜M, ˜K, ˜C are respectively the modal mass matrix, modal stiffness matrix, and modal damping matrix, while W = 1 − ˜M−1/2 B+Ais a convenient global transformation matrix (which is computed before the time loop), where Ais the modal constraint matrix, and Fext are the external modal forces applied on the system. In order to proceed to the computation of the vibratory response of the constraint system, for a given external force vector, we need to obtain the modal parameters of each unconstrained subsystem. For the strings, we consider the classical mode shapes that we find theoretically for a flexible string. We also use a theoretical formulation for the damping of the string [3]. For the simulation, we decide to take 50 modes for each strings, covering a frequency range up to 24.5 kHz. Concerning the modal parameters of the instrument soundboard, which were measured at the bridge, these were obtained through experimental modal identification, using 37 points for the discretization along the bridge. Once we measured the vibratory frequency response functions (between a reference location and each point of the bridge), we proceeded to the modal identification using a frequency-domain approach called LSRF (Least-squares rational function estimation method), implemented in Matlab [6]. This modal analysis was performed within a frequency band going from 40 to 800 Hz, leading to 12 identified modes. 32.3 Results and Conclusion To compare our model with experimental data, we used a vibrometer to measure the vibratory velocity of the RS of the C5 string, at two centimeters from the bridge, induced by the tangent excitation of PS of the F3 string (i.e. playing the F3 key), all the other strings being muffled. The vibratory response is only measured in the vertical polarization of the motion of the string, since the model developed gives the response in just one polarization of motion. Our first step was to model the F3 PS and the G4 and C5 RS being coupled with the bridge (see Fig. 32.1). We choose these two strings because their RS have harmonic frequency relations with the harmonics of the PS of the F3 string: therefore a significant vibratory coupling should be expected. We produced numerically a realistic string excitation such that the response of the played string was as close as possible to the experimental response. In Fig. 32.2, we compare the spectral response of the C5 RS given by the numerical simulation with the measured one. In both results, we see the fundamental frequency peak of the F3 string which is at 328 Hz and all its harmonics, which are the partials transmitted to the C5 RS by means of the coupling with the bridge. Also, we
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