( ) , ( , ) ( 0) ( ) , , 1,.., , 2 x j t T i j j i j j x j t a x A dx i j n n N (11) where ai,j(xj) is the integral of the component in the i th row and jth column of A(t) and is taken with respect to the jth component of the state vector for all the states within the original periodic orbits. The terms in the first N rows of the [ai,j] matrix define the identity relationships of the state vector components, but those in the lower N rows define the force relationship of the ith degree of freedom with respect to the jth state vector component. So this method allows these dynamic forces to be individually calculated from the identified time-periodic model. The total dynamic force that acts on a degree of freedom of the system is then equal to the sum of a single row of the matrix [aij], and is a function of the position and velocity states of the system. 1 ( , ) ( ) n d v j j j a x x a x (12) In the previous equation, aj is the j th column of the matrix, [a ij]. This function is directly related to the total restoring force of the system, since the following equation ˆ ( , ) ( , ) d v f f x u a x x du u (13) Fully defines the reconstructed nonlinear equations of motion. 2.2.5 Modified Restoring Force Surface Method for Periodic Response In this section a method is developed based on the restoring force surface approach [8] for solving for the net restoring forces over the periodic orbit from the measured periodic response. The restoring force surface method is based on the following equation of motion, which is valid for a broad range of structural systems. , ( ) a RF d v Mx g x x F t (14) Assuming that the acceleration of each of the nodes of the system has been measured as well as the applied force F(t), the restoring forces can be found as follows if one has an estimate for the mass matrix M. , ( ) RF d v a g x x F t Mx (15) The restoring forces are functions of the displacement and velocity, and since xa is known, xd and xv can be found by integrating xa. Since xa is the periodic response, it can be readily described by a Fourier series. As long as the constant term in the series is zero, then the Fourier series model can be integrated as in eq. (9) and then the restoring force can be plotted versus xd and xv or versus time over the periodic orbit. 3. Nonlinear cantilever beam system The proposed identification method was evaluated by applying it to measurements from a nonlinear beam. Figure 1 below shows a top view photograph of the actual experimental setup. An aluminum 6061 beam is bolted to a steel mounting block, shown on the left side of the image. A small strip of spring steel is bolted to the free end of the cantilever and clamped to another steel mounting block. Both of the mounting blocks are bolted to a massive steel tube. The steel tube and mounting blocks approximate the fixed support of an ideal cantilever. The whole setup rests on a foam pad on a massive table top. The beam is oriented such that the bending axis is parallel to the plane of the table top. Figure 2 shows a close top and front view of the spring steel between the tip of the beam and the right hand side support. Table 1 below provides the physical dimensions of the beam and the spring steel in millimeters. 107
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