10 Numerical and Experimental Modal Analysis of a Cantilever Beam. . . 109 This interaction is captured by the following boundary and connectivity conditions—the beam-tendon system is fixed at one end (x =0) w=0, (10.2a) w =0, (10.2b) wt =0. (10.2c) and free at the other end (for x =LN) EIw N =0, (10.3a) −EIw N −Pw N +mN ¨wN +Pw tN =0, (10.3b) wtN =wN. (10.3c) where mN is the tip mass. The origin of these boundary conditions is discussed in detail in [4] where they were also experimentally validated. In addition, for every attachment point (for x =Li where i =1,2, . . . ,N−1), w (R) i =w (L) i+1 , (10.4a) wi (R) =wi+1 (L) , (10.4b) EIwi (R) =EIwi+1 (L) , (10.4c) −EIwi (R) −Pwi (R) +Pwti (R) =−EIwi+1 (L) −Pwi+1 (L) +Pwt(i+1) (L) +mi ¨w (L) i+1 , (10.4d) w (R) ti =w (R) i , (10.4e) w (L) t(i+1) =w (L) i+1 , (10.4f) where the superscripts•(R) i and• (L) i mark the right and left side of the i th interval, respectively. The first four equations ensure the connectivity of beam’s displacements, slopes, moments and shear forces, respectively, on each side of the attachment fixture while the last two equations ensure the same displacement of the beam and the tendon. In order to evaluate the modal properties (natural frequencies and mode shapes) of the beam-tendon system, an assumption of the normal mode is used. A solution of any given dependent variable is expressed as a multiplication of the time-invariant mode shape and the time-varying harmonic function of the constant frequency in the following form w(t,x) =W(x)e iωt , wt(t,x) =Wt(x)e iωt . (10.5) Substituting the normal mode forms into the partial differential equations (PDEs) allows one to eliminate time and rewrite the PDEs into a system of first order ordinary differential equations (ODEs) that, together with the boundary conditions (BCs), define a boundary value problem. This boundary value problem can then be solved by a Matlab bvp4c solver [13] for unknown natural frequencies ωand corresponding mode shapes components W(x) and Wt(x). This solver is very versatile since it uses a collocation method but may suffer from a decreased numerical performance if an appropriate starting guess is not provided. 10.3 Numerical and Experimental Results In this section, numerical and experimental results are presented. The experimental set-up used can be seen in Fig. 10.2. A similar experimental set-up to test the beam-tendon system with no attachment fixtures was already used in [3, 4]. It consists of a cantilever beam-tendon system that is gravity-loaded using the masses placed on the hanging platform, and hardware required for experimental modal analysis. As in the previous studies, one attachment point was required at the clamp to ensure the same total length of the beam and the tendon. Unlike in the previous studies, the attachment fixture ensuring the
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