13 Mass Scaling of Mode Shapes Based on the Effect of Traffic on Bridges: A Numerical Study 97 ®i D˛i:§i (13.2) where ˛i is the scaling factor of mode shape i. The proposed methods existing in the literature for determining the scaling factors, are based on the eigenvalue changes of the structure before and after the addition of the masses. The eigenvalue problem of equation of motion in the absence of damping is m:®0:! 2 0 Dk:®0 (13.3) where mand kare mass and stiffness matrices of the structure, respectively, ®0 is the mass normalized mode shape and !0 is the natural frequency. Added masses are often considered lumped masses for simplification purposes and thus the mass change matrix mbecomes in general diagonal. The eigenvalue problem of equation of motion in the presence of additional masses is in the following form .mC m/ :®1:! 2 1 Dk:®1 (13.4) inwhich ®1 and !1 are modal parameters of the modified structure. It has been shown that the scaling factors of unscaled mode shapes can be derived from the following equation [8] ˛ Ds !2 0 !2 1 !2 1:§ T : m:§ (13.5) However, according to [3], if a full set of modes is used, the exact scale factors can be derived from the following equation and this equation is fulfilled for each value of i ˛j D vu uu t !2 0j !2 1i :Bji §0j T : !2 1i: m :§1i (13.6) in which matrixBcan be produced by the equation b BD b § 1 0 :§1 (13.7) where b § 1 0 is the pseudo-inverse of §0. 13.2.3 Modeling the Traffic Excitation Traffic load is shared between nodes of the system using a physics-based traffic excitation model. In this method, every vehicle induces a moving load when it is traversing the bridge and this load can be assumed to affect each node of the structure with a time history load [17]. Concentrated and constant vehicle forces are represented by Pduring vehicle movement over the bridge. The governing equation of motion for vertical deflection of the bridge is m @2 @t2 yb .x; t/ Cc @ @t yb .x; t/ CEI @4 @x4 yb .x; t/ D ı .x vt/P (13.8) inwhich, m, c, E and I are the mass per unit length, the damping coefficient, the Young’s modulus and the cross-sectional moment of inertia of the beam, respectively, and v is the velocity of the vehicle while yb is the vertical deflection of the bridge from the equilibrium position. Horizontal position of the forces is shown by x as illustrated in Fig. 13.1. Considering the forces induced by multiple vehicles on the bridge as a summation and assuming a lognormal distribution for the headways of the successive arrivals of the vehicles, one can replace the right hand side of Eq. (13.8) by
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