132 X. Liu et al. d Fig. 13.1 In-line, 3-DoF nonlinear oscillator. Three lumped masses are grounded via linear springs and linear viscous damper. Masses are also coupled with each other via linear and nonlinear springs. All three masses are excited by the single-frequency-sinusoidal forces their corresponding expressions are obtained. In Sect. 13.4, the certain system physical parameter values for both hardening and softening cases are chosen for computing the backbone curves results. Based on the results, the modal interactions are discussed. Conclusions are drawn in Sect. 13.5. 13.2 The Nonlinear 3-DoF Oscillators Considered and Normal Form Method Application The nonlinear 3-DoF oscillating system considered here is shown in Fig. 13.1. The system consists of three identical lumped masses which are forced sinusoidally at amplitudes P1, P2 and P3 respectively at the frequency . Their displacements are denotedbyx1, x2 andx3. For this system, all three masses are linked to the ground via linear viscous dampers, with a damping constant c, and linear springs, with a stiffness k or k Cı. The middle mass connects the two masses at the sides via linear viscous dampers, with damping coefficients c0, and nonlinear cubic springs with the linear stiffness k0 and cubic stiffness . The two side masses are also coupled via a linear spring, k0, and a viscous damper, c0. Here, the masses are weakly linear or nonlinear coupled with each other, thus the coupled spring stiffness is very small compared with the grounded one, i.e. k0 k and k. Meanwhile, a small part ı (ı k) is added to the grounded spring stiffness of the 2nd mass. This leads to the equivalent linear structure of this system to be mistuned. The equation of motion (EoM) for the 3-DoF system can be written in the general form, MRxCCPxCKxCNx.x/ DPcos. t/; (13.1) where M, Cand Kare matrices of mass, damping and stiffness respectively; x is a vector of physical displacement; Nx is a vector of nonlinear and damping terms andPis the external force amplitudes vector. Here the backbone curves are used to help illustrate the modal interactions of the nonlinear system. So to reach this, the second-order normal form method [13, 18] is chosen to obtain the backbone curves. Firstly based on the definition of the backbone curve that it describes the loci of dynamic responses of a system when unforced and undamped, Eq. (13.1) without damping and forcing terms, as, MRxCKxCNx.x/ D0; (13.2) is under consideration. Through the linear modal transformation for decoupling the linear terms, the EoM in terms of the modal coordinates qis obtained, RqCƒqCNq.q/ D0; (13.3) where qis a vector of modal displacements andƒis a diagonal matrix of the squares of the corresponding linearised natural frequencies !n1, !n2 and!n3, and the Nq is a vector of modal nonlinear terms. Here, expressions of the linear modal natural frequencies and linear modeshape matrix used are, !n1 D r 1C 1 2 . C3k0 p /; !n2 D p1 C3k0;
RkJQdWJsaXNoZXIy MTMzNzEzMQ==