By Ioannis T. Georgiou, Ira B. Schwartz (auth.), Alexander F. Vakakis (eds.)
The nonlinear common modes of a parametrically excited cantilever beam are developed through at once employing the tactic of a number of scales to the governing integral-partial differential equation and linked boundary stipulations. The impact of the inertia and curvature nonlin earities and the parametric excitation at the spatial distribution of the deflection is tested. the consequences are in comparison with these got by utilizing a single-mode discretization. within the absence of linear viscous and quadratic damping, it's proven that there are nonlinear basic modes, as outlined by means of Rosenberg, even within the presence of a relevant parametric excitation. in addition, the nonlinear mode form got with the direct method is in comparison with that got with the discretization process for a few values of the excitation frequency. within the single-mode discretization, the spatial distribution of the deflection is believed a priori to accept via the linear mode form ¢n, that is parametrically excited, as Equation (41). therefore, the mode form isn't prompted via the nonlinear curvature and nonlinear damping. however, within the direct strategy, the mode form isn't really assumed a priori; the nonlinear results regulate the linear mode form ¢n. for this reason, relating to large-amplitude oscillations, the single-mode discretization might yield erroneous mode shapes. References 1. Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. v., Pilipchuk, V. N., and Zevin A. A., Nonnal Modes and Localization in Nonlinear structures, Wiley, manhattan, 1996.
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Extra info for Normal Modes and Localization in Nonlinear Systems
3) In addition to the forces of constraint, the body B is assumed to be restrained by three sets of linear springs, attached to the center of mass of B, which respectively apply restoring forces in the x and y directions, and a restoring torque in the e direction, see Figure 1. These springs have net spring constants k, k, and Ke respectively. The identical x and y springs are assumed to have moveable anchors which permit smooth motion in a direction perpendicular to the length of the spring. This assumption keeps the restoring forces linear, avoiding the nonlinearities which would accompany fixed anchors.
Pontrjagin, L. , 'Asymptotic behavior of solutions of differential equations when the highcr derivatives contain a small parameter as a factor', /zvestija Akademia Nauk (Ser. ) 21,1975,605-621 [in Russian]. Tikbonov, A. , 'Systems of differential equations containing a small parameter with higher order derivatives', Matematicheskii Sbornik 31(73), 1952,575-584 [in Russian]. , 'Turbulence and the dynamics of coherent structures, pt. 1, Coherent structures', Quartely of Applied Mathematics 45, 1987,561-571.
1 1 1 2 2 2 2 2 -mi 2 + -m/ + -le 2 + -kx 2 + -kl + -Kee 2 = constant. 2 (13) Transforming to dimensionless variables (8), (9), and using (12) to eliminate e', Equation (13) can be written in the form: (3 - cos 2e)X,2 + (3 + cos 2e) y,2 - 2X' Y' sin 2e + 2X2 + 2y2 + 4Ke 2 = constant. (14) Equations (10), (11), (12) represent a flow on the five-dimensional phase space with coordinates X, Y, e, X', Y'. The energy manifold (14) represents an invariant codimension one surface in this phase space. Thus the system we are investigating may be thought of as a 2~-degrees-of-freedom nonlinear conservative dynamical system.