Bifurcations at Combination Resonance and Quasiperiodic Vibrations of Flexible Beams

The nonlinear dynamic behavior of flexible beams is described by nonlinear partial differential equations. The beam model accounts for the tension of the neutral axis under vibrations. The Bubnov–Galerkin method is used to derive a system of ordinary differential equations. The system is solved by the multiple-scale method. A system of modulation equations is analyzed

where cos Ω is the periodic transverse force, ρ is the density of the material, E is the elastic modulus, A and J are the area and moment of inertia of the cross section, and δ( ) is the delta function.
Introduce the following dimensionless parameters and variables: where ε << 1and r is the radius of inertia of the cross section. Omitting the asterisks in (2), we rearrange (1) in terms of the new variables: where f x t f x t ( , ) ( / ) ( ). = − 0 1 3 δ cos Ω Represent the vibrations in the form W t nx ). sin Applying the Bubnov-Galerkin method to (3), we obtain the following system of ordinary differential equations: Consider a combination resonance: 2 1 2 Ω = + + ω ω εσ. To analyze Eqs. (4), we will apply the multiple-scale method [6] based on the following change of variables: As a result, we obtain the system of modulation equations γ σ π π χ π χ π ′− + + + − +     5 16   The transition from (4) to (6) through the change of variables (5) is based on rigorous mathematical considerations. This procedure is fully described in [6]. The stability of the periodic motions of (6) corresponds to the stability of the periodic motions of (4). System (6) can be written in terms of the variables ( , , ) ( , , ) x y z a a a = 2 2 1 cos sin γ γ : To analyze the nonlinear vibrations of flexible beams, we will represent the solution of Eq.

Bifurcations of Fixed Points.
It is easy to verify that the dynamic system (6) The solutions of Eqs. (10) have the form where p f Equality (11) defines two groups of fixed points, denoted by the superscripts A and B. These points correspond to two groups of equilibrium states of the dynamic system (8): Let the above fixed points be associated with an amplitude surface (see Fig. 1) that represents the dependence a a f 1 1 0 This surface consists of joined sheets marked by A, B, and C in Fig. 1. The sheet C corresponds to the fixed point (a 1 , a 2 ) = (0, 0); and the sheets A and B represent fixed points marked by the same letters.
To analyze the fixed points for stability, let us determine the eigenvalues λ i of the Jacobian matrix of the vector field (8). The stability of the fixed points r A B , is defined by quantities λ i A B ( , ) as follows: , , where R ap To describe the stability of the fixed point (a 1 , a 2 ) = (0, 0), we will use the quantities Note that the nonhyperbolic fixed points of the sheet C satisfy the equation λ 2 = 0. Therefore, we have To analyze the dynamics of system (8) at C 1 , we will take advantage of the central-manifold method [3,14], which allows reducing the dimension of a dynamic system. Note that it is central manifolds on which bifurcation phenomena occur. In the neighborhood of the fixed point x = y = z = 0, Eqs. (10) can be represented as where ν π µ χ The central manifold of the dynamic system (18) can be represented as power series with undetermined coefficients. After determining these coefficients by the method described in [14], we represent the central manifold in the form The motions of the dynamic system (8) on the central manifold are described by the equation where a O a O It follows from this equation that the point C 1 is stable. Consider saddle-node bifurcation L 1 (Fig. 1) where 10 2 1 π δ δ = .
Rearranged in terms of the new variables, system (8)