Modal testing of nonlinear vibrating structures based on nonlinear normal modes: Experimental demonstration

Realizing that nonlinearity is a frequent occurrence in engineering structures and that linear experimental modal analysis (EMA) is of limited usefulness in this context, the present paper is an attempt to develop nonlinear EMA by targeting the extraction of nonlinear normal modes (NNMs) from time series of nonlinear mechanical systems. Based on a nonlinear extension of phase resonance testing, the proposed methodology excites the structure to isolate a single NNM during the experiments. Thanks to the invariance principle, the energy dependence of that nonlinear mode (i.e., the NNM modal curves and their oscillation frequencies) can be extracted from the resulting free decay response using time-frequency analysis. This paper is devoted to the experimental demonstration and robustness of this procedure. To this end, an experimental cantilever beam with a geometrical nonlinearity is considered, and the ability of the proposed methodology to extract its NNMs from the measured responses is assessed.     

高速旋转状态下的齿轮非线性模态分析

建立齿轮在高速旋转状态下的三维有限元分析模型,模拟某履带车辆传动系统齿轮旋转过程的实际工况,进行非线性模态分析,精确计算其在高速旋转时的应力-应变特性,得到齿轮非线性低阶固有频率和主振型,并通过与静止状态下分析得到的固有频率进行对比分析,所得结果既反映了动力学性能,又为系统的进一步动力学修改、噪声控制以及优化设计提供了有力的依据.     

On the distribution of nonlinear effects in locally nonlinear one-dimensional chain type structures

Nonlinear effect is a typical phenomenon for nonlinear systems, which is characterized as the energy of the system input transferred from the frequency modes in the input to the output frequency locations, which are either below or above the input frequency band. The present work is devoted to investigating the distribution of nonlinear effects in one-dimensional chain type structures with local nonlinearity in which only one or few components are of nonlinear property. The results reveal that the positions of both the nonlinear components and the external input play crucial roles in determining the distribution of nonlinear effects in the locally nonlinear structures, and provide a new insight into the complex dynamic of nonlinear systems in frequency domain. The results are of great significance to the analysis, design and control of the mechanical systems whose few components are of nonlinear property.     

A study on the nonlinear normal mode vibration using adelphic integral

Nonlinear normal mode (NNM) vibration, in a nonlinear dual mass Hamiltonian system, which has 6^th order homogeneous polynomial as a nonlinear term, is studied in this paper. The existence, bifurcation, and the orbital stability of periodic motions are to be studied in the phase space. In order to find the analytic expression of the invariant curves in the Poincare Map, which is a mapping of a phase trajectory onto 2 dimensional surface in 4 dimensional phase space, Whittaker’s Adelphic Integral, instead of the direct integration of the equations of motion or the Birkhoff-Gustavson (B-G) canonical transformation, is derived for small value of energy. It is revealed that the integral of motion by Adelphic Integral is essentially consistent with the one obtained from the B-G transformation method. The resulting expression of the invariant curves can be used for analyzing the behavior of NNM vibration in the Poincare Map.     

On the computation of the slow dynamics of nonlinear modes of mechanical systems

A novel method for the numerical prediction of the slowly varying dynamics of nonlinear mechanical systems has been developed. The method is restricted to the regime of an isolated nonlinear mode and consists of a two-step procedure: In the first step, a multiharmonic analysis of the autonomous system is performed to directly compute the amplitude-dependent characteristics of the considered nonlinear mode. In the second step, these modal properties are used to construct a two-dimensional reduced order model (ROM) that facilitates the efficient computation of steady-state and unsteady dynamics provided that nonlinear modal interactions are absent.The proposed methodology is applied to several nonlinear mechanical systems ranging form single degree-of-freedom to Finite Element models. Unsteady vibration phenomena such as approaching behavior towards an equilibrium point or limit cycles, and resonance passages are studied regarding the effect of various nonlinearities such as cubic springs, unilateral contact and friction. It is found that the proposed ROM facilitates very fast and accurate analysis of the slow dynamics of nonlinear systems. Moreover, the ROM concept offers a huge parameter space including additional linear damping, stiffness and near-resonant forcing.     

On modeling and vibration of gear drives influenced by nonlinear couplings

Gear drives are one of the most common parts in many rotating machinery. When the gear drive runs under lower torque load, nonlinear effects like gear mesh interruption can occur and vibration can be accompanied by impact motions of the gears. This paper presents an original method of the mathematical modeling of gear drive nonlinear vibrations using modal synthesis method with degrees of freedom number reduction. The model respects nonlinearities caused by gear mesh interruption, parametric gearing excitation caused by time-varying meshing stiffness and nonlinear contact forces acting between journals of the rolling-element bearings and the outer housing. The nonlinear model is then used for investigation of gear drive vibration, especially for detection of nonlinear phenomena like impact motions, bifurcation of solution and chaotic motions in case of small static load and in resonant states. The theoretical method is used for investigation of two-stage gearbox nonlinear vibration.     

Time-frequency characterization of nonlinear normal modes and challenges in nonlinearity identification of dynamical systems

Presented here is a new time-frequency signal processing methodology based on Hilbert-Huang transform (HHT) and a new conjugate-pair decomposition (CPD) method for characterization of nonlinear normal modes and parametric identification of nonlinear multiple-degree-of-freedom dynamical systems. Different from short-time Fourier transform and wavelet transform, HHT uses the apparent time scales revealed by the signal's local maxima and minima to sequentially sift components of different time scales. Because HHT does not use pre-determined basis functions and function orthogonality for component extraction, it provides more accurate time-varying amplitudes and frequencies of extracted components for accurate estimation of system characteristics and nonlinearities. CPD uses adaptive local harmonics and function orthogonality to extract and track time-localized nonlinearity-distorted harmonics without the end effect that destroys the accuracy of HHT at the two data ends. For parametric identification, the method only needs to process one steady-state response (a free undamped modal vibration or a steady-state response to a harmonic excitation) and uses amplitude-dependent dynamic characteristics derived from perturbation analysis to determine the type and order of nonlinearity and system parameters. A nonlinear two-degree-of-freedom system is used to illustrate the concepts and characterization of nonlinear normal modes, vibration localization, and nonlinear modal coupling. Numerical simulations show that the proposed method can provide accurate time-frequency characterization of nonlinear normal modes and parametric identification of nonlinear dynamical systems. Moreover, results show that nonlinear modal coupling makes it impossible to decompose a general nonlinear response of a highly nonlinear system into nonlinear normal modes even if nonlinear normal modes exist in the system.     

Dissipated energy and boundary condition effects associated to dry friction on the dynamics of vibrating structures

In turbomachines, dry friction devices (under platform dampers, shrouds, and tie-wire) are usually introduced to reduce resonant responses of bladed disks. Dry friction between rubbing elements induces a highly nonlinear dynamic behaviour which flattens the frequency response functions. It is clear that such behaviour requires an optimisation process to find the optimum parameters that lead to the minimum forced response amplitudes. However, different interpretations still remain concerning the explanation of the physical origin of this type of flattening. The most common one is based on dissipated energy. In this case, heat resulting from the relative frictional motion between contacting surfaces is supposed to bring sufficient dissipation to flatten response functions. On the other hand, a different approach considers that a decrease in vibrational amplitudes is explained by changes in boundary conditions induced by a stick/slip behaviour. In this study, a single degree-of-freedom system is used and analysed both in time and in frequency domains (Harmonic Balance Method) in order to show the contribution of respectively energy dissipation and change of contact state on peak levels.     

On the lateral instability of a thin beam under periodic bending loads

The lateral instability of a thin beam under periodic bending loads was investigated. Physical evidences of the instability were observed previously by experiments. But an analytical study has not been reported. The object of this study is to demonstrate the nature and existence of dynamic lateral instability. The harmonic balance method is applied to bifurcation modes which result from the stability change of torsional mode of a beam and then compared with numerical simulations. It is found, in a certain frequency range, that a small bending load results in the lateral instability when damping is small. Inha University     

Mixed-type finite element formulation of higher order shear deformation theory for the linear and nonlinear analyses of a laminated composite plate

For the linear and nonlinear analyses of a laminated composite plate structure, the mixed type finite element program is developed on the basis of higher order shear deformation theory of laminated plates. The accuracy of this program is checked by means of comparing with the existing results for laminated rectangular plates and is found to agree well with them. Deformations and interlaminar stresses of laminated plates are calculated according to the variation of layer numbers, fiber orientations, and plate thicknesses, so that the shear and nonlinear effects on their behaviors are studied. It is found that plate deformations are reduced by means of arranging the fiber direction into the angle-ply and increasing layer numbers.