Wave-based analysis of jointed elastic bars: nonlinear periodic response

Abstract

In this paper we develop two wave-based approaches for predicting the nonlinear periodic response of jointed elastic bars. First, we present a nonlinear wave-based vibration approach (WBVA) for studying jointed systems informed by re-usable, perturbation-derived scattering functions. This analytical approach can be used to predict the steady-state, forced response of jointed elastic bar structures incorporating any number and variety of nonlinear joints. As a second method, we present a nonlinear Plane-Wave Expansion (PWE) approach for analyzing periodic response in the same jointed bar structures. Both wave-based approaches have advantages and disadvantages when compared side-by-side. The WBVA results in a minimal set of equations and is re-usable following determination of the reflection and transmission functions, while the PWE formulation can be easily applied to new joint models and maintains solution accuracy to higher levels of nonlinearity. For example cases of two and three bars connected by linearly-damped joints with linear and cubic stiffness, the two wave-based approaches accurately predict the expected Duffing-like behavior in which multiple periodic responses occur in the near-resonant regime, in close agreement with reference finite element simulations. Lastly, we discuss extensions of the work to jointed structures composed of beam-like members, and propose follow-on studies addressing opportunities identified in the application of the methods presented.

Vibration Analysis of Locally Resonant Beams with L-Joint Using an Exact Wave-Based Vibration Approach

“…So far, researchers have proposed the exact wave-based vibration approach for analyzing beams and better simplifying the vibration analysis of such structures [ 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 ]. Based on the influence of discontinues and planar frame structure, especially on bending and longitudinal vibration, the further interpretation concerning the raised approach is shown by Mei [ 29 , 30 ], which greatly simplifies analysis of the vibration properties of planar frames.…”

“…The above studies illustrated the high efficiency of the wave-based vibration approach and made the contribution for further studies. Through incorporating a set of reusable equations as developed by Balaji et al [ 38 ], the above approach is still usable for analyzing the vibration of nonlinear structures. Furthermore, the wave-based vibration approach can relatively simplify vibration-based damage detection methods, similar to making an objective function minimized by an optimization algorithm during an iterative procedure in Refs.…”

Vibrations and energy distribution in inhomogeneous rods with elastic and viscous boundary conditions

“…Rods can be described by various continuum models as well [13][14][15][16][17][18][19]. Santo et al [20] investigated the harmonic response of vibrating homogeneous rods with nonlinear elastic boundary conditions.…”

Wave Propagation in Shear Beams Comprising Finite Periodic Lumped Masses and Resting on Elastic Foundation

“…In addition, this approach has been used to investigate wave propagation in 1D and two-dimensional (2D) periodic structures to find natural frequencies and mode shapes [32,33]. The wave-based transfer matrix method has also shown wide applicability in other studies to analyze the dynamic response of large mesh structures [34] to solve the nonlinear vibration problem [35] and to provide necessary vibration controls in the design and analysis of light beams [36][37][38].…”