Exact Nonlinear Dynamic Analysis of a Beam With a Nonlinear Vibration Absorber and With Various Boundary Conditions

Mohammad Bukhari and Oumar Barry (November 11, 2019). "Exact Nonlinear Dynamic Analysis of a Beam With a Nonlinear Vibration Absorber and With Various Boundary Conditions." ASME. J. Comput. Nonlinear Dynam. January 2020; 15(1): 011003. https://doi.org/10.1115/1.4045287

Beams are the basic component of many engineering applications. They are used in bridges, overhead transmission lines, pipelines, sensors, aircraft structures, and many others. To ensure safety and proper function, vibrations of beams need to be investigated for better prediction of the system dynamical response. When the vibration amplitude is small, linear theory can predict the response accurately. However, when the vibration amplitude becomes larger, nonlinearity must be considered to avoid erroneous results. This work investigates the nonlinear vibration of a beam with attached Nonlinear Vibration Absorber (NVA) consisting of a spring-mass system). The considered nonlinearity stems from mid-plane stretching due to immovable boundary conditions and from the nonlinear stiffness in the NVA. In addition, different types of immovable boundary are investigated. For weak nonlinearity, an approximate analytical solution is derived using the method of multiple scales. These analytical results are validated using direct numerical integration. Parametric studies demonstrate that the performance of the NVA does not only depend on its key design variables and location, but also on the beam boundary conditions, midplane stretching of the beam, and NVA configuration (i.e., grounded versus ungrounded). Our analysis also indicates that the common approach of employing approximate modes in estimating the nonlinear response of a loaded beam produces significant error, up to 1200% in some cases. These findings could contribute to the design improvement of NVAs, microelectromechanical systems (MEMS), energy harvesters, and metastructures. 

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