This paper presents a theoretical study on the influence of a discrete element in the nonlinear dynamics of a continuous mechanical system subject to randomness in the model parameters. This system is composed by an elastic bar, attached to springs and a lumped mass, with a random elastic modulus and subjected to a Gaussian white-noise distributed external force. One can note that the dynamic behavior of the bar is significantly altered when the lumped mass is varied, becoming, on the right extreme and for large values of the concentrated mass, similar to a mass-spring system. It is also observed that the system response is more influenced by the randomness for small values of the lumped mass. The study conducted also show an irregular distribution of energy through the spectrum of frequencies, asymmetries and multimodal behavior in the probability distributions of the lumped mass velocity.
Cunha, A., & Sampaio, R. (2015). On the nonlinear stochastic dynamics of a continuous system with discrete attached elements. Applied Mathematical Modelling, 39(2), 809–819. https://doi.org/10.1016/j.apm.2014.07.012