Free vibration analysis of viscoelastic non-locally damped Rayleigh beam using Galerkin method, null space of the matrix & Neumann series expansion

While studying large-scale systems, non-local damping definition can beneficially model contact shear and long-range forces resulting from adjacent and non-adjacent elements in the set of interconnected dampers, damping patches and foundations which are modeled as non-local domains. If two or three dimensional systems are considered one dimensional (e.g. analyzing three dimensional beams based on Euler-Bernoulli, Rayleigh or Timoshenko Theories which simplifies the behavior of structures), the concept of non-local damping models arises to improve the accuracy of numerical results. Even though defining dissipative forces which are dependent on more parameters and quantities helpfully boost the validity of results compared to experimental cases in labs and three dimensional numerical analysis done by software, many researchers have widely employed viscous damping model to demonstrate damped behavior of the structures in the sake of simplicity. Actually viscous damping does not model accurately the dissipative behavior of real systems and practical structures often demonstrate some kind of viscoelasticity while vibration.  In the recent study, external damping force at any point in the domain is influenced by the past history of velocity and long-range interactions through convolution integral over proper kernel functions. As a consequence of applying Laplace transformation and using Galerkin method, the integro-partial differential equation of Rayleigh beam as a distributed parameter system turns to an ordinary differential equation governing a discrete system with finite degrees of freedom. Galerkin method is established based on error minimization of assumed mode method and despite Rayleigh and Rayleigh-Ritz methods can suitably analyze nonconservative systems including damped beams. Corresponding undamped mode shapes of Rayleigh beam which satisfy essential boundary conditions are chosen as the best admissible functions to expand the trial response of equation of motion. In order to get  continuous functions and finite weighted residual integral, the equation is presented in the weak form. Afterwards stiffness, damping and two types of mass matrices are determined with respect to generalized coordinates. To get scaled mode shapes the mass change method is considered to evaluate scaling factor, then the results are mass normalized. By equating dynamic stiffness matrix to zero and solving the resulting algebraic equation, complex and real eigenvalues are obtained which are respectively elastic and non-viscous modes in stable systems. It’s noteworthy to announce that contrasted with the viscously damped systems the degree of algebraic equation in the case of viscoelastic non-locally damped Rayleigh beam would generally be more than 2N (which N stands for total degrees of freedom). Accordingly eigenvectors are found based on Gaussian elimination method and null space of the dynamic stiffness matrix. Additionally, Neumann series expansion is developed to determine eigenvectors of vibrating systems effected by inertia. Finally, numerical results of Rayleigh and Euler-Bernoulli beams are compared. Very thin Rayleigh beams are verified and show great accuracy but when the thickness increases and inertial effect becomes bold, results of Rayleigh and Euler-Bernoulli beams are not matched anymore. Furthermore it’s understood from the graphs and tables that when the characteristic parameter of nonlocality and relaxation constant increase and tend to infinity, the nonlocal and non-viscous effects decline.