Analysis of the lumped mass model for the cantilever beam subject to Grob’s swelling pressure

• Published in: Communications in nonlinear science & numerical simulation Vol. 85; p. 105230
• Main Authors: Skrzypacz, Piotr, Bountis, Anastasios, Nurakhmetov, Daulet, Kim, Jong
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.06.2020; Elsevier Science Ltd
• Subjects: Boundary value problems; Cantilever beam; Cantilever beams; Dynamical systems; Euler-Bernoulli beams; Finite difference method; Galerkin method; Lumped mass model; Numerical analysis; Periodic solution and resonances; Pressure; Quadratures; Reduced order models; Studies; Swelling pressure; Periodic solution and resonances; 74L10; 37N15; Swelling pressure; 65M60; Lumped mass model; Cantilever beam
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2020.105230

•Cantilever beam.•Swelling pressure.•Lumped mass model.•Periodic solution and Resonances. The lumped mass model is derived from a one-mode Galerkin discretization with the Gauss–Lobatto quadrature applied to the non-linear swelling pressure term. Our reduced-order model of the problem is then analyzed to study the essential dynamics of an elastic cantilever Euler–Bernoulli beam subject to the swelling pressure described by Grob’s law. The solutions to the initial value problem for the resulting nonlinear ODE are proved to be always periodic. The numerical solution to the derived lumped mass model satisfactorily matches the finite difference solution of the dynamic beam problem. Including the effect of oscillations at the base of the beam, we show that the model exhibits resonances that may crucially influence its dynamical behavior.