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

Piotr Skrzypacz; Anastasios Bountis; Daulet Nurakhmetov; Jong Kim

doi:10.1016/j.cnsns.2020.105230

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

Piotr Skrzypacz, Anastasios Bountis, Daulet Nurakhmetov, Jong Kim

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Article number	105230
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	85
DOIs	https://doi.org/10.1016/j.cnsns.2020.105230
Publication status	Published - Jun 2020

Keywords

Cantilever beam
Lumped mass model
Periodic solution and resonances
Swelling pressure

ASJC Scopus subject areas

Numerical Analysis
Modelling and Simulation
Applied Mathematics

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title = "Analysis of the lumped mass model for the cantilever beam subject to Grob's swelling pressure",

abstract = "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.",

keywords = "Cantilever beam, Lumped mass model, Periodic solution and resonances, Swelling pressure",

author = "Piotr Skrzypacz and Anastasios Bountis and Daulet Nurakhmetov and Jong Kim",

note = "Funding Information: This research was supported by the Horizon 2020 grant No 778360 . Dr. Piotr Skrzypacz is indebted to Prof. Dongming Wei from the Mathematics Department at the Nazarbayev University for fruitful discussions. ",

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AU - Skrzypacz, Piotr

AU - Bountis, Anastasios

AU - Nurakhmetov, Daulet

AU - Kim, Jong

N1 - Funding Information: This research was supported by the Horizon 2020 grant No 778360 . Dr. Piotr Skrzypacz is indebted to Prof. Dongming Wei from the Mathematics Department at the Nazarbayev University for fruitful discussions.

PY - 2020/6

Y1 - 2020/6

N2 - 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.

AB - 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.

KW - Cantilever beam

KW - Lumped mass model

KW - Periodic solution and resonances

KW - Swelling pressure

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

Abstract

Keywords

ASJC Scopus subject areas

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