Response variations of a cantilever beam–tip mass system with nonlinear and linearized boundary conditions

2018 ◽  
Vol 25 (3) ◽  
pp. 485-496 ◽  
Author(s):  
Vamsi C. Meesala ◽  
Muhammad R. Hajj
Keyword(s):  
Boundary Conditions ◽  
Cantilever Beam ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Boundary Conditions ◽  
Distributed Parameter ◽  
Governing Equations ◽  
Mass System ◽  
Tip Mass

The distributed parameter governing equations of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using a generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. The discretized governing equation considering the nonlinear boundary conditions assumes a simpler form. We solve the distributed parameter and discretized equations separately using the method of multiple scales. Through comparison with the direct approach, we show that accounting for the nonlinear boundary conditions boundary conditions is important for accurate prediction in terms of type of bifurcation and response amplitude.

The Influence of Nonlinear Boundary Conditions on the Nonplanar Autoparametric Responses of an Inextensible Beam

2001 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Springs ◽  
Out Of Plane

Abstract The nonplanar responses of a beam clamped at one end and restrained by nonlinear springs at the other end is investigated under a primary resonance base excitation. The beam’s geometry and the springs’ linear stiffnesses are such that the system possesses a one-to-one autoparametric resonance between the nth in-plane and out-of-plane modes. The beam is modeled using Euler-Bernoulli theory and includes cubic geometric and inertia nonlinearities. The objective is to assess the influence of the nonlinear boundary conditions on the beam’s oscillations. To this end, the method of multiple scales is directly applied to the integral-partial-differential equations of motion and associated boundary conditions. The result is a set of four nonlinear ordinary-differential equations that govern the slow dynamics of the system. Solutions of these modulation equations are then used to characterize the system’s nonlinear behavior.

scholarly journals Analysis of a Nonlinear ψ-Hilfer Fractional Integro-Differential Equation Describing Cantilever Beam Model with Nonlinear Boundary Conditions

2021 ◽  
Vol 5 (4) ◽  
pp. 177
Author(s):  
Kanoktip Kotsamran ◽  
Weerawat Sudsutad ◽  
Chatthai Thaiprayoon ◽  
Jutarat Kongson ◽  
Jehad Alzabut
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Conditions ◽  
Cantilever Beam ◽  
Point Theorem ◽  
Nonlinear Boundary ◽  
Sufficient Conditions ◽  
Nonlinear Boundary Conditions ◽  
Beam Model ◽  
Uniqueness Of Solutions

In this paper, we establish sufficient conditions to approve the existence and uniqueness of solutions of a nonlinear implicit ψ-Hilfer fractional boundary value problem of the cantilever beam model with nonlinear boundary conditions. By using Banach’s fixed point theorem, the uniqueness result is proved. Meanwhile, the existence result is obtained by applying the fixed point theorem of Schaefer. Apart from this, we utilize the arguments related to the nonlinear functional analysis technique to analyze a variety of Ulam’s stability of the proposed problem. Finally, three numerical examples are presented to indicate the effectiveness of our results.

Identification of Parameters in Nonlinear Boundary Conditions of Distributed Systems With Linear Fields

1973 ◽  
Vol 95 (4) ◽  
pp. 390-395 ◽  
Author(s):  
E. D. Ward ◽  
R. E. Goodson
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Simulated Data ◽  
Nonlinear Boundary Conditions ◽  
Distributed Parameter ◽  
Unknown Boundary ◽  
Unknown Parameters ◽  
Radiation Boundary ◽  
Unknown Boundary Conditions ◽  
Time Invariant

A method is presented for the formulation of an experimental procedure for the identification of unknown parameters in nonlinear boundary conditions in distributed parameter dynamic systems. In contrast to other available techniques, this method requires only as many measurement sensors within the field as there are unknown boundary conditions. Results are presented for simulated data from an example of heat conduction with a radiation boundary and for experimental data from a cantilever beam with a nonlinear moment at the boundary. The method may be applied to partial differential equations which are linear, one-dimensional, and have known time invariant coefficients. The nonlinear boundary conditions are specified up to a set of unknown constant parameters which appear linearly in the boundary conditions.

Nonlinear performances of an autoparametric vibration-based piezoelastic energy harvester

2016 ◽  
Vol 28 (2) ◽  
pp. 254-271 ◽  
Author(s):  
Zhimiao Yan ◽  
Muhammad R Hajj
Keyword(s):  
Cantilever Beam ◽  
Energy Harvester ◽  
Multiple Scales ◽  
Damping Ratio ◽  
Load Resistance ◽  
Governing Equations ◽  
Near Resonance ◽  
Base Displacement ◽  
Tip Mass ◽  
Analytical Expressions

Nonlinear characterizations of an autoparametric vibration-based energy harvester are investigated. The harvester consists of a base structure subjected to an external excitation and a cantilever beam with a tip mass. Two piezoelectric sheets bounded to both sides of the cantilever beam are used to harvest the energy. The governing equations accounting for the coupled effects of the base vibration, the response of the cantilever beam and the generated power are derived. Approximate analysis of the simplified governing equations is then performed by the method of multiple scales. The usefulness of this approach is demonstrated by deriving analytical expressions for the global frequency and damping ratio of the cantilever beam. Their dependence on the electrical load resistance is quantified. Analytical expressions for the amplitudes of the base displacement and the displacement of the tip mass are derived. An expression that relates the output power to the load resistance, global damping, and displacement of the tip mass is derived. The effects of the external force and electric load resistance on the nonlinear responses of the system are determined. The results show different responses for different operational electric loads. The broadening of the excitation regime over which energy can be harvested is analyzed. The effects of the load resistance on the types of bifurcations near resonance are determined.

Ordinary Differential Equations with Nonlinear Boundary Conditions

2002 ◽  
Vol 9 (2) ◽  
pp. 287-294
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Lower And Upper Solutions ◽  
Iterative Technique ◽  
Monotone Iterative

Abstract The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems.

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