Dynamic Behavior Analysis of an Axially Loaded Beam Supported by a Nonlinear Spring-Mass System

2021 ◽  
pp. 2150152
Author(s):  
Yuhao Zhao ◽  
Jingtao Du
Keyword(s):  
Dynamic Behavior ◽  
Stable Steady State ◽  
Bernoulli Beam ◽  
Lumped Mass ◽  
Mass System ◽  
Nonlinear Spring ◽  
Periodic State ◽  
Galerkin Truncation ◽  
Axially Loaded ◽  
Euler Bernoulli Beam

Dynamic analysis of an Euler–Bernoulli beam with nonlinear supports is receiving greater research interest in recent years. Current studies usually consider the boundary and internal nonlinear supports separately, and the system rotational restraint is usually ignored. However, there is little study considering the simultaneous existence of axial load, lumped mass and internal supports for such nonlinear problem. Motivated by this limitation, the dynamic behavior of an axially loaded beam supported by a nonlinear spring-mass system is solved and investigated in this paper. Modal functions of an axially loaded Euler–Bernoulli beam with linear elastic supports are taken as trail functions in Galerkin discretization of the nonlinear governing differential equation. Stable steady-state response of such axially loaded beam supported by a nonlinear spring-mass system is solved via Galerkin truncation method, which is also validated by finite difference method. Results show that parameters of nonlinear spring-mass system and boundary condition have a significant influence on system dynamic behavior. Moreover, appropriate nonlinear parameters can switch the system behavior between the single-periodic state and quasi-periodic state effectively.

Vibration of Beam with Elastically Restrained Ends and Rotational Spring-Lumped Rotary Inertia System at Mid-Span

2015 ◽  
Vol 15 (02) ◽  
pp. 1450040 ◽  
Author(s):  
Seyed Mojtaba Hozhabrossadati ◽  
Ahmad Aftabi Sani ◽  
Masood Mofid
Keyword(s):  
Technical Note ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Bernoulli Beam ◽  
Mass System ◽  
Rotational Spring ◽  
Beam System ◽  
Sample Frequency ◽  
Elastically Restrained ◽  
Euler Bernoulli Beam

This technical note addresses the free vibration problem of an elastically restrained Euler–Bernoulli beam with rotational spring-lumped rotary inertia system at its mid-span hinge. The governing differential equations and the boundary conditions of the beam are presented. Special attention is directed toward the conditions of the intermediate spring-mass system which plays a key role in the solution. Sample frequency parameters of the beam system are solved and tabulated. Mode shapes of the beam are also plotted for some spring stiffnesses.

A Wave-Based Analytical Solution to Free Vibrations in a Combined Euler–Bernoulli Beam/Frame and a Two-Degree-of-Freedom Spring–Mass System

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
C. Mei
Keyword(s):  
Analytical Solution ◽  
Coupling Effect ◽  
Free Vibrations ◽  
Degree Of Freedom ◽  
Transverse Motion ◽  
Bernoulli Beam ◽  
Combined System ◽  
Mass System ◽  
Euler Bernoulli Beam ◽  
Two Degree Of Freedom

In this paper, natural frequencies and modeshapes of a transversely vibrating Euler–Bernoulli beam carrying a discrete two-degree-of-freedom (2DOF) spring–mass system are obtained from a wave vibration point of view in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external forces applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam and 2DOF spring–mass system, the vibrating discrete spring–mass system injects waves into the distributed beam through the spring forces at the two spring attached points. Assembling these wave relations in the beam provides an analytical solution to vibrations of the combined system. Accuracy of the proposed wave analysis approach is validated through comparisons to available results. This wave-based approach is further extended to analyze vibrations in a planar portal frame that carries a discrete 2DOF spring–mass system, where in addition to the transverse motion, the axial motion must be included due to the coupling effect at the angled joint of the frame. The wave vibration approach is seen to provide a systematic and concise technique for solving vibration problems in combined distributed and discrete systems.

scholarly journals Nonlinear Dynamic Behavior of a Flexible Structure to Combined External Acoustic and Parametric Excitation

2006 ◽  
Vol 13 (4-5) ◽  
pp. 233-254 ◽  
Author(s):  
Paulo S. Varoto ◽  
Demian G. Silva
Keyword(s):  
Dynamic Response ◽  
Dynamic Behavior ◽  
Parametric Excitation ◽  
Accurate Determination ◽  
Longitudinal Axis ◽  
Equation Of Motion ◽  
Driving Mechanism ◽  
Lumped Mass ◽  
Mass System ◽  
Tip Mass

Flexible structures are frequently subjected to multiple inputs when in the field environment. The accurate determination of the system dynamic response to multiple inputs depends on how much information is available from the excitation sources that act on the system under study. Detailed information include, but are not restricted to appropriate characterization of the excitation sources in terms of their variation in time and in space for the case of distributed loads. Another important aspect related to the excitation sources is how inputs of different nature contribute to the measured dynamic response. A particular and important driving mechanism that can occur in practical situations is the parametric resonance. Another important input that occurs frequently in practice is related to acoustic pressure distributions that is a distributed type of loading. In this paper, detailed theoretical and experimental investigations on the dynamic response of a flexible cantilever beam carrying a tip mass to simultaneously applied external acoustic and parametric excitation signals have been performed. A mathematical model for transverse nonlinear vibration is obtained by employing Lagrange’s equations where important nonlinear effects such as the beam’s curvature and quadratic viscous damping are accounted for in the equation of motion. The beam is driven by two excitation sources, a sinusoidal motion applied to the beam’s fixed end and parallel to its longitudinal axis and a distributed sinusoidal acoustic load applied orthogonally to the beam’s longitudinal axis. The major goal here is to investigate theoretically as well as experimentally the dynamic behavior of the beam-lumped mass system under the action of these two excitation sources. Results from an extensive experimental work show how these two excitation sources interacts for various testing conditions. These experimental results are validated through numerically simulated results obtained from the solution of the system’s nonlinear equation of motion.

Imposing a Node at a Desired Location Along a Beam Under Harmonic Base Excitation: Theory and Experiment

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
J. H. Porter ◽  
P. S. Harvey
Keyword(s):  
Excitation Frequency ◽  
Vibration Absorber ◽  
Vibration Absorbers ◽  
Base Excitation ◽  
Bernoulli Beam ◽  
Structural Vibrations ◽  
Mass System ◽  
Arbitrary Boundary ◽  
Node Location ◽  
Euler Bernoulli Beam

Vibration absorbers are commonly used to reduce unwanted structural vibrations. In this paper, a vibration absorber composed of a sprung mass is used to enforce a location of zero displacement (or node) at a specified location on a Euler–Bernoulli beam under harmonic base excitation. Closed-form expressions for the optimal tuning of the auxiliary spring-mass system are found, and the results are presented for the cases of the attachment and node located at the same and different locations. The assumed modes method is used, so the results can be applied for arbitrary boundary conditions. To aid in the design process, this paper also characterizes the sensitivity of the displacement at the desired node location to parametric variations. Sensitivities are considered with respect to the base excitation frequency and the attachment mass, stiffness, and location. The sensitivities of the system highlight some feasible but less desirable attachment locations. Numerical and experimental results for a cantilever beam are presented to illustrate the proposed method and the effects of mistuning.

The assessment of soil depth sensitivity to dynamic behavior of the Euler-Bernoulli beam under accelerated moving load

2020 ◽  
Vol 6 (2) ◽  
pp. 84
Author(s):  
Amin Ghannadiasl ◽  
Hasan Rezaei Dolagh
Keyword(s):  
Dynamic Behavior ◽  
Dynamic Equilibrium ◽  
Soil Depth ◽  
Moving Load ◽  
Vertical Direction ◽  
Finite Depth ◽  
Bernoulli Beam ◽  
Transform Method ◽  
Beam Displacement ◽  
Euler Bernoulli Beam

Dynamic behavior is one of the most crucial characters in the railways structures. One of the items which leads to precise identification of the dynamic behavior of railways is the soil depth beneath them. In this paper, an Euler-Bernoulli beam on a finite depth foundation under accelerated moving load is presented. The dynamic equilibrium in the vertical direction is only regarded in accordance with the factor of finite beams. In this study, the dynamic equilibrium of the soil in the vertical direction and the sensitivity of soil depth are considered. The governing equations are simulated by using Fourier transform method. Eventually, by considering the sequences of shear waves, and different kinds of damping, displacement of the beam is obtained for the various acceleration, times and soil depth. As a result, it is shown that, higher acceleration is not dramatically effective on the beam displacement. Also, foundation inertia causes a significant reduction in critical velocity and can augment the beam response.

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