Non-Ideal System With Quadratic Nonlinearities Containing a Two-to-One Internal Resonance

2016 ◽  
Author(s):  
Rodrigo T. Rocha ◽  
Jose M. Balthazar ◽  
D. Dane Quinn ◽  
Angelo M. Tusset ◽  
Jorge L. P. Felix
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Coupled System ◽  
Dynamical Behaviour ◽  
Main System ◽  
Weakly Coupled System ◽  
Ideal System ◽  
Quadratic Nonlinearities

The dynamical behaviour of a non-ideal three-degrees-of-freedom weakly coupled system associated with the quadratic nonlinearities in the equations of motion is investigated. The main system consists of two nonlinear mechanical oscillators coupling with quadratic nonlinearities and in which possess a 2:1 internal resonance between their translational movements. Under these conditions, we analyzed the response when a DC unbalanced motor with limited power supply (non-ideal system) excites the main system. When the excitation frequency is near to second natural frequency of the main system, saturation and jump phenomena are presented. Then, this work will analyze some torques of the motor, which causes the phenomena, and due to high amplitudes of motion will be possible to look for a way to harvest energy in a future work.

NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS

2009 ◽  
Vol 19 (01) ◽  
pp. 225-243 ◽  
Author(s):  
D. X. CAO ◽  
W. ZHANG
Keyword(s):  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Coupled System ◽  
Dynamic Responses ◽  
Governing Equations ◽  
Chaotic Motions

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

A Fractional Derivative Model for Rubber Spring of Primary Suspension in Railway Vehicle Dynamics

2017 ◽  
Vol 3 (3) ◽  
Author(s):  
Dawei Zhang ◽  
Shengyang Zhu
Keyword(s):  
Fractional Derivative ◽  
Degrees Of Freedom ◽  
Excitation Frequency ◽  
Coupled System ◽  
Dynamic Responses ◽  
Viscous Force ◽  
Railway Vehicle ◽  
Spring Model ◽  
The Difference ◽  
Amplitude Dependency

This paper presents a nonlinear rubber spring model for the primary suspension of the railway vehicle, which can effectively describe the amplitude dependency and the frequency dependency of the rubber spring, by taking the elastic force, the fractional derivative viscous force, and nonlinear friction force into account. An improved two-dimensional vehicle–track coupled system is developed based on the nonlinear rubber spring model of the primary suspension. Nonlinear Hertz theory is used to couple the vehicle and track subsystems. The railway vehicle subsystem is regarded as a multibody system with ten degrees-of-freedom, and the track subsystem is treated as finite Euler–Bernoulli beams supported on a discrete–elastic foundation. Mechanical characteristic of the rubber spring due to harmonic excitations is analyzed to clarify the stiffness and damping dependencies on the excitation frequency and the displacement amplitude. Dynamic responses of the vehicle–track coupled dynamics system induced by the welded joint irregularity and random track irregularity have been performed to illustrate the difference between the Kelvin–Voigt model and the proposed model in the time and frequency domain.

Bifurcations in Dynamical Systems With Internal Resonance

1978 ◽  
Vol 45 (4) ◽  
pp. 895-902 ◽  
Author(s):  
P. R. Sethna ◽  
A. K. Bajaj
Keyword(s):  
Dynamical Systems ◽  
Internal Resonance ◽  
Excitation Frequency ◽  
The Other ◽  
Hopf Bifurcations ◽  
System Of Equations ◽  
Jump Phenomena ◽  
Quadratic Nonlinearities

Dynamical systems with quadratic nonlinearities and exhibiting internal resonance under periodic excitations are studied. Two types of transition from stable to unstable motions are shown to occur. One kind are shown to be associated with jump phenomena while the other kind are shown to be associated with Hopf bifurcations of the averaged system of equations. In the case of the latter, the motions are shown to be amplitude modulated motions at the excitation frequency with the amplitude of modulation determined by the motion of a point on a torus.

scholarly journals Decomposition of the Equations of Motion in the Analysis of Dynamics of a 3-DOF Nonideal System

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Roman Starosta ◽  
Grażyna Sypniewska-Kamińska
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Dynamical Behaviour ◽  
Engineering Systems ◽  
System Operation ◽  
Unbalanced Rotor ◽  
Nonideal System ◽  
Analysis Of Dynamics ◽  
Three Degrees Of Freedom

The dynamic response of a nonlinear system with three degrees of freedom, which is excited by nonideal excitation, is investigated. In the considered system the role of a nonideal source is played by a direct current motor, where the central axis of the rotor is not coincident with the axis of rotation. This translation generates a torque whose magnitude depends on the angular velocity. During the system operation a general coordinate assigned to the nonideal source grows rapidly as a result of rotation. We propose the decomposition of the equations of motion in such a way to extract the solution which is directly related to the rotation of an unbalanced rotor. The remaining part of the solution describes pure oscillation depending on the dynamical behaviour of the whole system. The decomposed equations are solved numerically. The influence of selected system parameters on the rotor vibration is examined. The presented approach can be applied to separate vibration and rotation of motions in many other engineering systems.

Dynamic Behavior of Damaged Bridge with Multi-Cracks Under Moving Vehicular Loads

2017 ◽  
Vol 17 (02) ◽  
pp. 1750019 ◽  
Author(s):  
Xinfeng Yin ◽  
Yang Liu ◽  
Lu Deng ◽  
Xuan Kong
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Coupled System ◽  
Interaction Force ◽  
Vehicle Model ◽  
Coupled Equations ◽  
Contact Points ◽  
Local Flexibility ◽  
Bridge Structures ◽  
Road Surface Roughness

When studying the vibration of a bridge–vehicle coupled system, most researchers mainly focus on the intact or original bridge structures. Nonetheless, a large number of bridges were built long ago, and most of them have suffered serious deterioration or damage due to the increasing traffic loads, environmental effect, material aging, and inadequate maintenance. Therefore, the effect of damage of bridges, such as cracks, on the vibration of vehicle–bridge coupled system should be studied. The objective of this study is to develop a new method for considering the effect of cracks and road surface roughness on the bridge response. Two vehicle models were introduced: a single-degree-of-freedom (SDOF) vehicle model and a full-scale vehicle model with seven degrees of freedom (DOFs). Three typical bridges were investigated herein, namely, a single-span uniform beam, a three-span stepped beam, and a non-uniform three-span continuous bridge. The massless rotational spring was adopted to describe the local flexibility induced by a crack on the bridge. The coupled equations for the bridge and vehicle were established by combining the equations of motion for both the bridge and vehicles using the displacement relationship and interaction force relationship at the contact points. The numerical results show that the proposed method can rationally simulate the vibrations of the bridge with cracks under moving vehicular loads.

Dynamic Stress Analysis of Exposed Pipes Subjected to Moving In-Line Inspection Tool

2020 ◽  
Author(s):  
Hamid Mostaghimi ◽  
Mohsen Hassani ◽  
Deli Yu ◽  
Ron Hugo ◽  
Simon Park
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Stress ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Beam Theory ◽  
Assessment Method ◽  
Integrity Assessment ◽  
Defect Assessment ◽  
Proposed Model ◽  
Assessment And Monitoring

Abstract In-line inspection is a non-destructive assessment method commonly used for defect assessment and monitoring of pipelines. The passage of an ILI tool through an excavated or exposed section of a pipe during an integrity assessment can excite vibrations and exert substantial forces, stress, and deflections on the pipe due to the weight and speed of the ILI tool. When the excitation frequency due to the ILI tool movement is close to the natural frequency of the overall structure, the dynamic stress generated within the pipe can be large enough to the extent that it imposes integrity concern on the line. This research aims to study effects of the ILI tool passage through floating and partially supported pipes under a variety of boundary and loading conditions. A finite element method is used to model the pipe with moving ILI tool. The model is developed based on Timoshenko beam theory with planar degrees-of-freedom and the differential equations of motion are solved numerically to predict displacement, strain, stress, and frequency responses of the pipe. The model is further validated using a lab-scale experimental setup. The comparison of the simulation to experimental results show how the proposed model is capable of predicting pipe dynamics, effectively.

Application of Generalized Newton-Euler Equations and Recursive Projection Methods to Dynamics of Deformable Multibody Systems

1989 ◽  
Author(s):  
R. A. Wehage ◽  
A. A. Shabana
Keyword(s):  
Euler Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Coupled System ◽  
Projection Methods ◽  
Open Loop ◽  
System Of Equations ◽  
Kinematic Chains ◽  
Deformable Bodies ◽  
Generalized Newton

Abstract A general symbolic-based method is presented for solving equations of motion for open-loop kinematic chains consisting of interconnected rigid and deformable bodies. The method utilizes matrix partitioning, recursive projection based on optimal block U-L factorization and generalized Newton-Euler equations to obtain an order n solution for the constrained equations of motion. Kinematic relationships between the absolute reference, joint and elastic coordinates are used with the generalized Newton-Euler equations for deformable bodies to obtain a large, loosely coupled system of equations. Taking advantage of the inertia matrix structure associated with elastic coordinates yields a recursive solution algorithm whose dimension is independent of the elastic degrees of freedom. The above solution techniques applied to this system of equations yield a much smaller operations count and can more effectively exploit vectorization and parallel processing. The algorithms presented in this paper are illustrated with the aid of cylindrical joints which are easily extended to revolute, prismatic, rigid and other joint types.

Non-Stationary Responses in Externally Excited Two-Degrees-of-Freedom Nonlinear Systems with 1: 2 Internal Resonance

2004 ◽  
Vol 10 (11) ◽  
pp. 1663-1697 ◽  
Author(s):  
Anil K. Bajaj ◽  
Patricia Davies ◽  
Bappaditya Banerjee
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Numerical Study ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Single Mode ◽  
Two Degrees Of Freedom ◽  
Analytical Technique ◽  
Bifurcation Points ◽  
Coupled Mode

The dynamics of two-degrees-of-freedom dynamical systems with weak quadratic nonlinearities is analyzed in the neighborhood of bifurcation points when the excitation frequency varies slowly through the region of primary resonance. The two modes of vibration are in 1: 2 subharmonic internal resonance. The slowly evolving averaged equations are numerically studied for motions initiated in the vicinity of stationary responses, and observations are made about the nature of responses of the system near the transition from single-mode to coupled-mode solutions (pitchfork points), and near jump and Hopf bifurcations in the coupled-mode solutions. An analytical technique based on the dynamic bifurcation theory is developed to explain the numerical observations for passage through the bifurcations. A numerical study is carried out to determine the effects of system parameters on the dynamics near the pitchfork bifurcation points and results are compared with analytical and numerical descriptions of dynamics.

Resonant Dynamics and Saturation in a Coupled System With Quadratic Nonlinearities

2004 ◽  
Author(s):  
Reddy Mankala ◽  
D. Dane Quinn
Keyword(s):  
Angular Velocity ◽  
Normal Modes ◽  
Coupled System ◽  
Angular Speed ◽  
Degree Of Freedom ◽  
Weakly Coupled System ◽  
Quadratic Nonlinearities ◽  
Resonant Dynamics ◽  
Three Degree Of Freedom ◽  
Dynamic Resonance

This work examines the behavior of a three-degree-of-freedom weakly coupled system. The system is composed of two components. The first is a two degree-of-freedom translational system that possesses an internal 2 : 1 resonance between the linear normal modes, which are coupled through quadratic nonlinearities. Under external forcing this component exhibits the saturation phenomena. The second is a rotational mass with a small imbalance, supported by the translational component. The angular speed of the rotor is not fixed, rather, the rotor is subject to a small torque and therefore its angular velocity slowly varies in time. A dynamic resonance occurs when the angular velocity of the rotor evolves to a neighborhood of one of the frequencies of the linear normal modes. Each of these resonances has been independently investigated previously in the literature. This work uncovers how the behavior of the dynamic resonance is modified by the mode coupling introduced by the 2 : 1 internal resonance and describes how the amplitudes of the linear normal modes are dependent on the properties of the dynamic resonance.

scholarly journals Modular Multibody Formulation for Simulating Off-Road Tracked Vehicles

2014 ◽  
Vol 1 (2) ◽  
pp. 77 ◽  
Author(s):  
Mohamed A Omar
Keyword(s):  
Degrees Of Freedom ◽  
Large Scale ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Dynamic Equations ◽  
Solution Stability ◽  
Simulation Code ◽  
Tracked Vehicles ◽  
Main System ◽  
6 Degrees Of Freedom

This paper presents a formulation and procedure for incorporating the multibody dynamics analysis capability of tracked vehicles in large-scale multibody system.  The proposed self-contained modular approach could be interfaced to any exiting multibody simulation code without need to alter the existing solver architecture.  Each track is modeled as a super-component that can be treated separate from the main system.  The super-component can be efficiently used in parallel processing environment to reduce the simulation time.  In the super-component, each track-link is modeled as separate body with full 6 degrees of freedom (DoF).  To improve the solution stability and efficiency, the joints between track links are modeled as complaint connection.  The spatial algebra operator is used to express the motion quantities and develop the link’s nonlinear kinematic and dynamic equations of motion.  The super-component interacts with the main system through contact forces between the track links and the driving sprocket, the support rollers and the idlers using self-contained force modules.  Also, the super-component models the interaction with the terrain through force module that is flexible to include different track-soil models, different terrain geometries, and different soil properties.  The interaction forces are expressed in the Cartesian system, applied to the link’s equation of motion and the corresponding bodies in the main system.  For sake of completeness, this paper presents dynamic equations of motion of the links as well as the main system formulated using joint coordinates approach.

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