Dynamics of a Simple Model for a Wind-Loaded Nonlinear Structure: Bifurcations of Codimension One and Two

1998 ◽  
Vol 65 (2) ◽  
pp. 505-512 ◽  
Author(s):  
K. Yagasaki
Keyword(s):  
Vortex Shedding ◽  
Degree Of Freedom ◽  
Force Characteristic ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Nonlinear Restoring Force ◽  
Codimension One ◽  
Theoretical Predictions ◽  
The Cross

The motion induced by vortex shedding of a structure with nonlinear restoring force is investigated. In particular, a conclusion about nonexistence of bounded motions obtained for a similar problem in the previous study is improved by taking into account the nonlinear restoring force characteristic. The vortex shedding frequency is assumed to be close to the natural frequency of the cross-wind oscillation and the along-wind oscillation is not excited, so that a single-degree-of-freedom model representing the cross-wind motions is obtained. The averaging method is applied to the single-degree-of-freedom system, and the normal form and center manifold theories are used to discuss bifurcations of codimension one, saddle-node and Hopf bifurcations. Moreover, it is shown that a multiple bifurcation of codimension two, called the Bogdanov-Takens bifurcation, occurs in the averaged system. The implications of the averaging results on the dynamics of the original single-degree-of-freedom system are described. Numerical examples are also given with numerical simulation results for both the averaged and original systems to demonstrate our theoretical predictions.

Ultraharmonic Resonance of a System with an Asymmetrical Restoring Force Characteristic

1969 ◽  
Vol 11 (6) ◽  
pp. 592-597 ◽  
Author(s):  
W. Carnegie ◽  
Z. F. Reif
Keyword(s):  
Theoretical Analysis ◽  
Averaging Method ◽  
Degree Of Freedom ◽  
Force Characteristic ◽  
Analogue Computer ◽  
Static Deflection ◽  
Restoring Force ◽  
Disturbing Force ◽  
Single Degree Of Freedom ◽  
Single Degree

The ultraharmonic resonance of order 2, excited by a centrifugal type disturbing force, is investigated for a single-degree-of-freedom system with a Duffing restoring force characteristic. The effect of gravity is taken into account. The resulting asymmetry of the restoring force is expressed in terms of the static deflection parameter. The Ritz averaging method is used for the theoretical analysis and the results are verified by means of an analogue computer.

On the Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems

1959 ◽  
Vol 26 (3) ◽  
pp. 377-385
Author(s):  
R. M. Rosenberg ◽  
C. P. Atkinson
Keyword(s):  
Free Vibrations ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Natural Modes ◽  
Theoretical Predictions ◽  
Stability Chart ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

Abstract The natural modes of free vibrations of a symmetrical two-degree-of-freedom system are analyzed theoretically and experimentally. This system has two natural modes, one in-phase and the other out-of-phase. In contradistinction to the comparable single-degree-of-freedom system where the free vibrations are always orbitally stable, the natural modes of the symmetrical two-degree-of-freedom system are frequently unstable. The stability properties depend on two parameters and are easily deduced from a stability chart. For sufficiently small amplitudes both modes are, in general, stable. When the coupling spring is linear, both modes are always stable at all amplitudes. For other conditions, either mode may become unstable at certain amplitudes. In particular, if there is a single value of frequency and amplitude at which the system can vibrate in either mode, the out-of-phase mode experiences a change of stability. The experimental investigation has generally confirmed the theoretical predictions.

Single Degree-of-Freedom Modeling of the Nonlinear Vibration Response of a Machining Robot

2020 ◽  
Vol 143 (5) ◽  
Author(s):  
Yaser Mohammadi ◽  
Keivan Ahmadi
Keyword(s):  
Control Strategies ◽  
Vibration Response ◽  
Industrial Robots ◽  
Degree Of Freedom ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Machining Forces ◽  
Sdof System ◽  
Single Degree ◽  
Machining Robot

Abstract Highly dynamic machining forces can cause excessive and unstable vibrations when industrial robots are used to perform high-force operations such as milling and drilling. Implementing appropriate optimization and control strategies to suppress vibrations during robotic machining requires accurate models of the robot’s vibration response to the machining forces generated at its tool center point (TCP). The existing models of machining vibrations assume the linearity of the structural dynamics of the robotic arm. This assumption, considering the inherent nonlinearities in the robot’s revolute joints, may cause considerable inaccuracies in predicting the extent and stability of vibrations during the process. In this article, a single degree-of-freedom (SDOF) system with the nonlinear restoring force is used to model the vibration response of a KUKA machining robot at its TCP (i.e., machining tool-tip). The experimental identification of the restoring force shows that its damping and stiffness components can be approximated using cubic models. Subsequently, the higher-order frequency response functions (HFRFs) of the SDOF system are estimated experimentally, and the parameters of the SDOF system are identified by curve fitting the resulting HFRFs. The accuracy of the presented SDOF modeling approach in capturing the nonlinearity of the TCP vibration response is verified experimentally. It is shown that the identified models accurately predict the variation of the receptance of the nonlinear system in the vicinity of well-separated peaks, but nonlinear coupling around closely spaced peaks may cause inaccuracies in the prediction of system dynamics.

Classification of Periodic Solutions in a Single-Degree-of-Freedom System With Backlash

2007 ◽  
Author(s):  
Bart Besselink ◽  
Amit Shukla ◽  
Rob Fey ◽  
Henk Nijmeijer
Keyword(s):  
Piecewise Linear ◽  
Continuation Method ◽  
Degree Of Freedom ◽  
Multiple Shooting ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Boundary Crossings ◽  
Single Degree ◽  
Piecewise Linear System

In this paper a single degree-of-freedom system with backlash is studied for its periodic response. This system is modeled as a piecewise linear system with discontinuity in the net restoring force, caused by additional damping in the contact-zone. The periodic orbits are classified by their number of subspace boundary crossings and Floquet multipliers. For this classification, the known analytical solutions in the different subspaces are used in the multiple shooting algorithm and a continuation method. Some observations are also presented about the qualitative features (such as symmetry, rigid body solutions) exhibited by this class of systems.

Dynamic analysis of a ball screw feed system with time-varying and piecewise-nonlinear stiffness

2019 ◽  
Vol 233 (18) ◽  
pp. 6503-6518 ◽  
Author(s):  
Jianguo Gu ◽  
Yimin Zhang
Keyword(s):  
Dynamic Characteristics ◽  
Ball Screw ◽  
Time Varying ◽  
Feed System ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Nonlinear Restoring Force ◽  
Abrupt Changes ◽  
Screw Feed

In this study, a single-degree-of-freedom model is established to investigate the dynamic characteristics of a single-nut double-cycle ball screw feed system by considering the contact states of the nonlinear kinematic joints. Based on fully considering the parameters of the ball screw feed system, the axial deformations and forces of the key components are calculated to construct a set of piecewise-nonlinear restoring force functions of the system displacement and worktable position. The variations of the contact stiffnesses of the kinematic joints and transmission stiffness of the system with different boundary conditions are analyzed and the results indicate that they all have abrupt changes when the system displacement reaches a critical value. The changing law of the system transmission stiffness in the whole stoke is discussed. Additionally, the effects of excitation force, worktable position and system mass on the dynamic characteristics of the system and its correlative components are analyzed.

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