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.

Effect of Static Deflection on the Natural Frequency of a System with a Hardening Non-Linear Spring

1967 ◽  
Vol 9 (3) ◽  
pp. 177-181 ◽  
Author(s):  
Z. F. Reif
Keyword(s):  
Natural Frequency ◽  
Averaging Method ◽  
Analogue Computer ◽  
Static Deflection ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
First Approximation ◽  
Non Linear ◽  
Linear Spring

The effect of the static deflection on the natural frequency response is investigated for a single-degree-of-freedom system with a positive cubic non-linearity in the restoring force. The Ritz averaging method is used for the theoretical analysis, and by comparison with analogue computer solutions the first approximation, consisting of two terms, is shown to be satisfactory for most applications. Results indicate that in systems with a non-linear restoring force the existence of a static deflection produces significant effects and cannot be neglected in a theorectical analysis.

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.

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.

scholarly journals Higher order stochastic averaging method for mechanical systems having two degrees of freedom

2004 ◽  
Vol 26 (2) ◽  
pp. 103-110
Author(s):  
Nguyen Duc Tinh
Keyword(s):  
Degrees Of Freedom ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
Degree Of Freedom ◽  
Approximation Solution ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Two Degree Of Freedom

Higher order stochastic averaging method is widely used for investigating single-degree-of-freedom nonlinear systems subjected to white and coloured random noises.In this paper the method is further developed for two-degree-of-freedom systems. An application to a system with cubic damping is considered and the second approximation solution to the Fokker-Planck (FP) equation is obtained.

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