scholarly journals Nonlinear Vibration Characteristics of a Flexible Blade with Friction Damping due to Tip-Rub

2011 ◽  
Vol 18 (1-2) ◽  
pp. 105-114 ◽  
Author(s):  
Dengqing Cao ◽  
Xiaochun Gong ◽  
Dong Wei ◽  
Shiming Chu ◽  
Ligang Wang
Keyword(s):  
Nonlinear Vibration ◽  
Piecewise Linear ◽  
Equation Of Motion ◽  
Vibration Characteristics ◽  
Vibration Characteristic ◽  
Linear Vibration ◽  
Friction Damping ◽  
The Galerkin Method ◽  
Three Degree Of Freedom ◽  
Frequency Curves

An approximate approach is proposed in this paper for analyzing the two-dimensional friction contact problem so as to compute the dynamic response of a structure constrained by friction interfaces due to tip-rub. The dynamical equation of motion for a rotational cantilever blade in a centrifugal force field is established. Flow-induced distributed periodic forces and the internal material damping in the blade are accounted for in the governing equation of motion. The Galerkin method is employed to obtain a three-degree-of-freedom oscillator with friction damping due to tip-rub. The combined motion of impact and friction due to tip-rub produced a piecewise linear vibration which is actually nonlinear. Thus, a complete vibration cycle is divided into successive intervals. The system possesses linear vibration characteristic during each of these intervals, which can be determined using analytical solution forms. Numerical simulation shows that the parameters such as gap of the tip and the rotational speed of the blades have significant effects on the dynamical responses of the system. Finally, the nonlinear vibration characteristics of the blade are investigated in terms of the Poincare graph, and the frequency spectrum of the responses and the amplitude-frequency curves.

Mathematical Modeling and Analysis of Vibration Characteristics of Smart-Phone

2013 ◽  
Vol 284-287 ◽  
pp. 800-805
Author(s):  
Suk Min Yoon ◽  
Jae Hee Kim ◽  
Jin Ho Kim
Keyword(s):  
Mathematical Modeling ◽  
Mathematical Analysis ◽  
Smart Phone ◽  
Vibration Characteristics ◽  
Linear Actuator ◽  
Vibration Characteristic ◽  
Linear Vibration ◽  
Modeling And Analysis ◽  
Key Technologies

Slimming of a smart phone is one of the key technologies in the current smart-phone industry. Because the vibration actuator is the thickest part of a smart phone, a slim linear vibration actuator has been developed. For the analysis of the vibration characteristic of a smart phone, experiment has been used one of the most popular techniques, but it consumes a lot of cost and time. Accordingly, as an alternative to experiments, mathematical modeling and analysis can be applied. In this paper, a mathematical modeling is created for analysis of the vibration characteristics of a smart-phone applied with a horizontally vibrating linear actuator. In addition, the result of the mathematical analysis is compared with that of the experiment for verification.

Transient Resonance Passage With Respect to Friction

2012 ◽  
Author(s):  
Marius Bonhage ◽  
Lars Panning-von Scheidt ◽  
Jörg Wallaschek ◽  
Christoph Richter
Keyword(s):  
Power Plants ◽  
Piecewise Linear ◽  
Linear Equations ◽  
Time Integration ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Friction Contact ◽  
Friction Damping ◽  
Tangential Stiffness ◽  
Multi Body

Gas and steam turbine applications are exposed to high vibration amplitudes. Friction damping elements are commonly used to prevent blades from fatigue failure. Most of the work done so far in this field is dealing with steady state vibration occurring due to harmonic excitation. Regarding the growing importance of renewable energy, the number of run-ups and rundowns in power plants is increasing continuously. Furthermore, also the speed of jet engines is changed regularly during operation. To give an analytical approach on how to calculate and create most effective friction damping during instationary excitation, this paper discusses the computation of transient nonlinear vibration. Following the description of a single-degree-of-freedom system comprising a friction contact, it is shown how to assemble the equation of motion using piecewise linear equations. An analytical ansatz is illustrated to solve the ODEs of the system. In the next step, two simple multi-body-systems are created that are coupled by a friction contact. To validate the analytical computation of the vibration response of the system, a numeric time integration of the system is done. The calculations are compared, and it is shown that the analytical ansatz is valid to compute the vibration of the system. Using the solution of the equation of motion of the multi-body-system, it is shown how the optimal friction damping of the system can be determined while varying important parameters like e.g. the normal force of the friction contact and the tangential stiffness of the contact model.

Refined Modeling and Free Vibration of Inextensional Beams on the Elastic Foundation

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Lianhua Wang ◽  
Jianjun Ma ◽  
Yueyu Zhao ◽  
Qijian Liu
Keyword(s):  
Nonlinear Equation ◽  
Free Vibration ◽  
Nonlinear Vibration ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Equation Of Motion ◽  
Rotary Inertia ◽  
Second Law ◽  
Modal Interaction ◽  
Linear Vibration

In this study, the nonlinear equation of motion of the beam on the elastic foundation is obtained via the Newton's second law of motion, and its free vibration nature is investigated. Considering the inextensional condition, the planar model of the beam accounting for the effects of the rotary inertia is derived. Then, the linear vibration and nonlinear vibration of the beam on the elastic foundation are examined. It is shown that the cut-off frequency can be observed in the frequency spectrum of the beam response. The effects of the rotary inertia on the natural frequencies are systematically investigated. Finally, the frequency differences, due to the different foundation models, and the possible modal interaction of the beam are discussed.

Research on Vibration Characteristics of the Pedestal and Equipment Systems with Variable Parameter

2012 ◽  
Vol 479-481 ◽  
pp. 1273-1276
Author(s):  
Hai Long Chen ◽  
Zhong Hui Qiu ◽  
Fu Zhen Pang ◽  
Lv Zheng
Keyword(s):  
Differential Equation ◽  
Forced Vibration ◽  
Piecewise Linear ◽  
Variable Parameter ◽  
Vibration Characteristics ◽  
Exciting Force ◽  
Vibration Response ◽  
Single Frequency ◽  
Linear Vibration ◽  
Vibration Problem

According to the relation between exciting force and the pretightening force, the parameter of pedestal and equipment systems can be variable or invariable. The differential equation for forced vibration problem of the system has been established. The condition whether piecewise linear vibration will happen has been found. The result shows that the system will appear multi-frequency vibration response on the condition of single-frequency turbulence force. The results in this paper can be applied to the installation of the equipment.

Numerical Analysis of the Tapping Atomic Force Microscopy on a Viscoelastic Sample

2014 ◽  
Author(s):  
Ben Carmichael ◽  
Gary Frey ◽  
S. Nima Mahmoodi
Keyword(s):  
Numerical Analysis ◽  
Time History ◽  
Equation Of Motion ◽  
Viscous Effects ◽  
Force Microscopy ◽  
Forcing Function ◽  
Atomic Force ◽  
Cantilevered Beam ◽  
Soft Samples ◽  
The Galerkin Method

Mechanical characterization of thin samples is now routine due to the prominence of the Atomic Force Microscope. Advances in amplitude modulation techniques have allowed for accurate measurement of a sample’s elastic properties by interpreting the changes in the vibration of a cantilevered beam in intermittent contact. However, the nonlinearities associated with contact complicate attempts to find an accurate time-history for the beam. Furthermore, the inclusion of viscous effects, common to soft samples, puts an explicit solution even farther from reach. A numerical method is proposed that analyzes the time-history and frequency response of a microcantilever beam with a viscoelastic end-condition. The mathematics can be simplified by incorporating the viscoelastic end-condition into the equation of motion directly by modeling it as a distributed load. A forcing function can then be derived from the Standard Linear Solid model of viscoelasticity and implemented in the non-conservative work term of Hamilton’s principle. The Galerkin method can separate the resulting nonlinear equation of motion into time and space components. Performing a numerical analysis of the time factor equation provide the beam’s response over time. The results demonstrate the distinctive effects of viscoelasticity and periodic contact on the beam’s motion and provide the framework for the determination of viscous properties using dynamic techniques.

ASYMPTOTIC NUMERICAL METHOD FOR THE DYNAMIC STUDY OF NONLINEAR VIBRATION ABSORBERS

2014 ◽  
Vol 06 (05) ◽  
pp. 1450053 ◽  
Author(s):  
FATHI DJEMAL ◽  
FAKHER CHAARI ◽  
JEAN LUC DION ◽  
FRANCK RENAUD ◽  
IMAD TAWFIQ ◽  
...  
Keyword(s):  
Nonlinear Equation ◽  
Vibration Control ◽  
Numerical Method ◽  
Nonlinear Vibration ◽  
Degrees Of Freedom ◽  
Equation Of Motion ◽  
Vibration Absorbers ◽  
Asymptotic Numerical Method ◽  
Dynamic Absorber ◽  
Newton Raphson

Vibrations are usually undesired phenomena as they may cause discomfort, disturbance, damage, and sometimes destruction of machines and structures. It must be reduced or controlled or eliminated. One of the most common methods of vibration control is the use of the dynamic absorber. The paper is interested in the study of a nonlinear two degrees of freedom (DOF) model. To solve nonlinear equation of motion a high order implicit algorithm is proposed. It is based on the introduction of a homotopy, an implicit scheme of Newmark and the use of techniques of Asymptotic Numerical method (ANM). We propose also a regularization of the contact force to overcome the difficulty of the singularity in this model. A comparison will be presented between the results obtained by the proposed algorithm and those using the classical Newton–Raphson and Newmark time scheme.

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