scholarly journals Vibration Analysis of a Nonlinear System With Cyclic Symmetry

2010 ◽  
Vol 133 (2) ◽  
Author(s):  
Aurélien Grolet ◽  
Fabrice Thouverez
Keyword(s):  
Nonlinear Differential Equations ◽  
Geometrical Nonlinearity ◽  
Harmonic Balance Method ◽  
Dynamic Responses ◽  
Cyclic Symmetry ◽  
Force Amplitude ◽  
Closed Curves ◽  
Nonlinear Modes ◽  
Nonlinear Localization ◽  
Engine Order

This work is devoted to the study of nonlinear dynamics of structures with cyclic symmetry under geometrical nonlinearity using the harmonic balance method (HBM). In order to study the influence of the nonlinearity due to large deflection of blades, a simplified model has been developed. It leads to nonlinear differential equations of the second order, linearly coupled, in which the nonlinearity appears by cubic terms. Periodic solutions in both free and forced cases are sought by the HBM coupled with an arc length continuation and stability analysis. In this study, specific attention has been paid to the evaluation of nonlinear modes and to the influence of excitation on dynamic responses. Indeed, several cases of excitation have been analyzed: punctual one and tuned or detuned low engine order. This paper shows that for a localized, or sufficiently detuned, excitation, several solutions can coexist, some of them being represented by closed curves in the frequency-amplitude domain. Those different kinds of solution meet up when increasing the force amplitude, leading to forced nonlinear localization. As the closed curves are not tied with the basic nonlinear solution, they are easily missed. They were calculated using a sequential continuation with the force amplitude as a parameter.

scholarly journals Vibration Analysis of a Nonlinear System With Cyclic Symmetry

2010 ◽  
Author(s):  
Aure´lien Grolet ◽  
Fabrice Thouverez
Keyword(s):  
Nonlinear Differential Equations ◽  
Geometrical Nonlinearity ◽  
Harmonic Balance Method ◽  
Dynamic Responses ◽  
Cyclic Symmetry ◽  
Force Amplitude ◽  
Closed Curves ◽  
Nonlinear Modes ◽  
Nonlinear Localization ◽  
Engine Order

This work is devoted to the study of non linear dynamics of structures with cyclic symmetry under geometrical nonlinearity using the harmonic balance method (HBM). In order to study the influence of the non-linearity due to large deflection of blades a simplified model has been developed. It leads to nonlinear differential equations of the second order, linearly coupled, in which the nonlinearity appears by cubic terms. Periodic solutions in both free and forced cases are sought by the HBM coupled with an arc length continuation and stability analysis. In this study, a specific attention has been paid to the evaluation of nonlinear modes and to the influence of excitation on dynamic responses. Indeed, several cases of excitation have been analyzed: punctual one and tuned or detuned low engine order. The paper shows that for a localized, or sufficiently detuned, excitation, several solutions can coexist, some of them being represented by closed curves in the Frequency-Amplitude domain. Those different kinds of solution meet up when increasing the force amplitude, leading to forced nonlinear localization. As the closed curves are not tied with the basic nonlinear solution they are easily missed. They were calculated using a sequential continuation with the force amplitude as a parameter.

Dynamics of the Gear Train Set in Wind Turbine

2011 ◽  
Vol 697-698 ◽  
pp. 701-705
Author(s):  
D.D. Ji ◽  
Y.M. Song ◽  
J. Zhang
Keyword(s):  
Wind Turbine ◽  
Dynamic Model ◽  
Harmonic Balance Method ◽  
Planetary Gear ◽  
Gear Train ◽  
Dynamic Responses ◽  
Multi Stage ◽  
Lumped Parameter ◽  
Cartesian Coordinate ◽  
The Impact

A lumped-parameter dynamic model for gear train set in wind turbine is proposed to investigate the dynamics of the speed-increasing gear box. The proposed model is developed in a universal Cartesian coordinate, which includes transversal and torsional deflections of each component, time-varying mesh stiffness, gear profile errors and external excitations. By solving the dynamic model, a modal analysis is performed. The results indicate that the modal properties of the multi-stage gear train in wind turbine are similar to those of a single-stage planetary gear set. A harmonic balance method (HBM) is used to obtain the dynamic responses of the gearing system. The responses give insight into the impact of excitations on the vibrations.

scholarly journals Friction Damper Optimisation: Simulation of Rainbow Tests

1999 ◽  
Author(s):  
Kenan Y. Sanliturk ◽  
David J. Ewins ◽  
Robert Elliott ◽  
Jeff S. Green
Keyword(s):  
Harmonic Balance Method ◽  
Lumped Parameter Model ◽  
Friction Damper ◽  
Blade Vibration ◽  
Friction Dampers ◽  
Lumped Parameter ◽  
Engine Order ◽  
Resonance Response ◽  
Theoretical Predictions ◽  
Different Characteristics

Friction dampers have been used to reduce turbine blade vibration levels for a considerable period of time. However, optimal design of these dampers has been quite difficult due both to a lack of adequate theoretical predictions and to difficulties in conducting reliable experiments. One of the difficulties of damper weight optimisation via the experimental route has been the inevitable effects of mistuning. Also, conducting separate experiments for different damper weights involves excessive cost. Therefore, current practice in the turbomachinery industry has been to conduct so-called ‘rainbow tests’ where friction dampers with different weights are placed between blades with a predefined configuration. However, it has been observed that some rainbow test results have been difficult to interpret and have been inconclusive for determining the optimum damper weight for a given bladed-disc assembly. A new method of analysis — a combination of Harmonic Balance Method and structural modification approaches — is presented in this paper for the analysis of structures with friction interfaces and the method is applied to search for qualitative answers about the so-called ‘rainbow tests’ in turbomachinery applications. A simple lumped-parameter model of a bladed-disc model was used and different damper weights were modelled using friction elements with different characteristics. Resonance response levels were obtained for bladed discs with various numbers of blades under various engine-order excitations. It was found that rainbow tests, where friction dampers with different weights are used on the same bladed-disc assembly, can be used to find the optimum damper weight if the mode of vibration concerned has weak blade-to-blade coupling (the case where the disc is almost rigid and blades vibrate almost independently from each other). Otherwise, it is very difficult to draw any reliable conclusion from such expensive experiments.

scholarly journals Nonlinear Dynamic Analysis for Rectangular FGM Plates with Variable Thickness Subjected to Mechanical Load

2019 ◽  
Vol 35 (3) ◽  
Author(s):  
Khuc Van Phu ◽  
Le Xuan Doan ◽  
Nguyen Van Thanh
Keyword(s):  
Variable Thickness ◽  
Nonlinear Dynamic ◽  
Volume Fraction ◽  
Mechanical Load ◽  
Plate Theory ◽  
Geometrical Nonlinearity ◽  
Nonlinear Dynamic Analysis ◽  
Dynamic Responses ◽  
Rectangular Plates ◽  
Classical Plate Theory

 In this paper, the governing equations of rectangular plates with variable thickness subjected to mechanical load are established by using the classical plate theory, the geometrical nonlinearity in von Karman-Donnell sense. Solutions of the problem are derived according to Galerkin method. Nonlinear dynamic responses, critical dynamic loads are obtained by using Runge-Kutta method and the Budiansky–Roth criterion. Effect of volume-fraction index k and some geometric factors are considered and presented in numerical results.

scholarly journals Rattle dynamics of noncircular face gear under multifrequency parametric excitation

2021 ◽  
Vol 12 (1) ◽  
pp. 361-373
Author(s):  
Dawei Liu ◽  
Zhenzhen Lv ◽  
Guohao Zhao
Keyword(s):  
Dynamic Model ◽  
Parametric Excitation ◽  
Nonlinear Dynamic ◽  
Velocity Ratio ◽  
Harmonic Balance Method ◽  
Transmission Error ◽  
Dynamic Responses ◽  
Discrete Fourier Transformation ◽  
Nonlinear Dynamic Model ◽  
Face Gear

Abstract. A noncircular face gear (NFG) conjugated with a pinion is a new type of face gear which can transmit variable velocity ratio and in which two time-varying excitations exist, namely the meshing stiffness excitation and instantaneous center excitation. Considering the tooth backlash, static transmission error and multifrequency parametric excitation, a nonlinear dynamic model of the NFG pair is presented. Based on the harmonic balance method and discrete Fourier transformation, a semi-analytic approach for the nonlinear dynamic model is given to analyze the dynamic behaviors of the NFG. Results demonstrate that, with increase in the eccentric ratio, input velocity and error amplitude, the NFG will undergo a non-rattle, unilateral rattle and bilateral rattle state in succession, and a jump phenomenon will appear in the dynamic responses when the rattle state of the gears is transformed from unilateral rattle to bilateral rattle.

Nonlinear dynamic response analysis on gear-rotor-bearing transmission system

2016 ◽  
Vol 24 (9) ◽  
pp. 1632-1651 ◽  
Author(s):  
Shihua Zhou ◽  
Guiqiu Song ◽  
Mengnan Sun ◽  
Zhaohui Ren
Keyword(s):  
Friction Coefficient ◽  
Friction Force ◽  
Nonlinear Dynamic ◽  
Nonlinear Differential Equations ◽  
Transmission System ◽  
Dynamic Responses ◽  
Response Analysis ◽  
Light Load ◽  
Rotor Bearing ◽  
The Mean

A coupled lateral-torsional nonlinear dynamic model with 16-degree-of-freedom (16-DOF) of gear-rotor-bearing transmission system (GRBTS) is developed after comprehensive considering the nonlinear features associated with time-varying meshing stiffness, backlash, transmission error, friction force, input/output load, gravity and gear eccentricity. Based on the nonlinear differential equations, the coupled multi-body dynamic responses of the GRBTS are demonstrated using the Runge-Kutta numerical method, and the effects of friction coefficient and mean load on the dynamic characteristics are investigated. The results show that the friction force could enlarge the vibration amplitude and affect the low frequency components seriously. The mean load excitation has a complicated influence on the coupled GRBTS, and the torsional vibration is the dominate response. Whereas the mean load excitation has a certain extent vibration suppression, and light load and heavy load could no longer effectively control the nonlinear vibration of the GRBTS. With the increasing of rotational speed, the friction coefficient and mean load ranges of the chaotic behavior widen and the chaotic characteristics strengthens. It is shown that small parameter random perturbation might be propagated in the vibration system and lead to relatively large vibration of the system. The contribution to provide a reference for the design and study of gear system.

Analysis of the Non-Linear Dynamic Responses of an Elastic Rotor With a Slant Crack

2012 ◽  
Author(s):  
Xiangmin Zhang ◽  
Changping Chen ◽  
Liming Dai
Keyword(s):  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Rotor System ◽  
Dynamic Responses ◽  
Cracked Rotor ◽  
Lagrange Equations ◽  
Linear Dynamic ◽  
Slant Crack ◽  
Non Linear ◽  
Line Spring Model

Considering a rotor system with a slant crack, and using an equivalent line-spring model to simulate the slant crack of the rotor, the flexibility model of the slant-cracked rotor is derived. Then considered the geometric non-linearity and based on the Lagrange equations, the non-linear dimensionless differential equations of motion for the slant-cracked rotor are obtained. Further the non-linear dynamic responses of the single rotor system with a slant crack are discussed by the Galerkin method and the harmonic balance method. It’s detailedly studied that the angle, the depth and the position of the slant crack on the rotor all affect on the non-linear dynamic responses of the rotor system, and the conclusion is very significant to the design of the rotor system in the practical reference aspect.

Nonlinear Vibration Signature Analysis of a High Speed Rotor Bearing System Due to Race Imperfection

2011 ◽  
Vol 7 (1) ◽  
Author(s):  
P. K. Kankar ◽  
Satish C. Sharma ◽  
S. P. Harsha
Keyword(s):  
High Speed ◽  
Nonlinear Differential Equations ◽  
Mathematical Formulation ◽  
Nonlinear Damping ◽  
Dynamic Responses ◽  
Fast Fourier Transformation ◽  
Surface Waviness ◽  
Contact Points ◽  
Rotor Bearing ◽  
Bearing System

In this paper the nonlinear dynamic responses of a rigid rotor supported by ball bearings due to surface waviness of bearing races are analyzed. A mathematical formulation has been derived with consideration of the nonlinear springs and nonlinear damping at the contact points of rolling elements and races, whose stiffnesses are obtained by using Hertzian elastic contact deformation theory. The numerical integration technique Newmark-β with the Newton–Raphson method is used to solve the nonlinear differential equations, iteratively. The effect of bearing running surface waviness on the nonlinear vibrations of rotor bearing system is investigated. The results are mainly presented in time and frequency domains are shown in time-displacement, fast Fourier transformation, and Poincaré maps. The results predict discrete spectrum with specific frequency components for each order of waviness at the inner and outer races, also the excited frequency and waviness order relationships have been set up to prognosis the race defect on these bearing components. Numerical results obtained from the simulation are validated with respect to those of prior researchers.

A comparison between the friction and piezoelectric synchronized switch dampers for blisks

2018 ◽  
Vol 29 (12) ◽  
pp. 2693-2705 ◽  
Author(s):  
Jiuzhou Liu ◽  
Lin Li ◽  
Yu Fan
Keyword(s):  
Piezoelectric Materials ◽  
Vibration Reduction ◽  
Harmonic Balance Method ◽  
Single Mode ◽  
Excitation Level ◽  
State Response ◽  
Lumped Parameter ◽  
Engine Order ◽  
Lumped Parameter Models ◽  
Promising Solution

In this work, we explore the feasibility of applying the synchronized switch damping on negative capacitor damper to integrally bladed disks (blisks) by comparing its damping performance with that of the friction ring with the same added mass. Both dampers are coupled to the disk of the blisk. Namely, the friction ring is in contact with the underside of the wheel and the piezoelectric materials for synchronized switch damping on negative capacitor are bonded on the disk as well. The lumped parameter models are used for the blisk and dampers, and the multi-harmonic balance method is employed to obtain the steady-state response under engine-order excitations. The vibration-reduction performance of the dampers are compared with respect to the excitation level, the engine order, and multi-mode. This leads to a discussion concerning the parameter design for the synchronized switch damping on negative capacitor damper to achieve a comparable single-mode damping as the friction ring. We show that the synchronized switch damping on negative capacitor damper has a better performance for multiple modes and at a wider range of excitation level. These results indicate that the synchronized switch damping on negative capacitor damping can be a promising solution for the vibration reduction of blisks where the spatial and spectral distributions of the excitation are rich.

The Effects of Gravity on the Response of Centrifugal Pendulum Vibration Absorbers

2021 ◽  
pp. 1-61
Author(s):  
Darryl Tchokogoue ◽  
Ming Mu ◽  
Brian F. Feeny ◽  
Bruce K. Geist ◽  
Steven W. Shaw
Keyword(s):  
Equations Of Motion ◽  
Operating Conditions ◽  
Vibration Absorbers ◽  
Cyclic Symmetry ◽  
Passive Devices ◽  
Engine Order ◽  
Gravity Effects ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
Automotive Powertrain ◽  
Linearized Model

Abstract This paper describes the effects of gravity on the response of systems of identical, cyclically arranged, centrifugal pendulum vibration absorbers (CPVAs). CPVAs are passive devices composed of movable masses suspended on a rotor, suspended such that they reduce torsional vibrations at a given engine order. These absorbers are becoming prevalent in automotive powertrain components in order to expand fuel-efficient engine operating conditions. Gravitational effects acting on the absorbers can be important for a horizontal rotor/CPVA system spinning at relatively low rotation speeds, for example, during engine idle conditions. The main goal of this investigation is to predict the response of a CPVA/rotor system in the presence of gravity. A linearized model which includes the effects of gravity and an order n torque acting on the rotor is analyzed by exploiting the cyclic symmetry of the system. The results show that the N absorbers respond in one or more groups, where the absorbers in each group respond with identical waveforms but shifted phases. The number of groups depends on the engine order n and the ratio Nn. It is shown that there are special resonant effects if the engine order is n = 1 or n = 2, the latter of which is particularly important in applications. In addition, it is shown that for N > 1 the rotor response is not affected by gravity, due to the symmetry of the gravity effects. The analytical predictions are verified by direct simulations of the equations of motion.

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