Prediction of Periodic Response of Flexible Mechanical Systems With Nonlinear Characteristics

1990 ◽  
Vol 112 (4) ◽  
pp. 501-507 ◽  
Author(s):  
Ting-Nung Shiau ◽  
An-Nan Jean
Keyword(s):  
Nonlinear Differential Equations ◽  
System Response ◽  
Algebraic Equations ◽  
Nonlinear Characteristics ◽  
Periodic Response ◽  
Rotor Systems ◽  
Nonlinear Algebraic Equations ◽  
Numerical Analytical Method ◽  
Harmonic Components ◽  
Nonlinear Components

A numerical-analytical method for the prediction of steady state periodic response of large order nonlinear rotordynamic systems is addressed. Using this method, the set of nonlinear differential equations governing the motion of the rotor systems is transformed to a set of nonlinear algebraic equations. A condensation technique is proposed to reduce the nonlinear algebraic equations to those only related to the physical coordinates associated with nonlinear components. The method allows for the inclusion of searching for sub, super, ultra-sub and ultra-super harmonic components of the system response. Furthermore it can be used to locate limit cycles of an autonomous system. Three examples are employed to demonstrate the accuracy and the efficiency of the present method.

Prediction of Periodic Response of Rotor Dynamic Systems With Nonlinear Supports

1995 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Dynamic Systems ◽  
Nonlinear Differential Equations ◽  
Squeeze Film ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Periodic Response ◽  
Nonlinear Algebraic Equations ◽  
Nodal Displacements ◽  
The Time Domain ◽  
Rotor Dynamic

Abstract A numerical-analytical method for estimating steady-state periodic behavior of nonlinear rotordynamic systems is presented. Based on a finite element formulation in the time domain, this method transforms the nonlinear differential equations governing the motion of large rotor dynamic systems with nonlinear supports into a set of nonlinear algebraic equations with unknown temporal nodal displacements. A procedure is proposed to reduce the resulting problem to solving nonlinear algebraic equations in terms of the coordinates associated with the nonlinear supports only. The result is a simple and efficient approach for predicting all possible fundamental and sub-harmonic responses. Stability of the periodic response is readily determined by a direct use of Floquet’s theory. The feasibility and advantages of the proposed method are illustrated with two examples of rotor-bearing systems of deadband supports and squeeze film dampers, respectively.

Prediction of Periodic Response of Rotor Dynamic Systems With Nonlinear Supports

1997 ◽  
Vol 119 (3) ◽  
pp. 346-353 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Dynamic Systems ◽  
Nonlinear Differential Equations ◽  
Squeeze Film ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Periodic Response ◽  
Nonlinear Algebraic Equations ◽  
Nodal Displacements ◽  
The Time Domain ◽  
Rotor Dynamic

A numerical-analytical method for estimating steady-state periodic behavior of nonlinear rotordynamic systems is presented. Based on a finite element formulation in the time domain, this method transforms the nonlinear differential equations governing the motion of large rotor dynamic systems with nonlinear supports into a set of nonlinear algebraic equations with unknown temporal nodal displacements. A procedure is proposed to reduce the resulting problem to solving nonlinear algebraic equations in terms of the coordinates associated with the nonlinear supports only. The result is a simple and efficient approach for predicting all possible fundamental and sub-harmonic responses. Stability of the periodic response is readily determined by a direct use of Floquet’s theory. The feasibility and advantages of the proposed method are illustrated with two examples of rotor-bearing systems of deadband supports and squeeze film dampers, respectively.

scholarly journals Approximate method of solving one periodic optimal regulated boundary value problem

2019 ◽  
pp. 66-69
Author(s):  
I. Askerov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Small Parameter ◽  
General Problem ◽  
Asymptotic Method ◽  
Nonlinear Differential Equations ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Lagrange Equations ◽  
Nonlinear Algebraic Equations

In the present work we considered the solution of one periodic optimal regulated boundary value problem by the asymptotic method. For the solution of the problem with extended functional writing, boundary conditions and Euler-Lagrange equations were found. The approach to the solution of the problem depending on a small parameter by seeking a system of nonlinear differential equations and solving Euler-Lagrange equations, the solution of the general problem in the first approach comes down to solving two nonlinear algebraic equations.

Nonlinear Transient Analysis of Large Rotordynamic Systems

1991 ◽  
Vol 58 (2) ◽  
pp. 596-598 ◽  
Author(s):  
T. N. Shiau ◽  
A. N. Jean
Keyword(s):  
Degrees Of Freedom ◽  
Substantial Reduction ◽  
Computational Time ◽  
Solution Technique ◽  
Flexible Rotor ◽  
Nonlinear Characteristics ◽  
Rotor Systems ◽  
Nonlinear Transient ◽  
Nonlinear Components ◽  
Flexible Rotor Systems

A solution technique based on implicit numerical integration combined with a condensation technique is presented to predict the transient response of large flexible rotor systems with nonlinear characteristics. The analysis directly tackles the nonlinear second-order differential equations which describe the system motion. The condensation technique can lead to a reduced model in which only the coordinates associated with nonlinear components of the system are considered. Thus, a substantial reduction of computation can be expected if the nonlinear components of system are sparse. A flexible rotor system is studied to illustrate the merits of the procedures. The results show that, if the system is of a small number of coordinates associated with nonlinear components compared to that of entire system degrees-of-freedom, the computational time can be considerably reduced using this technique.

scholarly journals Study on Aerodynamic Nonlinear Characteristics of Semiclosed Box Deck Based on Variation of Motion Parameters

2022 ◽  
Vol 2022 ◽  
pp. 1-15
Author(s):  
Jie Jia ◽  
Haoyang Lu ◽  
Xiaobo Li ◽  
Qian Chen
Keyword(s):  
Wind Tunnel ◽  
High Order ◽  
The Self ◽  
Aerodynamic Forces ◽  
Nonlinear Characteristics ◽  
High Order Harmonic ◽  
Aerodynamic Moment ◽  
Motion Parameters ◽  
Harmonic Components ◽  
Nonlinear Components

In order to study the nonlinear characteristics of self-excited aerodynamic forces of bluff body bridge section with the change of motion parameters, a numerical wind tunnel is established by the dynamic mesh technique of computational fluid dynamics (CFD). A state-by-state forced vibration method is used to identify the self-excited aerodynamic forces of single degree-of-freedom (DOF) heaving and pitching motion. Fast Fourier transform (FFT) is adopted to obtain frequency-domain data for analysis. The reliability of the obtained aerodynamic results is verified by wind tunnel tests. The results show that the high-order harmonic components are found in the self-excited aerodynamic forces of semiclosed box deck section, which are more significant in aerodynamic lift than in aerodynamic moment. The proportion of aerodynamic nonlinear components increases with amplitude. The effect of amplitude on the nonlinear components of heaving motion is generally higher than that of pitching motion, and aerodynamic moment is highly sensitive to the increase of vertical amplitude. The variation of the nonlinear components of the deck section with frequency is not a simple monotonic relationship, and there is a stationary point at 10 Hz frequency. The existence of wind attack angle makes the proportion of nonlinear components reach more than 30% and greatly increases the proportion of second harmonic. In addition, the high-order harmonic components, which are not integer multiples, are found at large amplitude and positive angle of attack.

Nonlinear Dynamic Analysis of an Asymmetric Ball Bearing Rotor System

2019 ◽  
Author(s):  
Doğancan Bahan ◽  
Ender Ciğeroğlu
Keyword(s):  
Ball Bearing ◽  
Nonlinear Dynamic Analysis ◽  
Harmonic Balance Method ◽  
Beam Theory ◽  
Hertzian Contact ◽  
Radial Clearance ◽  
Ball Bearings ◽  
Algebraic Equations ◽  
Rotor Systems ◽  
Nonlinear Algebraic Equations

Abstract Performance of ball bearing–rotor systems are highly dependent on and often limited by characteristics of ball bearings. Several studies are available in the literature, investigating varying compliance and subharmonic resonances of ball bearings. Most of the studies are carried out with rigid rotors to focus on modelling of the bearings. There exist few studies which take flexibility of rotors into account. Furthermore, even if the rotor flexibility is modelled, most of the time symmetrical rotors are considered. However, rotors are rarely symmetric in realistic applications due to different locations of bearings and different weights of rotor components (compressors, turbines etc.). In this study, an asymmetric, balanced, flexible rotor supported by ball bearings considering Hertzian contact and radial clearance is investigated. Rotor shaft is modelled with Nelson finite rotor elements using Timoshenko beam theory and disks are considered as rigid masses. Harmonic Balance Method (HBM) is used to obtain nonlinear algebraic equations in the frequency domain and Alternating Frequency Time (AFT) method is utilized to find Fourier coefficients of nonlinear bearing forces. In order to decrease the number of nonlinear equations to be solved, Receptance Method (RM) is applied. Resulting set of nonlinear algebraic equations is solved by using Newton’s method with arclength continuation. Several case studies are performed and effects of asymmetry on nonlinear periodic vibration response of rotors are studied.

Application of Finite Element Method in Time Domain to a Rotor System With Nonlinear Supports

2001 ◽  
Author(s):  
Liguo Wang ◽  
Wenhu Huang ◽  
Chao Hu
Keyword(s):  
Finite Element ◽  
Time Domain ◽  
Nonlinear Differential Equations ◽  
Element Formulation ◽  
Algebraic Equations ◽  
Periodic Response ◽  
Jeffcott Rotor ◽  
Rotor Bearing ◽  
The Time Domain ◽  
Bearing System

Abstract A new method for analyzing periodic response of rotor dynamic system with nonlinear supports is presented in this paper. Based on a finite element formulation in the time domain, this method transforms nonlinear differential equations governing the dynamic behavior of rotor-bearing system into a set of nonlinear algebraic equations that can be reduced and calculated by the characteristic set of Wu elimination method. The analytic solution of the nodal displacement has been obtained finally. According to this result the behavior of periodic response is analyzed. The feasibility and advantage of the proposed method are illustrated with an example of flexible Jeffcott rotor-bearing system with nonlinear supports.

scholarly journals A Numerical Approach Based on Taylor Polynomials for Solving a Class of Nonlinear Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
C. Guler ◽  
S. O. Kaya
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Numerical Approach ◽  
The Other ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Taylor Coefficients ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
Taylor Polynomials

In this study, a matrix method based on Taylor polynomials and collocation points is presented for the approximate solution of a class of nonlinear differential equations, which have many applications in mathematics, physics and engineering. By means of matrix forms of the Taylor polynomials and their derivatives, the technique we have used reduces the solution of the nonlinear equation with mixed conditions to the solution of a matrix equation which corresponds to a system of nonlinear algebraic equations with the unknown Taylor coefficients. On the other hand, to illustrate the validity and applicability of the method, some numerical examples together with residual error analysis are performed and the obtained results are compared with the existing results in literature.

The limit state of concrete and reinforced concrete rods at complex and longitudinal-transverse bending

2020 ◽  
pp. 60-73
Author(s):  
Yu V Nemirovskii ◽  
S V Tikhonov
Keyword(s):  
General Solution ◽  
Least Squares ◽  
Nonlinear Differential Equations ◽  
Limit State ◽  
Algebraic Equations ◽  
Method Of Least Squares ◽  
Elasticity Limit ◽  
Limit Values ◽  
Symbolic Calculations ◽  
Deformation Diagrams

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

scholarly journals Frequency–Amplitude Relationship of a Nonlinear Symmetric Panel Absorber Mounted on a Flexible Wall

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1188
Author(s):  
Yiu-Yin Lee
Keyword(s):  
Elliptic Integral ◽  
Algebraic Equations ◽  
Cavity Depth ◽  
Nonlinear Algebraic Equations ◽  
Flexible Panel ◽  
Flexible Wall ◽  
Elliptic Integral Method ◽  
Flexible Panels ◽  
Relationship Of ◽  
Panel Absorber

This study addresses the frequency–amplitude relationship of a nonlinear symmetric panel absorber mounted on a flexible wall. In many structural–acoustic works, only one flexible panel is considered in their models with symmetric configuration. There are very limited research investigations that focus on two flexible panels coupled with a cavity, particularly for nonlinear structural–acoustic problems. In practice, panel absorbers with symmetric configurations are common and usually mounted on a flexible wall. Thus, it should not be assumed that the wall is rigid. This study is the first work employing the weighted residual elliptic integral method for solving this problem, which involves the nonlinear multi-mode governing equations of two flexible panels coupled with a cavity. The reason for adopting the proposed solution method is that fewer nonlinear algebraic equations are generated. The results obtained from the proposed method and finite element method agree reasonably well with each other. The effects of some parameters such as vibration amplitude, cavity depth and thickness ratio, etc. are also investigated.

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