scholarly journals Analytical Study of Nonlinear Vibrations of Marine Risers by Newton Harmonic Balance Method

2020 ◽  
Vol 1 (2) ◽  
Author(s):  
Masoud Rahmani ◽  
◽  
Amin Moslemi Petrudi ◽  
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Beam Theory ◽  
Balance Method ◽  
Design Parameters ◽  
Nonlinear Frequency ◽  
Stability Range ◽  
Marine Risers ◽  
Fluid Damping ◽  
The Stability

In this paper, the nonlinear motions of marine risers are studied using the Newton's Harmonic Balance Method (NHBM). The nonlinear vibrational equations of the marine risers were obtained in the present study using the Hamilton principle and the Euler–Bernoulli beam theory. The Galerkin's decomposition technique is used to convert the partial differential governing equation (PDE) of the riser vibrations to the ordinary differential equation (ODE). By using the NHBM method, an analytical formulation has been obtained to express the natural nonlinear frequency of the riser. The effect of design parameters such as riser length and initial static displacement of high support has been investigated on riser frequency, which shows acceptable accuracy after comparing the results with previous research. The results show that fluid damping coefficient has a great effect on system instability and reducing this coefficient increases the stability range of the system. Examining the effect of nonlinear parameters shows that the effect of these parameters is greater in large amplitude of motion.

scholarly journals Analytical Study of Nonlinear Vibrations of Marine Risers by Newton Harmonic Balance Method

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Masoud Rahmani ◽  
◽  
Amin Moslemi Petrudi ◽  
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Beam Theory ◽  
Balance Method ◽  
Design Parameters ◽  
Nonlinear Frequency ◽  
Stability Range ◽  
Marine Risers ◽  
Fluid Damping ◽  
The Stability

In this paper, the nonlinear motions of marine risers are studied using the Newton's Harmonic Balance Method (NHBM). The nonlinear vibrational equations of the marine risers were obtained in the present study using the Hamilton principle and the Euler–Bernoulli beam theory. The Galerkin's decomposition technique is used to convert the partial differential governing equation (PDE) of the riser vibrations to the ordinary differential equation (ODE). By using the NHBM method, an analytical formulation has been obtained to express the natural nonlinear frequency of the riser. The effect of design parameters such as riser length and initial static displacement of high support has been investigated on riser frequency, which shows acceptable accuracy after comparing the results with previous research. The results show that fluid damping coefficient has a great effect on system instability and reducing this coefficient increases the stability range of the system. Examining the effect of nonlinear parameters shows that the effect of these parameters is greater in large amplitude of motion.

Multiple Harmonic Balance Method for Aperiodic Vibration of a Piecewise-Linear System

1998 ◽  
Vol 120 (1) ◽  
pp. 181-187 ◽  
Author(s):  
Y. B. Kim
Keyword(s):  
Linear System ◽  
Harmonic Balance ◽  
Piecewise Linear ◽  
Harmonic Balance Method ◽  
Steady State Solution ◽  
Balance Method ◽  
Piecewise Linear System ◽  
Higher Dimensional ◽  
Computational Procedures ◽  
The Stability

A multiple harmonic balance method is presented in this paper for obtaining the aperiodic steady-state solution of a piecewise-linear system. As the method utilizes general and systematic computational procedures, it can be applied to analyze the multi-tone or combination-tone responses for the higher dimensional nonlinear systems such as rotors. Moreover, it is capable of informing the stability of the obtained solution using Floquet theory. To demonstrate the systematic approach of the new method, the almost periodic forced vibration of an articulated loading platform (ALP) with a piecewise-linear stiffness is computed as an example.

scholarly journals Calculating periodic vibrations of nonlinear autonomous multi-degree-of-freedom systems by the incremental harmonic balance method

2004 ◽  
Vol 26 (3) ◽  
pp. 157-166
Author(s):  
Nguyen Van Khang ◽  
Thai Manh Cau
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Van Der Pol ◽  
Incremental Harmonic Balance ◽  
The Stability

In this paper the incremental harmonic balance method is used to calculate periodic vibrations of nonlinear autonomous multip-degree-of-freedom systems. According to Floquet theory, the stability of a periodic solution is checked by evaluating the eigenvalues of the monodromy matrix. Using the programme MAPLE, the authors have studied the periodic vibrations of the system multi-degree van der Pol form.

Nonlinear Frequency Response Analysis of a Multistage Clutch Damper With Multiple Nonlinearities

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Jong-Yun Yoon ◽  
Hwan-Sik Yoon
Keyword(s):  
Frequency Response ◽  
Harmonic Balance ◽  
Piecewise Linear ◽  
Dynamic Performance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Response Analysis ◽  
Nonlinear Frequency ◽  
System Equation ◽  
Nonlinear Frequency Response

This paper presents the nonlinear frequency response of a multistage clutch damper system in the framework of the harmonic balance method. For the numerical analysis, a multistage clutch damper with multiple nonlinearities is modeled as a single degree-of-freedom torsional system subjected to sinusoidal excitations. The nonlinearities include piecewise-linear stiffness, hysteresis, and preload all with asymmetric transition angles. Then, the nonlinear frequency response of the system is numerically obtained by applying the Newton–Raphson method to a system equation formulated by using the harmonic balance method. The resulting nonlinear frequency response is then compared with that obtained by direct numerical simulation of the system in the time domain. Using the simulation results, the stability characteristics and existence of quasi-harmonic response of the system are investigated. Also, the effect of stiffness values on the dynamic performance of the system is examined.

On the Extension of a Harmonic Balance Method for the Simulation of Compressor Casing Treatments

2016 ◽  
Author(s):  
Christian Voigt ◽  
Graham Ashcroft
Keyword(s):  
Time Domain ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Nonlinear Frequency ◽  
Transonic Compressor ◽  
Frequency Domain Methods ◽  
Blade Tip ◽  
The Time Domain ◽  
Casing Treatments

In recent years both linear and nonlinear frequency domain methods have become increasingly popular in the simulation and investigation of time-periodic flows in turbomachinery. In this work the extension of an alternating frequency/time domain Harmonic Balance method to support arbitrary inter-domain block interfaces, with possibly different frames of reference, is described in detail. The approach outlined is based on the time-domain, area-based interpolation algorithm originally developed for the investigation of casing treatments. In this paper, it is shown that by solving the domain coupling problem in the time-domain it is possible to accurately and efficiently capture the flow physics of such complex, nonlinear problems as blade tip interaction with casing treatments in transonic compressors. To demonstrate and verify the basic algorithm the advection of a simple entropy disturbance in a subsonic duct flow is first computed. Secondly, unsteady flow due to rotor-stator interaction in a transonic compressor stage is simulated and the data compared with reference numerical methods. Finally, to validate the method a single stage transonic axial compressor with casing treatments is simulated and the results are compared with previously published time-domain simulations as well as experimental data based on particle image velocimetry measurements in the blade tip region.

FEED FORWARD RESIDUE HARMONIC BALANCE METHOD FOR A QUADRATIC NONLINEAR OSCILLATOR

2011 ◽  
Vol 21 (06) ◽  
pp. 1783-1794 ◽  
Author(s):  
AYT LEUNG ◽  
ZHONGJIN GUO
Keyword(s):  
Fourier Series ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
High Order ◽  
Balance Method ◽  
Quadratic Nonlinearity ◽  
Analytical Prediction ◽  
Original Solution ◽  
Feed Forward ◽  
The Stability

The harmonic balance method truncates the Fourier series in a finite number of terms. In this paper we show that the truncated residues may be important to determine the stability of the approximated solution and that the truncated residues in the stability analysis can fully be considered without increasing the number of equations in the original solution. Therefore, the high order superharmonic and subharmonic responses and the cascade of bifurcations to irregular attractor can be accurately approximated by just the first few terms of the Fourier series so that analytical prediction is possible. A harmonically driven oscillator with quadratic nonlinearity is taken as examples. The explicitly analytical solutions are obtained for the steady state solutions and for the high order superharmonic approximation. The stabilities of the solutions are determined by the Floquet theory. It is shown that the predicted stability of the solution can be qualitatively different with and without the consideration of the feed forward residues. The second-, fourth- and eighth-order subharmonic analytical bifurcation solutions are calculated to obtain the cascades of bifurcations to irregular attractor. The improved analytical harmonic approximations are compared with other results and with numerical solutions. It is proved that a two superharmonic expansion with appropriate subharmonic is sufficient for determining the characteristics of the solutions of a harmonically driven oscillator with quadratic nonlinearity.

Period-1 Evolutions to Chaos in a Periodically Forced Brusselator

2018 ◽  
Vol 28 (14) ◽  
pp. 1830046 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyu Guo
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Good Method ◽  
Balance Method ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Stable Period ◽  
The Stability ◽  
Generalized Harmonic Balance Method

In this paper, analytical solutions of periodic evolutions of the Brusselator with a harmonic diffusion are obtained through the generalized harmonic balance method. The stability and bifurcation of the periodic evolutions are determined. The bifurcation tree of period-1 to period-8 evolutions of the Brusselator is presented through frequency-amplitude characteristics. To illustrate the accuracy of the analytical periodic evolutions of the Brusselator, numerical simulations of the stable period-1 to period-8 evolutions are completed. The harmonic amplitude spectrums are presented for the accuracy of the analytical periodic evolution, and each harmonics contribution on the specific periodic evolution can be achieved. This study gives a better understanding of periodic evolutions to chaos in the slowly varying Brusselator system, and the bifurcation tree of period-1 evolution to chaos are clearly demonstrated, which can help one understand a route of periodic evolution to chaos in chemical reaction oscillators. From this study, the generalized harmonic balance method is a good method for slowly varying systems, and such a method provides very accurate solutions of periodic motions in such nonlinear systems.

Dynamic vibratory motion analysis of a multi-degree-of-freedom torsional system with strongly stiff nonlinearities

2014 ◽  
Vol 229 (8) ◽  
pp. 1399-1414 ◽  
Author(s):  
Jong-yun Yoon ◽  
Hyeongill Lee
Keyword(s):  
Numerical Simulation ◽  
Nonlinear Dynamic ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Nonlinear Frequency ◽  
Dynamic Behaviors ◽  
Resonance Frequencies ◽  
Torsional System

Physical driveline systems have inherent nonlinearities such as multiple piecewise linear springs, gear backlashes, and drag torques. The multi-staged clutch dampers, in particular, cause severe problems in simulating the nonlinear dynamic behaviors of multi-degree-of-freedom systems. In order to analyze the nonlinear dynamic behaviors of the system, the harmonic balance method has been employed. This study suggests a method to overcome the convergence problems with strong nonlinearities by employing two distinct smoothening factors for stiffness and hysteresis. First, the dynamic behaviors of the multi-degree-of-freedom torsional system are investigated by employing multi-staged clutch dampers subjected to a sinusoidal excitation. Second, the effects of system parameters are examined with respect to dynamic characteristics of torsional vibration. The regimes of resonance frequencies along with the relevant parameters of the system are investigated by calculating backbone curves, which reduce the calculation time significantly. In order to validate harmonic balance method simulation, the simulated results are compared with those of numerical simulation. Harmonic balance method is shown to be more efficient than numerical simulation in calculating the nonlinear frequency response, as well as in simulating the steady-state responses without transient response effect.

Trained Harmonic Balance Method for Parametrically Excited Jeffcott Rotor Analysis

2020 ◽  
Vol 16 (1) ◽  
Author(s):  
Ghasem Ghannad Tehrani ◽  
Chiara Gastaldi ◽  
Teresa M. Berruti
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Periodic Variation ◽  
Balance Method ◽  
Rolling Bearings ◽  
Parametrically Excited ◽  
Axial Forces ◽  
Catastrophic Failures ◽  
Jeffcott Rotor ◽  
The Stability

Abstract Being able to identify instability regions is an important task for the designers of rotating machines. It allows discarding, since the early design stages, those configurations which may lead to catastrophic failures. Instability can be induced by different occurrences such as an unbalanced disk, torsional, and axial forces on the shaft or periodic variation of system parameters known as “parametric excitation.” In this paper, the stability of a Jeffcott rotor, parametrically excited by the time-varying stiffness of the rolling bearings, is studied. The harmonic balance method (HBM) is here applied as an approximate procedure to obtain the well-known “transition curves (TCs)” which separate the stable from the unstable regions of the design parameter space. One major challenge in the HBM application is identifying an adequate harmonic support (i.e., number of harmonics in the Fourier formulation), necessary to produce trustworthy results. A procedure to overcome this issue is here proposed and termed “trained HBM” (THBM). The results obtained by THBM are compared to those computed by Floquet theory, here used as a reference. The THBM proves to be able to produce reliable TCs in a timely manner, compatible with the design process.

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