A fast Galerkin method to obtain the periodic solutions of a nonlinear oscillator

Miralles, Juan Peña; Olivo, Pedro José Jiménez; Peiro, Damián Ginestar; Martín, Gumersindo Verdú; Muñoz-Cobo, J.L.

doi:10.1016/s0096-3003(96)00193-2

Cited by 6 publications

(7 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equation (27) can be written as where ℱ represents the operator of discrete Fourier transform, which can be calculated by the fast Fourier transform technique Eq. [23].…”

Section: Fast Calculation Methods For Itmmmentioning

confidence: 99%

“…By adding a convert factor in the iteration, ITMM is successfully applied to the rotor system with strong nonlinear support such as SFD. Besides, retaining the advantage of ITMM that reduces the order of multidegree-of-freedom system, by employing the fast Fourier transform technique [23,24], the disadvantage of slow calculation speed of numerical integrations has been overcome.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Improved incremental transfer matrix method for nonlinear rotor-bearing system

et al. 2020

View full text Add to dashboard Cite

Rotor system supported by nonlinear bearing such as squeeze film damper (SFD) is widely used in practice owing to its wide range of damping capacity and simplicity in structure. In this paper, an improved and effective Incremental transfer matrix method (ITMM) is first presented by combining ITMM and fast Fourier transform (FFT). Afterwards this method is applied to calculate the dynamic characteristics of a Jeffcott rotor system with SFD. The convergence difficulties incurred caused by strong nonlinearities of SFD has been dealt by adopting a control factor. It is found that for the more general boundary problems where the boundary conditions are not at input and output ends of a chain system, the supplementary equation is necessarily added. Additionally, the Floquet theory is used to analyze the stability and bifurcation type of the obtained periodic solution. The semi-analytical results, including the periodic solutions of the system, the bifurcation points and their types, are in good agreement with the numerical method. Furthermore, the involution mechanism of the quasi-periodic and chaotic motions near the first-order translational mode and the second order bending mode of this system is also clarified by this method with the aid of Floquet theory. Keywords Incremental transfer matrix method • Squeeze film damper • Jeffcott rotor system • Bifurcation • Nonlinear characteristics Abbreviations a, a si , a ci Fourier series, dimensionless , Displacement and angular components in the state vector c Damping coefficient of linear elastic support, N ⋅ s/m C Clearance of squeeze film damper, m

show abstract

“…Equation (27) can be written as where ℱ represents the operator of discrete Fourier transform, which can be calculated by the fast Fourier transform technique Eq. [23].…”

Section: Fast Calculation Methods For Itmmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Improved incremental transfer matrix method for nonlinear rotor-bearing system

et al. 2020

View full text Add to dashboard Cite

show abstract

“…The harmonic balance method may date back to [28][29][30] and it has been widely used as an efficient method for computing periodic and steady-state responses of nonlinear systems and, hence to gain insight into system nonlinear characteristics. Furthermore, the introduction of the Fast Fourier Transform (FFT) and the Alternating Frequency/Time (AFT) technique [31][32][33][34] enables the use of multiple (high-order) harmonics and accurate consideration of strong nonlinearity such as friction. With the AFT technique, it has been shown that the Jacobian matrix of the nonlinear algebraic equations resulting from multi-HB analysis can be formulated analytically, even for stiff systems with friction interfaces, which guarantees the computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

“…In solving the PDEs using iterative method, to consider the seabed effect, w 0 in Eqs. (29)(30)(31)(32)(33)(34) is replaced by w n e which is evaluated based on the cable nodal position obtained in the previous iteration step. After solving the equations in the Lagrangian coordinate system, the cable nodal displacement and position can be integrated node by node from the seabed anchor using s, φ and θ [13,15].…”

mentioning

confidence: 99%

Nonlinear periodic response analysis of mooring cables using harmonic balance method

Chen

Basu

Nielsen

2019

Journal of Sound and Vibration

View full text Add to dashboard Cite

Mooring cables are critical components of ocean renewable energy systems including offshore floating wind turbines and wave energy converters. Mooring cable dynamics is strongly nonlinear resulting from the geometric effect, hydrodynamic loads and probably seabed interactions. Time-domain methods are commonly used for numerical simulation. This study formulates a nonlinear frequency domain multi-harmonic balance method for efficient analysis of a mooring cable subjected to periodic fairlead motions. The periodic responses are of particular interest to investigate the mooring effect on the platform. In the formulation, the governing equations of the three-dimensional cable motions are spatially discretized using the finite difference method; the nonlinear ordinary differential equations are subsequently transformed into frequency domain by expanding both the structural responses and the nonlinear nodal forces using truncated Fourier series, leading to a set of nonlinear algebraic equations of the Fourier coefficients. The equations are eventually solved using Newton's method where the alternating frequency/time domain method is used to handle the nonlinearity effect. The presented method is then compared to a time-domain method by numerical studies of a mooring cable. The results show that the method is of comparable accuracy as the time-domain method while it is generally more efficient. The proposed method shows promising results even when the cable tension becomes non-positive for a period of time during the cable motion, which is a known ill-posed problem for time-domain methods.

show abstract

“…Thus, to solve it two ways are possible: a semi-analytical approach or a numerical approach. The semi-analytical approach is based on Galerkin-Urabe's method [3] and allows to determine only regime solutions (both stable and unstable solutions) in a very fast and efficient way; the numerical approach (e.g. Runge-Kutta's method),instead, allows to determine both the transient response and the stable regime solution but requires a much higher computational effort…”

Section: Numerical Modelmentioning

confidence: 99%

Nonlinear vibrations of a mems translational device

Leo¹,

Braghin²,

Resta³

Research in Microelectronics and Electronics, 2005 PhD

View full text Add to dashboard Cite

When significantly displacing a proof mass, the nonlinear hardening characteristic of the supporting beams becomes visible. Thus, the resonance peak of the structure is no longer vertical but bends towards the higher frequencies. This property is useful to easily synchronise sense and drive resonances thus increasing the sensibility of the MEMS device. Through a test structure designed to investigate the high deformation range of the supporting beams, its nonlinear vibrations were investigated both experimentally and numerically.

show abstract

A fast Galerkin method to obtain the periodic solutions of a nonlinear oscillator

Cited by 6 publications

References 9 publications

Improved incremental transfer matrix method for nonlinear rotor-bearing system

Improved incremental transfer matrix method for nonlinear rotor-bearing system

Nonlinear periodic response analysis of mooring cables using harmonic balance method

Nonlinear vibrations of a mems translational device

Contact Info

Product

Resources

About