scholarly journals Efficient Model Order Reduction of Structural Dynamic Systems with Local Nonlinearities under Periodic Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
M. Mohammadali ◽  
H. Ahmadian
Keyword(s):  
Harmonic Balance Method ◽  
Order Reduction ◽  
Nonlinear Effects ◽  
Discrete Systems ◽  
Algebraic Equations ◽  
Structural Dynamic ◽  
Model Order ◽  
Modal Expansion ◽  
Expansion Technique ◽  
Nonlinear Algebraic Equations

In many nonlinear structural systems, compared with the local regions with induced nonlinear effects, the main portions of the structures are linear. An exact condensation technique based on the harmonic balance method (HBM) in conjunction with the modal expansion technique is employed to convert the motion equations of such a system to a set of nonlinear algebraic equations that are considerably small and adequately accurate to determine periodic responses. To demonstrate the capability of the suggested method, few case studies consisting of discrete systems with weak and essential nonlinearities are studied, and the results are compared to other methodologies results.

Equivalent Circuit Parametrization Utilizing FE Model Order Reduction and its Application to Piezoelectric Generators and Actuators

2017 ◽  
Vol 4 (3) ◽  
pp. 115-129 ◽  
Author(s):  
Wiebold Wurpts ◽  
Jens Twiefel ◽  
Francois Brouet
Keyword(s):  
Equivalent Circuit ◽  
Model Order Reduction ◽  
Experimental Parameter ◽  
Equivalent Circuit Model ◽  
Order Reduction ◽  
Fe Model ◽  
Model Order ◽  
Modal Expansion ◽  
First Choice ◽  
Discrete Finite Element

Abstract Equivalent circuits are often the first choice for the modeling of piezoelectric systems, as they allow for the consideration of the complete electro-mechanical system with one or even more modes. The parameters of the equivalent circuit model are identified by a measured or simulated frequency response. In this contribution a method for a direct modal condensation of the equivalent parameters for arbitrary FE structures and loads is described and discussed. First the proposed method is demonstrated for a continuous piezoelectric rod and then applied to discrete finite element models. The derived equivalent circuit has an identical appearance to the classical solution, but additionally allows arbitrarily load conditions. Furthermore, the structure of the derived equivalent circuit depends on whether short- or open-circuited modes are used for the modal expansion. The influence of truncated modes is discussed utilizing residual terms, leading to a better understanding of the circuit parameters. Additionally the model based approaches in the third part an experimental parameter identification procedure for many modes is presented as well. The influence of the load and the quality of the model order reduction are discussed for piezoelectric rods. The methods are demonstrated for a base excited energy harvesting system an ultrasonic grubber.

scholarly journals A Robust Computational Technique for Model Order Reduction of Two-Time-Scale Discrete Systems via Genetic Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Othman M. K. Alsmadi ◽  
Zaer S. Abo-Hammour
Keyword(s):  
Genetic Algorithms ◽  
Time Scale ◽  
Model Order Reduction ◽  
Fitness Function ◽  
Order Reduction ◽  
Discrete Systems ◽  
Computational Technique ◽  
Model Order ◽  
New Approach ◽  
Reduced Order

A robust computational technique for model order reduction (MOR) of multi-time-scale discrete systems (single input single output (SISO) and multi-input multioutput (MIMO)) is presented in this paper. This work is motivated by the singular perturbation of multi-time-scale systems where some specific dynamics may not have significant influence on the overall system behavior. The new approach is proposed using genetic algorithms (GA) with the advantage of obtaining a reduced order model, maintaining the exact dominant dynamics in the reduced order, and minimizing the steady state error. The reduction process is performed by obtaining an upper triangular transformed matrix of the system state matrix defined in state space representation along with the elements ofB,C, andDmatrices. The GA computational procedure is based on maximizing the fitness function corresponding to the response deviation between the full and reduced order models. The proposed computational intelligence MOR method is compared to recently published work on MOR techniques where simulation results show the potential and advantages of the new approach.

scholarly journals A modified harmonic balance method for solving forced vibration problems with strong nonlinearity

2020 ◽  
pp. 146134842092343 ◽  
Author(s):  
M W Ullah ◽  
M S Rahman ◽  
M A Uddin
Keyword(s):  
Forced Vibration ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Strong Nonlinearity ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Vibration Problems ◽  
Nonlinear Forced Vibration

In this paper, a modified harmonic balance method is presented to solve nonlinear forced vibration problems. A set of nonlinear algebraic equations appears among the unknown coefficients of harmonic terms and the frequency of the forcing term. Usually a numerical method is used to solve them. In this article, a set of linear algebraic equations is solved together with a nonlinear one. The solution obtained by the proposed method has been compared to those obtained by variational and numerical methods. The results show good agreement with the results obtained by both methods mentioned above.

A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
R. Ju ◽  
W. Fan ◽  
W. D. Zhu ◽  
J. L. Huang
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Response Surfaces ◽  
Algebraic Equations ◽  
Incremental Harmonic Balance Method ◽  
Calculation Time ◽  
High Order Harmonic ◽  
Nonlinear Algebraic Equations ◽  
Incremental Harmonic Balance ◽  
Ihb Method

A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations of two incommensurate base frequencies. Truncated Fourier coefficients of residual vectors of nonlinear algebraic equations are obtained by a frequency mapping-fast Fourier transform procedure, and complex two-dimensional (2D) integration is avoided. Jacobian matrices are approximated by Broyden's method and resulting nonlinear algebraic equations are solved. These two modifications lead to a significant reduction of calculation time. To automatically calculate amplitude–frequency response surfaces of quasi-periodic responses and avoid nonconvergent points at peaks, an incremental arc-length method for one timescale is extended for quasi-periodic responses with two timescales. Two examples, Duffing equation and van der Pol equation with quadratic and cubic nonlinear terms, both with two external excitations, are simulated. Results from the modified two-timescale IHB method are in excellent agreement with those from Runge–Kutta method. The total calculation time of the modified two-timescale IHB method can be more than two orders of magnitude less than that of the original quasi-periodic IHB method when complex nonlinearities exist and high-order harmonic terms are considered.

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.

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