scholarly journals Modal Analysis Problem Solution for a Mathematical Model Formed by the Extended Nodal Method

2021 ◽  
pp. 33
Author(s):  
V. A. Trudonoshin ◽  
V. A. Ovchinnikov ◽  
V. G. Fedoruk
Keyword(s):  
Mathematical Model ◽  
Normal Form ◽  
Modal Analysis ◽  
Jacobi Matrix ◽  
Problem Solution ◽  
State Variables ◽  
Equilibrium Equations ◽  
Algebraic Form ◽  
Nodal Method ◽  
The Time Domain

The article proposes an option for transforming a mathematical model of the object, formed by the extended nodal method in the time-domain solution for modal analysis. Since finding the eigenvalues ​​and eigenvectors for systems of ordinary equations given in the Cauchy normal form is possible, calculations are presented that allow us to obtain a system of equations in the Cauchy normal form from a mathematical model in a differential-algebraic form through linearization. The extended nodal method contains derivatives of state variables in the vector of unknown, and the Jacobi matrix obtained at each Newton iteration of each step of numerical integration can be used to obtain a linearized mathematical model, but the equilibrium equations, as a rule, contain several derivatives with respect to time. By introducing additional variables, it is possible to reduce the linearized mathematical model to the Cauchy normal form, while the Jacobi matrix structure remains essentially unchanged.The proposed solution is implemented in the mathematical core of the PRADIS Gen2 PA-8 software package, which made it possible to expand its functionality by an operator of modal analysis.The presented calculations of test schemes have shown the correctness of the method proposed.

Towards a Technique for Nonlinear Modal Analysis

2012 ◽  
Author(s):  
Simon A. Neild ◽  
Andrea Cammarano ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Transformation Process ◽  
Second Order ◽  
Identity Transformation ◽  
Modal Testing ◽  
System Response ◽  
Weakly Nonlinear ◽  
Nonlinear Modal Analysis

In this paper we discuss a theoretical technique for decomposing multi-degree-of-freedom weakly nonlinear systems into a simpler form — an approach which has parallels with the well know method for linear modal analysis. The key outcome is that the system resonances, both linear and nonlinear are revealed by the transformation process. For each resonance, parameters can be obtained which characterise the backbone curves, and higher harmonic components of the response. The underlying mathematical technique is based on a near identity normal form transformation. This is an established technique for analysing weakly nonlinear vibrating systems, but in this approach we use a variation of the method for systems of equations written in second-order form. This is a much more natural approach for structural dynamics where the governing equations of motion are written in this form as standard practice. In fact the first step in the method is to carry out a linear modal transformation using linear modes as would typically done for a linear system. The near identity transform is then applied as a second step in the process and one which identifies the nonlinear resonances in the system being considered. For an example system with cubic nonlinearities, we show how the resulting transformed equations can be used to obtain a time independent representation of the system response. We will discuss how the analysis can be carried out with applied forcing, and how the approximations about response frequencies, made during the near-identity transformation, affect the accuracy of the technique. In fact we show that the second-order normal form approach can actually improve the predictions of sub- and super-harmonic responses. Finally we comment on how this theoretical technique could be used as part of a modal testing approach in future work.

Theoretical Investigation of a Structure for Active Vibration Control with Fuzzy Logic Approach

2014 ◽  
Vol 598 ◽  
pp. 529-533
Author(s):  
Erdi Gülbahçe ◽  
Mehmet Çelik ◽  
Mustafa Tinkir
Keyword(s):  
Mathematical Model ◽  
Fuzzy Logic ◽  
Vibration Control ◽  
Modal Analysis ◽  
Prediction Models ◽  
Block Diagram ◽  
Active Vibration Control ◽  
Control Block ◽  
Inference System ◽  
Active Vibration

The main purpose of this study is to prepare mathematical model for active vibration control of a structure. This paper presents the numerical and experimental modal analysis of aluminum cantilever beam in order to investigate the dynamic characteristics of the structure. The results will be used for active vibration control of structure’s experimental setup. Experimental natural frequencies are obtained and compared to verify the proposed numerical model by using modal analysis results. MATLAB System Identification Toolbox and ANSYS harmonic response function are used together to estimate beam’s equations of motion which include its amplitude, frequency and phase angle values. Moreover, the mathematical model of beam is simulated in MATLAB/Simulink software to determine the dynamic behavior of the proposed system. Furthermore, another prediction model approach with multiple input and single output is used to find the realistic behavior of beam via an adaptive neural-network-based fuzzy logic inference system, in addition, impulse responses of the proposed models are compared and the control block diagram for active vibration control is implemented. As a first iteration, PID type controller is designed to suppress vibrations against the disturbance input. The results of modal analysis, the prediction models, controlled and uncontrolled system responses are presented in graphics and tables for obtaining a sample numerical active vibration control.

scholarly journals Power System Nonlinear Modal Analysis Using Computationally Reduced Normal Form Method

Energies ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 1249 ◽  
Author(s):  
Nnaemeka Sunday Ugwuanyi ◽  
Xavier Kestelyn ◽  
Bogdan Marinescu ◽  
Olivier Thomas
Keyword(s):  
Normal Form ◽  
Power Systems ◽  
Power System ◽  
Modal Analysis ◽  
Computation Time ◽  
Approximate Model ◽  
Polynomial Coefficients ◽  
Normal Form Method ◽  
Nonlinear Modal Analysis ◽  
The One

Increasing nonlinearity in today’s grid challenges the conventional small-signal (modal) analysis (SSA) tools. For instance, the interactions among modes, which are not captured by SSA, may play significant roles in a stressed power system. Consequently, alternative nonlinear modal analysis tools, notably Normal Form (NF) and Modal Series (MS) methods are being explored. However, they are computation-intensive due to numerous polynomial coefficients required. This paper proposes a fast NF technique for power system modal interaction investigation, which uses characteristics of system modes to carefully select relevant terms to be considered in the analysis. The Coefficients related to these terms are selectively computed and the resulting approximate model is computationally reduced compared to the one in which all the coefficients are computed. This leads to a very rapid nonlinear modal analysis of the power systems. The reduced model is used to study interactions of modes in a two-area power system where the tested scenarios give same results as the full model, with about 70% reduction in computation time.

Study on the Dynamic Models of Systems Biology from the View of Engineering

2012 ◽  
Vol 220-223 ◽  
pp. 952-957
Author(s):  
Chen Liu ◽  
Xiao Yan Liu
Keyword(s):  
Mathematical Model ◽  
Systems Biology ◽  
Dynamic Process ◽  
Biological System ◽  
Dynamic Models ◽  
Model Parameters ◽  
State Variables ◽  
Biochemical Pathway ◽  
The Right ◽  
The Mathematical Model

From the view of engineering, based on expatiating the features of systems biology, the paper discusses the workflows and the research emphasis of systems biology. It also explains how to model and analyze the dynamic process of signal transmitting network for a biological system by an example. Based on the complexity and uncertainty of the mathematical model, the right methods are chosen to realize the effective estimation of state variables and model parameters for the biochemical pathway.

Efficient Techniques for the Computation of the Nonlinear Dynamics of a Foil-Air Bearing Rotor System

2013 ◽  
Author(s):  
Hai Pham ◽  
Philip Bonello
Keyword(s):  
Harmonic Balance ◽  
High Speed ◽  
Nonlinear Effects ◽  
Air Bearing ◽  
State Variables ◽  
Computational Burden ◽  
Rotor Systems ◽  
Symmetric Rotor ◽  
The Time Domain ◽  
Air Film

The foil-air bearing (FAB) plays a key role in the development of high speed, economical and environmentally friendly oil-free turbomachinery. However, FABs are known to be capable of introducing undesirable nonlinear effects into the dynamic response of a rotor-bearing system. This necessitates a means for calculating the nonlinear response of rotor systems with FABs. Up to now, the computational burden introduced by the interaction of the dynamics of the rotor, air film and foil structure has been overcome by uncoupling these three subsystems, introducing the potential for significant error. This paper performs the time domain solution of a simple rotordynamic system without uncoupling the state variables. This solution is then used as a reference for the verification of two proposed novel methods for reducing the computational burden: (a) use of harmonic balance; (b) use of Galerkin transformation. The applicability and accuracy of these two methods is illustrated on a simple symmetric rotor-FAB system.

Diffusion Effects Induced by Dislocations in Crystalline Materials Subjected to Large Strains

2015 ◽  
Vol 651-653 ◽  
pp. 89-95
Author(s):  
Raisa Paşcan ◽  
Sanda Cleja-Ţigoiu
Keyword(s):  
Evolution Equations ◽  
Plane Stress State ◽  
Large Strains ◽  
Slip Systems ◽  
State Variables ◽  
Equilibrium Equations ◽  
Crystalline Materials ◽  
Dislocation Densities ◽  
Differential Type ◽  
Non Local

Abstract. We reconsider here the FEM-algorithm for solving the initial and boundary value problems performed within the viscoplastic constitutive framework and proposed in our paper [1]. The problems concerning the deformation of a sheet composed of a single fcc-crystal, generated by different slip systems simultaneously activated, are solved numerically for an in-plane stress state. The variational formulation is associated to the incremental equilibrium equations and is coupled with an update procedure for the state variables, which are described by the differential type equations, as well as for the non-local evolution equations of the dislocation densities. The length scale parameter is introduced into the model through the diffusion-like parameter which enters the evolution equations for dislocation densities. For more accuracy of the simulation, the shape functions have been chosen polynomials with higher than one degree. We do not consider that once a slip system was activated it remains active for the rest of simulation. The activation condition is a key point in the numerical algorithm. As a numerical example, we perform a tensile test of a rectangular and non-rectangular metallic sheet, comparring the results of the simulation when two, respectively eight slip systems are considered.

scholarly journals Heavy-Duty Gas Turbine Plant Aerothermodynamic Simulation Using Simulink

1996 ◽  
Author(s):  
G. Crosa ◽  
F. Pittaluga ◽  
A. Trucco Martinengo ◽  
F. Beltrami ◽  
A. Torelli ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Gas Turbine ◽  
Dynamic Responses ◽  
State Variables ◽  
Algebraic Equations ◽  
Heavy Duty ◽  
Combustion Chambers ◽  
Turbine Plant ◽  
Gas Turbine Plant ◽  
Heavy Duty Gas Turbine

This paper presents a physical simulator for predicting the off-design and dynamic behaviour of a single shaft heavy-duty gas turbine plant, suitable for gas-steam combined cycles. The mathematical model, which is non linear and based on the lumped parameter approach, is described by a set of first-order differential and algebraic equations. The plant components are described adding to their steady state characteristics the dynamic equations of mass, momentum and energy balances. The state variables are mass flow rates, static pressures, static temperatures of the fluid, wall temperatures and shaft rotational speed. The analysis has been applied to a 65 MW heavy-duty gas turbine plant with two off-board silo-type combustion chambers. To model the compressor, equipped with variable inlet guide vanes, a subdivision into five partial compressors is adopted, in serial arrangement, separated by dynamic blocks. The turbine is described using a one dimensional row by row mathematical model, that takes into account both the air bleed cooling effect and the mass storage among the stages. The simulation model considers also the air bleed transformations from the compressor down to the turbine. Both combustion chambers have been modelled utilising a sequence of several sub-volumes, to simulate primary and secondary zones in presence of three hybrid burners. A code has been created in Simulink environment. Some dynamic responses of the simulated plant, equipped with a proportional-integral speed regulator, are presented.

scholarly journals Comparative Simulation Study on Synchronous Generators Sudden Short Circuits

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Lucian Lupşa-Tătaru
Keyword(s):  
Mathematical Model ◽  
Time Domain ◽  
Short Circuit ◽  
Structural Equations ◽  
Process Models ◽  
State Variables ◽  
Synchronous Generators ◽  
State Equations ◽  
Starting Point ◽  
Short Circuits

Although of a great extent in time, the research works directed at studying transients in synchronous generators have not yet provided fully sufficient comparative studies in respect to sudden short circuits of the machine. The present paper puts forward novel and comprehensive process models for dynamic simulation of short circuit faults of initially unloaded synchronous generators, using the generalizedd-q-0 mathematical model as starting point in derivation. Distinct from the time-domain analysis, the technique proposed here allows an effective comparative overview by employing a specialized procedure to perform repeated time-domain simulations accompanied by peak values recording for the various circumstances. The time consuming matrix numerical inversion at each step of integration, usually performed when selecting currents as state variables, is eliminated by advancing the process models in a convenient split matrix form that allows the symbolic processing. Also, the computational efficiency is being increased by introducing a set of auxiliary variables common to different state equations. The models derivation is carried out without altering the structural equations of the generalizedd-q-0 mathematical model of synchronous generators whilst the simulation results are both compared and discussed in detail.

Elastic-Kinematics Analysis and Numerical Method for Five-Rod Suspension

2011 ◽  
Vol 308-310 ◽  
pp. 27-34 ◽  
Author(s):  
Mo Wu Lu ◽  
Wei Qiang Zhao
Keyword(s):  
Mathematical Model ◽  
Numerical Method ◽  
Matrix Method ◽  
Road Surface ◽  
Equilibrium Theory ◽  
Kinematic Pair ◽  
Kinematics Analysis ◽  
Equilibrium Equations ◽  
Computer Based ◽  
Equilibrium Location

This paper presents a numerical method for elastic-kinematics analysis of five-rod suspension based on displacement matrix method and spatial body equilibrium theory. A mathematical model for elastic-kinematics analysis of five-rod suspension is established and equilibrium equations of suspension are derived. The method for calculating equilibrium location of suspension when counterforce of road surface acts on the wheel is discussed on the precondition of considering the influence of the elasticity of each kinematic pair on suspension and not considering it. This method simplifies the solving process of elastic-kinematics analysis of five-rod suspension and is efficient especially for computer-based solving process.

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