Numerical computation of second-order system response with variable parameters

1969 ◽  
Vol 20 (1) ◽  
pp. 131-138
Author(s):  
Josef F. Ury
Keyword(s):  
Numerical Computation ◽  
Second Order ◽  
Order System ◽  
System Response ◽  
Variable Parameters

Analytical Calculation of the Position Loop Gain for Linear Motor CNC Machine Tool

2012 ◽  
Vol 186 ◽  
pp. 182-187 ◽  
Author(s):  
Zoran Pandilov ◽  
Vladimir Dukovski
Keyword(s):  
Machine Tool ◽  
Linear Motor ◽  
Second Order ◽  
Order System ◽  
System Response ◽  
Loop Gain ◽  
Cnc Machine ◽  
Servo Drive ◽  
Simple Method ◽  
Following Error

One of the most important factors which influence on the dynamical behavior of the linear motor servo drives for CNC machine tools is position loop gain or Kv factor. From the magnitude of the Kv-factor depends tracking or following error. In multi-axis contouring the following errors along the different axes may cause form deviations of the machined contours. Generally position loop gain Kv should be high for faster system response and higher accuracy, but the maximum gains allowable are limited due to undesirable oscillatory responses at high gains and low damping factor. Usually Kv factor is experimentally tuned on the already assembled machine tool. This paper presents a simple method for analytically calculation of the position loop gain Kv. A combined digital-analog model of the 4-th order of the position loop is presented. In order to ease the calculation, the 4-th order system is simplified with a second order model. With this approach it is very easy to calculate the Kv factor for necessary position loop damping. The difference of the replacement of the 4-th order system with second order system is presented with the simulation program MATLAB. Analytically calculated Kv factor is function of the nominal angular frequency  and damping D of the linear motor servo drive electrical parts (motor and regulator), as well as sampling period T. :The influence of nonlinearities was taken with the correction factor. Our investigations have proven that experimentally tuned Kv factor differs from analytically calculated Kv factor less than 10%, which is completely acceptable

scholarly journals Forensic Engineering Analysis Of Trailer Sway Accidents

2008 ◽  
Vol 25 (1) ◽  
Author(s):  
Mark A.M. Ezra ◽  
Landiss Danel J.
Keyword(s):  
Data Reduction ◽  
Damping Ratio ◽  
Experimental System ◽  
Second Order ◽  
Coupled Systems ◽  
Order System ◽  
System Response ◽  
Statistical Validation ◽  
Response Data ◽  
Data Reduction Method

This Paper Describes A Method For The Reduction Of System Response Data For Second Order Electrical Or Mechanical Systems When That Data Is Available Only In Graphical Format. The Method Of Data Reduction Described Allows Quantitative Evaluation Of Generally Accepted Second Order System Parameters Such As: System Time Constant, Damped Natural Frequency, Damping Ratio, And Exponential Decay Time. The Discussion Includes The Application Of The Described Graphical Technique To Experimental System Response Data Of Coupled Systems, But Whose Experimental Response Approximates That Of An Isolated Second Order System. A Practical Application Of The Described Data Reduction Method Is Covered In Detail. The Described Technique Is Applied To The Analysis Of Data Obtained Experimentally For The Response Of A Tow Vehicle And Trailer System To A Standardized Steering Disturbance. Finally, The Statistical Validation For Experimental System Response Data And The Results Obtained From The Analysis Of Such Data, Using The Described Graphical Method, Is Discussed.

Closeness of the Solutions of Approximately Decoupled Damped Linear Systems to Their Exact Solutions

1992 ◽  
Vol 114 (3) ◽  
pp. 369-374 ◽  
Author(s):  
S. M. Shahruz ◽  
G. Langari
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Linear Systems ◽  
Equations Of Motion ◽  
Second Order ◽  
Order System ◽  
System Response ◽  
Damping Matrix

The simplest technique of decoupling the normalized equations of motion of a linear nonclassically damped second-order system is to neglect the off-diagonal elements of the normalized damping matrix. In this paper, the error introduced in the system response due to this decoupling technique is studied. Conditions are derived under which the solution of an approximately decoupled system is “close” to the exact solution of the system.

scholarly journals Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems

2021 ◽  
Vol 11 (8) ◽  
pp. 3430
Author(s):  
Erik Cuevas ◽  
Héctor Becerra ◽  
Héctor Escobar ◽  
Alberto Luque-Chang ◽  
Marco Pérez ◽  
...  
Keyword(s):  
Convergence Rates ◽  
Optimization Problems ◽  
Search Space ◽  
Second Order ◽  
Order System ◽  
Search Patterns ◽  
Multiple Behaviors ◽  
Complex Optimization ◽  
Second Order Systems ◽  
Order Algorithm

Recently, several new metaheuristic schemes have been introduced in the literature. Although all these approaches consider very different phenomena as metaphors, the search patterns used to explore the search space are very similar. On the other hand, second-order systems are models that present different temporal behaviors depending on the value of their parameters. Such temporal behaviors can be conceived as search patterns with multiple behaviors and simple configurations. In this paper, a set of new search patterns are introduced to explore the search space efficiently. They emulate the response of a second-order system. The proposed set of search patterns have been integrated as a complete search strategy, called Second-Order Algorithm (SOA), to obtain the global solution of complex optimization problems. To analyze the performance of the proposed scheme, it has been compared in a set of representative optimization problems, including multimodal, unimodal, and hybrid benchmark formulations. Numerical results demonstrate that the proposed SOA method exhibits remarkable performance in terms of accuracy and high convergence rates.

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.

Second Order Averaging Study of an Autoparametric System

1993 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj ◽  
Patricia Davies
Keyword(s):  
Dynamical Behavior ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Pitchfork Bifurcation ◽  
Second Order ◽  
System Response ◽  
Response Curves ◽  
Vibratory System ◽  
First Order

Abstract The autoparametric vibratory system consisting of a primary spring-mass-dashpot system coupled with a damped simple pendulum serves as an useful example of two degree-of-freedom nonlinear systems that exhibit complex dynamic behavior. It exhibits 1:2 internal resonance and amplitude modulated chaos under harmonic forcing conditions. First-order averaging studies of this system using AUTO and KAOS have yielded useful information about the amplitude dynamics of this system. Response curves of the system indicate saturation and the pitchfork bifurcation sets are found to be symmetric. The period-doubling route to chaotic solutions is observed. However questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid cannot be answered by a first-order study. Some observed dynamical behavior, like saturation, may not persist when higher-order nonlinear effects are taken into account. Second-order averaging of the system, using Mathematica (Maeder, 1991; Wolfram, 1991) is undertaken to address these questions. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. The dynamics of the system is found to be very sensitive to damping, with extremely complicated behavior arising for low values of damping. For large ε second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response.

Development of a Second-Order System for Rapid Estimation of Maximum Brain Strain

2018 ◽  
Vol 47 (9) ◽  
pp. 1971-1981 ◽  
Author(s):  
Lee F. Gabler ◽  
Jeff R. Crandall ◽  
Matthew B. Panzer
Keyword(s):  
Second Order ◽  
Order System ◽  
Rapid Estimation
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