Eigensolution Derivatives for Arbitrarily Normalized Modes
|
|
- Kellie Cornelia Rice
- 5 years ago
- Views:
Transcription
1 B. P. Wang Department of Mechanical Aerospace Engineering The University of Texas at Arlington Box 19023, Arlington, TX Eigensolution Derivatives for Arbitrarily Normalized Modes Methods for computing eigenvector derivatives for arbitrarily normalized modes are derived. Formulations are presented for modal methods, Nelson's method, a methodfor computing eigenvalue eigenvector derivatives simultaneously. A simple numerical example is provided to illustrate various formulations John Wiley & Sons, Inc. INTRODUCTION The derivatives of eigenvalue eigenvectors with respect to design parameters are essential in design sensitivity analysis, structural optimization, test/analysis correlation studies. Methods of computing eigensolution derivatives have been reported by many researchers. These methods have a common feature: they all use modes normalized to unit generalized mass. In this article, methods for computing eigensolution derivatives for arbitrarily normalized modes are presented. The relationship of eigenvector derivatives computed by various normalizations were studied. We present a simple numerical example to illustrate the various formulations. The eigenvalue problem for undamped systems in structural dynamics is (1) K M are stiffness mass matrices, 'A is an eigenvalue, 1J is the associated eigenvector. Assume'll. is not a repeated eigenvalue. Let ~ be mass-normalized, 1J be arbitrarily normalized. That is Received November 27, 1993; Revised January 25,1994 Shock Vibration, Vol. 1, No.4, pp (1994) 1994 John Wiley & Sons, Inc. (2) pt1j = c. (3) Note that the normalization scheme presented in Eq. (3) is very general. For example, if the eigenvector is normalized so that the ph component is 1, then P = ej = a unit vector c = 1. (4a) (4b) If the eigenvector is normalized to unit length then P=1J c = 1. (5a) (5b) In the literature, methods for eigensolution derivatives are derived based on Eqs. (1) (2). In this article, formulations are based on Eqs. (1) (3). CCC /94/
2 396 Wang BASIC EQUATIONS FOR EIGENSOlUTION DERIVATIVES Differentiating Eqs. (1) (3) with respect to a design variable yields (K - AM)lJ>' + (K' - A'M - AM')lJ> = 0 (6) ptlj>' + pltlj> = 0 (7) a prime in the above equations indicates differentiation with respect to a design variable. Equations (6) (7) are basic equations for computing eigensolution derivatives. The formula for eigenvalue derivative can be derived by premultiplying Eq. (6) by lj>t. This yield, after some simplification, This is a well known result. n Cl = - ~ Cj(PTlJ>j)/(PTlJ>t>. j=l j"'l Approximate Mode-Superposition Method (13) If there are only L modes available, L < n, then an approximation of lj>/ can be computed by changing the summation limit in Eq. (9) from n to L, that is L lj>la = ~ CjlJ>j (14) j=i Cj is given by Eq. (11) Cl is given by Eq. (13) with the upper limit of summation changed to L. The approximate eigenvector derivative may not be accurate enough in application. Its accuracy can be improved by a modeacceleration approach. Using the procedures of Wang (1991), the following improved approximation to eigenvector derivatives are obtained. METHODS OF COMPUTING EIGENVECTOR DERIVATIVES (15) Several methods for computing eigenvector derivatives of arbitrarily normalized modes include the mode superposition method, improved modal method, Nelson's method, simultaneous method. Mode Superposition Method Following the procedures of Fox Kapoor (1968), 1, the eigenvector derivative for a system with n degrees of freedom (DOF) can be computed by the following equation. n lj>/ = ~ CjlJ>j j=i (9) L lj>/l = ~ CjlJ>j j=l HI Cj = Eq. (11) for j +- I Cll = -PT(lJ>/L + YI)/(PTlJ>I). (16a) (16b) (16c) Equation (15) is the explicit formulation of Wang (1991). Further improvement can be obtained by using implicit formulation from the same reference: lj>/l is given in Eq. (16a) (17a),n, i +- I (11) Cy = ytft!(ytky - AlyTMY) Cl2 = _PT(lJ>/L + cyy)/(ptlj>i). (17b) (17c) Fl = (K' - A/M - Gl = lj>tmlj>1 AIM')lJ>1 (12a) (12b) Nelson's Method The modal methods require all modes for an exact solution. Alternatively, the popular Nelson
3 Eigensolution Derivatives for Arbitrarily Normalized Modes 397 method (1976) requires only the mode itself for computing derivatives. In the traditional Nelson method this is achieved by computing a particular solution of eigenvector derivative by introducing a constraint in Eq. (6). The complete solution can then be calculated using the mass normalization condition. We will derive similar solution procedures for arbitrarily normalized modes. Rewrite Eq. (6) for the lth mode (18) zt, Ft have the same form as given in Eqs. (23a, b). Substituting Eq. (24) into (7) yields (26) In summary, Eqs. (24)-(26) constitute the procedures for computing derivatives of arbitrarily normalized modes when P is a constant matrix. If P is a function of the design variable, then it can be shown that F, is given by Eq. (11) (19) Next, our development of a Nelson type approach depends on the normalization scheme used Max-Normalized Modes. For max-normalized modes (20) because 4>j1 = 1 by definition. The eigenvector derivative can then be calculated from equations (18) (20). Let 4>; = TV (21) V is an (n - 1) x 1 vector T is an n x (n - 1) matrix constructed by deleting the jth column of an n x n identity matrix. The vector V can be computed from ztv = Ft zt = rrz,t Ft = TTF, (22) (23a) (23b) Simultaneous Computing of Eigenvalue Eigenvector Derivatives Another method for computing 4>;, using only 4>1, is to compute it with Al at the same time, Murthy Haftke (1988). This is achieved by putting Eqs. (6) (7) together treating 4>; Al as variables, that is Ax = b (28) A = [Zl -M4>/] pt 0 b = [-(K1 - AM 1)4>,]. -PIT4>1 (29a) (29b) (29c) Note that in the above formulation, we have assumed that P is an explicit function of the design variables. The advantage of this formulation is that 4>1 Al are computed simultaneously. Arbitrarily Normalized Modes. For this case, let 4>; = Tw + c,4>, (24) w is an (n - 1) x 1 vector. T is formed by deleting a column in the n x n identity matrix. Substituting Eq. (24) into Eq. (18), we can compute w from ztw = Ft (25) RELATIONSHIP OF EIGENVECTOR DERIVATIVES Let cbl 4>1 be a mass-normalized mode an arbitrarily normalized mode, respectively. Then by definition (30)
4 398 Wang Table 1. First Eigenvector With Different Normalization Mass Max Length mass system with the following mass stiffness matrices It follows that (31) The eigensolution sensitivities at a = 0 are to be computed. For this case, Thus, given cf>;, cf>; can be computed from Eqs. (31) (33). Alternatively, cf>1 = cf>li r (34) r = max( cf>jl) = maximum elements in cf>1. Assume r > O. Then I,i,1 Equation (36) relates cf>; to cf>1. Numerical Example (35) (36) r = 'l'jl. (37) A simple 3-DOF system is used to illustrate the above formulation. Consider a 3-DOF spring- Table 2. Derivatives of First Eigenvector With Different Normalization Mass Max Length o The eigenvalues for the system are: , , The corresponding mass-normalized eigenvectors are: cf> [ = J Table 1 summarizes the first mode with three different normalization methods. The corresponding eigenvector sensitivities are given in Table 2. U sing the relationship between cf>; cf>1, the same cf>; can be computed from cf>lmax cf> ;,Iength Note that Table 2 shows very different eigenvector derivatives for modes normalized by different methods. However, when these derivatives are used to study the change of eigenvector Table 3. Comparison of Eigenvectors for a Mass Max Length Values were predicted by eigenvector derivatives using different modes.
5 Eigensolution Derivatives for Arbitrarily Normalized Modes 399 Table 4. Comparison of Eigenvectors for IX Exact Mass Max Length Values were predicted by different eigenvector derivatives. with respect to design change the results are independent of normalization schemes. For example, let us predict the new eigenvector for the system when a = 0.1. The predict eigenvectors for various normalizations are given in Table 3. Mter max-normalization, these modes become those shown in Table 4. Also shown in Table 4 are the exact mode shape of the modified system. It is clear that all eigenvector derivatives provided the same results. CONCLUSIONS Methods for computing eigensolution derivatives for arbitrarily normalized modes are similar to the conventional eigenvector derivatives based on mass-normalized modes. The relationship between these derivatives are presented. Because in practice the computed modes are often maxnormalized, the proposed formulation can be used to compute the required derivatives. Additionally, from their relationships, we can convert eigenvector derivatives for modes normalized by different methods. It should be noted that in Haftka, Gurdal, Kamat (1989), methods for computing eigenvector derivatives are presented for simultaneous methods for mode shapes normalized with a general symmetric weight matrix W such that evwcp = 1. The contribution of this study is to present a direct formulation of computing eigenvector derivatives for various methods, including an improved modal method, for arbitrarily normalized modes. REFERENCES Fox, R. L., Kapoor, M. P., 1968, "Rate of Change of Eigenvalue Eigenvectors," AIAA Journal, Vol. 6, No. 12, pp Haftka, R. T., Gurdal, Z., Kamat, M. P., 1989, Elements of Structural Optimization, Kluwer Academic Publishers, Boston. Murthy, D. V., Haftka, R. T., 1988, "Derivatives of Eigenvalues Eigenvectors of General Complex Matrix," International Journal for Numerical Methods in Engineering, Vol. 26, pp Nelson, R. B., 1976, "Simplified Calculation of Eigenvector Derivative," AIAA Journal, Vol. 14, No.9, pp Wang, B. P., 1991, "Improved Approximate Methods for Computing Eigenvector Derivatives in Structural Dynamics," AIAA Journal, Vol. 29, No.6, pp
6 Rotating Machinery Engineering The Scientific World Journal Distributed Sensor Networks Sensors Control Science Engineering Advances in Civil Engineering Submit your manuscripts at Electrical Computer Engineering Robotics VLSI Design Advances in OptoElectronics Navigation Observation Chemical Engineering Active Passive Electronic Components Antennas Propagation Aerospace Engineering Volume 2010 Modelling & Simulation in Engineering Shock Vibration Advances in Acoustics Vibration
Iterated Dynamic Condensation Technique and Its Applications in Modal Testing
Zhikun Hou Shiyu Chen Department of Mechanical Engineering Worcester Polytechnic Institute Worcester, MA 01609 Iterated Dynamic Condensation Technique and Its Applications in Modal Testing This article
More informationA Parameter Study of Localization
Sandor Stephen Mester Haym Benaroya Department of Mechanical and Aerospace Engineering Rutgers The State University of New Jersey New Brunswick NJ A Parameter Study of Localization Etensive work has been
More informationPerformance of various mode indicator functions
Shock and Vibration 17 (2010) 473 482 473 DOI 10.3233/SAV-2010-0541 IOS Press Performance of various mode indicator functions M. Radeş Universitatea Politehnica Bucureşti, Splaiul Independenţei 313, Bucureşti,
More informationAnalysis of Cylindrical Shells Using Generalized Differential Quadrature
C. T. Loy K. Y. Lam C. Shu Department of Mechanical and Production Engineering ational University of Singapore 10 Kent Ridge Crescent Singapore 0511 Analysis of Cylindrical Shells Using Generalized Differential
More informationDesign and control of nonlinear mechanical systems for minimum time
Shock and Vibration 15 (2008) 315 323 315 IOS Press Design and control of nonlinear mechanical systems for minimum time J.B. Cardoso a,, P.P. Moita b and A.J. Valido b a Instituto Superior Técnico, Departamento
More informationMETAMAGNETIC MATERIALS WITH VOLUME-DEPENDENT
Active and Passive Elec. Comp., 2001, Vol. 24, pp. 63-67 Reprints available directly from the publisher Photocopying permitted by license only (C) 2001 OPA (Overseas Publishers Association) N.V. Published
More informationDynamic Response of Structures With Frequency Dependent Damping
Dynamic Response of Structures With Frequency Dependent Damping Blanca Pascual & S Adhikari School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/
More informationAnalysis of Vibrating Timoshenko Beams Using the Method of Differential Quadrature
P. A. A. Laura R. H. Gutierrez Institute of Applied Mechanics (CONICET-SENID-ACCE) and Department of Engineering Universidad Nacional del Sur BOOO-Bahia Blanca, Argentina Analysis of Vibrating Timoshenko
More informationApplication of Lanczos and Schur vectors in structural dynamics
Shock and Vibration 15 (2008) 459 466 459 IOS Press Application of Lanczos and Schur vectors in structural dynamics M. Radeş Universitatea Politehnica Bucureşti, Splaiul Independenţei 313, Bucureşti, Romania
More informationAn Expansion Method Dealing with Spatial Incompleteness of Measured Mode Shapes of Beam Structures
Appl. Math. Inf. Sci. 8, o. 2, 651-656 (2014) 651 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080223 An Expansion Method Dealing with Spatial Incompleteness
More informationFractal two-level finite element method for free vibration of cracked beams
61 Fractal two-level finite element method for free vibration of cracked beams A.Y.T. Leung School of Engineering, University of Manchester, Manchester M13 9PL, UK R.K.L. Su Ove Arup & Partners, Hong Kong
More informationModel tests and FE-modelling of dynamic soil-structure interaction
Shock and Vibration 19 (2012) 1061 1069 1061 DOI 10.3233/SAV-2012-0712 IOS Press Model tests and FE-modelling of dynamic soil-structure interaction N. Kodama a, * and K. Komiya b a Waseda Institute for
More informationTwo Visually Meaningful Correlation Measures for Comparing Calculated and Measured Response Histories
Benjamin Whang William E. Gilbert Stephen Zilliacus Carderock Division Naval Surface Warfare Center Bethesda, MD 20084 Two Visually Meaningful Correlation Measures for Comparing Calculated and Measured
More informationFREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS
Lecture Notes: STRUCTURAL DYNAMICS / FALL 2011 / Page: 1 FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS : : 0, 0 As demonstrated previously, the above Equation of Motion (free-vibration equation) has a solution
More informationCOMPLEX MODES IN LINEAR STOCHASTIC SYSTEMS
Proceedings of VETOMAC-I October 25-27, 2, Bangalore, INDIA COMPLEX MODES IN LINEAR STOCHASTIC SYSTEMS S. Adhikari Cambridge University Engineering Department Trumpington Street Cambridge CB2 1PZ (U.K.)
More informationResearch Article Simplified Robotics Joint-Space Trajectory Generation with a via Point Using a Single Polynomial
Robotics Volume, Article ID 75958, 6 pages http://dx.doi.org/.55//75958 Research Article Simplified Robotics Joint-Space Trajectory Generation with a via Point Using a Single Polynomial Robert L. Williams
More information3. Mathematical Properties of MDOF Systems
3. Mathematical Properties of MDOF Systems 3.1 The Generalized Eigenvalue Problem Recall that the natural frequencies ω and modes a are found from [ - ω 2 M + K ] a = 0 or K a = ω 2 M a Where M and K are
More informationThe use of transmissibility properties to estimate FRFs on modified structures
Shock and Vibration 7 (00) 56 577 56 DOI 0./SAV-00-058 IOS Press The use of transmissibility properties to estimate FRFs on modified structures R.A.B. Almeida a,, A.P.V. Urgueira a and N.M.M. Maia b a
More informationResearch Article Experimental Parametric Identification of a Flexible Beam Using Piezoelectric Sensors and Actuators
Shock and Vibration, Article ID 71814, 5 pages http://dx.doi.org/1.1155/214/71814 Research Article Experimental Parametric Identification of a Flexible Beam Using Piezoelectric Sensors and Actuators Sajad
More informationEstimation of dynamic characteristics of a spring-mass-beam system
Shock and Vibration 14 (2007) 271 282 271 IOS Press Estimation of dynamic characteristics of a spring-mass-beam system Ding Zhou a,b, and Tianjian Ji b a College of Civil Engineering, Nanjing University
More informationResearch Article Numerical Study of Flutter of a Two-Dimensional Aeroelastic System
ISRN Mechanical Volume 213, Article ID 127123, 4 pages http://dx.doi.org/1.1155/213/127123 Research Article Numerical Study of Flutter of a Two-Dimensional Aeroelastic System Riccy Kurniawan Department
More informationResearch Article Partial Pole Placement in LMI Region
Control Science and Engineering Article ID 84128 5 pages http://dxdoiorg/11155/214/84128 Research Article Partial Pole Placement in LMI Region Liuli Ou 1 Shaobo Han 2 Yongji Wang 1 Shuai Dong 1 and Lei
More informationMulti Degrees of Freedom Systems
Multi Degrees of Freedom Systems MDOF s http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano March 9, 07 Outline, a System
More informationQUANTIZATION OF THE ELECTRIC
Active and Passive Elec. Comp., 2001, Vol. 24, pp. 165--168 () 2001 OPA (Overseas Publishers Association) N.V. Reprints available directly from the publisher Published by license under Photocopying permitted
More informationResearch Article A Mathematical Images Group Model to Estimate the Sound Level in a Close-Fitting Enclosure
Advances in Acoustics and Vibration, Article ID 284362, 7 pages http://dx.doi.org/10.1155/2014/284362 Research Article A Mathematical Images Group Model to Estimate the Sound Level in a Close-Fitting Enclosure
More informationStructural Dynamics A Graduate Course in Aerospace Engineering
Structural Dynamics A Graduate Course in Aerospace Engineering By: H. Ahmadian ahmadian@iust.ac.ir The Science and Art of Structural Dynamics What do all the followings have in common? > A sport-utility
More informationLOW FREQUENCY NOISE OF TANTALUM CAPACITORS*
Active and Passive Elec. Comp., 2002, Vol. 25, pp. 161 167 LOW FREQUENCY NOISE OF TANTALUM CAPACITORS* J. SIKULA a,{, J. HLAVKA a, J. PAVELKA a, V. SEDLAKOVA a, L. GRMELA a, M. TACANO b and S. HASHIGUCHI
More informationCalculation of Eigensolution Derivatives for Nonviscously Damped Systems Using Nelson s Method
AIAA JOURNAL Vol. 44, No. 8, August 006 Calculation of Eigensolution Derivatives for Nonviscously Damped Systems Using Nelson s Method S. Adhikari and M. I. Friswell University of Bristol, Bristol, England
More informationReduction in number of dofs
Reduction in number of dofs Reduction in the number of dof to represent a structure reduces the size of matrices and, hence, computational cost. Because a subset of the original dof represent the whole
More informationA Robust Semi-Analytical Method for Calculating the Response Sensitivity of a Time Delay System
A Robust Semi-Analytical Method for Calculating the Response Sensitivity of a Time Delay System Mohammad H. Kurdi 1 Postdoctoral Associate e-mail: mhkurdi@gmail.com Raphael T. Haftka Distinguished Professor
More informationTransient Vibration Prediction for Rotors on Ball Bearings Using Load-Dependent Nonlinear Bearing Stiffness
Rotating Machinery, 10(6): 489 494, 2004 Copyright c Taylor & Francis Inc. ISSN: 1023-621X print / 1542-3034 online DOI: 10.1080/10236210490504102 Transient Vibration Prediction for Rotors on Ball Bearings
More informationResearch Article The Microphone Feedback Analogy for Chatter in Machining
Shock and Vibration Volume 215, Article ID 976819, 5 pages http://dx.doi.org/1.1155/215/976819 Research Article The Microphone Feedback Analogy for Chatter in Machining Tony Schmitz UniversityofNorthCarolinaatCharlotte,Charlotte,NC28223,USA
More informationDeflections and Strains in Cracked Shafts due to Rotating Loads: A Numerical and Experimental Analysis
Rotating Machinery, 10(4): 283 291, 2004 Copyright c Taylor & Francis Inc. ISSN: 1023-621X print / 1542-3034 online DOI: 10.1080/10236210490447728 Deflections and Strains in Cracked Shafts due to Rotating
More informationMagnetic damping for maglev 1,2
119 Magnetic damping for maglev 1,2 S. Zhu, Y. Cai, D.M. Rote and S.S. Chen Argonne National Laboratory, Argonne, IL 60439, USA Received 3 March 1998 Magnetic damping is one of the important parameters
More informationResearch Article Influence of the Parameterization in the Interval Solution of Elastic Beams
Structures Volume 04, Article ID 3953, 5 pages http://dx.doi.org/0.55/04/3953 Research Article Influence of the Parameterization in the Interval Solution of Elastic Beams Stefano Gabriele and Valerio Varano
More informationParametric Study of Greitzer s Instability Flow Model Through Compressor System Using the Taguchi Method
Rotating Machinery, 10: 91 97, 2004 Copyright c Taylor & Francis Inc. ISSN: 1023-621X print DOI: 10.1080/10236210490276683 Parametric Study of Greitzer s Instability Flow Model Through Compressor System
More informationCOMPUTER SIMULATION OF HYBRID INTEGRATED CIRCUITS INCLUDING COMBINED ELECTRICAL AND THERMAL EFFECTS
Electrocomponent Science and Technology, 1983, Vol. 10, pp. 171-176 (C) 1983 Gordon and Breach Science Publishers, Inc. 0305-3091/83/1003-0171 $18.50/0 Printed in Great Britain COMPUTER SIMULATION OF HYBRID
More informationResearch Article Emissivity Measurement of Semitransparent Textiles
Advances in Optical Technologies Volume 2012, Article ID 373926, 5 pages doi:10.1155/2012/373926 Research Article Emissivity Measurement of Semitransparent Textiles P. Bison, A. Bortolin, G. Cadelano,
More informationResearch Article Travel-Time Difference Extracting in Experimental Study of Rayleigh Wave Acoustoelastic Effect
ISRN Mechanical Engineering, Article ID 3492, 7 pages http://dx.doi.org/.55/24/3492 Research Article Travel-Time Difference Extracting in Experimental Study of Rayleigh Wave Acoustoelastic Effect Hu Eryi
More informationJUST THE MATHS UNIT NUMBER 9.9. MATRICES 9 (Modal & spectral matrices) A.J.Hobson
JUST THE MATHS UNIT NUMBER 9.9 MATRICES 9 (Modal & spectral matrices) by A.J.Hobson 9.9. Assumptions and definitions 9.9.2 Diagonalisation of a matrix 9.9.3 Exercises 9.9.4 Answers to exercises UNIT 9.9
More informationIdentification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016
Prof. Dr. Eleni Chatzi Lecture 4-09. March, 2016 Fundamentals Overview Multiple DOF Systems State-space Formulation Eigenvalue Analysis The Mode Superposition Method The effect of Damping on Structural
More informationResearch Article Trapped-Mode Resonance Regime of Thin Microwave Electromagnetic Arrays with Two Concentric Rings in Unit Cell
Microwave Science and Technology Volume 2, Article ID 3688, 6 pages doi:.55/2/3688 Research Article Trapped-Mode Resonance Regime of Thin Microwave Electromagnetic Arrays with Two Concentric Rings in Unit
More informationHorizontal bulk material pressure in silo subjected to impulsive load
Shock and Vibration 5 (28) 543 55 543 IOS Press Horizontal bulk material pressure in silo subjected to impulsive load Radosław Tatko a, and Sylwester Kobielak b a The Faculty of Environmental Engineering
More informationResearch Article Two Mathematical Models for Generation of Crowned Tooth Surface
e Scientific World Journal, Article ID 6409, 6 pages http://dx.doi.org/0.55/204/6409 Research Article Two Mathematical Models for Generation of Crowned Tooth Surface Laszlo Kelemen and Jozsef Szente University
More informationA GENERALIZED APPROACH FOR CALCULATION OF THE EIGENVECTOR SENSITIVITY FOR VARIOUS EIGENVECTOR NORMALIZATIONS
A GENERALIZED APPROACH FOR CALCULATION OF THE EIGENVECTOR SENSITIVITY FOR VARIOUS EIGENVECTOR NORMALIZATIONS A Thesis presented to the Faculty of the Graduate School University of Missouri - Columbia In
More informationDamage Assessment of the Z24 bridge by FE Model Updating. Anne Teughels 1, Guido De Roeck
Damage Assessment of the Z24 bridge by FE Model Updating Anne Teughels, Guido De Roeck Katholieke Universiteit Leuven, Department of Civil Engineering Kasteelpark Arenberg 4, B 3 Heverlee, Belgium Anne.Teughels@bwk.kuleuven.ac.be
More informationClustered natural frequencies in multi-span beams with constrained characteristic functions
Shock and Vibration 18 (2011) 697 707 697 DOI 10.3233/SAV-2010-0592 IOS Press Clustered natural frequencies in multi-span beams with constrained characteristic functions Khodabakhsh Saeedi and Rama B.
More informationSensitivity of Parameter Changes in Structural Damage Detection
C. H. Jenkins L. Kjerengtroen H. Oestensen Mechanical Engineering Department South Dakota School of Mines and Technology Rapid City, SD 57701 Sensitivity of Parameter Changes in Structural Damage Detection
More informationStability of Mass-Point Systems
Simulation in Computer Graphics Stability of Mass-Point Systems Matthias Teschner Computer Science Department University of Freiburg Demos surface tension vs. volume preservation distance preservation
More informationVibration control systems for sensitive equipment: Limiting performance and optimal design
Shock and Vibration 12 (2005) 37 47 37 IOS Press Vibration control systems for sensitive equipment: Limiting performance and optimal design V.M. Ryaboy Newport Corporation, 1791 Deere Avenue, Irvine, CA
More informationTEMPERATURE DEPENDENCE OF THE HALL COEFFICIENT OF THIN FILMS
Electrocomponent Science and Technology 1979, Vol. 6, pp. 19-22 0305-3091/79/0601-0019 $04.50/0 (C) 1979 Gordon and Breach Science Publishers, Inc. Printed in Great Britain TEMPERATURE DEPENDENCE OF THE
More informationResearch Letter An Algorithm to Generate Representations of System Identification Errors
Research Letters in Signal Processing Volume 008, Article ID 5991, 4 pages doi:10.1155/008/5991 Research Letter An Algorithm to Generate Representations of System Identification Errors Wancheng Zhang and
More informationComparison of Non-Parabolic Hydrodynamic Models
VLSI DESIGN 1998, Vol. 6, Nos. (1--4), pp. 177-180 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by
More informationSolution of Vibration and Transient Problems
. 9 Solution of Vibration and Transient Problems 9 Chapter 9: SOLUTION OF VIBRATION AND TRANSIENT PROBLEMS 9 9. MODAL APPROACH TO TRANSIENT ANALYSIS Consider the following large-order finite element model
More informationTransient Response Analysis of Structural Systems
. 21 Transient Response Analysis of Structural Systems 21 1 Chapter 21: TRANSIENT RESPONSE ANALYSIS OF STRUCTURAL SYSTEMS 21 2 21.1 MODAL APPROACH TO TRANSIENT ANALYSIS Consider the following large-order
More informationResearch Article On the Security of a Novel Probabilistic Signature Based on Bilinear Square Diffie-Hellman Problem and Its Extension
e Scientific World Journal, Article ID 345686, 4 pages http://dx.doi.org/10.1155/2014/345686 Research Article On the Security of a Novel Probabilistic Signature Based on Bilinear Square Diffie-Hellman
More informationResearch Article A New Flexibility Based Damage Index for Damage Detection of Truss Structures
Shock and Vibration Volume 214, Article ID 46692, 12 pages http://dx.doi.org/1.1155/214/46692 Research Article A New Flexibility Based Damage Index for Damage Detection of Truss Structures M. Montazer
More informationSensitivity analysis of viscoelastic structures
Shock and Vibration 13 (26) 545 558 545 IOS Press Sensitivity analysis of viscoelastic structures A.M.G. de Lima a, M.H. Stoppa b, D.A. Rade a, and V. Steffen. Jr. a a Federal University of Uberlândia,
More informationThe sensitivity analysis of the translation and the rotation angle of the first-order mode shape of the joints in frame structures
The sensitivity analysis of the translation and the rotation angle of the first-order mode shape of the joints in frame structures Yi Chen Yuan 1, Lin Li 2, Hongping Zhu 3 School of Civil Engineering and
More informationResearch Article The Application of Baum-Welch Algorithm in Multistep Attack
e Scientific World Journal, Article ID 374260, 7 pages http://dx.doi.org/10.1155/2014/374260 Research Article The Application of Baum-Welch Algorithm in Multistep Attack Yanxue Zhang, 1 Dongmei Zhao, 2
More informationInterval model updating: method and application
Interval model updating: method and application H. Haddad Khodaparast, J.E. Mottershead, K.J. Badcock University of Liverpool, School of Engineering, Harrison-Hughes Building, The Quadrangle, L69 3GH,
More informationWell pin Solar Cells: Dark Behavior
VLSI DESIGN 1998, Vol. 8, Nos. (1-4), pp. 419-422 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by
More informationA consistent dynamic finite element formulation for a pipe using Euler parameters
111 A consistent dynamic finite element formulation for a pipe using Euler parameters Ara Arabyan and Yaqun Jiang Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, AZ 85721,
More informationAPPROXIMATE DYNAMIC MODEL SENSITIVITY ANALYSIS FOR LARGE, COMPLEX SPACE STRUCTURES. Timothy S. West, Senior Engineer
APPROXIMATE DYNAMIC MODEL SENSITIVITY ANALYSIS FOR LARGE, COMPLEX SPACE STRUCTURES Timothy S. West, Senior Engineer McDonnell Douglas Aerospace Space Station Division, Houston, Texas ABSTRACT During the
More informationMatrix Iteration. Giacomo Boffi.
http://intranet.dica.polimi.it/people/boffi-giacomo Dipartimento di Ingegneria Civile Ambientale e Territoriale Politecnico di Milano April 12, 2016 Outline Second -Ritz Method Dynamic analysis of MDOF
More informationDesign Optimization of Gas Turbine Blades with Geometry and Natural Frequency Constraints
ES THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 E. 47 St., New York, N.Y. 10017 The Boger, mail riot be resaonslble for statements or opinions advanced in papers or in disousa n a! meetings o 4 the
More informationResearch Article Weather Forecasting Using Sliding Window Algorithm
ISRN Signal Processing Volume 23, Article ID 5654, 5 pages http://dx.doi.org/.55/23/5654 Research Article Weather Forecasting Using Sliding Window Algorithm Piyush Kapoor and Sarabjeet Singh Bedi 2 KvantumInc.,Gurgaon22,India
More informationA Compound Semiconductor Process Simulator and its
VLSI DESIGN 1998, Vol. 6, Nos. (1--4), pp. 393-397 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by
More information1792. On the conditions for synchronous harmonic free vibrations
1792. On the conditions for synchronous harmonic free vibrations César A. Morales Departamento de Mecánica, Universidad Simón Bolívar, Apdo. 89000, Caracas 1080A, Venezuela E-mail: cmorales@usb.ve (Received
More informationResearch Article Doppler Velocity Estimation of Overlapping Linear-Period-Modulated Ultrasonic Waves Based on an Expectation-Maximization Algorithm
Advances in Acoustics and Vibration, Article ID 9876, 7 pages http://dx.doi.org/.55//9876 Research Article Doppler Velocity Estimation of Overlapping Linear-Period-Modulated Ultrasonic Waves Based on an
More informationTWO-STAGE ISOLATION FOR HARMONIC BASE EXCITATION Revision A. By Tom Irvine February 25, 2008
TWO-STAGE ISOLATION FOR HARMONIC BASE EXCITATION Revision A By Tom Irvine Email: tomirvine@aol.com February 5, 008 Introduction Consider a base plate mass m and an avionics mass m modeled as two-degree-of-freedom.
More informationSoftware Verification
SAP000 EXAMPLE 6-00 LINK LINEAR LINK WITH RAMP LOADING PROBLEM DESCRIPTION In this example a ramp load is applied to an undamped single degree of freedom structure. The ramp loading has a finite rise time
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationFree vibration and stability of axially functionally graded tapered Euler-Bernoulli beams
Shock and Vibration 18 211 683 696 683 DOI 1.3233/SAV-21-589 IOS Press Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams Ahmad Shahba a,b, Reza Attarnejad a,b, and
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationA -SiC MOSFET Monte Carlo Simulator Including
VLSI DESIGN 1998, Vol. 8, Nos. (1-4), pp. 257-260 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by
More informationResearch Article Electromagnetic and Mechanical Characteristics Analysis of a Flat-Type Vertical-Gap Passive Magnetic Levitation Vibration Isolator
Shock and Vibration Volume 6, Article ID 577, pages http://dx.doi.org/.55/6/577 Research Article Electromagnetic and Mechanical Characteristics Analysis of a Flat-Type Vertical-Gap Passive Magnetic Levitation
More informationResearch Article A New Type of Magnetic Actuator Capable of Wall-Climbing Movement Using Inertia Force
Engineering Volume 14, Article ID 93178, 6 pages http://dx.doi.org/1.1155/14/93178 Research Article A New Type of Magnetic Actuator Capable of Wall-Climbing Movement Using Inertia Force H. Yaguchi, S.
More informationResearch Article Propagation Characteristics of Oblique Incident Terahertz Wave in Nonuniform Dusty Plasma
Antennas and Propagation Volume 216, Article ID 945473, 6 pages http://dx.doi.org/1.1155/216/945473 Research Article Propagation Characteristics of Oblique Incident Terahert Wave in Nonuniform Dusty Plasma
More informationStudy on the Performance of a Sirocco Fan (Flow Around the Runner Blade)
Rotating Machinery, 10(5): 415 424, 2004 Copyright c Taylor & Francis Inc. ISSN: 1023-621X print / 1542-3034 online DOI: 10.1080/10236210490474629 Study on the Performance of a Sirocco Fan (Flow Around
More information1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction
1. Multiple Degree-of-Freedom (MDOF) Systems: Introduction Lesson Objectives: 1) List examples of MDOF structural systems and state assumptions of the idealizations. 2) Formulate the equation of motion
More informationOptimal design and control of mechanical systems with uncertain input
hock and Vibration 19 (2012) 1019 1025 1019 DOI 10.3233/AV-2012-0708 IO Press Optimal design and control of mechanical systems with uncertain input P.P. Moita a, *, J.B. Cardoso b and A. Barreiros b a
More informationResearch Article On Counting and Embedding a Subclass of Height-Balanced Trees
Modelling and Simulation in Engineering, Article ID 748941, 5 pages http://dxdoiorg/101155/2014/748941 Research Article On Counting and Embedding a Subclass of Height-Balanced Trees Indhumathi Raman School
More informationTHICK FILM RESISTORS *
Electrocomponent Science and Technology, 1984, Vol. 11, pp. 185-190 (C) 1984 Gordon and Breach Science Publishers, Inc. 0305-3091/84/1102-0185 $18.50/0 Printed in Great Britain MODEL FOR RELIABILITY PREDICTION
More informationIdentification of damage in a beam structure by using mode shape curvature squares
Shock and Vibration 17 (2010) 601 610 601 DOI 10.3233/SAV-2010-0551 IOS Press Identification of damage in a beam structure by using mode shape curvature squares S. Rucevskis and M. Wesolowski Institute
More informationAA 242B / ME 242B: Mechanical Vibrations (Spring 2016)
AA 242B / ME 242B: Mechanical Vibrations (Spring 206) Solution of Homework #3 Control Tab Figure : Schematic for the control tab. Inadequacy of a static-test A static-test for measuring θ would ideally
More informationResearch Article Band Structure Engineering in 2D Photonic Crystal Waveguide with Rhombic Cross-Section Elements
Advances in Optical Technologies Volume 214, Article ID 78142, 5 pages http://dx.doi.org/1155/214/78142 Research Article Band Structure Engineering in 2D Photonic Crystal Waveguide with Rhombic Cross-Section
More informationSecond Order Newton Iteration Method
VLSI DESIGN 1998, Vol. 6, Nos. (1-4), pp. 141--145 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by
More informationIn-Structure Response Spectra Development Using Complex Frequency Analysis Method
Transactions, SMiRT-22 In-Structure Response Spectra Development Using Complex Frequency Analysis Method Hadi Razavi 1,2, Ram Srinivasan 1 1 AREVA, Inc., Civil and Layout Department, Mountain View, CA
More informationThe Quantum Hydrodynamic Smooth Effective Potential
VLSI DESIGN 1998, Vol. 6, Nos. (1-4), pp. 17-2 Repnnts available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by license
More informationTHE STATIC SUBSTRUCTURE METHOD FOR DYNAMIC ANALYSIS OF STRUCTURES. Lou Menglin* SUMMARY
264 THE STATIC SUBSTRUCTURE METHOD FOR DYNAMIC ANALYSIS OF STRUCTURES Lou Mengl* SUMMARY In this paper, the static substructure method based on the Ritz vector direct superposition method is suggested
More informationTheory of Vibrations in Stewart Platforms
Theory of Vibrations in Stewart Platforms J.M. Selig and X. Ding School of Computing, Info. Sys. & Maths. South Bank University London SE1 0AA, U.K. (seligjm@sbu.ac.uk) Abstract This article develops a
More informationResearch Article Study on Zero-Doppler Centroid Control for GEO SAR Ground Observation
Antennas and Propagation Article ID 549269 7 pages http://dx.doi.org/1.1155/214/549269 Research Article Study on Zero-Doppler Centroid Control for GEO SAR Ground Observation Yicheng Jiang Bin Hu Yun Zhang
More informationAA242B: MECHANICAL VIBRATIONS
AA242B: MECHANICAL VIBRATIONS 1 / 50 AA242B: MECHANICAL VIBRATIONS Undamped Vibrations of n-dof Systems These slides are based on the recommended textbook: M. Géradin and D. Rixen, Mechanical Vibrations:
More information2108. Free vibration properties of rotate vector reducer
2108. Free vibration properties of rotate vector reducer Chuan Chen 1, Yuhu Yang 2 School of Mechanical Engineering, Tianjin University, Tianjin, 300072, P. R. China 1 Corresponding author E-mail: 1 chenchuan1985728@126.com,
More informationVIBRATION PROBLEMS IN ENGINEERING
VIBRATION PROBLEMS IN ENGINEERING FIFTH EDITION W. WEAVER, JR. Professor Emeritus of Structural Engineering The Late S. P. TIMOSHENKO Professor Emeritus of Engineering Mechanics The Late D. H. YOUNG Professor
More informationOrder Reduction of Parametrically Excited Linear and Nonlinear Structural Systems
Venkatesh Deshmukh 1 Eric A. Butcher e-mail: ffeab@uaf.ediu Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775 S. C. Sinha Nonlinear Systems Research Laboratory,
More informationDifferential equations
Differential equations Math 7 Spring Practice problems for April Exam Problem Use the method of elimination to find the x-component of the general solution of x y = 6x 9x + y = x 6y 9y Soln: The system
More informationFIRST-ORDER PERTURBATION SOLUTION TO THE COMPLEX EIGENVALUES. Li Ji-ming (~ji:l~fj) Wang Wei (3_ ~ ) (Zhengzhou Institute of Technology, Zhengzhou)
Applied Mamatics Mechanics (English Edition, Vol.8, No.6, June 1987) Published by SUT, Shanghai, China FIRST-ORDER PERTURBATION SOLUTION TO THE COMPLEX EIGENVALUES Li Ji-ming (~ji:l~fj) Wang Wei (3_ ~
More informationShell-Filling Effects in Circular Quantum Dots
VLSI DESIGN 1998, Vol. 8, Nos. (1-4), pp. 443-447 Reprints available directly from the publisher Photocopying permitted by license only (C) 1998 OPA (Overseas Publishers Association) N.V. Published by
More information