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Journal ArticleDOI

Reduction of stiffness and mass matrices

01 Feb 1965-AIAA Journal (American Institute of Aeronautics and Astronautics (AIAA))-Vol. 3, Iss: 2, pp 380-380
TL;DR: In this article, the authors proposed a method for reducing the size of the stiffness matrix by eliminating coordinates at which no forces are applied, based on the procedure used in Ref. 1 for stiffness matrix reduction.
Abstract: Just as it is often necessary to reduce the size of the stiff­ness matrix in statical structural analysis, the simulta­neous reduction of the nondiagonal mass matrix for natural mode analysis may also be required. The basis for one such reduction technique may follow the procedure used in Ref. 1 for the stiffness matrix, namely, the elimination of coordinates at which no forces are applied.
Citations
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Journal Article
TL;DR: A review of the development of the original modal assurance criterion (MAC) together with other related assurance criteria that have been proposed over the last twenty years can be found in this paper.
Abstract: This article reviews the development of the original modal assurance criterion (MAC) together with other related assurance criteria that have been proposed over the last twenty years. Some of the other assurance criteria that will be discussed include the coordinate modal assurance criterion (COMAC), the frequency response assurance criterion (FRAC), coordinate orthogonality check (CORTHOG), frequency scaled modal assurance criterion (FMAC), partial modal assurance criterion (PMAC), scaled modal assurance criterion (SMAC), and modal assurance criterion using reciprocal modal vectors (MACRV). In particular, the thought process that relates the original MAC development to ordinary coherence and to orthogonality computations will be explained. Several uses of MAC that may not be obvious to the casual observer (modal parameter estimation consistency diagrams and model updating are two examples) will be identified. The common problems with the implementation and use of modal assurance criterion computations will also be identified. The development of the modal assurance criterion 1-2 over twenty years ago has led to a number of similar assurance criteria used in the area of experimental and analytical structural dynamics. It is important to recognize the mathematical similarity of these varied criteria in order to be certain that conclusions be correctly drawn from what is essentially a squared, linear regression correlation coefficient. The modal assurance criterion is a statistical indicator, just like ordinary coherence, which can be very powerful when used correctly but very misleading when used incorrectly. This article will first review the historical development of the modal assurance criterion. Other similar assurance criteria will then be identified although the list is not intended to be comprehensive. Typical uses of the modal assurance criterion will be discussed and finally, typical abuses will be identified.

1,117 citations

Journal ArticleDOI
TL;DR: Four decades after the development of the first dynamic substructuring techniques, there is a necessity to classify the different methods in a general framework that outlines the relations between them, and a certain vision on substructureuring methods is proposed.
Abstract: Four decades after the development of the first dynamic substructuring techniques, there is a necessity to classify the different methods in a general framework that outlines the relations between them. In this paper, a certain vision on substructuring methods is proposed, by recalling important historical milestones that allow us to understand substructuring as a domain decomposition concept. Thereafter, based on the dual and primal assembly of substructures, a general framework for the classification of the methods is presented. This framework allows us to indicate how the various classes of methods, proposed along the years, can be derived from a clear mathematical description of substructured problems. Current bottlenecks in experimental dynamic substructuring, as well as solutions found in literature, will also be briefly discussed.

668 citations

Journal ArticleDOI
TL;DR: In this paper, a new structure (shell or plate) containing an integrated distributed piezoelectric sensor and actuator is proposed, where the distributed sensing layer monitors the structural oscillation due to the direct PDE and the distributed actuator layer suppresses the oscillation via the converse PDE.

642 citations

Book
01 Jan 1990
TL;DR: In this article, the finite element displacement method was used for the analysis of free vibration of plates and shells, and for the simulation of forced response and forced response analysis of rigid and flexible plates.
Abstract: 1 Formulation of the equations of motion 2 Element energy functions 3 Introduction to the finite element displacement method 4 In-plane vibration of plates 5 Vibration of solids 6 Flexural vibration of plates 7 Vibration of stiffened plates and folded plate structures 8 Vibration of shells 9 Vibration of laminated plates and shells 10 Hierarchical finite element method 11 Analysis of free vibration 12 Forced response 13 Forced response II 14 Computer analysis technique

592 citations

Journal ArticleDOI
Alex Berman1, E. J. Nagy1
TL;DR: In this paper, a method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure, which directly identifies, without iteration, a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements.
Abstract: A method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure. The method directly identifies, without iteration, a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements. An application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure. The resulting changes in the model are judged to be small compared to expectations of error in the analysis. Thus, the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics. In addition, it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure. Nomenclature - analytical matrix = matrix of changes = identity matrix = full improved stiffness and mass matrices (n x n) = full analytical K, M matrices (n x n) = partitions of KA,MA corresponding to test coordinates = partitions of KA,MA corresponding to coupling elements = partitions of KA,MA corresponding to unmea- sured coordinates = number of measured modes = number of degrees of freedom in the model = measures of changes, Eqs. (15-17) = matrix norm, sum of the squares of all the elements = rectangular modal matrix, normalized (n x m) = /th mode, /th column of $ = measured and unmeasured partitions of ,- = diagonal matrix of measured natural frequencies (m xm) = natural frequency of /th mode = 12 17 = sum of the squares of all elements of matrix ( )

568 citations

References
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Journal ArticleDOI
John S. Archer1
TL;DR: In this article, a technique was developed for constructing a consistent mass matrix for vibration analysis of general structural configurations, consistent with standard procedures for stiffness matrix formulation of structural problems, but accounts for the actual mass distribution within the structure in a manner similar to the Rayleigh-Ritz formulation.
Abstract: A technique is developed for constructing a consistent mass matrix for vibration analysis of general structural configurations. The approach is consistent with standard procedures for stiffness matrix formulation of structural problems, but accounts for the actual mass distribution within the structure in a manner similar to the Rayleigh-Ritz formulation. The basic equation for computing the mass matrix coefficients is given a physical interpretation. It is shown that the natural frequencies obtained by using the consistent mass matrix are upper bounds to the exact solution. The procedure is applicable to general dynamic response analysis and is demonstrably superior to the usual procedure of physical mass lumping by application to frequency analysis of free-free and simply-supported prismatic beams with uniformly distributed mass. It is observed that if lateral translation coordinates alone are used to represent the distortion of a uniform beam, the number of individual beam segments must exceed by one the number of natural mode frequencies it is desired to closely approximate.

132 citations

Journal ArticleDOI
TL;DR: In this article, a collimated total radiation probe for arcjet measurements was proposed for high temperature measurement, which was used to measure the radiation from an argon arc jet using a partially ionized gas.
Abstract: 1 Forstall, W, Jr and Shapiro, A H, "Momentum and mass transfer in coaxial gas jets, J Appl Mech 10, 399-408 (December 1950) 2 Pai, S I , Fluid Dynamics of Jets (D Van Nostrand Co , Inc , Princeton, N J , 1954), Chaps 5-7, pp 96-173 3 Willis, D R and Glassman, I , "The mixing of unbounded coaxial compressible streams/ Jet Propulsion 27, 1241-1248 (1957) 4 Pitkin, E T and Glassman, I , "Experimental mixing profiles of a Mach 2 6 free jet," J Aerospace Sci 25, 791-792 (1958) 5 Grey, J , Jacobs, P F , and Sherman, M P , "Calorimetric probe for the measurement of extremely high temperatures, ' Rev Sci Instr 33, 738-741 (July 1962) 6 Grey, J , "Thermodynamic methods of high temperature measurement," Am Soc Mech Engrs Annual Meeting, Philadelphia, Pa (November 18, 1963); also Instr Soc Am (submitted for publication) 7 Sherman, M P , Jacobs, P F , and Grey, J , "Analytical and experimental study of radiation from an argon arc jet," Princeton Univ Aeronaut Eng Rept 658 (July 1963) 8 Grey, J , Sherman, M P , and Jacobs, P F , "A collimated total radiation probe for arcjet measurements,' Inst Elec Electron Engrs Intern Symp Plasma Phenomena and Measurements, San Diego, Calif (October 31, 1963); also Inst Elec Electron Engrs Trans Nucl Sci (January 1964) 9 Grey, J and Jacobs, P F , ' Turbulent mixing in a partially ionized gas," Princeton Univ Aeronaut Eng Rept 625 (September 1962) 10 Sherman, M P and Grey, J , "The degree of approach to equilibrium in an atmospheric-pressure arcjet using argon, ' Princeton Univ Aeronaut Eng Rept 645 (April 1963) 11 Cann, G L and Ducati, A C , "Argon mollier chart, ' Air Force Office Sci Res AF 49(638)-54 (February 1959) 12 Bosnjakovic, R , Springe, W, Knoche, K F , and Burghoi te, P , "Mollier Enthalpie—Entropie Diagramm fur Argon Plasma im Gleichgewicht, ' transl by J F Gross, Warmetechnische Institut, Technische Hochschule, Braunschweig (August 1958) 13 Squire, H B and Trouncer, J, "Round jets in a general stream, ' Brit Aeronaut Res Comm Repts and Memo 1974, pp 9-31 (January 1944) 14 Corrsin, S and Uberoi, M S , "Further experiments on the flow and heat transfer in a heated turbulent air jet," NACA Rept 998(1950)

38 citations

Journal ArticleDOI
TL;DR: In this article, the influence of viscous drag on the lift-drag ratio of slender blunt cones is shown experimentally and theoretically, and the significant contributions to the zero-lift, viscous charge rise of cold-wall cones are identified as the usual "similar" or Blasius friction drag and transverse curvature effects.
Abstract: Experimental drag data from a series of cone models are presented over a wide range of Reynolds numbers at hypersonic flow conditions. The data include Mach numbers from 9 to 22 with Reynolds numbers based on model length ranging from 600 to 500,000. Most data were obtained with test model surface temperature cold relative to the stagnation temperature; a limited amount of hot-wall data were obtained. The influence of viscous drag on the lift-drag ratio of slender blunt cones is shown experimentally and theoretically. Drag reduction by boundary-layer gas injection is shown experimentally for both coldand hot-wall test conditions. Theoretical estimates based on existing theories are given, and the significant contributions to the zero-lift, viscous drag rise of cold-wall cones are identified as the usual "similar" or Blasius friction drag and transverse curvature effects. The theoretical estimates are shown to offer a good engineering approximation to the hypersonic viscous drag of coldwall, blunt slender cones within the con tint; um flow regime.

35 citations

Journal ArticleDOI
TL;DR: In this article, the drag, stability, and wake characteristics of slender cones and spheres at very high speeds were investigated using a mean viscous interaction parameter and good agreement was obtained between the experimentally determined drag and stability coefficients.
Abstract: Tests have been conducted in the Naval Ordnance Laboratory's ballistics ranges to investigate the drag, stability, and wake characteristics of slender cones and spheres at very high speeds Drag coefficients were found to be linear functions of the mean squared angle of attack Zero angle of attack drag coefficients were correlated using a mean viscous interaction parameter Stability derivatives were found to be independent of the mean squared angle of attack Good agreement was obtained between the experimentally determined drag and stability coefficients and theory The width of the turbulent viscous core behind both cones and spheres was found to be proportional to the cube root of (xCDA) The constant of proportionality was the same for cones and spheres and varied with flight Mach number Wake transition Reynolds numbers were found to be independent of the freestream body Reynolds number for a constant Mach number, but strongly dependent on flight Mach number

31 citations