Optimal control of reduced-order finite element models of rotor-bearing-support systems

Active tilting-pad journal bearings supporting flexible rotors: Part II–The model-based feedback-controlled lubrication

Abstract: This is part II of a twofold paper series dealing with the design and implementation of model-based controllers meant for assisting the hybrid and developing the feedback-controlled lubrication regimes in active tilting pad journal bearings (active TPJBs). In both papers theoretical and experimental analyses are presented with focus on the reduction of rotor lateral vibration. This part is devoted to synthesising model-based LQG optimal controllers (LQR regulator + Kalman Filter) for the feedback-controlled lubrication and is based upon the mathematical model of the rotor-bearing system derived in part I. Results show further suppression of resonant vibrations when using the feedback-controlled or active lubrication, overweighting the reduction already achieved with hybrid lubrication, thus improving the whole machine dynamic performance.

Vibration performance evaluation of planar flexible multibody systems with joint clearance

Abstract: Clearances are necessary in assemblage of mechanisms to allow the relative motion between the members. This clearance is due to machining tolerances, wear, material deformations, and imperfections, and it can worsen mechanism performance such as precision and vibration. As a new study in this topic, the effect of joint stiffness on the variation of instantaneous natural frequencies and mode shapes of a flexible four-bar mechanism with a clearance between coupler and follower is studied in this paper. To model the clearance, the continuous contact approach is used. The Lankarani’s and Nikravesh’s continuous contact force model is used to model the contact force arising from contact between journal and bearing. Finite element method is used to determine the instantaneous natural frequencies and their corresponding mode shapes. The stiffness of the clearance is modeled as a linear spring added to the assembled stiffness matrix. To validate the clearance model in rigid mechanism, the dynamic response is compared with the results in the literature. To show the validity of the formulation which calculates the instantaneous natural frequencies, two methods are used and compared with each other in the case no clearance exists. The results show that taking the joint stiffness into account has a considerable effect on the instantaneous natural frequencies and their corresponding mode shapes of a flexible multibody system.

Active vibration control of a rotor bearing system using flexible piezoelectric patch actuators

Abstract: Rotor vibration control is crucial for the reliability of rotating machines. This article applies active vibration control to reduce the vibration of a rotor bearing system using flexible piezoelec...

Dynamic behaviors of helical geared multishaft rotor systems by modal synthesis

Abstract: A theoretical investigation of dynamic behaviors of helical geared multishaft rotor systems is performed in this paper. A modal synthesis is developed to calculate free and forced vibrations for geared rotor systems including more than two shafts and gears. Degrees of freedom can be reduced to save the computing time in this method. Results of reduced model are compared with those of full degrees of freedom model, to obtain the acceptable reduction of degrees of freedom without significant loss of accuracy in predicting free and forced vibrations. Then, with the help of this method, dynamic behaviors of helical geared multishaft rotor systems are investigated. Parametric studies are conducted to reveal the effects of several system parameters on the vibration characteristics. Results show the reliability and accuracy of the modal synthesis as well as its limitations of calculating responses due to high-frequency excitations, and provide some references to designers attempting to obtain desirable dynamic b...

Bending fault evaluation for the HP-IP rotor system of the nuclear steam turbine based on the dynamic model

Abstract: With considering the unbalance mass-fluid film bearings-rotor elements, a dynamics model for the nuclear half-speed 1000 MW saturated steam turbine with No. Dongfang HN1089 is constructed. By solving of the Reynolds Equation and the dynamics model, the oil pressure distribution and dynamic coefficients of the fluid film bearings, and the unbalance response of the rotor, are obtained. The method for evaluating the bending fault based on the dynamics model is proposed, in which the bending parameter is transformed as the unbalance mass. A case on the bending fault evaluation for HN1089 is carried out. The results show that the response sensitivity of HN1089 on the unbalance mass is about 1/6 that of the thermal power units with the same capacity (1000 MW); and it is difficult to decrease the excited response from the bending fault even to add the maximum unbalance mass. In actual, the removing stress in the partial zone and turning method are applied to deal with the HP-IP rotor bending fault, and the response of the repaired rotor is 0.033 m by the actual field test. The results show the model and the method for evaluating the bending fault are accurate and reasonable, which will provide the important theoretical guide for fast and accurately dealing with such bending fault in the steam turbine rotor system.

Coupling of substructures for dynamic analyses.

Abstract: A method for treating a complex structure as an assemblage of distinct regions, or substructures, is presented. Using basic mass and stiffness matrices for the substructures, together with conditions of geometrical compatibility along substructure boundaries, the method employs two forms of generalized coordinates. Boundary generalized coordinates give displacements and rotations of points along substructure boundaries and are related to the displacement modes of the substructures known as "constraint modes." All constraint modes are generated by matrix operations from substructure input data. Substructure normal-mode generalized coordinates are related to free vibration modes of the substructures relative to completely restrained boundaries. The definition of substructure modes and the requirement of compatibility along substructure boundaries lead to coordinate transformation matrices that are employed in obtaining system mass and stiffness matrices from the mass and stiffness matrices of the substructures. Provision is made, through a RayleighRitz procedure, for reducing the total number of degrees of freedom of a structure while retaining accurate description of its dynamic behavior. Substructure boundaries may have any degree of redundancy. An example is presented giving a free vibration analysis of a structure having a highly indeterminate substructure boundary.

Reduction of stiffness and mass matrices

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.

A First Course in the Finite Element Method

Abstract: 1 INTRODUCTION Brief History Introduction to Matrix Notation Role of the Computer General Steps of the Finite Element Method Applications of the Finite Element Method Advantages of the Finite Element Method Computer Programs for the Finite Element Method 2 INTRODUCTION TO THE STIFFNESS (DISPLACEMENT) METHOD Definition of the Stiffness Matrix Derivation of the Stiffness Matrix for a Spring Element Example of a Spring Assemblage Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) Boundary Conditions Potential Energy Approach to Derive Spring Element Equations 3 DEVELOPMENT OF TRUSS EQUATIONS Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates Selecting Approximation Functions for Displacements Transformation of Vectors in Two Dimensions Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane Computation of Stress for a Bar in the x-y Plane Solution of a Plane Truss Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space Use of Symmetry in Structure Inclined, or Skewed, Supports Potential Energy Approach to Derive Bar Element Equations Comparison of Finite Element Solution to Exact Solution for Bar Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations Other Residual Methods and Their Application to a One-Dimensional Bar Problem Flowchart for Solutions of Three-Dimensional Truss Problems Computer Program Assisted Step-by-Step Solution for Truss Problem 4 DEVELOPMENT OF BEAM EQUATIONS Beam Stiffness Example of Assemblage of Beam Stiffness Matrices Examples of Beam Analysis Using the Direct Stiffness Method Distribution Loading Comparison of the Finite Element Solution to the Exact Solution for a Beam Beam Element with Nodal Hinge Potential Energy Approach to Derive Beam Element Equations Galerkin's Method for Deriving Beam Element Equations 5 FRAME AND GRID EQUATIONS Two-Dimensional Arbitrarily Oriented Beam Element Rigid Plane Frame Examples Inclined or Skewed Supports - Frame Element Grid Equations Beam Element Arbitrarily Oriented in Space Concept of Substructure Analysis 6 DEVELOPMENT OF THE PLANE STRESS AND STRAIN STIFFNESS EQUATIONS Basic Concepts of Plane Stress and Plane Strain Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations Treatment of Body and Surface Forces Explicit Expression for the Constant-Strain Triangle Stiffness Matrix Finite Element Solution of a Plane Stress Problem Rectangular Plane Element (Bilinear Rectangle, Q4) 7 PRACTICAL CONSIDERATIONS IN MODELING: INTERPRETING RESULTS AND EXAMPELS OF PLANE STRESS/STRAIN ANALYSIS Finite Element Modeling Equilibrium and Compatibility of Finite Element Results Convergence of Solution Interpretation of Stresses Static Condensation Flowchart for the Solution of Plane Stress-Strain Problems Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress-Strain Problems 8 DEVELOPMENT OF THE LINEAR-STRAIN TRAINGLE EQUATIONS Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations Example of LST Stiffness Determination Comparison of Elements 9 AXISYMMETRIC ELEMENTS Derivation of the Stiffness Matrix Solution of an Axisymmetric Pressure Vessel Applications of Axisymmetric Elements 10 ISOPARAMETRIC FORMULATION Isoparametric Formulation of the Bar Element Stiffness Matrix Isoparametric Formulation of the Okabe Quadrilateral Element Stiffness Matrix Newton-Cotes and Gaussian Quadrature Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature Higher-Order Shape Functions 11 THREE-DIMENSIONAL STRESS ANALYSIS Three-Dimensional Stress and Strain Tetrahedral Element Isoparametric Formulation 12 PLATE BENDING ELEMENT Basic Concepts of Plate Bending Derivation of a Plate Bending Element Stiffness Matrix and Equations Some Plate Element Numerical Comparisons Computer Solutions for Plate Bending Problems 13 HEAT TRANSFER AND MASS TRANSPORT Derivation of the Basic Differential Equation Heat Transfer with Convection Typical Units Thermal Conductivities K and Heat-Transfer Coefficients, h One-Dimensional Finite Element Formulation Using a Variational Method Two-Dimensional Finite Element Formulation Line or Point Sources Three-Dimensional Heat Transfer by the Finite Element Method One-Dimensional Heat Transfer with Mass Transport Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method Flowchart and Examples of a Heat-Transfer Program 14 FLUID FLOW IN POROUS MEDIA AND THROUGH HYDRAULIC NETWORKS AND ELECTRICAL NETWORKS AND ELECTROSTATICS Derivation of the Basic Differential Equations One-Dimensional Finite Element Formulation Two-Dimensional Finite Element Formulation Flowchart and Example of a Fluid-Flow Program Electrical Networks Electrostatics 15 THERMAL STRESS Formulation of the Thermal Stress Problem and Examples 16 STRUCTURAL DYNAMICS AND TIME-DEPENDENT HEAT TRANSFER Dynamics of a Spring-Mass System Direct Derivation of the Bar Element Equations Numerical Integration in Time Natural Frequencies of a One-Dimensional Bar Time-Dependent One-Dimensional Bar Analysis Beam Element Mass Matrices and Natural Frequencies Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices Time-Dependent Heat-Transfer Computer Program Example Solutions for Structural Dynamics APPENDIX A - MATRIX ALGEBRA Definition of a Matrix Matrix Operations Cofactor of Adjoint Method to Determine the Inverse of a Matrix Inverse of a Matrix by Row Reduction Properties of Stiffness Matrices APPENDIX B - METHODS FOR SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS Introduction General Form of the Equations Uniqueness, Nonuniqueness, and Nonexistence of Solution Methods for Solving Linear Algebraic Equations Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods APPENDIX C - EQUATIONS FOR ELASTICITY THEORY Introduction Differential Equations of Equilibrium Strain/Displacement and Compatibility Equations Stress-Strain Relationships APPENDIX D - EQUIVALENT NODAL FORCES APPENDIX E - PRINCIPLE OF VIRTUAL WORK APPENDIX F - PROPERTIES OF STRUCTURAL STEEL AND ALUMINUM SHAPES ANSWERS TO SELECTED PROBLEMS INDEX

A hybrid method of component mode synthesis

Abstract: A method is described for representing a structural component by means of its vibration modes. The modes used to describe the component may have the connection points to the remainder of the structure free, or fixed, or some points free and some points fixed. The modes may either be calculated or experimentally measured. Statically determined deflection influence coefficients may be used to improve the accuracy of the representation. The advantages claimed for the method derive from the generality of the conditions under which the component modes are calculated (or measured). Thus the boundary conditions may be selected to optimize accuracy or, in the event that the modes have already been obtained, the method permits the available data to be used. Examples are presented that illustrate use of the method, and the significance of the improvements derived from static calculations.

Turbomachinery Rotordynamics: Phenomena, Modeling, and Analysis

Abstract: Structural-Dynamic Models and Eigenanalysis for Undamped Flexible Rotors. Rotordynamic Introduction to Hydrodynamic Bearings and Squeeze-Film Dampers. Rotordynamic Models for Liquid Annular Seals. Rotordynamic Models for Annular Gas Seals. Rotordynamic Models for Turbines and Pump Impellers. Developing and Analyzing a System Rotordynamics Model. Example Rotor Analysis. Appendices. Index.