Free and forced vibrations of stepped rods and coupled systems
TL;DR: In this paper, a new two-way state-flow model based on the dynamics of the rod is developed and the exact frequency response of both the stepped rod and the coupled system can be directly written out according to the graph theory.
About: This article is published in Journal of Sound and Vibration.The article was published on 1999-10-07. It has received 12 citations till now. The article focuses on the topics: Frequency response & Boundary value problem.
Citations
More filters
TL;DR: In this paper, a unified analytical solution of the wave equation governing the propagation of the longitudinal stress wave in an elastic rod is proposed, which can be regarded as an exact interpolation function in the time domain.
21 citations
TL;DR: In this paper, a new analytical method is developed for transient vibration analysis of stepped systems composed of distributed components like elastic bars, flexible shafts and taut strings, and lumped masses.
19 citations
TL;DR: In this paper, an analytical and closed-form vibration transmissibility of a general unidirectional multi-degree-of-freedom (MDOF) system with multiple dynamic absorbers is studied.
14 citations
TL;DR: In this paper, the propagation characteristics of longitudinal wave are investigated when the signals pass through a multi-step rod with variable cross-section, and several numerical examples for a step-rod with three segments are given by the present theoretical predictions and the finite element analysis, from which the performance of the three wave theories are compared with each other.
Abstract: Vibration signals collected after propagation are usually different from the real ones of some vibration source due to the corresponding stop band of a particular propagation media. Here, the propagation characteristics of longitudinal wave are investigated when the signals pass through a multi-step rod with variable cross-section. Firstly, the transfer matrices for a single rod are derived based on the three wave theories, i.e. the elementary wave theory, the Love wave theory and the Mindlin-Herrmann wave theory, respectively. Next, the matching conditions for any two adjacent segments are analyzed, from which the transfer matrix for any step-rod with two or more segments are presented. Moreover, possible elastic constraint is also taken into consideration. Finally, several numerical examples for a step-rod with three segments are given by the present theoretical predictions and the finite element analysis, from which the performance of the three wave theories are compared with each other. The results sh...
14 citations
Cites background from "Free and forced vibrations of stepp..."
...Hsueh (1999) proposed the analytical and closed forms of the frequency responses of longitudinal vibration for an N-stepped rod and a system coupled with multi-stepped rods and lumped elements under nonclassical boundary conditions....
[...]
TL;DR: In this paper, large amplitude forced vibration behavior of thin beams under harmonic excitation was studied, incorporating the effect of geometric nonlinearity, and large amplitude free vibration analysis of the same beam was performed.
Abstract: Large amplitude forced vibration behavior of thin beams under harmonic excitation is studied, incorporating the effect of geometric nonlinearity. Large amplitude free vibration analysis of the same...
12 citations
Cites methods from "Free and forced vibrations of stepp..."
...Hsueh (1999) obtained Department of Mechanical Engineering, Jadavpur University, Kolkata, India Corresponding author: Prasanta Sahoo, Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India Email: psjume@gmail.com analytical and closed-form frequency responses for the…...
[...]
References
More filters
01 Jan 1975
TL;DR: In this article, a single-degree-of-freedom (SDF) dynamic system is considered, and the effect of different degrees of freedom on the dynamics of the system is investigated.
Abstract: TABLE OF CONTENTS PREFACE 1 INTRODUCTION 1.1 Objectives of the Study of Structural Dynamics 1.2 Importance of Vibration Analysis 1.3 Nature of Exciting Forces 1.4 Mathematical Modeling of Dynamic Systems 1.5 Systems of Units 1.6 Organization of the Text PART I 2 FORMULATION OF THE EQUATIONS OF MOTION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS 2.1 Introduction 2.2 Inertia Forces 2.3 Resultants of Inertia Forces on a Rigid Body 2.4 Spring Forces 2.5 Damping Forces 2.6 Principle of Virtual Displacement 2.7 Formulation of the Equations of Motion 2.8 Modeling of Multi Degree-of-Freedom Discrete Parameter System 2.9 Effect of Gravity Load 2.10 Axial Force Effect 2.11 Effect of Support Motion 3 FORMULATION OF THE EQUATIONS OF MOTION: MULTI-DEGREE-OF-FREEDOM SYSTEMS 3.1 Introduction 3.2 Principal Forces in Multi Degree-of-freedom Dynamic System 3.3 Formulation of the Equations of Motion 3.4 Transformation of Coordinates 3.5 Static Condensation of Stiffness matrix 3.6 Application of Ritz Method to Discrete Systems 4 PRINCIPLES OF ANALYTICAL MECHANICS 4.1 Introduction 4.2 Generalized coordinates 4.3 Constraints 4.4 Virtual Work 4.5 Generalized Forces 4.6 Conservative Forces and Potential Energy 4.7 Work Function 4.8 Lagrangian Multipliers 4.9 Virtual Work Equation For Dynamical Systems 4.10 Hamilton's Equation 4.11 Lagrange's Equation 4.12 Constraint Conditions and Lagrangian Multipliers 4.13 Lagrange's Equations for Discrete Multi-Degree-of-Freedom Systems 4.14 Rayleigh's Dissipation Function PART II 5 FREE VIBRATION RESPONSE: SINGLE-DEGREE-OF-FREEDOM SYSTEM 5.1 Introduction 5.2 Undamped Free Vibration 5.3 Free Vibrations with Viscous Damping 5.4 Damped Free vibration with Hysteretic Damping 5.5 Damped Free vibration with Coulomb Damping 6 FORCED HARMONIC VIBRATIONS: SINGLE-DEGREE-OF-FREEDOM SYSTEM 6.1 Introduction 6.2 Procedures for the Solution of Forced Vibration Equation 6.3 Undamped Harmonic Vibration 6.4 Resonant Response of an Undamped System 6.5 Damped Harmonic Vibration 6.6 Complex Frequency Response 6.7 Resonant Response of a Damped System 6.8 Rotating Unbalanced Force 6.9 Transmitted Motion due to Support Movement 6.10 Transmissibility and Vibration Isolation 6.11 Vibration Measuring Instruments 6.12 Energy Dissipated in Viscous Damping 6.13 Hysteretic Damping 6.14 Complex Stiffness 6.15 Coulomb Damping 6.16 Measurement of Damping 7 RESPONSE TO GENERAL DYNAMIC LOADING AND TRANSIENT RESPONSE 7.1 Introduction 7.2 Response to an Impulsive force 7.3 Response to General Dynamic Loading 7.4 Response to a Step Function Load 7.5 Response to a Ramp Function Load 7.6 Response to a Step Function Load With Rise Time 7.7 Response to Shock Loading 7.8 Response to a Ground Motion Pulse 7.9 Analysis of Response by the Phase Plane Diagram 8 ANALYSIS OF SINGLE-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 8.1 Introduction 8.2 Conservation of Energy 8.3 Application of Rayleigh Method to Multi Degree of Freedom Systems 8.4 Improved Rayleigh Method 8.5 Selection of an Appropriate Vibration Shape 8.6 Systems with Distributed Mass and Stiffness: Analysis of Internal Forces 8.7 Numerical Evaluation of Duhamel's Integral 8.8 Direct Integration of the Equations of Motion 8.9 Integration Based on Piece-wise Linear Representation of the Excitation 8.10 Derivation of General Formulae 8.11 Constant Acceleration Method 8.12 Newmark's beta Method 8.13 Wilson-theta Method 8.14 Methods Based on Difference Expressions 8.15 Errors involved in Numerical Integration 8.16 Stability of the Integration Method 8.17 Selection of a Numerical Integration Method 8.18 Selection of Time Step 9 ANALYSIS OF RESPONSE IN THE FREQUENCY DOMAIN 9.1 Transform Methods of Analysis 9.2 Fourier Series Representation of a Periodic Function 9.3 Response to a Periodically Applied Load 9.4 Exponential Form of Fourier Series 9.5 Complex Frequency Response Function 9.6 Fourier Integral Representation of a Nonperiodic Load 9.7 Response to a Nonperiodic Load 9.8 Convolution Integral and Convolution Theorem 9.9 Discrete Fourier Transform 9.10 Discrete Convolution and Discrete Convolution Theorem 9.11 Comparison of Continuous and Discrete Fourier Transforms 9.12 Application of Discrete Inverse Transform 9.13 Comparison Between Continuous and Discrete Convolution 9.14 Discrete Convolution of an Infnite and a Finite duration Waveform 9.15 Corrective Response Superposition Methods 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution PART III 10 FREE VIBRATION RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEM 10.1 Introduction 10.2 Standard Eigenvalue Problem 10.3 Linearized Eigenvalue Problem and its Properties 10.4 Expansion Theorem 10.5 Rayleigh Quotient 10.6 Solution of the Undamped Free-Vibration Problem 10.7 Mode Superposition Analysis of Free-Vibration Response 10.8 Solution of the Damped Free-Vibration Problem 10.9 Additional Orthogonality Conditions 10.10 Damping Orthogonality 11 NUMERICAL SOLUTION OF THE EIGENPROBLEM 11.1 Introduction 11.2 Properties of Standard Eigenvalues and Eigenvectors 11.3 Transformation of a Linearized Eigenvalue Problem to the Standard Form 11.4 Transformation Methods 11.5 Iteration Methods 11.6 Determinant Search Method 11.7 Numerical Solution of Complex Eigenvalue Problem 11.8 Semi-definite or Unrestrained Systems 11.9 Selection of a Method for the Determination of Eigenvalues 12 FORCED DYNAMIC RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEMS 12.1 Introduction 12.2 Normal Coordinate Transformation 12.3 Summary of Mode Superposition Method 12.4 Complex Frequency Response 12.5 Vibration Absorbers 12.6 Effect of Support Excitation 12.7 Forced Vibration of Unrestrained System 13 ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 13.1 Introduction 13.2 Rayleigh-Ritz Method 13.3 Application of Ritz Method to Forced Vibration Response 13.4 Direct Integration of the Equations of Motion 13.5 Analysis in the Frequency Domain PART IV 14 FORMULATION OF THE EQUATIONS OF MOTION: CONTINUOUS SYSTEMS 14.1 Introduction 14.2 Transverse Vibrations of a Beam 14.3 Transverse Vibrations of a Beam: Variational Formulation 14.4 Effect of Damping Resistance on Transverse Vibrations of a Beam 14.5 Effect of Shear Deformation and Rotatory Inertia on the Flexural Vibrations of a Beam 14.6 Axial Vibrations of a Bar 14.7 Torsional Vibrations of a Bar 14.8 Transverse Vibrations of a String 14.9 Transverse Vibration of a Shear Beam 14.10 Transverse Vibrations of a Beam Excited by Support Motion 14.11 Effect of Axial Force on Transverse Vibrations of a Beam 15 CONTINUOUS SYSTEMS: FREE VIBRATION RESPONSE 15.1 Introduction 15.2 Eigenvalue Problem for the Transverse Vibrations of a Beam 15.3 General Eigenvalue Problem for a Continuous System 15.4 Expansion Theorem 15.5 Frequencies and Mode Shapes for Lateral Vibrations of a Beam 15.6 Effect of Shear Deformation and Rotatory Inertia on the Frequencies of Flexural Vibrations 15.7 Frequencies and Mode Shapes for the Axial Vibrations of a Bar 15.8 Frequencies and Mode Shapes for the Transverse Vibration of a String 15.9 Boundary Conditions Containing the 15.10 Free-Vibration Response of a Continuous System 15.11 Undamped Free Transverse Vibrations of a Beam 15.12 Damped Free Transverse Vibrations of a Beam 16 CONTINUOUS SYSTEMS: FORCED-VIBRATION RESPONSE 16.1 Introduction 16.2 Normal Coordinate Transformation: General Case of an Undamped System 16.3 Forced Lateral Vibration of a Beam 16.4 Transverse Vibrations of a Beam Under Traveling Load 16.5 Forced Axial Vibrations of a Uniform Bar 16.6 Normal Coordinate Transformation, Damped Case 17 WAVE PROPAGATION ANALYSIS 17.1 Introduction 17.2 The Phenomenon of Wave Propagation 17.3 Harmonic Waves 17.4 One Dimensional Wave Equation and its Solution 17.5 Propagation of Waves in Systems of Finite Extent 17.6 Reection and Refraction of Waves at a Discontinuity in the System Properties 17.7 Characteristics of the Wave Equation 17.8 Wave Dispersion PART V 18 FINITE ELEMENT METHOD 18.1 Introduction 18.2 Formulation of the Finite Element Equations 18.3 Selection of Shape Functions 18.4 Advantages of the Finite Element Method 18.5 Element Shapes 18.6 One-dimensional Bar Element 18.7 Flexural Vibrations of a Beam 18.8 Stress-strain Relationship for a Continuum 18.9 Triangular Element in Plane Stress and Plane Strain 18.10 Natural Coordinates 19 COMPONENT MODE SYNTHESIS 19.1 Introduction 19.2 Fixed Interface Methods 19.3 Free Interface Method 19.4 Hybrid Method 20 ANALYSIS OF NONLINEAR RESPONSE 20.1 Introduction 20.2 Single-degree-of-freedom System 20.3 Errors involved in Numerical Integration of Nonlinear Systems 20.4 Multiple Degree-of-freedom System ANSWERS TO SELECTED PROBLEMS INDEX
5,044 citations
Book•
07 Jun 1995
TL;DR: Striking a balance between theory and applications, Linear System Theory and Design, 3/e, is ideal for use in advanced undergraduate/first-year graduate courses in linear systems and multivariable system design in electrical, mechanical, chemical, and aeronautical engineering departments.
Abstract: From the Publisher:
An extensive revision of the author's highly successful text, this third edition of Linear System Theory and Design has been made more accessible to students from all related backgrounds. After introducing the fundamental properties of linear systems, the text discusses design using state equations and transfer functions. The two main objectives of the text are to: use simple and efficient methods to develop results and design procedures; enable students to employ the results to carry out design. Striking a balance between theory and applications, Linear System Theory and Design, 3/e, is ideal for use in advanced undergraduate/first-year graduate courses in linear systems and multivariable system design in electrical, mechanical, chemical, and aeronautical engineering departments. It assumes a working knowledge of linear algebra and the Laplace transform and an elementary knowledge of differential equations.
4,017 citations
Book•
01 Jan 1975
TL;DR: In this article, a comprehensive study of elastic wave propagation in solids is presented, ranging from the theory of waves and vibrations in strings to the three-dimensional theory of elastic waves in thick plates.
Abstract: The book presents a comprehensive study of elastic wave propagation in solids. Topics covered range from the theory of waves and vibrations in strings to the three-dimensional theory of waves in thick plates. The subject is covered in the following chapters: (1) waves and vibrations in strings, (2) longitudinal waves in thin rods, (3) flexural waves in thin rods, (4) waves in membranes, thin plates and shells, (5) waves in infinite media, (6) waves in semi-infinite media, (7) scattering and diffraction of elastic waves, and (8) wave propagation in plates and rods. Appendices contain introductory information on elasticity, transforms and experimental techniques. /TRRL/
3,359 citations
Book•
01 Jan 1975
TL;DR: In this paper, an overview of structural dynamics analysis of free vibrations response to harmonic loading response, periodic loading response to impulse loading response and general dynamic loading -step by step methods, superposition methods generalized single degree-of-freedom systems.
Abstract: Part 1 Single-degree-of-freedom systems: overview of structural dynamics analysis of free vibrations response to harmonic loading response to periodic loading response to impulse loading responses to general dynamic loading - step by step methods, superposition methods generalized single degree-of-freedom systems. Part 2 Multi-degree-of-freedom systems: formulation of the MDOF equations of motion evaluation of structural-property matrices undamped free vibrations analysis of dynamic response using superposition vibration analysis by matrix iteration selection of dynamic degrees of freedom analysis of MDOF dynamic response - step by step methods variational formulation of the equations of motion. Part 3 Distributed parameter systems: partial differential equations of motion analysis of undamped free vibrations analysis if dynamic response. Part 4 Random vibrations: probability theory random processes stochastic response of linear SDOF systems stochastic response of non-linear MDOF systems. Part 5 Earthquake engineering: seismological background free-field surface ground motions deterministic structural response - including soil-structure interaction stochastic structural response.
1,627 citations
01 Jul 1956
TL;DR: A way to enhance writing gain at a glance as discussed by the authors is to use the phrase "pray not frustrative" as a way to describe the goal of a sentence in a paragraph.
Abstract: A way to enhance Writing gain at a glance. Dr. Tustin extended Proof appended. Examples illustrative Pray not frustrative.
643 citations