Numerical Analysis of Vibrations of Structures under Moving Inertial Load

Numerical Analysis of Vibrations of Structures under Moving Inertial Load
Czeslaw I. Bajer and Bartlomiej Dyniewicz
Contents
1 Introduction .
1.1 Literature Review
1.2 Solution Methods
1.3 Approximate Methods
1.4 Review of Analytical-Numerical Methods in Moving Load
Problems .
1.4.1 d’Alembert Method
1.4.2 Fourier Method
1.4.3 Lagrange Formulation
1.5 Examples .
2 Analytical Solutions
2.1 A Massless String under a Moving Inertial Load .
2.1.1 Case of ? = 1 .
2.1.2 Case of ? = 1 .
2.2 Discontinuity of the Solution .
2.3 Conclusions .
3 Semi-analytical Methods
3.1 String
3.1.1 Fourier Analysis .
3.1.2 The Lagrange Equation .
3.2 Bernoulli–Euler Beam
3.2.1 Fourier Solution .
3.2.2 The Lagrange Equation of the Second Kind
3.2.3 Conclusions .
3.3 Timoshenko Beam
3.3.1 Fourier Solution .
3.3.2 The Lagrange Equation .
3.3.3 Examples .
3.3.4 Conclusions and Discussion .
3.4 Bernoulli–Euler Beam vs. Timoshenko Beam .
3.5 Plate .
3.6 The Renaudot Approach vs. The Yakushev Approach
3.6.1 The Renaudot Approach
3.6.2 The Yakushev Approach
4 Review of Numerical Methods of Solution
4.1 Oscillator .
4.1.1 String Vibrations under a Moving Oscillator
4.1.2 Beam Vibrations under a Moving Oscillator
4.2 Inertial Load .
4.2.1 A Bernoulli–Euler Beam Subjected to an Inertial Load .
4.2.2 A Timoshenko Beam Subjected to an Inertial Load
5 Classical Numerical Methods of Time Integration .
5.1 Integration of the First Order Differential Equations
5.2 Single-Step Method SSpj
5.3 Central Difference Method .
5.3.1 Stability of the Method .
5.3.2 Accuracy of the Method
5.4 The Adams Methods
5.4.1 Explicit Adams Formulas (Open) .
5.4.2 Implicit Adams Formulas (Closed) .
5.5 The Newmark Method
5.6 The Bossak Method
5.7 The Park Method .
5.8 The Park–Housner Method
5.8.1 Stability of the Park–Housner Method .
5.9 The Trujillo Method
6 Space–Time Finite Element Method
6.1 Formulation of the Method—Displacement Approach .
6.1.1 Space–Time Finite Elements in the Displacement
Description .
6.2 Properties of the Integration Schemes
6.2.1 Accuracy of Methods .
6.3 Velocity Formulation of the Method .
6.3.1 One Degree of Freedom System
6.3.2 Discretization of the Differential Equation of String
Vibrations
6.3.3 General Case of Elasticity .
6.3.4 Other Functions of the Virtual Velocity
6.4 Space–Time Element Method and Other Time Integration
Methods
6.4.1 Convergence
6.4.2 Phase Error .
6.4.3 Non-inertial Problems
6.5 Space–Time Finite Element Method vs. Newmark Method
6.6 Simplex Elements
6.6.1 Property of Space Division
6.6.2 Numerical Efficiency .
6.7 Simplex Elements in the Displacement Description
6.7.1 Triangular Element of a Bar Vibrating Axially .
6.7.2 Space–Time Finite Element of the Beam of Moderate
Height .
6.7.3 Tetrahedral Space–Time Element of a Plate
6.8 Triangular Elements Expressed in Velocities
7 Space–Time Finite Elements and a Moving Load
7.1 Space–Time Finite Element of a String
7.1.1 Discretization of the String Element Carrying a Moving
Mass .
7.1.2 Numerical Results .
7.1.3 Conclusions .
7.2 Space–Time Elements for a Bernoulli–Euler Beam Carrying
a Moving Mass
7.2.1 Numerical Results .
7.3 Space–Time Element of Timoshenko Beam Carrying a Moving
Mass .
7.3.1 Conclusions .
7.4 Space–Time Finite Plate Element Carrying a Moving Mass .
7.4.1 Thin Plate
7.4.2 Thick Plate .
7.4.3 Plate Placed on an Elastic Foundation .
7.5 Problems with Zero Mass Density .
8 The Newmark Method and a Moving Inertial Load
8.1 The Newmark Method in Moving Mass Problems .
8.2 The Newmark Method in the Vibrations of String
8.3 The Newmark Method in Vibrations of the Bernoulli–Euler
Beam
8.4 The Newmark Method in Vibrations of a Timoshenko Beam
8.5 Numerical Results
8.6 Accelerating Mass—Numerical Approach
8.6.1 Mathematical Model .
8.6.2 The Finite Element Carrying the Moving Mass Particle
8.6.3 Accelerating Mass—Examples .
8.7 Conclusions .
9 Meshfree Methods in Moving Load Problems
9.1 Meshless Methods (Element-Free Galerkin Method)
9.2 Results
10 Examples of Applications
10.1 Dynamics of the Classical Vehicle–Track System .
10.2 Dynamics of the System Vehicle—Y-Type Track .
10.3 Dynamics of Subway Track .
10.4 Vibrations of Airport Runways
Appendix .
A Computer Programs .
A.1 String—Space–Time Element Method
A.2 Timoshenko Beam—Newmark Method .
A.3 Mindlin Plate—Space–Time Element Method
A.4 Kirchhoff Plate — Space-Time Element Method .
References
Index
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