A Contribution to Moving Force Identification in Bridge Dynamics

Acknowledgement
Abstract
Zusammenfassung
Contents
List of Figures
List of Tables
Glossaries
	Abbreviations
	Symbols: bridge dynamics
	Symbols: inverse problems
	Symbols: microwave interferometry
	Mathematical Operators
1 Introduction
	1.1 Motivation
	1.2 Objectives
	1.3 Original contributions
	1.4 Outline of the text
2 Force identification methods – state of the art and research
	2.1 Direct measurement
	2.2 Indirect measurement: Bridge Weigh-In-Motion
	2.3 Dynamic load identification
		2.3.1 Moving force identification
		2.3.2 Identification of non-moving forces
	2.4 Conclusion
3 Fundamentals of bridge dynamics under moving loads
	3.1 Generals
	3.2 Analytical solution for a simply supported beam with uniformly distributed mass
		3.2.1 Undamped free vibrations of a beam
		3.2.2 Damped vibrations of a beam subjected to moving loads
	3.3 Finite Element solution for dynamic analysis
		3.3.1 Equations of motion
		3.3.2 Natural frequencies and mode shapes
		3.3.3 Solution using mode superposition
		3.3.4 Solution using direct numerical integration
	3.4 Conclusion
4 Introduction to inverse problems
	4.1 Generals
	4.2 Inverse and ill-posed problems
	4.3 Direct methods for solving inverse problems
		4.3.1 Normal equations
		4.3.2 Tikhonov regularization
		4.3.3 Singular value decomposition
	4.4 Solutions based on optimization algorithms
		4.4.1 General form of least squares problems
		4.4.2 Concept of unconstrained optimization
		4.4.3 Trust region strategy
			4.4.3.1 Outline of the trust region concept
			4.4.3.2 Cauchy point
			4.4.3.3 Dogleg method
			4.4.3.4 Two-dimensional subspace minimization
		4.4.4 Sensitivity analysis
			4.4.4.1 Finite difference method
			4.4.4.2 Direct sensitivity
	4.5 Conclusion
5 Formulation of an inverse problem for moving force identification
	5.1 Objective function
		5.1.1 Vector of unknown variables
		5.1.2 Measured and computed data
		5.1.3 Main optimization criterion
		5.1.4 Penalty functions
			5.1.4.1 Penalty function 1
			5.1.4.2 Penalty function 2
			5.1.4.3 Weighting factors of the penalty functions
	5.2 Sensitivity analysis
		5.2.1 Gradient of the objective function
			5.2.1.1 Gradient of the main optimization criterion
			5.2.1.2 Gradients of the penalty functions
			5.2.1.3 Total derivative of the objective function
		5.2.2 Hessian of the objective function
			5.2.2.1 Hessian of the main optimization criterion
			5.2.2.2 Hessian of the penalty functions
			5.2.2.3 Total Hessian of the objective function
	5.3 Optimization algorithm
	5.4 Moving window approach
		5.4.1 Forward sliding window
		5.4.2 Backward sliding window
	5.5 Conclusion
6 Numerical validation with simulated measurement data
	6.1 System description
	6.2 Simulated measurements
		6.2.1 Reference load scenario 1
		6.2.2 Reference load scenario 2
		6.2.3 Reference load scenario 3
		6.2.4 Generation of the simulated measurements
	6.3 Computational parameters
		6.3.1 Basic computational parameters
		6.3.2 Initial values of the optimization process
		6.3.3 Termination criteria of the optimization
	6.4 Verification of the analytical sensitivities
		6.4.1 Gradient check
			6.4.1.1 Gradient of the main optimization criterion
			6.4.1.2 Gradient of the total objective function
		6.4.2 Hessian check
			6.4.2.1 Hessian of the main optimization criterion
			6.4.2.2 Hessian of the total objective function
	6.5 Optimization results without penalty functions
		6.5.1 Analysis with noisy data
			6.5.1.1 Results of the load scenario 1 without penalty functions
			6.5.1.2 Results of the load scenario 2 without penalty functions
			6.5.1.3 Results of the load scenario 3 without penalty functions
		6.5.2 Analysis with noise-free data
			6.5.2.1 Solvability condition 1
			6.5.2.2 Solvability condition 2
			6.5.2.3 Additional remarks
	6.6 Optimization results including penalty functions
		6.6.1 Weighting factors of the penalty functions
			6.6.1.1 Weighting factors determination based on load scenario 1
			6.6.1.2 Weighting factors determination based on load scenario 2
			6.6.1.3 Weighting factors determination based on load scenario 3
			6.6.1.4 Trade-off selection of the weighting parameters for the penalty functions
		6.6.2 Effect of the backward sliding window approach
		6.6.3 Selected results of the optimization with noisy data
			6.6.3.1 Results of load scenario 1
			6.6.3.2 Results of load scenario 2
			6.6.3.3 Results of load scenario 3
		6.6.4 Solution behavior with violated solvability conditions
	6.7 Influence factors on the identification accuracy
		6.7.1 Effect of noise
			6.7.1.1 Different noise samples with the same statistical characteristics
			6.7.1.2 Different noise levels
		6.7.2 Effect of computational parameters
			6.7.2.1 Window length
			6.7.2.2 Smoothing radius
			6.7.2.3 Length of the overlapping interval
		6.7.3 Effect of the amount of modal information
		6.7.4 Effect of the modal truncation
		6.7.5 Effect of the sensor configuration
		6.7.6 Effect of modeling errors
		6.7.7 Effect of an inexact speed
	6.8 Conclusion
7 Experimental validation: railway bridge in operation
	7.1 Framework of the investigation
	7.2 Investigation object
	7.3 Measurement set-up
		7.3.1 Acceleration measurements
		7.3.2 Displacement measurements
			7.3.2.1 Microwave interferometry
		7.3.3 Axle detection
	7.4 Measurement results
		7.4.1 Modal analysis
		7.4.2 Train passages
		7.4.3 Resonance curves of accelerations
		7.4.4 Displacement measurements
	7.5 Structural model
	7.6 Force identification with real measurement data
		7.6.1 Computational parameters
		7.6.2 Weighting factors of the penalty functions
		7.6.3 Results and discussion of the experimental validation
			7.6.3.1 Jump discontinuities of the identified forces
			7.6.3.2 Significant differences of the static axle loads within a bogie
			7.6.3.3 Dynamic force components
			7.6.3.4 Improved weighting factors of the penalty functions by means of calibration passages
			7.6.3.5 Effect of the decay phase on the identification accuracy
			7.6.3.6 Application of the method on further train passages
			7.6.3.7 Effect of the measurement configuration
			7.6.3.8 Effect of the damping ratio
			7.6.3.9 Effect of the number of modes considered in the analysis
			7.6.3.10 Effect of an inexact train speed
	7.7 Conclusion
8 Experimental validation: pedestrian bridge
	8.1 Framework of the investigation
	8.2 Ground reaction forces: problem statement
	8.3 Investigation object
	8.4 Measurement set-up
	8.5 Structural model
	8.6 Force identification with real measurement data
		8.6.1 Computational parameters
		8.6.2 Weighting factors of the penalty functions
		8.6.3 Preprocessing of the measurement data
		8.6.4 Results and discussion of the experimental validation
	8.7 Conclusion
9 Conclusions and outlook
	9.1 Summary and conclusions
	9.2 Outlook
Bibliography
	Literature
	Standards and technical guidelines
Table of appendices
A Derivations of solutions for inverse problems
	A.1 Derivation of the normal equations
	A.2 Solution of inverse problems using Tikhonov regularization
	A.3 Trust region solution using iterative approach
B Additional force identification results for the Schmutter bridge
	B.1 Re-built responses for different passages
	B.2 Results for different passages using unrefined weighting parameters
	B.3 Results for different measurement set-ups
	B.4 Results for different damping ratios
	B.5 Results for different number of modes
	B.6 Results for a perturbed speed