Nonlinear Model Updating in Structural Dynamics

Identification of nonlinear structural dynamics has received a significant attention during last decades. Yet, there are many aspects of the identification methods of nonlinear structural models to be improved. The main objective of this study is to introduce novel identification approaches for nonlinear structures. The first step in identifying nonlinear structural elements is to detect their exact location. Hence, the first section of this study focuses on the localization of nonlinear elements in structural dynamics utilizing base excitation measured data. To this end, a localization approach is used to find the location of nonlinear electromagnetic restoring force applied to the tip of a cantilever beam.Inferring the exact location of nonlinear elements, identification methods are utilized to identify and characterize the mathematical model of nonlinear structures. However, various sources of noise and error may affect the accuracy of the identified model. Therefore, in the second part of the thesis, the effect of various sources of inaccuracy on the results of nonlinear model identification is investigated. It is shown that measurement noise, expansion error, modelling error, and neglecting the effect of higher harmonics may lead to an erroneously identified model.An optimization-based framework for the identification of nonlinear systems is proposed in this work in order to avoid the bottlenecks mentioned above. The introduced method is applied to a test rig composed of a base-excited cantilever beam subjected to an electromagnetic force at the tip. According to the nonlinear response of the system, four different functions are assumed as candidate models for the unknown nonlinear electromagnetic force. The measured response is compared with the reconstructed response using various models and the most appropriate mathematical model is selected.Utilizing optimization-based identification method to characterize complex mathematical models with large number of unknown parameters would be computationally expensive. Therefore, this study introduces a harmonic-balance-based parameter estimation method for the identification of nonlinear structures in the presence of multi-harmonic response and force. For this purpose, a method with two different approaches are introduced: Analytical Harmonic-Balance-based (AHB) approach and the Alternating Frequency/Time approach using Harmonic Balance (AFTHB). The method is applied to five simulated examples of nonlinear systems to highlight different features of the method. The method can be applied to all forms of both smooth and non-smooth nonlinear functions. The computational cost is relatively low since a dictionary of candidate basis functions is avoided. The results illustrate that neglecting higher harmonics, particularly in systems with multi-harmonic response and force, may lead to an inaccurate identified model. The AFTHB approach benefits from including significant harmonics of the response and force. Applying this method leads to accurate algebraic equations for each harmonic, including the effect of higher harmonics without truncated error. In the last part of this study, the AFTHB method is applied to two experimental case studies and identifies the nonlinear mathematical model of the structures. The first case is composed of a cantilever beam with a nonlinear electromagnetic restoring force applied to the tip which is excited by a multi-harmonic external force. In the second experimental case study, a configuration of linear springs applies a geometric nonlinear restoring force to the tip of a cantilever beam resulting in internal resonance in the dynamics of the system. The good performance of the AFTHB approach in estimating the unknown parameters of the structure is illustrated by the results of identification.

50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference ◽

Modeling the Nonlinear Structural Dynamics of a Plunging Membrane Airfoil Using a High Dimensional Harmonic Balance Approach

10.2514/6.2009-2474 ◽

2009 ◽

Cited By ~ 1

Author(s):

Allen LaBryer ◽

Peter Attar

Keyword(s):

Structural Dynamics ◽

Harmonic Balance ◽

High Dimensional ◽

Balance Approach ◽

Mathematical model of one flexible transport category aircraft

Aircraft Engineering and Aerospace Technology ◽

10.1108/aeat-12-2013-0230 ◽

2017 ◽

Vol 89 (3) ◽

pp. 384-396 ◽

Cited By ~ 2

Author(s):

Marcelo Santiago Sousa ◽

Pedro Paglione ◽

Roberto Gil Annes Silva ◽

Flavio Luiz Cardoso-Ribeiro ◽

Sebastião Simões Cunha

Keyword(s):

Mathematical Model ◽

Structural Dynamics ◽

Degrees Of Freedom ◽

Flight Dynamics ◽

Content Type ◽

Nonlinear Structural ◽

Structural Deformations ◽

Dynamics Simulations ◽

The Mathematical Model

Purpose The purpose of this paper is to present a mathematical model of one very flexible transport category airplane whose structural dynamics was modeled with the strain-based formulation. This model can be used for the analysis of couplings between the flight dynamics and structural dynamics. Design/methodology/approach The model was developed with the use of Hamiltonian mechanics and strain-based formulation. Nonlinear flight dynamics, nonlinear structural dynamics and inertial couplings are considered. Findings The mathematical model allows the analysis of effects of high structural deformations on airplane flight dynamics. Research limitations/implications The mathematical model has more than 60 degrees of freedom. The computational burden is too high, if compared to the traditional rigid body flight dynamics simulations. Practical implications The mathematical model presented in this work allows a detailed analysis of the couplings between flight dynamics and structural dynamics in very flexible airplanes. The better comprehension of these couplings will contribute to the development of flexible airplanes. Originality/value This work presents the application of nonlinear flight dynamics-nonlinear structural dynamics-strain-based formulation (NFNS_s) methodology to model the flight dynamics of one very flexible transport category airplane. This paper addresses also the way as the analysis of results obtained in nonlinear simulations can be made. Comparisons of the NFNS_s and nonlinear flight dynamics-linear structural dynamics methodologies are presented in this work.

A harmonic balance approach for large-scale problems in nonlinear structural dynamics

Computers & Structures ◽

10.1016/j.compstruc.2010.06.003 ◽

2010 ◽

Vol 88 (17-18) ◽

pp. 1002-1014 ◽

Cited By ~ 22

Author(s):

A. LaBryer ◽

P.J. Attar

Keyword(s):

Structural Dynamics ◽

Harmonic Balance ◽

Large Scale ◽

Large Scale Problems ◽

Balance Approach ◽

Harmonic-Balance-Based parameter estimation of nonlinear structures in the presence of Multi-Harmonic response and force

Mechanical Systems and Signal Processing ◽

10.1016/j.ymssp.2021.108057 ◽

2022 ◽

Vol 162 ◽

pp. 108057

Author(s):

Javad Taghipour ◽

Hamed Haddad Khodaparast ◽

Michael I. Friswell ◽

Alexander D. Shaw ◽

Hassan Jalali ◽

...

Keyword(s):

Parameter Estimation ◽

Harmonic Balance ◽

Nonlinear Structures ◽

Harmonic Response

Input shaping control of a flexible cantilever beam excited by a moving hub

Science & Technology Development Journal - Engineering and Technology ◽

10.32508/stdjet.v3i2.605 ◽

2020 ◽

Vol 3 (2) ◽

pp. First

Author(s):

Quoc Chi Nguyen ◽

Nguyen Tri Giang Vu ◽

Phuong Tung Pham

Keyword(s):

Mathematical Model ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Cantilever Beam ◽

High Speed ◽

Beam Theory ◽

Flexible Beam ◽

Structural Element ◽

Unknown Parameters ◽

The Mathematical Model

Introduction: A cantilever beam is a well-known structural element in engineering, which is only fixed at one end. This structure can be used to describe a manipulator, whose stiffness is large to ensure rigidity and stability of the system. A flexible cantilever beam provides a light-weight structure and high cost efficiency but generates vibration under high-speed positioning. In this paper, we aim to control the vibratory behavior of a flexible cantilever beam attached to a moving hub. Method: The mathematical model of the flexible beam is described by partial differential equations (PDEs) using Euler-Bernoulli beam theory. Then, The PDE model is approximated by using the Galerkin method, which is resulted in a set of ordinary differential equations (ODEs). Experiment is used to determine unknown parameters of the system to complete the model. The ODE model enables the control design of three input shapers: (i) Zero-Vibration (ZV), (ii) Zero-Vibration-Derivative (ZVD), and (iii) Zero-Vibration-Derivative-Derivative (ZVDD), which are employed to drive the flexible beam to the desired position and to reduce vibrations of the beam. Results and conclusion: The dynamic model is obtained in term of ordinary differential equations. Unknown parameters of the system are determined by experimental process. Various controllers are designed to eliminate the vibration of the beam. The simulation is applied to predict the dynamic response of the beam to verify the designed controllers numerically. Experiment shows the validity of the mathematical model through the consistency between the simulation and experimental data and the effectiveness of the controllers for the real system. These controllers show several advantages such as no need of extra equipment; the positioning controller is intact, which means it may be applied to many existing systems.

Toward a Fundamental Understanding of the Hilbert-Huang Transform in Nonlinear Structural Dynamics

Journal of Vibration and Control ◽

10.1177/1077546307079381 ◽

2008 ◽

Vol 14 (1-2) ◽

pp. 77-105 ◽

Cited By ~ 51

Author(s):

G. Kerschen ◽

A.F. Vakakis ◽

Y.S. Lee ◽

D.M. McFarland ◽

L.A. Bergman

Keyword(s):

Structural Dynamics ◽

Hilbert Huang Transform ◽

Nonlinear Structural ◽

Fundamental Understanding

Algorithms for Nonlinear Structural Dynamics

Journal of the Structural Division ◽

10.1061/jsdeag.0004855 ◽

1978 ◽

Vol 104 (2) ◽

pp. 263-280 ◽

Cited By ~ 2

Author(s):

Hojjat Adeli ◽

James M. Gere ◽

William Weaver

Keyword(s):

Structural Dynamics ◽

An Improved Sensitivity Method for the Simultaneous Identification of Unknown Parameters and External Loads of Nonlinear Structures

Shock and Vibration ◽

10.1155/2018/1737594 ◽

2018 ◽

Vol 2018 ◽

pp. 1-12

Author(s):

Chongwen Wang ◽

Chengbin Du

Keyword(s):

Regularization Parameter ◽

Parameter Distribution ◽

Nonlinear Structures ◽

Noise Levels ◽

Unknown Parameters ◽

Key Issues ◽

The Difference ◽

Sensitivity Method ◽

Simultaneous Identification ◽

External Loads

Because structures may be subject to unknown loads and may simultaneously involve unknown parameters and because simple load identification or parameter identification algorithms cannot be applied under such conditions, it is necessary to seek algorithms that can simultaneously identify unknown parameters and external loads of structures. The sensitivity method is one of them, and this paper extends this method to nonlinear structures. In addition, the key issues associated with the sensitivity method are systematically studied, and suggestions for improvement are put forward, including the use of the difference method instead of the derivative method to calculate the sensitivity, the use of a fixed regularization parameter instead of the traditional regularization parameter calculation methods, and measures for guarantee of iterative convergence. The improved sensitivity method is applied to two types of nonlinear structures, and the effects of the regularization parameter, distribution of measured points, response types, noise levels, and the magnitude of the perturbation on the identified results are discussed.

On the Simulation of Unsteady Turbulence and Transition Effects in a Multistage Low Pressure Turbine: Part III — Comparison of Harmonic Balance and Full Wheel Simulation

Volume 2C: Turbomachinery ◽

10.1115/gt2018-76749 ◽

2018 ◽

Author(s):

Edmund Kügeler ◽

Georg Geiser ◽

Jens Wellner ◽

Anton Weber ◽

Anselm Moors

Keyword(s):

Harmonic Balance ◽

Leading Edge ◽

Suction Side ◽

Low Pressure ◽

Transition Model ◽

Higher Harmonics ◽

Low Pressure Turbine ◽

The Third ◽

Transition Effects ◽

Transition Models

This is the third part of a series of three papers on the simulation of turbulence and transition effects in a multistage low pressure turbine. The third part of the series deals with the detailed comparison of the Harmonic Balance calculations with the full wheel simulations and measurements for the two-stage low-pressure turbine. The Harmonic Balance simulations were carried out in two confingurations, either using only the 0th harmonic in the turbulence and transition model or additional in all harmonics. The same Menter SST two-equation k–ω turbulence model along with Menter and Langtrys two-equation γ–Reθ transition model is used in the Harmonic Balance simulation as in the full wheel simulations. The measurements on the second stator ofthe low-pressure turbine have been carried out separately for downstream and upstream influences. Thus, a dedicated comparison of the downstream and upstream influences of the flow to the second stator is possible. In the Harmonic Balance calculations, the influences of the not directly adjacent blade, i.e. the first stator, were also included in the second stator In the first analysis, however, it was shown that the consistency with the full wheel configuration and the measurement in this case was not as good as expected. From the analysis ofthe full wheel simulation, we found that there is a considerable variation in the order ofmagnitude ofthe unsteady values in the second stator. In a further deeper consideration of the configuration, it is found that modes are reflected in upstream rows and influences the flow in the second stator. After the integration of these modes into the Harmonic Balance calculations, a much better agreement was reached with results ofthe full wheel simulation and the measurements. The second stator has a laminar region on the suction side starting at the leading edge and then transition takes place via a separation or in bypass mode, depending on the particular blade viewed in the circumferential direction. In the area oftransition, the clear difference between the calculations without and with consideration ofthe higher harmonics in the turbulence and transition models can be clearly seen. The consideration ofthe higher harmonics in the turbulence and transition models results an improvement in the consistency.