Stochastic Dynamics of Structures

By Abdelkhalak El Hami and Bouchaib Radi

This book is dedicated to the general study of the dynamics of mechanical structures with consideration of uncertainties. The goal is to get the appropriate forms of a part in minimizing a given criterion. In all fields of structural mechanics, the impact of good design of a room is very important to its strength, its life and its use in service. The development of the engineer's art requires considerable effort to constantly improve structural design techniques.

• Language: English
• Publisher: Wiley
• Release date: Nov 22, 2016
• ISBN: 9781119377658

Abdelkhalak El Hami

Abdelkhalak El Hami is Professeur des universités at the Institut National des Sciences Appliquées (INSA-Rouen) in France and is in charge of the Normandy Conservatoire National des Arts et Metiers (CNAM) Chair of Mechanics and Head of the department of mechanical engineering of INSA Normandy, as well as several European pedagogical projects. He is an expert in fluid–structure interaction studies, reliability and optimization.

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Preface

Over the past few years, engineers, scientists and authorities have shown a real interest in stochastic methods and the optimization of mechanical systems. These two areas have received increasing attention because of their theoretical complexities and industrial applications. At present, deterministic models do not take into account the variability of factors, often falsely identified, and show results that do not correspond to the reality of the specific problem.

This book includes the most recent ideas from research and industry in the domain of stochastic dynamics and optimization of mechanical structures. It contains 11 chapters that focus on different tools including uncertainties, stochastic methods, reliability and the optimization of systems. The chapters discuss the interaction between optimization and reliability of structural dynamics in order to consider the uncertainties in modeling and resolve the problems that are encountered. Each chapter clearly sets out the technological methods used and developed along with illustrative and relevant examples.

Chapter 1 explains the problems in mechanical structure dynamics. The works used include the exposition of uncertainties with regard to the problems of optimization. The last section is dedicated to describing the use of different tools to analyze structures that consist of uncertain factors. Chapter 2 presents decoupled mechanical structures. Issues regarding structure and liquid are studied separately. In each area, initial theories are used. Equations are expressed in the variational form and then discretized by finite element in order to display the matrix systems required to resolve the separate problem numerically. Chapter 3 presents the equations that show a fluid–structure coupled system and are put into a variational form. They are then discretized using the finite element method to obtain the matrix systems so they can be solved numerically. Finally, the application of modal reduction methods to coupled systems is considered. Modal reduction methods are then applied to vibro-acoustic and hydro-elastic issues.

Chapter 4 demonstrates the modeling methods of mesh-free structures and the formal theory of the element-free Galerkin method. Different mesh-free methods will be presented, together with the approximation of lower movable squares and the utilization of the weak form of elasticity equations in order to determine the unknown nodal values in the mesh-free Galerkin approximation. Chapter 5 aims to combine the modal reduction methods with non-frequentative stochastic methods and apply them to the study of (paired and unpaired) mechanical systems dynamics with uncertain parameters. It then evaluates the impact of modal reduction with regard to saving calculation time when reliably analyzing non-deterministic systems. Chapter 6 presents general dynamic equations by aiming to apply them to stochastic modal synthesis methods. It recalls the sub-structuring method, initially formulated for static problems. This involves considering a structure like a network of interconnected sub-structures. Synthesis methods differ in terms of the choice of method, in order to represent the dynamics of each sub-structure, and in assembly procedures. Then, a DDL junction reduction strategy will be proposed after assembly. This strategy is based on using calculations of modes of interface, which are obtained via Guyan condensation at the interfaces of the complete structure. Chapter 7 explores frequencies and appropriate modes for a dynamic conservative system, in which mass and rigid matrices are functions of random parameters. Two perturbation methods are used. The first method uses a second-order Taylor series expansion. The second is a method proposed by Muscolino, which uses first-order expansion. The objective is to highlight the advantages of methods of modal synthesis in predicting the dynamic behavior of stochastic structures. The traditional solution to the stochastic problem will be compared with that which uses sub-structuring methods.

Chapter 8 is devoted to the dynamic response of a structure with an uncertain variable to a given agitation. Two stochastic methods are presented. The first is the perturbation method. The second involves projecting a solution onto polynomial chaos. Both methods are used to calculate the first two moments of the response (mean value and variance) using knowledge of the laws of probability in relation to the distribution of structural parameters. The use of modal synthesis methods will allow us to reduce the dimensions of the model before integrating the equation of motion. Then, the extension of modal synthesis methods will be presented to evaluate the stochastic response of a dynamic system to a given agitation using the perturbation method. The end of this chapter attempts to show the interaction between dynamic sub-structuring methods and the method for predicting homogeneous chaos in order to evaluate the variability of the response with respect to the variability of parameters of a large-scale model. Chapter 9 presents the development of the homogeneous chaos projection method to determine the function of the stochastic response in terms of frequency. Two methods are presented. In the first method, the calculation is performed directly. The second method is based on the use of modes specific to the structure. This is followed by the extension of modal synthesis methods, which is put forward to reduce the size of the mechanical model, allowing us to calculate the function of the response in terms of the frequency of a large-scale structure that has uncertain parameters. In general, the size of the model is reduced using transformation matrices that are constructed from the modes of each sub-structure, which can be normal modes of vibration, static modes, connection modes or rigid body modes. These modes contain the uncertain parameters of each sub-structure. Finally, numerical applications will be explored to show the efficiency and accuracy of using a homogeneous chaos with a mechanical model reduced by modal synthesis methods. Chapter 10 presents a methodology combining modal synthesis techniques with reliability optimization in design. This chapter presents an algorithm that allows us to incorporate modal synthesis methods in a reliability optimization process. Finally, it evaluates this algorithm using different applications to show the effectiveness and robustness of the proposed method. Chapter 11 aims to study the dynamic behavior of gear transmission in a wind turbine with uncertain parameters.

Finally, this book constitutes a useful source of information for teachers and researchers. It may also be informative for engineering students, engineers and students who are pursuing Master’s degree.

Acknowledgments

We would like to thank all who have contributed to the creation of this book, our families, and in particular the PhD students at INSA de Rouen who have helped us over the past few years.

Abdelkhalak EL HAMI

Bouchaib RADI

September 2016

1

Introduction to Structural Dynamics

The aim of this chapter is to convey a non-exhaustive image of all areas considered, from near or far, in this work.

Section 1.1 is dedicated to the general study of structural dynamics. This study intends to attach the essential evaluations to the calculations of dynamic responses, frequencies, appropriate methods and their response functions. All of these aspects are consequently tackled using practical applications.

The dynamic balance equation system of a structure can be solved by using one of the traditional strategies [MOH 05]. The most frequent resolution strategy in dynamics is modal superposition, which is suited to linear structures whose first methods are only the ones that are agitated. In contrast, direct resolution methods incorporate movement equations in order to handle nonlinear structures. These structures can also be applied when the frequency contents of the disturbance cover a large number of methods of the mechanical structure studied.

In section 1.2, a non-exhaustive bibliographic study is put forward regarding the optimization of structures. The objective is to obtain suitable forms from an article by minimizing a given criterion. In every area of structural mechanics, knowing the impact of effective object design is very important in determining its resistance, lifetime and operation. This is one of the challenges faced by industries daily. The development of engineering requires considerable effort to constantly improve the techniques for designing structures. Optimization plays an important role in increasing performance and significantly reducing aerospace and motoring engineering equipment, while simultaneously substantially saving energy.

The last section of this chapter is devoted to describing the different tools that analyze structures with uncertain parameters. The uncertainty of parameters is particularly dangerous in vibratory mechanics. However, consideration of this effect has the ability to respond to different sorts of needs, among which one can identify two categories: analysis and design. In general, modeled objects and structures respond to a design brief, such as safeguarding security, reliability or comfort guidelines.

When creating a deterministic design, one tends to search for the best possible design from among all potential solutions. This choice is based on cost as well as improvement in product quality. In this case, the objectives of the designer to produce the optimal design are hampered despite the accuracy of the mechanical characteristics of the materials, the geometry and the loading (effects of uncertainties). The resulting optimal design can thus have an unsatisfactory level of reliability. The process that incorporates reliability analysis with the named problem of optimization (Reliability based design optimization or RBDO) aims to envisage structures while establishing the best compromise between cost and effective functioning.

1.1. Composition of problems relating to dynamic structures

The composition of a dynamic problem of small disturbances using Ω of the boundary (Figure 1.1) and in a [0, T] time interval is:

[1.1]

[1.2]

Figure 1.1. Structure Ω

Initial conditions:

[1.3]

[1.4]

Limited conditions:

[1.5]

[1.6]

Here, u is the displacement vector, σ and ε are the constrained and deformation tensors, respectively, and ρ is the volumetric density. The vectors and ū represent volumetric strength, exterior strength and imposed movement, respectively, and is the normal vector at the surface.

In terms of isotropic elasticity, the behavior law is written as follows:

[1.7]

where λ and μ are the functions of Young’s modulus and Poisson’s coefficient ν , respectively:

[1.8]

[1.9]

The dynamic problem presented above in the case of elasticity can be represented can by the Navier equation as follows:

[1.10]

where ∇² denotes the Laplacian operator: and ∇⋅ is the notation for the divergence operator:

1.1.1. Finite element method

In the case of complex geometric structures, numerical methods like the finite element method are used. In problems concerning elastodynamics, generally movements are expressed by a combination of vectors [GMÜ 97]:

[1.11]

where [B(x)] is the matrix form of functions and {q(t)} is the vector of discrete real movements, whose components are discrete unknowns of approximation.

After discretization of the problem, a second-order equation system was obtained:

[1.12]

where N is the number of degrees of freedom of the system; M (N×N) is the mass symmetrical matrix, which is defined as positive; C (N×N) and K (N×N) are the matrices of viscous shock absorption and rigidity, which are symmetrically defined as being non-negative; and F represents the vector of all forces applied.

Equation [1.12] represents a system of differential second-order equations that can be solved by either a direct incorporation method or superposition method.

1.1.2. Modal superposition method

If one applies the following transformation to the system presented in equation [1.12]:

[1.13]

where {p} is the vector of generalized coordinates, [Φ] is the modal matrix that verifies the attributes of orthogonality: [Φ]T [M][Φ] = I and [Φ]T [K][Φ] = [w²] with Equpg19.jpg , where wi is the specific vibration, equation [1.12] becomes:

[1.14]

where {P} = [Φ]T{F} is the vector of modal force.

The shock absorption matrix can be proposed as being proportional to the mass and stiffness matrix. This hypothesis was made by Rayleigh and is relatively frequently employed in structural calculations. One can write:

[1.15]

[1.16]

which can be transformed into:

[1.17]

The unpaired system becomes:

[1.18]

[1.19]

where ζi is the coefficient of reduced shock absorption and the values of α and β are initially unknown, which are calculated using ζi.

Figure 1.2. Graph of the shock absorption coefficient

Figure 1.2 shows the shock absorption coefficient ζ in graphical form. It can be noted that the sum of the two functions is almost a constant to the shock absorption on the frequency band chosen. Therefore, given the modal shock absorption (ζ) and a frequency interval (f1 and f2), the two equations can be simultaneously solved to determine α and β:

[1.20]

and

[1.21]

1.1.3. Direct integration

There are many methods of integration for differential equations. The general process is to discretize time and formulate what is occurring at the given instance "t + Δt in terms of what happens at instance t" using Taylor developments. The Newmark method will be presented in this section, as well as that of Wilson [KLE 92, EL 13].

1.1.3.1. Newmark method

Newmark proposed a method in which speed and movement of t + Δt are estimated in terms of and acceleration . In addition, movement and speed are developed in a Taylor series with the help of two independent parameters, β and γ, together with time [KLE