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 O of the boundary (Figure 1.1) and in a [0, T] time interval is:

Figure 1.1. Structure O

Initial conditions:

Limited conditions:

Here, u is the displacement vector, s and e are the constrained and deformation tensors, respectively, and ? is the volumetric density. The vectors and u 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:

where ? and µ are the functions of Young's modulus and Poisson's coefficient ? , respectively:

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

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]:

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:

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]:

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

where {P} = [F]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:

which can be transformed into:

The unpaired system becomes:

where ?i is the coefficient of reduced shock absorption and the values of a and ß are initially unknown, which are calculated using ?i.

Figure 1.2. Graph of the shock absorption coefficient

Figure 1.2 shows the shock absorption coefficient ? 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 (?) and a frequency interval (f1 and f2), the two equations can be simultaneously solved to determine a and ß:

and

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 + ?t" 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 + ?t 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, ß and ?, together with time [KLE 92]:

where are the approximations of respectively, and tn+1 = tn + ?t , with ?t being time. The two independent parameters, ß and ?, assure the accuracy and stability of the solution. When ?=1/2, ß=(?+0.5)/4.

By transferring these equations onto the movement equation, the following matrix relation can be obtained:

with

Acceleration at the moment t = 0 is created by the balancing conditions and the initial conditions on {q} and . The solution of equation [1.24] requires the solution of a linear system at each time interval.

1.1.3.2. The Wilson method, ?

The Wilson method is the one in which acceleration varies linearly in the interval [n ?t, (n+1) ?t]. Wilson supposed that this linear variation occurs in the interval [n ?t, (n+1) ?t]. The value of ? recommended by Wilson is 1.4.

If t denotes time in interval [0, ? ?t], then acceleration in the interval [t, t +...