Aeroelasticity and Structural Dynamics

Modeling and Simulating Medium and High Frequency Vibrations of Complex Structures

The problem of medium and high frequencies

Realistic, complex industrial structures exhibit typical transport and diffusive behaviours in their high-frequency (HF) range of vibration. Their dynamics are physically well represented by global standing waves (the eigenmodes) in the low-frequency (LF) range. But this approach fails dramatically as frequency increases because of the multiple interactions of waves with material heterogeneities at the wavelength scale, and reflections on boundaries and at interfaces between structural elements with different stiffnesses (stiffeners on plates and shells, bulkheads, stringers, etc.). In the "intermediate" or mid-frequency (MF) range, a transition zone between the LF and HF behaviours is observed.

Mid-frequency vibrations with structural complexity

Mechanical modelling strategies and numerical methods are now extremely efficient for the low-frequency modal domain. Structural complexity, namely all equipment or secondary substructures attached to a parent structure, is usually taken into account through equivalent isolated or distributed masses. Reduced models are constructed by a Ritz-Galerkin projection onto a finite number of eigenmodes associated to the first lowest eigenfrequencies. They can be used to estimate the response of the parent structure to any type of loads, whether deterministic or random.

The construction of reduced models becomes problematic in the mid-frequency range. It requires the computation of a large number of eigenmodes for stiffness matrices which are dependent on the frequency, because viscous memory effects of the materials have to be taken into account in this frequency range. Furthermore, structural complexity has now a great influence on the dynamics of the complete system. As frequency increases, the fundamental eigenfrequency of each substructure may be reached, indeed exceeded, thus loading a large number of internal degrees of freedom linked to these subsystems. Part of the total mechanical energy is transferred to the secondary substructures, introducing substantial apparent damping in the vibratory levels of the parent structure. By definition, a conventional deterministic model of this structural complexity – for example using the finite element method – is neither efficient nor even accessible. Hence it was proposed to introduce a probabilistic model, essentially linked to the substructure attachment area. The theory of structural fuzzy (Soize, 1986) has been developed at ONERA in this context, and validated on a number of significant examples in structural dynamics and structural acoustics.

The next step consists in validating the numerical methods developed previously by comparing them with the experimental results obtained for a real system which is representative of a complex industrial structure equipped with numerous substructures. The term "complex structure" is used to refer to a mechanical system made up of assemblies of simple structures such as beams, plates, shells, etc., or more complex sub-systems which are modelled by means of their impedances seen from their boundaries with the main structure (fuzzy structure theory), such as electromechanical or hydraulic equipments for example.

The system thus built for this study is a 3D assembly of some 300 plates made of duralumin, stiffened or not, including 50 acoustic cavities, weighing 825 kg, and being 5.3 m long, 2.5 m wide and 1.4 m high (see Figure 1 below). The secondary substructures attached to the parent structure weighed about 200 kg. The finite element model of the parent structure has been updated on the basis of modal identifications performed for the "unloaded" (no subsystem) configuration. The validity of the proposed approach has been tested by comparing the measured and computed values of the mechanical energy propagated by the main structure for two possible configurations with the subsystems. A complete experimental database has been compiled for the vibratory response of the main structure.

Figure 1: CAD view of the "unloaded" experimental complex structure

Some results of these tests for the unloaded and full configurations are shown below. Figure 2 plots the frequency responses of the "unloaded" structure, measured and computed at various points. Figure 3 plots the measured and computed mechanical energies estimated for three parts of the "full" structure. While these results are satisfactory up to about 600 Hz, they also demonstrate that the numerical models developed so far are still insufficient to cover the complete MF band of analysis (100-1,200 Hz for this structure). Advanced research must be conducted to enhance the available models, by multi-scale or multi-physics approaches for example, or alternative methods need be developed, with particular emphasis on an asymptotic transport theory.

Figure 2: Comparison of the measured
and computed accelerations at different points

Figure 3: Comparison of the mechanical energy
estimated by measurement and computation for different parts of the structure

High frequency vibrations

Two approaches are considered for predicting high-frequency vibrations of structures. Statistical Energy Analysis (SEA) is a global approach insofar as it only provides the mean vibratory energy estimated globally for each subsystem constituting the overall complex structure. The main difficulty of the method is the determination of some specific parameters introduced in its formulation, namely loss factors, coupling loss factors, modal densities and input powers. The Vibrational Conductivity Analogy (VCA) is a local approach, since its purpose is to estimate energy and power flow densities. However it has only been used so far for simple structures (beams and plates), as it is based on some assumptions which are extremely difficult to fulfil – or are even wrong – in the case of more complex structures.

Statistical Energy Analysis

The DADS department at ONERA has gained an extensive experience in this field, both from the experimental and theoretical points of view. SEA basic formulation is founded on sufficiently weakened assumptions to open up a relatively vast application field. The main difficulties, as outlined above, are those currently encountered by engineers and researchers seeking to apply SEA to practical industrial cases. The task is essentially to identify the parameters involved in these applications. Nevertheless, its theoretical core remains relatively stable. SEA is the subject of comprehensive research works worldwide, and is routinely used in the automotive or aerospace industries at a design stage.

Vibrational conductivity and energy diffusion

VCA models have been studied at ONERA in the mid 1990’s for the case of bending vibrations of Euler-Bernouilli beams and Kirchhoff-Love plates. The main contribution of this work is a proposal of a new "energetic behaviour law", or "transmission relationship" in the dedicated literature, linking the power flow vector to the mechanical energy density gradient of a plate. Injecting this relation into the local energy conservation equation, a diffusion equation is obtained for the latter. This is the reason why such models are referred to as energy diffusion, in analogy with temperature diffusion.

VCA models were initiated in the 1970’s by some Russian researchers before they were resumed in the early 1990s by several teams in the United States and Europe. All these works are limited to simple structures such as cables, beams or plates. The validity of the energetic behaviour law, although a crucial ingredient in VCA, is widely questioned in the structural acoustics literature, thus limiting its generalization to more complex systems. In fact, this law is generally incorrect, or is only accurate within the framework of extremely restrictive conditions. This inconsistency partially explains the limitations of VCA models when they are confronted with "exact" reference solutions or experimental results. Nevertheless, they have been implemented in various computational softwares, and are sometimes used at a design stage.

The proposed approach: asymptotic energy analysis

Using a number of recent mathematical results dealing with energy estimators (H-measure, Wigner measure) for the oscillating solutions of wave equations, it can be proved that the vibratory energy density of any heterogeneous elastic structure satisfies a transient transport equation. This approach also allows us to track the energy flow paths. It generalises the two methods described above for the prediction of high-frequency vibrations. In order to illustrate it, the theory is applied to the case of plates or shells with random mechanical parameters. Numerical simulations performed for a thick cylindrical shell outline the transport and diffusive regimes of energy propagation and their transition (see Figures 4 and 5 below where stands for the ratio of the correlation length of material heterogeneities to the wavelength). Application of this research concerns the prediction of the dynamic response of structures to mechanical or acoustical impact loads, such as pyrotechnic shocks for example, in the transient domain.

Figure 4: Propagation of bending energy in a heterogeneous thick cylindrical shell,
k₀ = 1,0

Figure 5: Propagation of bending energy in a heterogeneous thick cylindrical shell,
k₀ = 10,0

Extension of the asymptotic energy approach to more complex structures (bounded media, viscoelastic materials) is not straightforward. At present, two essential aspects are not taken into account in the theory, namely the effects of damping and boundary conditions.

Regarding the influence of damping memory effects, it can be shown that transport equations are globally unchanged from the purely elastic case. This trend is confirmed by the observations done on the high-frequency response of viscoelastic materials.
Regarding the influence of boundary conditions, the main task is to obtain energetic boundary conditions for transport (or radiative transfer) equations from the natural and essential boundary conditions applied to the solutions of wave equations in bounded domains.

These results may be adapted to some practical cases in structural dynamics and acoustics, namely the analysis of the transport properties of porous materials, or structural joints. External acoustic fluid-structure interaction may also be addressed, bearing in mind that the results obtained thus far apply directly to unbounded acoustic media.

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