Khaled M. Ahmida

Spectral element-based prediction of active power flow in Timoshenko beams

The analysis of standing waves, which correspond to the reactive part of the power in structures, is not a sufficient tool for studying structural vibration problems. Indeed, the active power component (structural intensity) has shown to be of great importance in studying damped structural vibration problems. One of the most common numerical discretization methods used in structural mechanics is the finite element method. Although this procedure has its advantages in solving dynamic problems, it also has disadvantages mainly when dealing with high frequency problems and large complex spatial structures due to the prohibitive computational cost. On the other hand, the spectral element method has the potential to overcome this kind of problem. In this paper, the formulation of the Timoshenko beam spectral element is reviewed and applied to the prediction of the structural intensity in beams. A structure of two connected beams is used. One of the beams has a higher internal dissipation factor. This factor is used to indicate damping effect and therefore causes structural power to flow through the structure. The total power flow through a cross-section of the beam is calculated and compared to the input power. The spectral element method is shown to be more suitable to model higher frequency propagation problems when compared to the finite element method.

Estimation of the SEA coupling loss factors by means of spectral element modeling

The coupling loss factors are of critical importance when building and solving Statistical Energy Analysis (SEA) models. This paper proposes a methodology to numerically estimate these factors for frame-type structures. The estimated factors are compared with those obtained through analytical expressions for frame structures, where members are joined at right angles. The example used to verify the proposed technique consists of two infinite beams connected at a right angle modeled via the Spectral Element Method (SEM) using throw-off elements. It is shown that the obtained coupling loss factors compare very well with the analytical expressions that may be derived for this simple right-angle connection case. By using the SEM approach, the coupling loss factors can be obtained for arbitrary frame structure connections, thus facilitating the analysis via SEA.

Design and modeling of an acoustically excited double-paddle scanner

Dynamic analysis is an essential factor in the design, fabrication and optimization of micro-systems. Micro-scanners are currently subjected to wide research work. In this paper, the dynamic behavior of a monolithic single-crystal silicon microstructure is investigated. The microstructure used is a double-paddle scanning mirror for laser applications. It consists of two similar plates (wings) connected to another plate (mirror) and is suspended by one torsion bar. The dynamic analysis is conducted numerically, using finite element analysis. The numerical modeling is described. The numerical results are validated experimentally by measuring the frequency response functions collected at some points on the scanner surface. The experimental modal analysis is performed using a laser Doppler vibrometer and an acoustic excitation device. The excitation device consists of a polyester resin mount with two conic-shaped ducts which give access to the back of the two wings from one side and to two mini loudspeakers on the other side. This excitation device was used and good agreement was found between the numerically predicted and the experimentally identified modal parameters. The non-intrusive excitation mechanism and the optical measurement techniques used in the experiments are discussed. A high quality factor is identified for the chosen operational mode shape.

Dynamic Analysis of Silicon Micromachined Double-Rotor Scanning Mirror

The use of MEMS-based technologies for producing scanning mirrors enables its batch production with a consequent increase in the throughput and a decrease in the manufacturing costs per device. However, the use of Silicon as a structural material could introduce non-linearities in the device behavior due to the variation of its mechanical properties according to the crystalline orientation. The orthotropic properties when taken into account in the finite element model of the device could enhance the accuracy in the design of micromachined scanning mirrors. The model used in this paper does not take into account the orthotropic behavior, however, satisfactory results were obtained. To validate the finite element model, a modal analysis of the device was performed using the Laser Doppler Vibrometry method. The normal modes of the structure were identified and the results agree well with the finite element model. This work presents the FE model and experimental modal analysis results of a Silicon micromachined double-rotor scanning mirror.

Analyzing the Total Structural Intensity in Beams Using a Homodyne Laser Doppler Vibrometer

The total structural intensity in beams can be considered as composed of three types of waves: bending, longitudinal, and torsional. In passive and active control applications, it is useful to separate each of these components in order to evaluate their contribution to the total structural power flowing through the beam. In this paper, a twisted z-shaped beam is used in order to allow the three types of waves to propagate. The contributions of the structural intensity, due to these waves, are computed from measurements taken over the surface of the beam with a simple homodyne interferometric laser vibrometer. The optical sensor incorporates some polarizing optics, additional to a Michelson type interferometer, to generate two optical signals in quadrature, which are processed to display velocities and/or displacements. This optical processing scheme is used to remove the directional ambiguity from the velocity measurement and allows nearly all back-scattered light collected from the object to be detect. This paper investigates the performance of the laser vibrometer in the estimation of the different wave components. The results are validated by comparing the total structural intensity computed from the laser measurements, with the measured input power. Results computed from measurements using PVDF sensors are also shown, and compared with the non-intrusive laser measurements.

Electromagnetic and structural FEA modeling of induction actuated scanning mirrors

The prediction of the behavior of the induction actuated scanners is a problem that involves the modeling of different physical domains as structural and electromagnetic. The Finite Element Approach is a highly viable alternative to obtain reliable predictions for its behavior over other available methods as the analytic, or circuit equivalent methods. In this owrk a finite element model for the structural an electromagnetic domains of the induction actuated scanning mirror was presented. To validate these models two experiments were performed, a laser doppler vibrometry of the double-rotor scanner to identify its modes shapes and natural frequencies and a magnetic field mapping of the actuator to obtain the spatial characteristic of the AC and DC magnetic fields generated by the actuator in the device armature region. There is a good agreement between the FEA models and the experimental results.

Computing structural energy density and power flow using spectral elements

The large deterministic models obtained using numerical methods are frequently inadequate to deal with structural dynamics problems in the audio frequency range. An alternative approach that has been used for many years is the spectral approach. After transformation to the frequency domain, structural dynamics problems are described by partial differential equations that can be solved analytically in an exact way. In the case of one‐dimensional waveguides, the spectral approach may be combined with a mobility approach yielding a spectral element methodology that, in many ways, resembles the finite element method. The major drawback of this approach is the difficulty in treating two‐ and three‐dimensional waveguides. Exact solutions exist only for plates and shells with particular geometry and boundary conditions. An approximate method has been proposed for flat plates with arbitrary conditions, but the element assembling is still restricted to one dimension. In this paper, different formulations for flat‐...

Analyzing the total structural intensity in beams using a homodyne laser Doppler vibrometer

The total structural intensity in beams can be considered as composed of three kinds of waves: bending, longitudinal, and torsional. In passive and active control applications, it is useful to separate each of these components in order to evaluate its contribution to the total structural intensity flowing through the beam. In this paper, a z-shaped beam is used in order to allow the three kinds of waves to propagate. The contributions of the structural intensity due to the three kinds of waves are computed from measurements made over the surface of the beam with a simple homodyne interferometric laser vibrometer. The optical sensor incorporates some additional polarizing optics to a Michelson type interferometer to generate two optical signals in quadrature, which are processed to display velocities and/or displacements. This optical processing scheme is used to remove the directional ambiguity from the velocity measurement and allows to detect nearly all backscattered light collected from the object. This paper investigates the performance of the laser vibrometer in the estimation of the different wave components. The results are validated by comparing the total structural intensity computed from the laser measurements with the measured input power. Results computed from measurements using PVDF sensors are also shown, and compared with the non-intrusive laser measurements.