Modal Cross Coupling Research Articles

Modal wavefront reconstruction to obtain Zernike coefficient with no cross coupling in lateral shearing measurement

Modal cross coupling frequently occurs in modal approaches from wavefront gradient data such as lateral shearing measurement through Zernike circle polynomials, since the gradients of Zernike circle polynomials are not orthogonal. We use a modal approaches incorporating the Gram matrix, using the orthogonality of angular derivative of m≠0 modes with respect to weight function w(ρ) = ρ (polar coordinates), and the orthogonality of radial derivative of m = 0 modes with respect to weight function w(ρ) = ρ(1-ρ2) (polar coordinates). The Gram matrix method needs no auxiliary vector functions. The Zernike coefficients can be obtained with no modal cross coupling. The simulation results are given, which indicate that the modal cross coupling is avoided by using Gram matrix method. This method can be easily extended to annulus, and the coefficients of Zernike annular polynomials with no modal cross coupling can be obtained.

Estimation of atmospheric turbulence parameters from Shack–Hartmann wavefront sensor measurements

The estimation of atmospheric turbulence parameters is of relevance for: a) site evaluation & characterisation; b) prediction of the point spread function; c) live assessment of error budgets and optimisation of adaptive optics performance; d) optimisation of fringe trackers for long baseline optical interferometry. The ubiquitous deployment of Shack-Hartmann wavefront sensors in large telescopes makes them central for atmospheric turbulence parameter estimation via adaptive optics telemetry. Several methods for the estimation of the Fried parameter and outer scale have been developed, most of which are based on the fitting of Zernike polynomial coefficients variances reconstructed from the telemetry. The non-orthogonality of Zernike polynomial derivatives introduces modal cross coupling, which affects the variances. Furthermore, the finite resolution of the sensor introduces aliasing. In this article the impact of these effects on atmospheric turbulence parameter estimation is addressed with simulations. It is found that cross coupling is the dominant bias. An iterative algorithm to overcome it is presented. Simulations are conducted for typical ranges of the outer scale (4 to 32m), Fried parameter (10 cm) and noise in the variances (signal-to-noise ratio of 10 and above). It is found that, using the algorithm, both parameters are recovered with sub-percent accuracy.

On modal cross-coupling in the asymptotic modal limit

The conditions under which significant modal cross-coupling occurs in dynamical systems responding to high-frequency, broadband forcing that excites many modes is studied. The modal overlap factor plays a key role in the analysis of these systems as the modal density (the ratio of number of modes to the frequency bandwidth) becomes large. The modal overlap factor is effectively the ratio of the width of a resonant peak (the damping ratio times the resonant frequency) to the average frequency interval between resonant peaks (or rather, the inverse of the modal density). It is shown that this parameter largely determines whether substantial modal cross-coupling occurs in a given system's response. Here, two prototypical systems are considered. The first is a simple rectangular plate whose significant modal cross-coupling is the exception rather than the norm. The second is a pair of rectangular plates attached at a point where significant modal cross-coupling is more likely to occur. We show that, for certain cases of modal density and damping, non-negligible cross coupling occurs in both systems. Under similar circumstances, the constraint force between the two plates in the latter system becomes broadband. The implications of this for using Asymptotic Modal Analysis (AMA) in multi-component systems are discussed.

Quasi-resonance Behaviour of Viscoelastic Rods Subjected to Free and Forced Longitudinal Vibrations

A classical viscoelastic rod subjected to longitudinal vibrations is considered. The boundary conditions are so that the left end of the rod is fixed and the right end is free. Moreover it is assumed that the rod is growing and hence, its length is changing in time. The function of growth is assumed to be twice continuously differentiable with respect to time. A particular case of rod's growth proportional to time is of special interest. The change of variables is introduced so that in new variables the rod length becomes constant. The new partial differential equation describing the rod's dynamics is derived in these variables. This equation is simplified using assumptions on slow rate growth constant and small viscoelastic damping factor. A special representation of solution is introduced which uses eigenfunctions of the generating problem, when growth and damping are neglected, and satisfy the boundary conditions. By means of this representation the governing partial differential equation is converted in infinite system of ordinary differential equations. It is shown that solutions of truncated systems converge to solution of the original system of equations. Three major problems of the growing rod vibrations are formulated and solved. In the first problem free undamped vibrations are considered. It is shown that at linear growth of rod length amplitudes of all its modes are also growing linearly in time. The simplified model neglecting the modes cross-coupling is composed for the explanation of this effect. The corresponding differential equation is solved exactly in elementary functions and it is shown that amplitudes of vibration of any modes grow linearly and almost-periods of vibrations have logarithmic dependence on time. In the second problem free damped vibrations of linearly growing rod are considered. It is shown that time behaviour of the rod has two characteristic domains: in the first the vibration amplitudes decay exponentially due to domination of the viscoelastic damping effects; in the second domain these amplitudes start to grow linearly in time due to domination of the effects considered in the first problem. The simplified model describing this effect and neglecting the modal cross-coupling is developed. The exact solution of the corresponding differential equation is obtained in the confluent hypergeometric functions, which qualitatively explain the abovementioned behaviour of the rod. In the third problem the forced damped vibrations of the rod are considered. It is shown that at fixed frequency of excitation the resonant effects are manifested subsequently in all modes of the rod in the process of its growing.

Modal wavefront reconstruction with Zernike polynomials and eigenfunctions of Laplacian

Modal cross coupling between basis functions in modal wavefront reconstruction is discussed. Eigenfunctions of Laplacian are proposed in modal approach for wavefront reconstruction. As the gradients of the eigenfunctions are orthogonal to each other, the modal cross coupling can be avoided theoretically. Wavefront reconstructions by use of Zernike polynomials and eigenfunctions of Laplacian with different sampling densities are compared. The results show that modal cross coupling between eigenfunctions of Laplacian is much smaller than that between Zernike polynomials.

Eigenfunctions of Laplacian for phase estimation from wavefront gradient or curvature sensing

Modal cross coupling usually exists in wavefront estimation through Zernike polynomials. In order to cope with the problem, the eigenfunctions of Laplacian with Neumann boundary condition are proposed instead of Zernike polynomials to reconstruct phase from wavefront gradient or curvature sensing. It is proved theoretically that these modals can avoid modal cross coupling in both wavefront gradient sensing and curvature sensing. In wavefront gradient sensing, the coefficients of eigenfunctions of Laplacian can be obtained from the integral of the scalar product between the gradient of Laplacian's eigenfunctions and wavefront gradient signal. In wavefront curvature sensing, the coefficients of eigenfunctions of Laplacian can be calculated from the integral of the product of Laplacian's eigenfunctions and wavefront curvature signal. This approach is applicable on arbitrary apertures as long as eigenfunctions of Laplacian on apertures of arbitrary shape can be obtained.

Acoustic scattering by a sphere with a hemispherically split boundary condition

A general analytical model is developed for the scattering of sound by a sphere with a nonuniform impedance boundary condition that is divided into two uniformly distributed hemispheres. In addition to the overall solution for the time harmonic pressure, the analytical result gives insight into the modal contributions and coupling for different cases of source incidence and boundary impedance. Modal cross coupling is shown to exist between incoming and scattered wave modes of equi-order and nonequal degree when the degrees are opposite in parity (odd-even or even-odd coupling). This cross coupling is strongest between modes of adjacent degree, and decreases as the degrees become dissimilar. The overall magnitude of the cross coupling is dependent on the extent of the impedance mismatch between the two surface hemispheres. Simulation and discussion are given for several specific cases of source incidence and impedance (each hemisphere is given a different constant impedance value). These results are consistent with expectations from the scattering of sound by a sphere with a uniformly distributed surface boundary. The broad scattering characteristics of the hemispherically divided sphere are shown to be analogous to connecting the appropriate sectors from the corresponding uniformly distributed spheres.

Variational solution for modal wave-front projection functions of minimum-error norm.

Common wave-front sensors such as the Hartmann or curvature sensor provide measurements of the local gradient or Laplacian of the wave front. The expression of wave fronts in terms of a set of orthogonal basis functions thus generally leads to a linear wave-front-estimation problem in which modal cross coupling occurs. Auxiliary vector functions may be derived that effectively restore the orthogonality of the problem and enable the modes of a wave front to be independently and directly projected from slope measurements. By using variational methods, we derive the necessary and sufficient condition for these auxiliary vector functions to have minimum-error norm. For the specific case of a slope-based sensor and a basis set comprising the Zernike circular polynomials, these functions are precisely the Gavrielides functions.

On acoustic and structural modal cross-couplings in plate-cavity systems

Modal cross-couplings are sometimes neglected in the prediction of sound field and structural responses of vibroacoustic systems where an enclosed sound field is coupled to a vibrating boundary structure. In such systems, there are two types of modal cross-couplings and they are commonly referred to as acoustic modal cross-coupling (ACC) and structural modal cross-coupling (SCC). The prediction errors generated from neglecting either of these cross-couplings are dependent not only on the modal properties of the vibroacoustic system (e.g., modal densities, dampings, etc.), but also on whether the sound field or the structure is directly driven. However, the physical mechanisms and characteristics of both cross-couplings are not well understood and, consequently, the conditions when ACC or SCC has a significant contribution to the system responses become unknown. This paper presents a mathematical description which allows the two types of modal cross-couplings to be studied independently. This description is then used to obtain the physical mechanisms and features of both cross-couplings. The effects of each type of cross-couplings on the system responses are then investigated and the general conditions under which these modal cross-couplings may be ignored are underlined.

Hartmann sensing with Albrecht grids

The modal coefficients of a wavefront can be estimated by suitably weighted integrals of the wavefront slopes. When the basis functions chosen to expand the wavefront are orthogonal (e.g. Zernike polynomials), the reconstruction problem becomes orthogonal itself, so that each coefficient can be estimated independently from the others. Modal cross-coupling can in this way be avoided. The obtained coefficients correspond to a direct least-squares fit between the real and estimated wavefronts. The evaluation of the modal integrals from the finite data set of measurements supplied by Shack-Hartmann sensors or shearing interferometers requires the use of efficient numerical integration methods. In this paper it is shown that Albrecht cubatures are a good candidate to perform this task. The wavefront slopes have to be measured at the nodal points of the chosen cubature, which in general are located in an unevenly spaced grid. Numerical results show that the method of orthogonal reconstruction with Albrecht cubatures allow to recover the modal coefficients with better accuracy than other usual approaches, including the standard non-orthogonal least-squares fit between the gradients of the orginal and reconstructed wavefronts.

Coupled FEM-BEM approach for mean flow effects on vibro-acoustic behavior of planar structures

The originality of the present paper lies in the development of a formulation accounting for mean flow effects on the forced vibro-acoustic response of a baffled plate. The importance of these effects on the vibrational behavior of the plate, as well as on its acoustic radiation pattern, is investigated for a baffled plate with different kinds of boundary conditions. The analysis is based on a finite element method for the calculation of the plate transverse vibrations and the use of the extended Kirchhoff's integral equation to account for fluid loading with mean flow. A variational boundary element method is used to compute the acoustic radiation impedance. The formulation shows explicitly the effects of mean flow in terms of added mass, stiffness, and radiation damping. Details of the formulation as well as its numerical implementation are expounded, and results showing the effect of mean flow hi light and heavy fluid on the vibro-acoustic quantities, such as mean square velocity and radiated acoustic power, are presented. It is seen that the effects of a mean flow amount to a decrease of the natural frequencies of the plate, a small damping effect on the vibrations, and a change in the radiated acoustic power when compared with the no-flow case. Besides, these effects increase with the flow speed. The negative stiffness added by the flow is shown to be mainly responsible for the natural frequency shift effect. The changes in the radiated acoustic power are explained in terms of important changes in the radiation efficiencies and modal cross-coupling induced by the flow.

Response of clamped sandwich panels with viscoelastic core under random acoustic excitation

Both theoretical and experimental studies have been made to evaluate the acceleration response of square sandwich panels with clamped boundaries under a specific random acoustic excitation. Generalized harmonic analysis was used to compute the response spectrum. The modal frequencies and associated loss factors of the composite used in the analytical estimation of the response were obtained by applying Galerkin's procedure. The effects of modal cross coupling and of the core thickness on the response have been studied. The theoretical estimates of the response are compared with the results from the experiments for certain core thicknesses.