Asymptotic representations for root vectors of nonselfadjoint operators and pencils generated by an aircraft wing model in subsonic air flow. (English) Zbl 1068.76045

Summary: This paper is the second in a series of several works devoted to the asymptotic and spectral analysis of an aircraft wing in a subsonic air flow [see also IMA J. Appl. Math. 66, No. 4, 319–356 (2001; Zbl 1053.76034)]. This model has been developed in the Flight Systems Research Center of UCLA and is presented in the works by A. V. Balakrishnan. The model is governed by a system of two coupled integro-differential equations and a two-parameter family of boundary conditions modeling the action of the self-straining actuators. The differential parts of the above equations form a coupled linear hyperbolic system; the integral parts are of the convolution type. The system of equations of motion is equivalent to a single operator evolution-convolution equation in the energy space. The Laplace transform of the solution of this equation can be represented in terms of the so-called generalized resolvent operator, which is an operator-valued function of the spectral parameter. This generalized resolvent operator is a finite-meromorphic function on the complex plane having the branch cut along the negative real semi-axis. Its poles are precisely the aeroelastic modes and the residues at these poles are the projectors on the generalized eigenspaces. In the first paper and in the present one, our main object of interest is the dynamics generator of the differential parts of the system. It is a nonselfadjoint operator in the energy space with a purely discrete spectrum. In the first paper, we have shown that the spectrum consists of two branches and have derived their precise spectral asymptotics. In the present paper, we derive the asymptotical approximations for the mode shapes. Based on the asymptotical results of these first two papers, in the next paper, we will discuss the geometric properties of the mode shapes such as minimality, completeness, and the Riesz basis property in the energy space.


76G25 General aerodynamics and subsonic flows
47N20 Applications of operator theory to differential and integral equations
35P20 Asymptotic distributions of eigenvalues in context of PDEs


Zbl 1053.76034
Full Text: DOI


[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Ashley, H., Engineering analysis of flight vehicles, (1992), Dover New York
[3] Ashley, H.; Landahl, M., Aerodynamics of wings and bodies, (1985), Dover New York · Zbl 0161.22502
[4] Balakrishnan, A.V., Aeroelastic control with self-straining actuators: continuum models, (), 44-54
[5] Balakrishnan, A.V., Subsonic flutter suppression using self-straining actuators, J. franklin inst., 338, 149-171, (2001) · Zbl 0981.74015
[6] Balakrishnan, A.V., Vibrating systems with singular mass-inertia matrices, (), 23-32
[7] Balakrishnan, A.V., Theoretical limits of damping attainable by smart beams with rate feedback, (), 204-215
[8] Balakrishnan, A.V., Damping performance of strain actuated beams, Comput. appl. math., 18, 31-86, (1999) · Zbl 0931.74030
[9] Balakrishnan, A.V.; Edwards, J.W., Calculation of the transient motion of elastic airfoils forced by control surface motion and gusts, Nasa tm, 81, (1980)
[10] Balakrishnan, A.V., Dynamics and control of articulated anisotropic Timoshenko beams, Dynamics and control of distributed systems, (1998), Cambridge Univ. Press Cambridge, p. 121-202
[11] Balakrishnan, A.V., Control of structures with self-straining actuators: coupled Euler/Timoshenko model, Nonlinear problems in aviation and aerospace, (1998), Gordon Breach Reading
[12] Bisplinghoff, R.L.; Ashley, H.; Halfman, R.L., Aeroelasticity, (1996), Dover New York
[13] Chen, G.; Krantz, S.G.; Ma, D.W.; Wayne, C.E.; West, H.H., The euler – bernoulli beam equations with boundary energy dissipation, Oper. meth. for opt. contr. problems, Lecture notes in math., 108, (1987), Dekker New York, p. 67-96
[14] Istratescu, V., Introduction to linear operator theory, Pure appl. math series of monog., (1981), Dekker New York
[15] Lee, C.K.; Chiang, W.W.; O’Sullivan, T.C., Piezoelectric modal sensor/actuator pairs for critical active damping…, J. acoust. soc. amer., 90, 384-394, (1991)
[16] Magnus, W.; Oberhettinger, F.; Soni, R.P., Formulas and theorems for the special functions of mathematical physics, (1966), Springer-Verlag New York · Zbl 0143.08502
[17] M. A. Shubov, Mathematical analysis of problem arising in modelling of flutter phenomenon in aircraft wing in subsonuc airflow, IMA. J. Appl. Math, in press.
[18] Shubov, M.A., Spectral operators generated by Timoshenko Bean model, Systems control lett., 38, 249-258, (1999) · Zbl 0985.93012
[19] Shubov, M.A., Timoshenko beam model: spectral properties and control, (), 140-152
[20] Shubov, M.A., Exact controllability of Timoshenko beam, IMA J. math. control inform., 17, 375-395, (2000) · Zbl 0991.93016
[21] M. A. Shubov, Asymptotics and spectral analysis of Timoshenko beam model, Math. Nachr, in press. · Zbl 1037.47032
[22] Shubov, M.A., Asymptotics of resonances and geometry of resonance states in the problem of scattering of acoustical waves by a spherically symmetric inhomogeneity of the density, Differential integral equations, 8, 1073-1115, (1995) · Zbl 0827.34075
[23] Shubov, M.A., Asymptotics of spectrum and eigenfunctions for nonselfadjoint operators generated by radial nonhomogeneous damped wave equations, Asymptotic anal., 16, 245-272, (1998) · Zbl 0938.35113
[24] Shubov, M.A., Nonselfadjoint operators generated by the equation of nonhomogeneous damped string, Trans. amer. math. soc., 349, 4481-4499, (1997) · Zbl 0889.47004
[25] Shubov, M.A., Spectral operators generated by 3-dimensional damped wave equation and application to control theory, (), 177-188 · Zbl 0901.35065
[26] Tzou, H.S.; Gadre, M., Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls, J. sound vibration, 132, 433-450, (1989)
[27] Weidmann, J., Spectral theory of ordinary differential operators, Lecture notes in math., 1258, (1987), Springer-Verlag New York/Berlin
[28] Yang, S.M.; Lee, Y.J., Modal analysis of stepped beams with peizoelectric materials, J. sound vibration, 176, 289-300, (1994) · Zbl 0945.74584
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.