Abstract
The nonlinear eigenvalue problem describing eigenvibrations of a beam with elastically attached load is investigated. The existence of an increasing sequence of positive simple eigenvalues with limit point at infinity is established. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. The problem is approximated by the finite element method with Hermite finite elements of arbitrary order. The convergence and accuracy of approximate eigenvalues and eigenfunctions are investigated.
Similar content being viewed by others
References
Yu. P. Zhigalko and S. I. Solov’ev, “Natural oscillations of a beam with a harmonic oscillator,” Russ. Math. 45 (10), 33–35 (2001).
S. I. Solov’ev, Eigenvibrations of a Plate with Elastically Attached Load, Preprint SFB393/03-06 (Technische Universität Chemnitz, 2003).
S. I. Solov’ev, Vibrations of Plates with Masses, Preprint SFB393/03-18 (Technische Universität Chemnitz, 2003).
S. I. Solov’ev, Nonlinear Eigenvalue Problems: Approximate Methods (Lambert Acad. Publ., Saarbrücken, 2011) [in Russian].
A. V. Gulin and S. V. Kartyshov, “Numerical study of stability and nonlinear eigenvalue problems,” Surv. Math. Ind. 3, 29–48 (1993).
T. Betcke, N. J. Higham, V. Mehrmann, C. Schröder, and F. Tisseur, “NLEVP: A collection of nonlinear eigenvalue problems,” ACMTrans.Math. Software. 39 (2), Article number 7 (2013).
V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations (AmericanMathematical Society, Providence, 2001).
Th. Apel, A.-M. Sändig, and S. I. Solov’ev, “Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes,” Math. Model. Numer. Anal. 36 (6), 1043–1070 (2002).
A. D. Lyashko and S. I. Solov’ev, “Fourier method of solution of FE systems with Hermite elements for Poisson equation,” Sov. J. Numer. Anal.Math. Modelling. 6 (2), 121–129 (1991).
S. I. Solov’ev, “Fast direct methods of solving finite-element grid schemes with bicubic elements for the Poisson equation,” J. Math. Sciences. 71 (6), 2799–2804 (1994).
S. I. Solov’ev, “A fast directmethod of solving Hermitian fourth-order finite-element schemes for the Poisson equation,” J. Math. Sciences. 74 (6), 1371–1376 (1995).
E. M. Karchevskii and S. I. Solov’ev, “Investigation of a spectral problem for the Helmholtz operator on the plane,” Differ. Equations 36 (4), 631–634 (2000).
A. V. Gulin and A. V. Kregzhde, On the Applicability of the Bisection Method to Solve Nonlinear Difference Eigenvalue Problems, Preprint no. 8 (Inst. Appl. Math., USSR Science Academy, Moscow, 1982).
A. V. Gulin and S. A. Yakovleva, “On a numerical solution of a nonlinear eigenvalue problem,” in Computational Processes and Systems, (Nauka, Moscow, 1988), Vol. 6, pp. 90–97 [in Russian].
R. Z. Dautov, A. D. Lyashko, and S. I. Solov’ev, “The bisection method for symmetric eigenvalue problems with a parameter entering nonlinearly,” Russ. J. Numer. Anal. Math. Modelling. 9 (5), 417–427 (1994).
A. Ruhe, “Algorithms for the nonlinear eigenvalue problem,” SIAM J. Numer. Anal. 10, 674–689 (1973).
F. Tisseur and K. Meerbergen, “The quadratic eigenvalue problem,” SIAM Rev. 43, 235–286 (2001).
V. Mehrmann and H. Voss, “Nonlinear eigenvalue problems: A challenge for modern eigenvalue methods,” GAMM–Mit. 27, 1029–1051 (2004).
S. I. Solov’ev, “Preconditioned iterativemethods for a class of nonlinear eigenvalue problems,” Linear Algebra Appl. 415 (1), 210–229 (2006).
D. Kressner, “A block Newton method for nonlinear eigenvalue problems,” Numer. Math. 114 (2), 355–372 (2009).
X. Huang, Z. Bai, and Y. Su, “Nonlinear rank-one modification of the symetric eigenvalue problem,” J. Comput.Math. 28 (2), 218–234 (2010).
H. Schwetlick and K. Schreiber, “Nonlinear Rayleigh functionals,” Linear Algebra Appl. 436 (10), 3991–4016 (2012).
W.-J. Beyn, “An integral method for solving nonlinear eigenvalue problems,” Linear Algebra Appl. 436 (10), 3839–3863 (2012).
A. Leblanc and A. Lavie, “Solving acoustic nonlinear eigenvalue problems with a contour integral method,” Eng. Anal. Bound. Elem. 37 (1), 162–166 (2013).
X. Qian, L. Wang, and Y. Song, “A successive quadratic approximations method for nonlinear eigenvalue problems,” J.Comput. Appl. Math. 290, 268–277 (2015).
A. V. Gulin and A. V. Kregzhde, Difference Schemes for Some Nonlinear Spectral Problems, Preprint no. 153 (Inst. Appl.Math., USSR Science Academy, Moscow, 1981).
A. V. Kregzhde, “On difference schemes for the nonlinear Sturm–Liouville problem,” Differ. Uravn. 17 (7), 1280–1284 (1981).
S. I. Solov’ev, “The error of the Bubnov–Galerkin method with perturbations for symmetric spectral problems with a non-linearly occurring parameter,” Comput.Math. Math. Phys. 32 (5), 579–593 (1992).
S. I. Solov’ev, “The finite element method for symmetric nonlinear eigenvalue problems,” Comput. Math. Math. Phys. 37 (11), 1269–1276 (1997).
R. Z. Dautov, A. D. Lyashko, and S. I. Solov’ev, “Convergence of the Bubnov–Galerkin method with perturbations for symmetric spectral problems with parameter entering nonlinearly,” Differ. Equations. 27 (7), 799–806 (1991).
V. S. Zheltukhin, S. I. Solov’ev, P. S. Solov’ev, and V. Yu. Chebakova, “Computation of the minimum eigenvalue for a nonlinear Sturm–Liouville problem,” Lobachevskii J.Math. 35 (4), 416–426 (2014).
S. I. Solov’ev, “Approximation of differential eigenvalue problems with a nonlinear dependence on the parameter,” Differ. Equations. 50 (7), 947–954 (2014).
S. I. Solov’ev, “Approximation of nonlinear spectral problems in a Hilbert space,” Differ. Equations. 51 (7), 934–947 (2015).
O. O. Karma, “Approximation in eigenvalue problems for holomorphic Fredholm operator functions. I,” Numer. Funct. Anal. Optimiz. 17, 365–387 (1996).
O. O. Karma, “Approximation in eigenvalue problems for holomorphic Fredholm operator functions. II. Convergence rate,” Numer. Funct. Anal. Optimiz. 17, 389–408 (1996).
I. Babushka and A. K. Aziz, “Survey lectures on the mathematical foundations of the finite element method,” in TheMathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Academic Press, New York, 1972), pp. 3–359.
G. M. Vainikko, “Asymptotic evaluations of the error of projection methods for the eigenvalue problem,” USSRComput. Math.Math. Phys. 4 (3), 9–36 (1964).
G. M. Vainikko, “Evaluation of the error of the Bubnov-Galerkin method in an eigenvalue problem,” USSR Comput.Math. Math. Phys. 5 (4), 1–31 (1965).
G. M. Vainikko, “On the speed of convergence of approximate methods in the eigenvalue problem,” USSR Comput.Math. Math. Phys. 7 (5), 18–32 (1967).
S. I. Solov’ev, “Superconvergence of finite element approximations of eigenfunctions,” Differ. Equations. 30 (7), 1138–1146 (1994).
S. I. Solov’ev, “Superconvergence of finite element approximations to eigenspaces,” Differ. Equations. 38 (5), 752–753 (2002).
S. I. Solov’ev, “Approximation of variational eigenvalue problems,” Differ. Equations. 46 (7), 1030–1041 (2010).
S. I. Solov’ev, “Approximation of positive semidefinite spectral problems,” Differ. Equations. 47 (8), 1188–1196 (2011).
S. I. Solov’ev, “Approximation of sign-indefinite spectral problems,” Differ. Equations. 48 (7), 1028–1041 (2012).
S. I. Solov’ev, “Approximation of differential eigenvalue problems,” Differ. Equations. 49 (7), 908–916 (2013).
S. I. Solov’ev, “Finite element approximationwith numerical integration for differential eigenvalue problems,” Appl. Numer. Math. 93, 206–214 (2015).
V. P.Mikhaylov, Partial Differential Equations (Nauka, Moscow, 1983) [in Russian].
I. Babushka and J. E. Osborn, “Eigenvalue problems,” in Handbook of Numerical Analysis (North- Holland, Amsterdam, 1991), Vol. 2, pp. 642–787 (1991).
V. S. Zheltukhin, P. S. Solov’ev, and V. Yu. Chebakova, “Boundary conditions for electron balance equation in the stationary high-frequency induction discharges,” Research Journal of AppliedSciences 10 (10), 658–662 (2015).
Author information
Authors and Affiliations
Corresponding author
Additional information
Submitted by A. V. Lapin
Rights and permissions
About this article
Cite this article
Solov’ev, S.I. Eigenvibrations of a beam with elastically attached load. Lobachevskii J Math 37, 597–609 (2016). https://doi.org/10.1134/S1995080216050115
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080216050115