On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate
ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 1, pp. 63-87.

We consider a partially hinged rectangular plate and its normal modes. There are two families of modes, longitudinal and torsional. We study the variation of the corresponding eigenvalues under domain deformations. We investigate the possibility of finding a shape functional able to quantify the torsional instability of the plate, namely how prone is the plate to transform longitudinal oscillations into torsional ones. This functional should obey several rules coming from both theoretical and practical evidences. We show that a simple functional obeying all the required rules does not exist and that the functionals available in literature are not reliable.

Received:
Accepted:
DOI: 10.1051/cocv/2016076
Classification: 35J40, 35P15, 74K20
Keywords: Shape variation, eigenvalues, plates, torsional instability, suspension bridges
Berchio, Elvise 1; Buoso, Davide 1; Gazzola, Filippo 2

1 Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy.
2 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.
@article{COCV_2018__24_1_63_0,
     author = {Berchio, Elvise and Buoso, Davide and Gazzola, Filippo},
     title = {On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {63--87},
     publisher = {EDP-Sciences},
     volume = {24},
     number = {1},
     year = {2018},
     doi = {10.1051/cocv/2016076},
     mrnumber = {3764134},
     zbl = {1400.35092},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv/2016076/}
}
TY  - JOUR
AU  - Berchio, Elvise
AU  - Buoso, Davide
AU  - Gazzola, Filippo
TI  - On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2018
SP  - 63
EP  - 87
VL  - 24
IS  - 1
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv/2016076/
DO  - 10.1051/cocv/2016076
LA  - en
ID  - COCV_2018__24_1_63_0
ER  - 
%0 Journal Article
%A Berchio, Elvise
%A Buoso, Davide
%A Gazzola, Filippo
%T On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2018
%P 63-87
%V 24
%N 1
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv/2016076/
%R 10.1051/cocv/2016076
%G en
%F COCV_2018__24_1_63_0
Berchio, Elvise; Buoso, Davide; Gazzola, Filippo. On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 1, pp. 63-87. doi : 10.1051/cocv/2016076. http://archive.numdam.org/articles/10.1051/cocv/2016076/

O.H. Ammann, T. von Kármán and G.B. Woodruff, The failure of the Tacoma Narrows Bridge, Federal Works Agency (1941).

P.R.S. Antunes and F. Gazzola, Convex shape optimization for the least biharmonic Steklov eigenvalue. ESAIM: COCV 19 (2013) 385–403 | Numdam | MR | Zbl

J.R. Banerjee, A simplified method for the free vibration and flutter analysis of bridge decks. J. Sound Vibration 260 (2003) 829–845. | DOI | Zbl

U. Battisti, E. Berchio, A. Ferrero and F. Gazzola, Periodic solutions and energy transfer between modes in a nonlinear beam equation. To appear in: J. Math. Pures Appl. (2018).

F. Bleich, Dynamic instability of truss-stiffened suspension bridges under wind action. In vol. 74 of Proc. of ASCE(1948) 1269–1314.

F. Bleich, C.B. McCullough, R. Rosecrans and G.S. Vincent, The mathematical theory of vibration in suspension bridges. U.S. Dept. of Commerce, Bureau of Public Roads, Washington D.C. (1950).

E. Berchio, A. Ferrero and F. Gazzola, Structural instability of nonlinear plates modelling suspension bridges: mathematical answers to some long-standing questions. Nonlin. Anal. Real World Appl. 28 (2016) 91–125. | DOI | MR | Zbl

E. Berchio and F. Gazzola, A qualitative explanation of the origin of torsional instability in suspension bridges. Nonlin. Anal. TMA 121 (2015) 54–72. | DOI | MR | Zbl

E. Berchio, F. Gazzola and C. Zanini, Which residual mode captures the energy of the dominating mode in second order Hamiltonian systems? SIAM J. Appl. Dyn. Syst. 15 (2016) 338–355. | DOI | MR | Zbl

G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures. Calc. Var. 5 (1997) 37–54. | DOI | MR | Zbl

G. Bouchitté and I. Fragalà, Second-order energies on thin structures: variational theory and non-local effects. J. Funct. Anal. 204 (2003) 228–267. | DOI | MR | Zbl

G. Bouchitté, W. Gangbo and P. Seppecher, Michell trusses and lines of principal action. Math. Models Methods Appl. Sci. 18 (2008) 1571–1603. | DOI | MR | Zbl

H.W. Broer and M. Levi, Geometrical aspects of stability theory for Hill’s equations. Arch. Ration. Mech. Anal. 131 (1995) 225–240. | DOI | MR | Zbl

H.W. Broer and C. Simó, Resonance tongues in Hill’s equations: a geometric approach. J. Differ. Equ. 166 (2000) 290–327. | DOI | MR | Zbl

D. Bucur and G. Buttazzo, Variational methods in shape optimization problems. In vol. 65 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser (2005). | MR | Zbl

D. Buoso, Shape sensitivity analysis of the eigenvalues of polyharmonic operators and elliptic systems. Ph.D. Dissertation, Università degli Studi di Padova, Padova (2015).

D. Buoso, Analyticity and criticality for the eigenvalues of the biharmonic operator. Geometric Properties for Parabolic and Elliptic PDE’2019s. In vol. 176 of Springer INdAM Series (2016) 65–85. | MR

D. Buoso and P.D. Lamberti, Eigenvalues of polyharmonic operators on variable domains. ESAIM COCV 19 (2013) 1225–1235. | DOI | Numdam | MR | Zbl

D. Buoso and P.D. Lamberti, Shape deformation for vibrating hinged plates. Math. Methods Appl. Sci. 37 (2014) 237–244. | DOI | MR | Zbl

D. Buoso and L. Provenzano, A few shape optimization results for a biharmonic Steklov problem. J. Differ. Eq. 259 (2015) 1778–1818 | DOI | MR | Zbl

E.I. Butikov, Subharmonic resonances of the parametrically driven pendulum. J. Phys. A: Math. Gen. 35 (2002) 6209–6231. | DOI

T. Cazenave and F.W. Weissler, Asymptotically periodic solutions for a class of nonlinear coupled oscillators. Portugal. Math. 52 (1995) 109–123. | MR | Zbl

T. Cazenave and F.W. Weissler, Unstable simple modes of the nonlinear string, Quart. Appl. Math. 54 (1996) 287–305. | DOI | MR | Zbl

L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations. Springer, Berlin (1971). | MR | Zbl

L.M. Chasman, An isoperimetric inequality for fundamental tones of free plates with nonzero Poisson’2019s ratio. Appl. Anal. 95 (2016) 1700–1735. | DOI | MR | Zbl

C. Chicone, Ordinary differential equations with applications. In vol. 34 of Texts in Applied Mathematics, 2nd Editions. Springer, New York (2006). | MR | Zbl

M. Como, S. Del Ferraro, A. Grimaldi, A parametric analysis of the flutter instability for long span suspension bridges. Wind and Structures 8 (2005) 1–12. | DOI

M.C. Delfour and J.P. Zolésio, Shapes and geometries. Analysis, differential calculus, and optimization. Advances in Design and Control. Society for Industrial and Applied Mathematics SIAM, Philadelphia, PA (2001). | MR | Zbl

A. Ferrero and F. Gazzola, A partially hinged rectangular plate as a model for suspension bridges. Discrete Contin. Dyn. Syst. A. 35 (2015) 5879–5908. | DOI | MR | Zbl

F. Gazzola, Hexagonal design for stiffening trusses. Ann. Mat. Pura Appl. 194 (2015) 87–108. | DOI | MR | Zbl

F. Gazzola, Mathematical models for suspension bridges. Nonlinear structural instability. In Vol. 15 of MS&A. Springer Cham (2015). | MR | Zbl

M. Haberland, S. Hass and U. Starossek, Robustness assessment of suspension bridges, 6th International Conference on Bridge Maintenance, Safety and Management, IABMAS12. Edited by Stresa, Biondini & Frangopol. Taylor & Francis Group, London (2012) 1617–1624.

A. Henrot, Extremum problems for eigenvalues of elliptic operators. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2006). | MR | Zbl

A. Henrot and E. Oudet, Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions, Arch. Ration. Mech. Anal. 169 (2003) 73–87. | DOI | MR | Zbl

A. Henrot and M. Pierre, Variation et optimisation de formes, une analyse géométrique. Math. Appl. 48 (2005). | MR | Zbl

D. Henry, Perturbation of the boundary in boundary-value problems of partial differential equations. With editorial assistance from Jack Hale and Antônio Luiz Pereira. In vol. 318 of London Math. Soc. Lect. Note Ser. Cambridge University Press, Cambridge (2005). | MR | Zbl

G. Herrmann and W. Hauger, On the interrelation of divergence, flutter and auto-parametric resonance. Vol. 42 of Ingenieur-Archiv (1973) 81–88.

J.D. Holmes, Wind loading of structures, 2nd edition. Taylor & Francis, London & New York (2007).

H.M. Irvine, Cable Structures, MIT Press Series in Structural Mechanics. Massachusetts (1981).

A. Jenkins, Self-oscillation. Phys. Reports 525 (2013) 167–222. | DOI | MR | Zbl

J.A. Jurado, S. Hernández, F. Nieto and A. Mosquera, Bridge aeroelasticity: sensitivity analysis and optimum design (high performance structures and materials). Computational Mechanics. WIT Press, Southampton (2011).

B. Kawohl, J. Stara and G. Wittum, Analysis and numerical studies of a problem of shape design. Arch. Ration. Mech. Anal. 114 (1991) 349–363 | DOI | MR | Zbl

W. Lacarbonara, Nonlinear structural mechanics. Theory, dynamical phenomena and modeling. Springer New York (2013). | MR | Zbl

P.D. Lamberti, M. Lanza De Cristoforis, A real analyticity result for symmetric functions of the eigenvalues of a domain dependent Dirichlet problem for the Laplace operator. J. Nonlin. Convex Anal. 5 (2004) 19–42. | MR | Zbl

A. Larsen, Aerodynamics of the Tacoma Narrows Bridge – 60 years later. Struct. Eng. Internat. 4 (2000) 243–248. | DOI

M. Matsumoto, H. Matsumiya, S. Fujiwara, Y. Ito, New consideration on flutter properties based on step-by-step analysis. J. Wind Eng. Ind. Aerodynamics 98 (2010) 429–437. | DOI

A.G.M. Michell, The limits of economy of material in framed structures. Vol. 8 of Philosophical Magazine Series 6 (1904) 589–597. | JFM

National Park Service, Trusses: a study by the historical American engineering record, Society for Industrial Archeology, available at: http://www.nps.gov/hdp/samples/HAER/truss%20poster.pdf (1976).

C.L. Navier, Extraits des recherches sur la flexion des plans élastiques. Bulletin des Sciences de la Société Philomathique de Paris (1823) 92–102.

E. Oudet, Numerical minimization of eigenmodes of a membrane with respect to the domain. ESAIM COCV 10 (2004) 315–335. | DOI | Numdam | MR | Zbl

Y. Rocard, Dynamic instability: automobiles, aircraft, suspension bridges. Crosby Lockwood, London (1957).

J.A. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, 2nd edition. Vol. 59 of Appl. Math. Sci. Springer, New York (2007). | MR | Zbl

R.H. Scanlan, Developments in low-speed aeroelasticity in the civil engineering field. AIAA Journal 20 (1982) 839–844. | DOI | Zbl

R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives. J. Eng. Mech. 97 (1971) 1717–1737.

R. Scott, In the wake of Tacoma. Suspension bridges and the quest for aerodynamic stability. ASCE Press (2001).

A. Selberg,Oscillation and aerodynamic instability of suspension bridges, Acta Polytechnica Scandinavica, in vol. 13 of Civil Engineering and Construction Series. Trondheim (1961).

F.C. Smith and G.S. Vincent, Aerodynamic stability of suspension bridges: with special reference to the Tacoma Narrows Bridge, Part II: Mathematical analysis, Investigation conducted by the Structural Research Laboratory. University of Washington, University of Washington Press, Seattle (1950).

J. Sokolowski and J.P. Zolésio, Vol. 16 of Introduction to shape optimization. Springer Series in Comput. Math. (1992). | MR | Zbl

F. Verhulst, Perturbation analysis of parametric resonance. In: Encyclopedia of Complexity and Systems Science. Springer (2009) 6625–6639. | MR

Wikipedia, The Free Encylopedia. Available at: https://en.wikipedia.org/wiki/Aizhai˙Bridge.

P.P. Xanthakos, Theory and design of bridges. John Wiley & Sons, New York (1994).

YouTube, Tacoma Narrows Bridge collapse. Available at: http://www.youtube.com/watch?v=3mclp9QmCGs (1940).

YouTube, Airfoil flutter, aricraft flutter, bridge flutter. Available at: https://www.youtube.com/watch?v=72cQgXw7_kI; https://www.youtube.com/watch?v=iTFZNrTYp3k; https://www.youtube.com/watch?v=1Oq8HuB7_tI.

Cited by Sources: