In this study, an analysis of mechanical vibrations influenced by external additive white Gaussian noise and colored noise is conducted using the principal component analysis. The principal component analysis is widely employed for encoding images in image processing, biology, economics, sociology, and political science. However, it is hereby applied to analyze nonlinear dynamics of continuous mechanical systems for the first time. A rich class of objects, including straight beams, beams on Winkler foundations and spherical shells, is investigated in the present paper. The basic differential equations are obtained based on the Bernoulli–Euler hypothesis, and solutions of the linear PDEs are analyzed by means of the principal component analysis. Results obtained with the principal component analysis are compared with those for the method of empirical modal decomposition and the wavelet-packet decomposition.
KerschenGGolinvalJCVakakisAF, et al.The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview. Nonlin Dyn2005; 41: 147–170.
2.
Loève M. Fonctions Aléatoires du Second Ordre. In: Levy P (ed) Processus Stochastiques et Mouvement Brownien. Paris: Gauthier-Villars, 1948, pp.299–352.
3.
LiangYCLeeHPLimSP, et al.Proper orthogonal decomposition and its applications. Part 1: theory. J Sound Vib2002; 252: 527–544.
4.
KosambiD. Statistics in function space. J Indian Math Soc1943; 7: 76–88.
5.
Watanabe S. Karhunen-Loève expansion and factor analysis: theoretical remarks and applications. In: Proceedings of the 4th conference on information theory, Prague, Czech Republic, 31 August–11 September 1965.
6.
MeesAIRappPEJenningsLS. Singular value decomposition and embedding dimension. Phys Rev A1987; 36: 340–346.
7.
RavindraB. Comments on ‘On the physical interpretation of proper orthogonal modes in vibrations’. J Sound Vib1999; 219: 189–192.
8.
WuCGLiangYCLinWZ, et al.A note on equivalence of proper orthogonal decomposition methods. J Sound Vib2003; 265: 1103–1110.
9.
HolmesPLumleyJLBerkoozG. Turbulence, coherent structures, dynamical systems and symmetry, New York: Cambridge, 1996.
Pardo MA, Sorenson L, Pan W, et al. Phase noise shaping via forced nonlinearity in piezoelectrically actuated silicon micromechanical oscillators. In: Proceedings of the IEEE international conference on micro electro mechanical systems (MEMS), Cancun, Mexico, 23–27 January 2011, art. no. 5734541, pp.780–783. IEEE.
12.
OcakHLoparoKADiscenzoFM. Online tracking of bearing wear using wavelet packet decomposition and probabilistic modelling: a method for bearing prognostic. J Sound Vib2007; 302: 951–961.
13.
GokhaleMYKhandujaDK. Time domain signal analysis using wavelet packet decomposition approach. Int J Comm Net Sys Sci2010; 3: 321–329.
14.
Jiang E, Zan P, Zhang S, et al. Optimal wavelet packet decomposition for rectal pressure signal feature extraction. In: IEE 5th international conference on advanced computational intelligence, Nanjing, Jiangsu, China, 18–20 October 2012, pp.1204–1208. IEEE.
15.
MaXVakakisA. Karhunen-Loeve decomposition of the transient dynamics of a multibay truss. AIAA J1999; 37: 939–946.
LiRGhanemR. Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration. Probab Eng Mech1998; 13: 125–136.
18.
SarkarAGhanemR. Mid frequency structural dynamics with parameter uncertainty. Comput Meth Appl Mech Eng2002; 191: 5499–5513.
19.
SchenkCASchuellerGI. Buckling analysis of cylindrical shells with random geometric imperfections. Mech Eng2002; 191: 5499–5513.
20.
HotAKerschenGFoltêteE, et al.Detection and quantification of non-linear structural behavior using principal component analysis. Mech Sys Sig Proc2012; 26: 104–116.
21.
OnozatoNNagaiKMaruyamaS, et al.Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. Nonlin Dyn2012; 67: 2363–2379.
22.
MaruyamaSNagaiKTsurutaY. Modal interaction in chaotic vibrations of a shallow double-curved shell-panel. J Sound Vib2008; 315: 607–625.
23.
NagaiKMaruyamaSSakaimotoK, et al.Experiments on chaotic vibrations of a post-buckled beam with an axial elastic constraint. J Sound Vib2007; 304: 541–555.
24.
YanagisawaDNagaiKMaruyamaS. Chaotic vibrations of a clamped-supported beam with a concentrated mass subjected to static axial compression and periodic lateral acceleration. J Sys Des Dyn2008; 2: 762–773.
25.
KryskoAVAwrejcewiczJPapkovaIV, et al.Principal component analysis in the nonlinear dynamics of beams: purification of the signal from noise induced by the nonlinearity of beam vibrations. Adv Math Phys2017; 2017: 1–9.
26.
HorsthemkeWLefeverR. Noise-induced transitions. Theory and applications in physics, chemistry and biology, Berlin: Springer-Verlag, 1984.
27.
FrauenfelderHWolynesPG. Rate theories and puzzles of hemeprotein kinetics. Science1985; 26: 337–345.
FengCSWuJYZhuWQ. Response of Duffing system with delayed feedback control under combined harmonic and real noise excitation. Comm Nonlin Sci Num Simul2009; 14: 2542–2550.
32.
WuYJZhuWQ. Stationary response of Duffing-Rayleigh-Mathieu system under colored noise excitation. J Vib Eng2009; 22: 2542–2550.
33.
GuoSSShiQ. Probabilistic solutions of nonlinear oscillators excited by combined colored and white noise excitations. Comm Nonlin Sci Num Simul2017; 44: 414–423.
34.
ZhangHXuYXuW. Approximate stationary solution and stochastic stability for a class of differential equations with parametric colored noise. Nonlin Dyn2009; 56: 213–221.
35.
KoharVKiaBLindnerJF, et al.Reduction of additive colored noise using coupled dynamics. Int J Bif Chaos2016; 26: 1650005–1650005.
36.
HorvathJS. Modulus of subgrade reaction: new perspective. J Geotech Eng ASCE1983; 109: 1591–1596.
37.
WangHThamLGCheungYK. Beams and plates on elastic foundations: a review. Prog Struct Eng Mater2005; 7: 174–182.
38.
AwrejcewiczJKryskoVAPapkovaIV, et al.Deterministic chaos in one-dimensional continuous systems, Singapore: World Scientific, 2016.
39.
SamarskyAA. Introduction in numerical methods, Moscow: Nauka, 1987. (in Russian).
40.
KryskoAVAwrejcewiczJKutepovIE, et al.Chaotic dynamics of flexible Euler-Bernoulli beams. Chaos2014; 34: 043130–1-043130-25.
41.
ShuaiqiLMingLPeifeiL, et al.SAR image denoising via sparse representation in shearlet domain based on continuous cycle spinning. IEEE Transac Geosci Remote Sens2017; 55: 2985–2992.
42.
KryskoVAKrylovaEYUPapkovaIV. Wavelet-based analysis for the regular and chaotic dynamics of rectangular flexible plates subjected to shear-harmonic loading. Shock Vib2012; 19: 979–994.
43.
KryskoVAYakovlevaTVDobriyanVV, et al.Wavelet-analysis-based chaotic synchronization of vibrations of multilayer mechanical structures. Int Appl Mech2014; 50: 706–720.
