To obtain approximate probability density functions for a class of axially moving viscoelastic plates under external and parametric white noise excitation

Abstract

The probability density function plays an essential role to investigate the behaviors of stochastic linear or nonlinear systems. This function can be evaluated by several approaches but due to its analytical theme, the Fokker–Planck–Kolmlgorov (FPK) approach is preferable. FPK equation is a nonlinear PDE gives the probability density function for a stochastic linear or nonlinear system. Many researches have been done in literature tried to specify the conditions, in which the FPK equation gives an exact solution. Although, the exact probability density function can be achieved by solving the FPK equation even for some nonlinear systems, many types of systems cannot satisfy the conditions for exact solution. In this article, the axially moving viscoelastic plates under both external and parametric white noise excitation as one of the newest and applicable research areas are studied. Due to strong nonlinearities recognized in the governing equation of the system, the exact probability density function cannot be obtained, however, via an approximate method; some precise approximate solutions for different but comprehensive case studies are evaluated, validated, and discussed.

Appendices

Appendix A: The equations needed to be solved for Case (1.a) Example (1)

$2a_5 +a_2^2 +10=0$

$100a_2 +a_8 +a_2 a_4 =0$

$200a_4 +2a_{12} +2a_2 a_7 +a_4^2 =0$

$45a_2 +100a_7 +a_2 a_{11} +a_4 a_7 =0$

$90a_4 +200a_{11} +2a_4 a_{11} +a_7^2 =0$

$-\,100a_1 +5a_2 +3a_9 +2a_2 a_5 =0$

$-\,100a_4 +10a_5 +6a_{14} +3a_2 a_9 +2a_5^2 =0$

$-\,100a_8 +15a_9 +4a_2 a_{14} +6a_5 a_9 =0$

$-\,200a_3 +5a_4 +200a_5 +3a_{13} +2a_2 a_8 +2a_4 a_5 =0$

$-\,200a_7 +10a_8 +300a_9 +3a_2 a_{13} +3a_4 a_9 +4a_5 a_8 =0$

$-\,200a_{12} +15a_{13} +400a_{14} +4a_4 a_{14} +6a_5 a_{13} +6a_8 a_9 =0$

$-\,300a_6 +5a_7 +200a_8 +2a_2 a_{12} +2a_4 a_8 +2a_5 a_7 =0$

$-\,300a_{11} \!+\!10a_{12} \!+\!300a_{13} \!+\!3a_4 a_{13} \!+\!4a_5 a_{12} \!+\!3a_7 a_9 \!-\!2a_8^2 =0$

$90a_5 -\,400a_{10} \!+\!5a_{11} \!+\!200a_{12} \!+\!2a_4 a_{12} \!+\!2a_5 a_{11} \!+\!2a_7 a_8 =0$

Appendix B: The values of unknown parameters $a$ in Example (2) for $n=6$

Case study (2.a)	$a_1 =a_2 =a_6 =a_7 =a_8 =a_9 =a_{15} =a_{16} =a_{17} =a_{18} =a_{19} =a_{20} =0$,
	$a_3 =-7.9975$,	$a_4 =-0.0201$,	$a_5 =-8.0000$,	$a_{10} =-\,2.1436$,	$a_{11} =0.0092$,
	$a_{12} =-\,0.6859$,	$a_{13} =0.1140$,	$a_{14} =-\,0.3357$,	$a_{21} =-\,0.1076$,	$a_{22} =-\,0.0041$,
	$a_{23} =-\,0.0144$,	$a_{24} =-\,0.0498$,	$a_{25} =0.1326$,	$a_{26} =-\,0.0179$,	$a_{27} =0.0438$
Case study (2.b)	$a_1 =a_2 =a_6 =a_7 =a_8 =a_9 =a_{15} =a_{16} =a_{17} =a_{18} =a_{19} =a_{20} =0$,
	$a_3 =7.9975$,	$a_4 =0.0204$,	$a_5 =-8.0000$,	$a_{10} =-1.4691$,	$a_{11} =0.0094$,
	$a_{12} =-0.6631$,	$a_{13} =-\,0.1155$,	$a_{14} =0.3395$,	$a_{21} =-0.1088$,	$a_{22} =0.0043$,
	$a_{23} =0.0277$,	$a_{24} =-\,0.0696$,	$a_{25} =0.1281$,	$a_{26} =0.0374$,	$a_{27} =-\,0.0460$
Case study (2.c)	$a_1 =0.4805$,	$a_2 =-\,0.0010$,	$a_3 =8.0019$,	$a_4 =0.0204$,	$a_5 =-\,8.0031$,
	$a_6 =0.0421$,	$a_7 =-\,0.0009$,	$a_8 =-\,0.0430$,	$a_9 =0.0066$,	$a_{10} =-\,1.4677$,
	$a_{11} =0.0094$,	$a_{12} =-\,0.6667$,	$a_{13} =-\,0.1155$,	$a_{14} =0.3410$,	$a_{15} =-\,0.0002$,
	$a_{16} =-\,0.0006$,	$a_{17} =-\,0.0196$,	$a_{18} =0.0065$,	$a_{19} =0.0105$,	$a_{20} =-\,0.0046$,
	$a_{21} =-\,0.1087$,	$a_{22} =0.0043$,	$a_{23} =0.0261$,	$a_{24} =-0.0695$,	$a_{25} =0.1304$,
	$a_{26} =0.0372$,	$a_{27} =-\,0.0467$

Appendix C: The values of unknown parameters $a$ in Example (3)

Case study (3.a)	$a_1 =-152.7729$,	$a_2 =-1.0$,	$a_3 =-1263.3921$,
	$a_4 =33.3333$,	$a_5 =-2545.7154$,	$a_6 =22307.2546$,
	$a_7 =-263.6060$,	$a_8 =42727.4406$,	$a_9 =2067.5139$,
	$a_{10} =-977080.1367$,	$a_{11} =-5429.5027$,	$a_{12} =-310635.3635$,
	$a_{13} =-103260.0029$,	$a_{14} =354511.3694$,	$a_{15} =-37643999.7723$,
	$a_{16} =283755.4916$,	$a_{17} =-54649119.9759$,	$a_{18} =1793834.4424$,
	$a_{19} =-16916856.7026$,	$a_{20} =-972009.1141$,	$a_{21} =-618635838.1771$,
	$a_{22} =8762930.1498$,	$a_{23} =-1317844219.5187$,	$a_{24} =48756550.8221$
	$a_{25} =-800119718.3269$,	$a_{26} =69851581.7298$,	$a_{27} =-93172529.6414$
Case study (3.b)	$a_1 =152.7729$,	$a_2 =1.0$,	$a_3 =-1263.3921$,
	$a_4 =33.3333$,	$a_5 =-2545.7153$,	$a_6 =-22307.2546$,
	$a_7 =263.6060$,	$a_8 =-42727.4406$,	$a_9 =-2067.5139$,
	$a_{10} =-977080.1367$,	$a_{11} =-5429.5027$,	$a_{12} =-310635.3635$,
	$a_{13} =-103260.0029$,	$a_{14} =354511.3694$,	$a_{15} =37643999.7723$,
	$a_{16} =-283755.4916$,	$a_{17} =54649119.9759$,	$a_{18} =-1793834.4424$,
	$a_{19} =16916856.7027$,	$a_{20} =972009.1141$,	$a_{21} =-618635838.1770$,
	$a_{22} =8762930.1498$,	$a_{23} =-1317844219.5187$,	$a_{24} =48756550.8221$,
	$a_{25} =-800119718.3270$,	$a_{26} =69851581.7298$,	$a_{27} =-93172529.6414$
Case study (3.c)	$a_1 =142.8932$,	$a_2 =1.0$,	$a_3 =434.4922$,
	$a_4 =-33.3333$,	$a_5 =-2381.0530$,	$a_6 =-67147.5624$,
	$a_7 =318.4935$,	$a_8 =64868.6926$,	$a_9 =-1216.9983$,
	$a_{10} =-678545.2791$,	$a_{11} =11010.3379$,	$a_{12} =1193845.8803$,
	$a_{13} =76734.7719$,	$a_{14} =-541485.1866$,	$a_{15} =-3766990.1854$,
	$a_{16} =30728.7128$,	$a_{17} =5497710.0682$,	$a_{18} =-1231894.5530$,
	$a_{19} =-1790374.0391$,	$a_{20} =-944753.2952$,	$a_{21} =-64841224.7283$,
	$a_{22} =813188.2951$,	$a_{23} =130455839.4146$,	$a_{24} =-26418263.8454$,
	$a_{25} =-78207280.2551$,	$a_{26} =46484737.9825$,	$a_{27} =9159228.0266$,
	$a_{28} =-740339672.2476$,	$a_{29} =16448970.5339$,	$a_{30} =2127420231.9701$,
	$a_{31} =542455975.5900$,	$a_{32} =-1856173147.6651$,	$a_{33} =-387163761.0702$,
	$a_{34} =602399461.3791$,	$a_{35} =-213505128.2748$,	$a_{36} =-3952823115.6985$,
	$a_{37} =162400529.6958$,	$a_{38} =16143716137.9273$,	$a_{39} =9597030428.9760$,
	$a_{40} =-23548224514.4540$,	$a_{41} =-18948051909.5750$,	$a_{42} =11975906633.5031$,
	$a_{43} =11320453919.5812$,	$a_{44} =-2696814742.9868$