Function, and as a result we set the coefficient with the prospective function
Function, and hence we set the coefficient with the potential function as an uncertain parameter, which can be expressed as = , (10)where is definitely the imply value of ; is usually a bounded random variable defined on [-1, 1] together with the probability density function p, which has been given in WZ8040 Technical Information Equation (4); is definitely the intensity of . Then, the dimensionless electromechanical equations of TEH with an uncertain parameter is usually written as follows x (t) x (t) x (t) x (t)three (t)5 – v(t) = A cos(t), (11) v(t) v(t) x (t) = 0. As outlined by the theory of orthogonal CFT8634 In Vivo polynomial approximation above, the responses of Equation (11) could be expressed by the following series x (t, ) = N x (t) H , i i i =0 N v(t, ) = v (t) H , i ii =0 N(12)exactly where xi (t) and vi (t) is usually obtained by Equation (9). Although N , xi (t) Hi i =and vi (t) Hi are strictly equivalent to x (t, ) and v(t, ). To meet the desires on the calculation efficiency, we take N = four in Equation (12), then x (t, ) 4 x (t) H , i i i =0 four v(t, ) v (t) H , i ii =0 i =N(13)that are approximate options with a minimal mean square residual error. Substituting Equation (13) into Equation (11), yieldsAppl. Sci. 2021, 11,6 ofi = xi (t) Hi xi (t) Hi xi (t) Hi xi (t) Hi i =0 four i =0 i =0 xi (t) Hi i = xi (t) Hi i =0 4- vi (t) Hi = A cos(t),i =0(14)i =vi (t) Hi vi (t) Hi xi (t) Hi = 0.i =0 i =Since the item of any two Chebyshev polynomials could be transformed into a linear combination of associated single Chebyshev polynomials, the cubic term and quantic term is often lowered asi =0xi (t) Hi =i =0Xi (t) Hi , Yi (t) Hi .(15)i =0xi (t) Hi =(16)i =Moreover, based on recurrent formulas (6) on the Chebyshev polynomials, xi (t) Hi i =0 4in Equation (14) is often lowered to3i =xi (t) Hi =i =Xi (t) Hi = 2 Xi (t)[ Hi-1 Hi1 ]i =1 12 = [ Xi-1 (t) Xi1 (t)] Hi , two i =(17)exactly where X-1 (t) and X5 (t) are supposed to be zero. Substituting expressions (15), (16) and (17) into Equation (14), we havei =0 1 two xi (t) Hi xi (t) Hi xi (t) Hi Xi (t) Hi i =0 i =0 i =i =[ Xi-1 (t) Xi1 (t)] Hi Yi (t) Hi – vi (t) Hi = A cos(t),i =0 four i =0(18)i =vi (t) Hi vi (t) Hi xi (t) Hi = 0.i =0 i =Multiplying both sides of Equation (18) by Hi in sequence and taking expectations with respect to , in accordance together with the orthogonality of the Chebyshev polynomials, the high-dimensional equivalent system of the TEH may be written asAppl. Sci. 2021, 11,7 of x0 (t) x0 (t) x0 (t) X0 (t) 1 X1 (t) 0 (t) – v0 (t) = A cos(t), two v0 (t) v0 (t) x0 (t) = 0,1 x1 (t) x1 (t) x1 (t) X1 (t) two [ X0 (t) X2 (t)] 1 (t) – v1 (t) = 0,v1 (t) v1 (t) x1 (t) = 0, x2 (t) x2 (t) x2 (t) X2 (t) 1 [ X1 (t) X3 (t)] two (t) – v2 (t) = 0, two v2 (t) v2 (t) x2 (t) = 0, x3 (t) x3 (t) x3 (t) X3 (t) 1 [ X2 (t) X4 (t)] three (t) – v3 (t) = 0, two v3 (t) v3 (t) x3 (t) = 0, x4 (t) x4 (t) x4 (t) X4 (t) 1 X3 (t) 4 (t) – v4 (t) = 0, 2 v4 (t) v4 (t) x4 (t) = 0, (19)which is a kind of weighted typical technique. So the ensemble imply response (EMR) in the TEH with an uncertain parameter can be introduced to reflect the average traits of your original system (11), which is E[ x (t, )] = E four x (t) H = x (t), i 0 i i =0 four E[v(t, )] = E v (t) H = v (t). 0 i ii =(20)3.three. Validation Within this section, we focus on to confirm the effectiveness on the Chebyshev polynomial approximation. The deterministic response (DR) of system (three) and the deterministic ensemble imply response (EMR0 ) of technique (19) by setting the stochastic intensity = 0.0 in Equation (19) could be obtained numerically. Right here t.
