# The interaction between two parametric excitations of the second and third degrees

## In the present paper, we study interaction between two parametric excitations of the second and third degrees. The asymptotic method of nonlinear mechanics in combination with a computer is used. | Vietnam Journal of Mechanics, NOST of Vietnam Vol. 22, 2000, No 2 (93 - 100) THE INTERACTION BETWEEN TWO PARAMETRIC EXCITATIONS OF THE SECOND AND THIRD DEGREES NGUYEN VAN DAO Vietnam National University, Hanoi ABSTRACT. The interaction between nonlinear oscillations is an interesting problem which has attracted many researches. The interaction between forced and parametric excitations and between two parametric excitations of first and second degrees and first and third degrees, is informed in [l ]. In the present paper, we study interaction between two parametric excitations of the second and third degrees . The asymptotic method of nonlinear mechanics in combination with a computer is used. 1. Stationary Oscillations Let us consider a dynamic system governed by the differential equation - x + w~--4 = .~ .~ e:{ ~x - ' hx -1x 3 + 2px 2 cos wt+ 2qx 3 cos(2wt + 2a) }, () where 2p ; '.::. 0, 2q > 0 are intensities of parametric excitations of second and third degrees, respectively, and 2a (0 :S 2a 0 is a small parameter, e:~ = w2 - 1 is the detuning parameter and 1 is the own frequency of the system under consideration; overdot denotes the derivative relative to time. The solution of equation () will be found in the form x = acos'l/;, ±= - awsin'I/;, 1/; =wt+ 0, () where a and (} are unknown functions of time, which satisfy the relationship acos 1/; - a(} sin 'ljJ = 0. () By substituting () into () and combining it with () we obtain the following equations for new variables a and 0: . . ·'· a= - -CF sin 'f'• w { aO. = - wc F cos 1/;, 93 () where eF denotes the right hand side of equation (). In the first approximation we can replace () by averaged equations ' a= -~(Fsin'lj;), w . () c = --(Fcost/J), aO w where (1) is an averaged value in time of the function 1. It is easy to verify the following form of equations (): a= - 4: { aO = - [2hwa + pa 2 sinO + qa3 sinO + qa 3 sin(20 - 2a)], : [ 2 (A-

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