1. A semi-active vibration control method for a nonlinear system of a large-span cable bridge structure is characterized by comprising the following steps:
(1) establishing a shaking force load model acting on a large-span cable bridge structure:
the harmonic waves formed by a plurality of trigonometric functions are mutually superposed to simulate the shaking force load acting on the large-span cable bridge structure, and the shaking force acting on the large-span cable bridge structure of the jth formed wave can be given by the following formula:
j=1,2,…,r
wherein A isj,ωjRespectively representing the amplitude, frequency and initial phase angle of the jth component wave,
order to To obtain
Wherein, order to Can obtain the product
Wherein,Iris an identity matrix, 0 ∈ Rr×rIs a zero matrix;
the shaking force load model acting on the large-span cable bridge structure is an external system described by formula 1:
in the formula 1, the compound is shown in the specification,
wherein: p (t) is the shaking force acting on the large-span cable bridge structure;
(2) establishing a large-span cable bridge structure vibration control system model:
the cable-bridge coupling vibration power system under the action of the shaking force load comprises:
wherein W (t) is the displacement from the equilibrium position; y (t) is the displacement of the cable end, namely the displacement of the bridge deck along the cable direction; omega1And ω2Natural frequency of cable and bridge deck, ξ damping ratio of bridge deck, ai(i =1,2, …,6) is a system parameterA number, u (t) is the semi-active control force, p (t) is the buffeting force load, generated by equation 1;
selecting a state variable:
then the model of the large-span cable bridge structure vibration control system is as follows:
formula 2;
wherein,
(3) selecting a composite performance index as described in a formula 3;
formula 3;
(4) the maximum principle is implemented for equations 2 and 3:
first, a Hamiltonian is constructed as formula 4:
H(·)=xTQx1+Ru2+λT(Ax+Bu+f(x)+Dpω) Formula 4;
and further converting the problem of solving the controller u (t) under the constraint of the performance index formula 3 in the formula 1 into a problem of solving the following nonlinear two-point boundary values in the formula 5 according to the extreme value condition:
formula 5;
(5) solving an optimal vibration control law:
solving the iterative solution of the nonlinear two-point boundary value problem described in equation 5 by using a successive approximation method, and enabling
λ(t)=P1x(t)+P2p(t)+P3pω(t)+g(t)
Where g (t) is a common mode vector, for λ (t) = P1x(t)+P2p(t)+P3Derivation of p ω (t) + g (t), followed by taking the first formula of formula 2 Substituted therein to obtain
From formula 2 and formula 5 to obtain
In view of
By comparisonThe matrix equation 7 and the sequence equations 8 and 9 are obtained, and the optimal vibration control law described by equation 6 can be obtained
Formula 7;
formula 8;
formula 9;
the approximate optimal vibration control law is obtained from the above equations 7, 8 and 9:
u(k)(t)=-R-1BTλ(k)(t)=-R-1BT[P1x(k)(t)+P2p(t)+P3pω(t)+g(k)(t)]formula 6;
wherein,and P is1,P2And P3The following equation 7: x is the number of(k)And g(k)(t) is determined from the following formulae 8 and 9.