VIBRATION CONTROL METHOD FOR FLAPPING-WING MICRO AIR VEHICLES

Abstract

The present invention provides a method for controlling the oscillation of flapping-wing air vehicle, which comprises the following steps: calculating the kinetic energy, potential energy and virtual work of the system using the flexible wing with the two-degree of freedom as the research object; establishing system dynamics model based on the Hamilton's principle; setting the boundary control rate according to said system dynamics model wherein said boundary control rate includes F(t) and M(t), said F(t) is the inputted boundary control force, and said M(t) is the inputted boundary torque; and controlling the flexible wings according to the system dynamics model in combination with the boundary control rate. The present invention establishes the system dynamics model based on the Hamilton's principle, set the boundary control rate according to said system dynamics model, sufficiently considers the situation of distributed disturbance occurring at the boundary and effectively prevents the flexible wings deformation caused by the external disturbances.

Claims

1. A method for controlling the oscillation of flapping-wing air vehicle, comprising the following steps: Calculating the kinetic energy, potential energy, and virtual work of the system using the flexible wing as the research object; Establishing the Hamilton's principle based system dynamics model; Setting the boundary control rate according to said system dynamics model wherein said boundary control rate includes F(t) and M(t), said F(t) is the inputted boundary control force, and M(t) is the inputted boundary torque; and Controlling the flexible wings according to the system dynamics model and combining the boundary control rate.

2. The method for controlling the oscillation of flapping-wing air vehicle of claim 1, characterized in that said calculating the kinetic energy, potential energy, and virtual work of the system using the flexible wing as the research object comprises: the kinetic energy of the system, E.sub.k(t) is expressed as follows: $\begin{array}{cc}{E}_k\left(t\right)=\frac{1}{2}.Math.m.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\frac{1}{2}.Math.I_p.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(1\right)\end{array}$ wherein the spatial variable of x is independent to the time variable of t, and m is the unitspan mass of the flexible wing; I.sub.p is inertial polar distance of the flexible wing; y(x, t) is the bending displacement at the position of x and at time of t in the x0y coordinate system; and θ(x, t) is the corresponding displacement of deflection angle; Potential energy of E.sub.p(t) is expressed as follows: $\begin{array}{cc}{E}_p\left(t\right)=\frac{1}{2}.Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\frac{1}{2}.Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(2\right)\end{array}$ wherein, EI.sub.b denotes the flexural rigidity, and GJ denotes the torsional rigidity; and the virtual work of δW.sub.c(t) caused by the above two rigidities is expressed as follows:
δW.sub.c(t)=mx.sub.oc∫.sub.0.sup.Lÿ(x, t)δθ(x, t)dx+mx.sub.oc∫.sub.0.sup.L{umlaut over (θ)}(x, t)δy(x, t)dx  (3), wherein x.sub.oc denotes the distance from the mass center of wing to the bending center; and the virtual work of δW.sub.d(t) provided by the Kelvin-Voigt damping force is expressed as follows:
δW.sub.d(t)=−ηEI.sub.b∫.sub.0.sup.L{dot over (y)}″(x, t)δy″(x, t)dx−ηGJ.sub.b∫.sub.0.sup.L{dot over (θ)}′(x, t)δθ′(x, t)dx  (4), wherein, η denotes the Kelvin-Voigtd damping coefficient; the virtual work of δW.sub.r(t) done by the distributed distraction is expressed as follows:
δW.sub.r(t)=∫.sub.0.sup.L[F.sub.b(x, t)δy(x, t)−x.sub.acF.sub.b(x, t)δθ(x, t)]dx  (5), wherein x.sub.ac denotes the distance from the aerodynamic center to the bending centre and F.sub.b is the unknown time varying distributed distraction along the wings; the virtual work of δW.sub.a(t) done by the boundary control force to the system is expressed as follows:
δW.sub.a(t)=F(t)δy(L, t)+M(t)δθ(L, t)  (6), In the above formula, F(t) is the inputted boundary control force and M(t) is the inputted boundary torque; Consequently, the total virtual work is:
δW(t)=δ[W.sub.c(t)+W.sub.d(t)+W.sub.r(t)+W.sub.a(t)]  (7).

3. The method for controlling the oscillation of flapping-wing air vehicle of claim 1, characterized in that, said establishing the system dynamics model based on the Hamilton's principle includes: utilizing the Hamilton's smooth action principle of ∫.sub.t.sub.1.sup.t.sup.2δ[E.sub.k(t)−E.sub.p(t)+W(t)]dt=0 Here δ denotes the variation symbol, and the governing equation for the system dynamics model is deduced as:
mÿ(x, t)+EI.sub.by″″(x, t)−mx.sub.oc{umlaut over (θ)}(x, t)+ηEI.sub.b{dot over (y)}″″(x, t)=F.sub.b(x, t)  (8)
I.sub.p{umlaut over (θ)}(x, t)−GJθ″(x, t)−mx.sub.ocÿ(x, t)−ηGJ{dot over (θ)}″(x, t)=−x.sub.acF.sub.b(x, t)  (9) And the boundary conditions for the system dynamics model are deduced as:
y(0, t)=y′(0, t)=y″(L, t)=θ(0, t)=0  (10),
EI.sub.by′″(L, t)+ηEI.sub.b{dot over (y)}′″(L, t)=−F(t)  (11) and
GJθ′(L, t)+ηGJ{dot over (θ)}′(L, t)=M(t)  (12).

4. The method for controlling the oscillation of flapping-wing air vehicle of claim 3, characterized in that, said setting the boundary controller based on the system dynamics model includes two controlling laws of F(t) and M(t) wherein said F(t) is the inputted boundary control force and said M(t) is the inputted boundary torque, and includes: Constructing the Lyapunov candidate function as follows:
V(t)=V.sub.1+Δ(t)  (13) Wherein, V.sub.1(t) and Δ(t) are respectively defined as: $\begin{array}{cc}{V}_1\left(t\right)=\frac{\beta }{2}.Math.m.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\frac{\beta }{2}.Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\hspace{1em}\frac{\beta }{2}.Math.I_p.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\frac{\beta }{2}.Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\theta }^{\prime }\left(x,t\right)]}^2.Math..Math.\mathrm{dx},& \left(14\right)\\ \Delta \left(t\right)=\alpha .Math..Math.m.Math.{\int }_0^L.Math.\stackrel{.}{y}\left(x,t\right).Math..Math.y\left(x,t\right).Math.\mathrm{dx}+\alpha .Math..Math.I_p.Math.{\int }_0^L.Math.\stackrel{.}{\theta }\left(x,t\right).Math.\theta \left(x,t\right).Math..Math.\mathrm{dx}-\alpha .Math..Math.{\mathrm{mx}}_e.Math.c.Math.{\int }_0^L.Math.\left[\stackrel{.}{y}\left(x,t\right).Math..Math.\theta \left(x,t\right)+y\left(x,t\right).Math.\stackrel{.}{\theta }\left(x,t\right)\right].Math.\mathrm{dx}-\beta .Math..Math.{\mathrm{mx}}_e.Math.c.Math.{\int }_0^L.Math.\stackrel{.}{y}\left(x,t\right).Math.\stackrel{.}{\theta }\left(x,t\right).Math..Math.\mathrm{dx};& \left(15\right)\end{array}$ In the above two equations, both α and β are the smaller positive weight coefficient; the boundary control rate is set by means of making the Lyapunov candidate function be positive definite, and making the derivative of Lyapunov candidate function of {dot over (V)}(t) to the time of t be negative definite.

5. The method for controlling the oscillation of flapping-wing air vehicle of claim 4, characterized in that, said calculating the boundary control rate when the Lyapunov candidate function is positive definite, and the derivative of Lyapunov candidate function of {dot over (V)}(t) to the time of t is negative definite includes: defining a new function as follows:
κ(t)=∫.sub.0.sup.L{[{dot over (y)}(x, t)].sup.2+[{dot over (θ)}(x, t)].sup.2+[y″(x, t)].sup.2+[θ′(x, t)].sup.2}dx  (16), Then V.sub.1(t) has the upper bound and lower bound which are defined as
γ.sub.2κ(t)≦V.sub.1(t)≦γ.sub.1κ(t)  (17), In the above formula, ${\gamma }_1=\frac{\beta }{2}.Math.\max \left(m,I_p,{\mathrm{EI}}_b,\mathrm{GJ}\right),.Math.{\gamma }_2=\frac{\beta }{2}.Math.\min \left(m,I_p,{\mathrm{EI}}_b,\mathrm{GJ}\right);$ Further, Δ(t) is magnified as: $\begin{array}{cc}.Math.\Delta .Math..Math.\left(t\right).Math.\le \left(\alpha .Math..Math.m+\alpha .Math..Math.{\mathrm{mx}}_e.Math.c+\beta .Math..Math.{\mathrm{mx}}_e.Math.c\right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\left(\alpha .Math..Math.I_p+\alpha .Math..Math.{\mathrm{mx}}_e.Math.c+\beta .Math..Math.{\mathrm{mx}}_e.Math.c\right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\left(\alpha .Math..Math.m+\alpha .Math..Math.{\mathrm{mx}}_e.Math.c\right).Math.L^4.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\left(\alpha .Math..Math.I_p+\alpha .Math..Math.{\mathrm{mx}}_e.Math.c\right).Math.L^2.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}\le y_3.Math.\kappa \left(t\right),& \left(18\right)\end{array}$ wherein γ.sub.3=max{αm+αmx.sub.oc+βmx.sub.oc, αI.sub.p+αmx.sub.oc+βmx.sub.oc, (αm+αmx.sub.oc)L.sup.4, (αI.sub.p+αmx.sub.oc)L.sup.2}, if the positive number of β satisfies $\beta >\frac{2.Math.{\gamma }_3}{\min \left(m,I_p,{\mathrm{EI}}_b,\mathrm{GJ}\right)},$ then
0≦λ.sub.2κ(t)≦V(t)≦λ.sub.3κ(t)  (19), which means that the constructed Lyapunov function is positive definite, wherein λ.sub.1=γ.sub.1+γ.sub.3 and λ.sub.2=γ.sub.2−γ.sub.3; by calculating the derivative of V(t) to t, we obtain:
{dot over (V)}(t)={dot over (V)}.sub.1(t)+{dot over (Δ)}(t)  (20),
{dot over (V)}.sub.1(t)=βm∫.sub.0.sup.L{dot over (y)}(x, t)ÿ(x, t)dx+βI.sub.p∫.sub.0.sup.L{dot over (θ)}(x, t){umlaut over (θ)}(x, t)dx +βGJ∫.sub.0.sup.Lθ′(x, t){dot over (θ)}′(x, t)dx+βEI.sub.b∫.sub.0.sup.Ly″(x, t){dot over (y)}″(x, t)dx  (21) by introducing the controlling equation (8) and (9) into the above formula, we obtain:
{dot over (V)}.sub.1(t)=A.sub.1+A.sub.2+A.sub.3+A.sub.4+A.sub.5+A.sub.6  (22), Wherein, A.sub.1˜A.sub.6 are respectively expressed as follows
A.sub.1=−βEI.sub.b∫.sub.0.sup.L{dot over (y)}(x, t)y″″(x, t)dx+βEI.sub.b∫.sub.0.sup.Ly″(x, t){dot over (y)}″(x, t)dx  (23),
A.sub.2=−βηEI.sub.b∫.sub.0.sup.L{dot over (y)}(x, t){dot over (y)}″″(x, t)dx  (24),
A.sub.3=βmx.sub.oc ∫.sub.0.sup.L[{dot over (y)}(x, t){umlaut over (θ)}(x, t)+ÿ(x, t){dot over (θ)}(x, t)]dx  (25),
A.sub.4=β∫.sub.0.sup.L{dot over (y)}(x, t)F.sub.b(x, t)dx−βx.sub.oc∫.sub.0.sup.L{dot over (θ)}(x, t)F.sub.b(x, t)dx  (26),
A.sub.5=βGJ∫.sub.0.sup.L{dot over (θ)}(x, t)θ″(x, t)dx+βGJ∫.sub.0.sup.Lθ′(x, t){dot over (θ)}′(x, t)dx  (27), and
A.sub.6=βηGJ∫.sub.0.sup.L{dot over (θ)}(x, t){dot over (θ)}″(x, t)dx  (28), By utilizing the integration by parts and the ba nary condition of (10), (11) and (12), we obtain $\begin{array}{cc}.Math.A_1=-\beta .Math..Math.{\mathrm{EI}}_b.Math.\stackrel{.}{y}\left(L,t\right).Math.y^{\mathrm{\prime \prime \prime }}\left(L,t\right)=-\beta .Math..Math.\stackrel{.}{y}\left(L,t\right).Math.F\left(t\right),& \left(29\right)\\ A_2\le -\beta .Math..Math.\eta .Math..Math.y\left(L,t\right).Math.\stackrel{.}{F}\left(t\right)-\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2.Math.L^4}.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math.\mathrm{dx}-\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},\mathrm{and}& \left(30\right)\\ A_4\le {\sigma }_1.Math.\beta .Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_2.Math.\beta .Math..Math.x_a.Math.c.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\left(\frac{\beta }{{\sigma }_1}+\frac{\beta .Math..Math.x_a.Math.c}{{\sigma }_2}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2,& \left(31\right)\end{array}$ wherein and σ.sub.1 and σ.sub.2 are the positive constant, F.sub.b max is the maximum value of the distributed disturbance of F.sub.b(x, t); $\begin{array}{cc}.Math.A_5=\beta .Math..Math.\mathrm{BJ}.Math..Math.\stackrel{.}{\theta }\left(L,t\right).Math.{\theta }^{\prime }\left(L,t\right)=\beta .Math..Math.\stackrel{.}{\theta }\left(L,t\right).Math.M\left(t\right)& \left(32\right)\\ A_6\le \beta .Math..Math.\eta .Math..Math.\stackrel{.}{\theta }\left(L,t\right).Math.\stackrel{.}{M}\left(t\right)-\frac{\beta .Math..Math.\eta .Math..Math.\mathrm{GJ}}{2.Math.L^2}.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\frac{\beta .Math..Math.\eta .Math..Math.\mathrm{GJ}}{2}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}& \left(33\right)\end{array}$ Based on the above A1˜A6, we acquire {dot over (V)}.sub.1(t) as follows: $\begin{array}{cc}{\stackrel{.}{V}}_1\left(t\right)\le -\left(\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2.Math.L^4}-{\sigma }_1.Math.\beta \right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\left(\frac{\beta .Math..Math.\eta .Math..Math.\mathrm{GJ}}{2.Math.L^2}-{\sigma }_2.Math.\beta .Math..Math.x_a.Math.c\right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\frac{\beta .Math..Math.\eta .Math..Math.\mathrm{GJ}}{2}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\beta .Math..Math.{\mathrm{mx}}_e.Math.c.Math.{\int }_0^L.Math.\left[\stackrel{.}{y}\left(x,t\right).Math.\stackrel{\mathrm{.Math.}}{\theta }\left(x,t\right)+\stackrel{\mathrm{.Math.}}{y}\left(x,t\right).Math.\stackrel{.}{\theta }\left(x,t\right)\right].Math..Math.\mathrm{dx}-\beta .Math..Math.\stackrel{.}{y}\left(L,t\right)\left[F\left(t\right)+\eta .Math..Math.\stackrel{.}{F}\left(t\right)\right]+\beta .Math..Math.\stackrel{.}{\theta }\left(L,t\right)\left[M\left(t\right)+\eta .Math..Math.\stackrel{.}{M}\left(t\right)\right]+\left(\frac{\beta }{{\sigma }_1}+\frac{\beta .Math..Math.x_a.Math.c}{{\sigma }_2}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2& \left(34\right)\end{array}$ similarly, by means of calculating the derivation of Δ(t) to t, we obtain:
{dot over (Δ)}(t)=B.sub.1+B.sub.2+. . . B.sub.8  (35),
B.sub.1=−αEI.sub.b∫.sub.0.sup.Ly(x, t)y″″(x, t)dx  (36),
B.sub.2=−αηEI.sub.b∫.sub.0.sup.Ly(x, t){dot over (y)}″″(x, t)dx  (37),
B.sub.3=αGJ∫.sub.0.sup.Lθ(x, t)θ″(x, t)dx  (38),
B.sub.4=αηGJ∫.sub.0.sup.Lθ(x, t){dot over (θ)}″(x, t)dx  (39),
B.sub.5=αm∫.sub.0.sup.L[{dot over (y)}(x, t)].sup.3dx+αI.sub.p∫.sub.0.sup.L[{dot over (θ)}(x, t)].sup.2dx  (40),
B.sub.6=−βmx.sub.oc∫.sub.0.sup.L[{dot over (y)}(x, t){umlaut over (θ)}(x, t)+ÿ(x, t){dot over (θ)}(x, t)]dx  (41),
B.sub.7=−2αmx.sub.oc∫.sub.0.sup.L{dot over (y)}(x, t){dot over (θ)}(x, t)dx  (42), and
B.sub.8=α∫.sub.0.sup.Ly(x, t)F.sub.b(x, t)dx−αx.sub.oc∫.sub.0.sup.Lθ(x, t)F.sub.b(x, t)dx  (43); By means of introducing the boundary conditions into the above formulas, we obtain: $\begin{array}{cc}.Math.B_1=-\alpha .Math..Math.y\left(L,t\right).Math.F\left(t\right)-\alpha .Math..Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(44\right)\\ B_2\le -.Math.\alpha .Math..Math.\eta .Math..Math.y\left(L,t\right).Math.\stackrel{.}{F}\left(t\right)+\frac{\alpha .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{{\sigma }_3}.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_3.Math..Math.\alpha .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(45\right)\\ .Math.B_3=\alpha .Math..Math.\theta \left(L,t\right).Math.M\left(t\right)-\alpha .Math..Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(46\right)\\ B_4\le .Math.\alpha .Math..Math.\eta .Math..Math.\theta \left(L,t\right).Math.\stackrel{.}{M}\left(t\right)+\frac{\alpha .Math..Math.\eta .Math..Math.\mathrm{GJ}}{{\sigma }_4}.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_4.Math..Math.\alpha .Math..Math.\eta .Math..Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(47\right)\\ B_7\le .Math.2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c.Math..Math.{\sigma }_5.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\frac{2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c}{{\sigma }_5}.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},\mathrm{and}& \left(48\right)\\ B_8\le {\sigma }_6.Math..Math.\alpha .Math..Math.L^4.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_7.Math..Math.\alpha .Math..Math.x_a.Math..Math.{\mathrm{cL}}^2.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\left(\frac{\alpha }{{\sigma }^6}+\frac{\alpha .Math..Math.x_a.Math.c}{{\sigma }_7}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2,& \left(49\right)\end{array}$ and All the above σ.sub.3-σ.sub.7 are the positive constant. Therefore, according to B.sub.1-B.sub.8, we obtain formula of (50): $\begin{array}{cc}\stackrel{.}{\Delta }\left(t\right)\le -.Math.\left(\alpha .Math..Math.{\mathrm{EI}}_b-\frac{\mathrm{\alpha \eta }.Math..Math.{\mathrm{EI}}_b}{{\sigma }_3}-{\sigma }_6.Math.\alpha .Math..Math.L^4\right).Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\left(\alpha .Math..Math.\mathrm{GJ}-\frac{\mathrm{\alpha \eta }.Math..Math.\mathrm{GJ}}{{\sigma }_4}-{\sigma }_3.Math..Math.\alpha .Math..Math.x_a.Math.c.Math..Math.L^2\right).Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\left(\alpha .Math..Math.m+2.Math..Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c.Math..Math.{\sigma }_5\right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+(.Math.\alpha .Math..Math.I_p+\frac{2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c}{{\sigma }_5}).Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_3.Math.\mathrm{\alpha \eta }.Math..Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_4.Math.\mathrm{\alpha \eta }.Math..Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\beta .Math..Math.{\mathrm{mx}}_e.Math.c.Math.{\int }_0^L.Math.\left[\stackrel{.}{y}\left(x,t\right).Math.\stackrel{\mathrm{.Math.}}{\theta }\left(x,t\right)+\stackrel{\mathrm{.Math.}}{y}\left(x,t\right).Math.\stackrel{.}{\theta }\left(x,t\right)\right].Math..Math.\mathrm{dx}+\alpha .Math..Math.\theta \left(L,t\right)\left[M\left(t\right)+\eta .Math..Math.\stackrel{.}{M}\left(t\right)\right]-\alpha .Math..Math.y\left(L,t\right)\left[F\left(t\right)+\eta .Math..Math.\stackrel{.}{F}\left(t\right)\right]+\left(\frac{\alpha }{{\sigma }_6}+\frac{\alpha .Math..Math.x_a.Math.c}{{\sigma }_7}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2,& \left(50\right)\end{array}$ Based on the formulas of (34) and (50), we can obtain: $\begin{array}{cc}\stackrel{.}{V}\left(t\right)\le -\left[\alpha .Math..Math.y\left(L,t\right)+\beta .Math..Math.\stackrel{.}{y}\left(L,t\right)\right]\left[F\left(t\right)+\eta .Math..Math.\stackrel{.}{F}\left(t\right)\right]+\left[\alpha .Math..Math.\theta \left(L,t\right)+\beta .Math.\stackrel{.}{.Math.\theta }.Math.\left(L,t\right)\right].Math.\hspace{1em}\left[M\left(t\right)+\eta .Math..Math.\stackrel{.}{M}\left(t\right)\right]-\left(\alpha .Math..Math.{\mathrm{EI}}_b-\frac{\mathrm{\alpha \eta }.Math..Math.{\mathrm{EI}}_b}{{\sigma }_3}-{\sigma }_6.Math.\alpha .Math..Math.L^4\right).Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\left(\alpha .Math..Math.\mathrm{GJ}-\frac{\mathrm{\alpha \eta }.Math..Math.\mathrm{GJ}}{{\sigma }_4}-{\sigma }_7.Math..Math.\alpha .Math..Math.x_a.Math.c.Math..Math.L^2\right).Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\left(\frac{\mathrm{\beta \eta }.Math..Math.{\mathrm{EI}}_b}{2.Math.L^4}-{\sigma }_1.Math.\beta -\alpha .Math..Math.m-2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c.Math..Math.{\sigma }_5\right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-(\frac{\mathrm{\beta \eta }.Math..Math.\mathrm{GJ}}{2.Math.L^2}-{\sigma }_2.Math.\beta .Math..Math.x_a.Math.c.Math..Math.\alpha .Math..Math.I_p-\alpha .Math..Math.I_p-\frac{2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c}{{\sigma }_5}.Math.).Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\left(\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2}-{\sigma }_3.Math.\alpha .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b\right).Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}-\left(\frac{\mathrm{\beta \eta }.Math..Math.\mathrm{GJ}}{2}-{\sigma }_4.Math.\mathrm{\alpha \eta }.Math..Math.\mathrm{GJ}\right).Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\left(\frac{\beta }{{\sigma }_1}+\frac{\beta .Math..Math.x_a.Math.c}{{\sigma }_2}+\frac{\alpha }{{\sigma }_6}+\frac{\alpha .Math..Math.x_a.Math.c}{{\sigma }_7}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2,& \left(51\right)\end{array}$ By setting U(t)=F(t)+η{dot over (F)}(t) and V(t)=M(t)+η{dot over (M)}(t) as the new controling variable, and their control rates are designed as follows:
U(t)=k.sub.1[αy(L, t)+β{dot over (y)}(L, t)]  (52),
V(t)=−k.sub.2[αθ(L, t)+β{dot over (θ)}(L, t)]  (53), Wherein k.sub.1≧0,k.sub.2≧0 is the control gain.

Description

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING

[0013] The invention will be described, by way of example, with reference to the accompanying drawings, in which:

[0014] FIG. 1. is the flow diagram of the method of this invention for controlling the oscillation of the flapping-wing air vehicle;

[0015] FIG. 2 is the figure of simulation of the bending displacement under the interference of the method of this invention for controlling the oscillation of the flapping-wing air vehicle;

[0016] FIG. 3. is the simulation of the torsional displacement under the interference of the method of this invention for controlling the oscillation of the flapping-wing air vehicle.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENT

[0017] In order to make the technical problem to be solved by this invention, the technical solution and the advantages clear, the invention will be described by way of example, with reference to the accompanying drawings, in which:

[0018] As shown in FIG. 1, the method for controlling the oscillation of flapping-wing air vehicle, in the example of this invention, which comprises the steps of:

[0019] Step 101: calculating the kinetic energy, potential energy, and virtual work of the system using the flexible wing with the two-degree of freedom as the research object;

[0020] Step 102: establishing the system dynamics model based on Hamilton's principle,

[0021] Step 103: setting up the boundary control rate according to said system dynamics model wherein said boundary control rate includes F(t) and M(t), said F(t) is the inputted boundary control force, and said M(t) is the inputted boundary torque; and

[0022] Step 104: controlling the oscillation of the flexible wing according to the system dynamics model and by combining the boundary control rate.

[0023] The method for controlling the oscillation of flapping-wing air vehicle in the example of this invention establishes the system dynamics model based on the Hamilton's principle, set the boundary control rate according to said system dynamics model, considers the situation of distributed disturbance occurring at the boundary sufficiently, and prevents the flexible wings deformation caused by the external disturbances effectively, thus is able to control the flexible wing accurately and stably.

[0024] Preferably, said calculating the kinetic energy, potential energy, and virtual work of the system using the flexible wing as the research object comprises:

[0025] Expressing the kinetic energy of the system, E.sub.k(t) as follows:

[00001]$\begin{array}{cc}{E}_k\left(t\right)=\frac{1}{2}.Math.m.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math.\mathrm{dx}+\frac{1}{2}.Math..Math.I_p.Math.{\int }_o^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math.\mathrm{dx}.Math.,& \left(1\right)\end{array}$

wherein the spatial variable of x is independent to the time variable of t, and m is the unitspan mass of the flexible wing; I.sub.p is inertial polar distance of the flexible wing; y(x, t) is the bending displacement at the position of x and at time of t in the x0y coordinate system; and θ(x, t) is the corresponding displacement of deflection angle;

[0026] The potential energy of E.sub.p(t) is expressed as follows:

[00002]$\begin{array}{cc}{E}_p\left(t\right)=\frac{1}{2}.Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\frac{1}{2}.Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(2\right)\end{array}$

wherein, EI.sub.b denotes the flexural rigidity, GJ denotes the torsional rigidity; and the virtual work of caused by the above two rigidities δW.sub.c(t) is expressed as follows:

δW.sub.c(t)=mx.sub.oc∫.sub.0.sup.Lÿ(x, t)δθ(x, t)dx+mx.sub.oc∫.sub.0.sup.L{umlaut over (θ)}(x, t)δy(x, t)dx  (3),

Wherein, x.sub.oc denotes the distance from the mass center of wing to the bending center; and the virtual work, δW.sub.d(t) done by the Kelvin-Voigt damping force is expressed as follows:

δW.sub.d(t)=−ηEI.sub.b∫.sub.0.sup.L{dot over (y)}″(x, t)δy″(x, t)dx−ηGJ.sub.b∫.sub.0.sup.L{dot over (θ)}′(x, t)δθ′(x, t)dx  (4),

Wherein, η denotes the Kelvin-Voigtd damping coefficient.

[0027] The virtual work of done by the distributed distraction, δW.sub.r(t) is as follows:

δW.sub.r(t)=∫.sub.0.sup.L[F.sub.b(x, t)δy(x, t)−x.sub.acF.sub.b(x, t)δθ(x, t)]dx  (5),

wherein x.sub.ac denotes the distance from the aerodynamic center to the bending centre and; F.sub.b is the unknown time varying distributed distraction along the wings;

[0028] The virtual work done by the boundary control force to the system, δW.sub.a(t) is expressed as follows:

δW.sub.a(t)=F(t)δy(L, t)+M(t)δθ(L, t)  (6),

[0029] In the above formula, F(t) is the inputted boundary control force and; M(t) is the inputted boundary torque;

[0030] Consequently, the total virtual work is:

δW(t)=δ[W.sub.c(t)+W.sub.d(t)+W.sub.r(t)+W.sub.a(t)]  (7).

[0031] Preferably, said establishing the system dynamics model based on the Hamilton's principle includes:

[0032] utilizing the Hamilton's smooth action principle of [0033] ∫.sub.t.sub.1.sup.t.sup.2δ[E.sub.k(t)−E.sub.p(t)+W(t)]dt=0

[0034] Here δ denotes the variation symbol, and the governing equation for the system dynamics model is deduced as:

mÿ(x, t)+EI.sub.by″″(x, t)−mx.sub.oc{umlaut over (θ)}(x, t)+ηEI.sub.b{dot over (y)}″″(x, t)=F.sub.b(x, t)  (8),

I.sub.p{umlaut over (θ)}(x, t)−GJθ″(x, t)−mx.sub.ocÿ(x, t)−ηGJ{dot over (θ)}″(x, t)=−x.sub.acF.sub.b(x, t)  (9),

[0035] And the boundary conditions for the system dynamics model are deduced as:

y(0, t)=y′(0, t)=y″(L, t)=θ(0, t)=0  (10),

EI.sub.by′″(L, t)+ηEI.sub.b{dot over (y)}′″(L, t)=−F(t)  (11) and

GJθ′(L, t)+ηGJ{dot over (θ)}′(L, t)=M(t)  (12).

[0036] Preferably, said setting the boundary controller based on the system dynamics model includes two controlling laws of [0037] F(t) and M(t), wherein said F(t) is the inputted boundary control force and M(t) is the inputted boundary torque, and which includes:

[0038] Constructing the Lyapunov candidate function as follows:

[0039] Preferably, said setting the boundary controller based on the system dynamics model includes two controlling laws of

V(t)=V.sub.1+Δ(t)  (13),

Wherein, V.sub.1(t) and Δ(t) are respectively defined as:

[00003]$\begin{array}{cc}{V}_1\left(t\right)=\frac{\beta }{2}.Math.m.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\frac{\beta }{2}.Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\frac{\beta }{2}.Math.I_p.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\frac{\beta }{2}.Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\theta }^{\prime }\left(x,t\right)].Math.2.Math..Math.\mathrm{dx}.Math..Math.\mathrm{and}& \left(14\right)\\ \Delta \left(t\right)=\alpha .Math..Math.m.Math.{\int }_0^L.Math.\stackrel{.}{y}\left(x,t\right).Math.y\left(x,t\right).Math.\mathrm{dx}+\alpha .Math..Math.I_p.Math.{\int }_0^L.Math.\stackrel{.}{\theta }\left(x,t\right).Math.\theta \left(x,t\right).Math.\mathrm{dx}-\alpha .Math..Math.{\mathrm{mx}}_e.Math.c.Math.{\int }_0^L.Math.\left[\stackrel{.}{y}\left(x,t\right).Math.\theta \left(x,t\right)+y\left(x,t\right).Math.\stackrel{.}{\theta }\left(x,t\right)\right].Math.\mathrm{dx}-\beta .Math..Math.{\mathrm{mx}}_e.Math.c.Math.{\int }_0^L.Math.\stackrel{.}{y}\left(x,t\right).Math.\stackrel{.}{\theta }\left(x,t\right).Math.\mathrm{dx}.& \left(15\right)\end{array}$

In the above two equations, both α and β are the smaller positive weight coefficient; [0040] the boundary control rate is set by means of making the Lyapunov candidate function be positive definite, and derivative of Lyapunov candidate function of to the time of t, {dot over (V)}(t) is negative definite.

[0041] Preferably, said calculating the boundary control rate when the Lyapunov candidate function is positive definite, and the derivative of Lyapunov candidate function the time of t, {dot over (V)}(t) is negative definite comprises:

defining a new function as follows:

κ(t)=∫.sub.0.sup.L{[{dot over (y)}(x, t)].sup.2+[{dot over (θ)}(x, t)].sup.2+[y″(x, t)].sup.2+[θ′(x, t)].sup.2}dx  (16),

[0042] Then V.sub.1(t) has the upper bound and lower bound which are defined as

γ.sub.2κ(t)≦V.sub.1(t)≦γ.sub.1κ(t)  (17),

[0043] In the above formula,

[00004]${\gamma }_1=\frac{\beta }{2}.Math.\max \left(m,I_p,{\mathrm{EI}}_b,\mathrm{GJ}\right),.Math.{\gamma }_2=\frac{\beta }{2}.Math.\min \left(m,I_p,{\mathrm{EI}}_b,\mathrm{GJ}\right).$

[0044] Further, Δ(t) is magnified as

[00005]$\begin{array}{cc}.Math.\Delta \left(t\right).Math.\le \left(\alpha .Math..Math.m+\alpha .Math..Math.{\mathrm{mx}}_e.Math.c+\beta .Math..Math.{\mathrm{mx}}_e.Math.c\right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\left(\alpha .Math..Math.I_p+\alpha .Math..Math.{\mathrm{mx}}_e.Math.c+\beta .Math..Math.{\mathrm{mx}}_e.Math.c\right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\left(\alpha .Math..Math.m+\alpha .Math..Math.{\mathrm{mx}}_e.Math.c\right).Math.L^4.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\left(\alpha .Math..Math.I_p+\alpha .Math..Math.{\mathrm{mx}}_e.Math.c\right).Math.L^2.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}\le {\gamma }_3.Math.\kappa \left(t\right),& \left(18\right)\end{array}$

Wherein,

[0045] γ.sub.3=max{αm+αmx.sub.oc+βmx.sub.oc, αI.sub.p+αmx.sub.oc+βmx.sub.oc, (αm+αmx.sub.oc)L.sup.4, (αI.sub.p+αmx.sub.oc)L.sup.2}, if the positive number, β satisfies

[00006]$\beta >\frac{2.Math.{\gamma }_3}{\min \left(m,I_p,{\mathrm{EI}}_b,\mathrm{GJ}\right)},$

then

0≦λ.sub.2κ(t)≦V(t)≦λ.sub.3κ(t)  (19).

This means that the constructed Lyapunov function is positive definite, wherein [0046] λ.sub.1=γ.sub.1+γ.sub.3 and λ.sub.2=γ.sub.2−γ.sub.3;

[0047] The derivative of V(t) to t is deduced as:

{dot over (V)}(t)={dot over (V)}.sub.1(t)+{dot over (Δ)}(t)  (20), and

{dot over (V)}.sub.1(t)=βm∫.sub.0.sup.L{dot over (y)}(x, t)ÿ(x, t)dx+βI.sub.p∫.sub.0.sup.L{dot over (θ)}(x, t){umlaut over (θ)}(x, t)dx +βGJ∫.sub.0.sup.Lθ′(x, t){dot over (θ)}′(x, t)dx+βEI.sub.b∫.sub.0.sup.Ly″(x, t){dot over (y)}″(x, t)dx  (21)

[0048] By introducing the controlling equations (8) and (9) into the above formula, we obtain:

{dot over (V)}.sub.1(t)=A.sub.1+A.sub.2+A.sub.3+A.sub.4+A.sub.5+A.sub.6  (22).

Wherein, A.sub.1-A.sub.6 are respectively expressed as follows:

A.sub.1=−βEI.sub.b∫.sub.0.sup.L{dot over (y)}(x, t)y″″(x, t)dx+βEI.sub.b∫.sub.0.sup.Ly″(x, t){dot over (y)}″(x, t)dx  (23),

A.sub.2=−βηEI.sub.b∫.sub.0.sup.L{dot over (y)}(x, t){dot over (y)}″″(x, t)dx  (24),

A.sub.3=βmx.sub.oc ∫.sub.0.sup.L[{dot over (y)}(x, t){umlaut over (θ)}(x, t)+ÿ(x, t){dot over (θ)}(x, t)]dx  (25),

A.sub.4=β∫.sub.0.sup.L{dot over (y)}(x, t)F.sub.b(x, t)dx−βx.sub.oc∫.sub.0.sup.L{dot over (θ)}(x, t)F.sub.b(x, t)dx  (26),

A.sub.5=βGJ∫.sub.0.sup.L{dot over (θ)}(x, t)θ″(x, t)dx+βGJ∫.sub.0.sup.Lθ′(x, t){dot over (θ)}′(x, t)dx  (27), and

A.sub.6=βηGJ∫.sub.0.sup.L{dot over (θ)}(x, t){dot over (θ)}″(x, t)dx  (28),

[0049] By utilizing the integration by parts and the boundary condition of (10) (11) and (12), we obtain

[00007]$\begin{array}{cc}.Math.A_1=-\beta .Math..Math.{\mathrm{EI}}_b.Math..Math.\stackrel{.}{y}\left(L,t\right).Math.y^{\mathrm{\prime \prime \prime }}\left(L,t\right)=\beta .Math..Math.\stackrel{.}{y}\left(L,t\right).Math.F\left(t\right),& \left(29\right)\\ A_2\le -\mathrm{\beta \eta }.Math..Math.\stackrel{.}{y}\left(L,t\right).Math.\stackrel{.}{F}\left(t\right)-\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2.Math.L^4}.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}-\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}.Math..Math..Math.\mathrm{and}& \left(20\right)\\ A_4\le {\sigma }_1.Math.\beta .Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+{\sigma }_2.Math.\beta .Math..Math.x_a.Math.c.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\left(\frac{\beta }{{\sigma }_1}+\frac{\beta .Math..Math.x_a.Math.c}{{\sigma }_2}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2,& \left(31\right)\end{array}$

wherein, σ.sub.1 and σ.sub.2 are the positive constants, F.sub.b max is the maximum value of the distributed disturbance, F.sub.b(x, t).

[00008]$\begin{array}{cc}.Math.A_5=\beta .Math..Math.\mathrm{GJ}.Math..Math.\stackrel{.}{\theta }\left(L,t\right).Math.{\theta }^{\prime }\left(L,t\right)=\beta .Math..Math.\stackrel{.}{\theta }\left(L,t\right).Math.M\left(t\right),& \left(32\right)\\ A_6\le \mathrm{\beta \eta }.Math..Math.\stackrel{.}{\theta }\left(L,t\right).Math.\stackrel{.}{M}\left(t\right)-\frac{\beta .Math..Math.\eta .Math..Math.\mathrm{GJ}}{2.Math.L^2}.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}-\frac{\beta .Math..Math.\eta .Math..Math.\mathrm{GJ}}{2}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}& \left(33\right)\end{array}$

[0050] Based on the above A1˜A6, we acquire {dot over (V)}.sub.1(t) as follows:

[00009]$\begin{array}{cc}{\stackrel{.}{V}}_1\left(t\right)\le -\left(\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2.Math.L^4}-{\sigma }_1.Math.\beta \right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-(\frac{\beta .Math..Math.\eta .Math..Math.\mathrm{GJ}}{2.Math.L^2}-{\sigma }_2.Math.\beta .Math..Math.x_a.Math.c.Math.).Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}-\frac{\beta .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{2}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\frac{\beta .Math..Math.\eta .Math..Math.\mathrm{GJ}}{2}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\beta .Math..Math.{\mathrm{mx}}_e.Math.c.Math..Math.{\int }_0^L.Math.\left[\stackrel{.}{y}\left(x,t\right).Math.\stackrel{\mathrm{.Math.}}{\theta }\left(x,t\right)+\stackrel{\mathrm{.Math.}}{y}\left(x,t\right).Math.\stackrel{.}{\theta }\left(x,t\right)\right].Math..Math.\mathrm{dx}-\beta .Math..Math.\stackrel{.}{y}\left(L,t\right)\left[F\left(t\right)+\eta .Math..Math.\stackrel{.}{F}\left(t\right)\right]+\beta .Math.\stackrel{.}{\theta }\left(L,t\right)\left[M\left(t\right)+\eta .Math..Math.\stackrel{.}{M}\left(t\right)\right]+\left(\frac{\beta }{{\sigma }_1}+\frac{\beta .Math..Math.x_a.Math.c}{{\sigma }_2}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2.& \left(34\right)\end{array}$

[0051] Similarly, by means of calculating the derivation of Δ(t) to t is deduced as

{dot over (Δ)}(t)=B.sub.1+B.sub.2+. . . B.sub.8  (35),

B.sub.1=−αEI.sub.b∫.sub.0.sup.Ly(x, t)y″″(x, t)dx  (36),

B.sub.2=−αηEI.sub.b∫.sub.0.sup.Ly(x, t){dot over (y)}″″(x, t)dx  (37),

B.sub.3=αGJ∫.sub.0.sup.Lθ(x, t)θ″(x, t)dx  (38),

B.sub.4=αηGJ∫.sub.0.sup.Lθ(x, t){dot over (θ)}″(x, t)dx  (39),

B.sub.5=αm∫.sub.0.sup.L[{dot over (y)}(x, t)].sup.3dx+αI.sub.p∫.sub.0.sup.L[{dot over (θ)}(x, t)].sup.2dx  (40),

B.sub.6=−βmx.sub.oc∫.sub.0.sup.L[{dot over (y)}(x, t){umlaut over (θ)}(x, t)+ÿ(x, t){dot over (θ)}(x, t)]dx  (41),

B.sub.7=−2αmx.sub.oc∫.sub.0.sup.L{dot over (y)}(x, t){dot over (θ)}(x, t)dx  (42) and

B.sub.8=α∫.sub.0.sup.Ly(x, t)F.sub.b(x, t)dx−αx.sub.oc∫.sub.0.sup.Lθ(x, t)F.sub.b(x, t)dx  (43).

[0052] By means of introducing the boundary conditions into the above formulas, we obtain:

[00010]$\begin{array}{cc}.Math.B_1=-\alpha .Math..Math.y\left(L,t\right).Math.F\left(t\right)-\alpha .Math..Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(44\right)\\ B_2\le -\alpha .Math..Math.\eta .Math..Math.y\left(L,t\right).Math.\stackrel{.}{F}\left(t\right)+\frac{\alpha .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{{\sigma }_3}.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_3.Math.\alpha .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(45\right)\\ .Math.B_3=\alpha .Math..Math.\theta \left(L,t\right).Math.M\left(t\right)-\alpha .Math..Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(46\right)\\ B_4\le \alpha .Math..Math.\eta .Math..Math.\theta \left(L,t\right).Math.\stackrel{.}{M}\left(t\right)+\frac{\alpha .Math..Math.\eta .Math..Math.\mathrm{GJ}}{{\sigma }_4}.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_4.Math.\alpha .Math..Math.\eta .Math..Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},& \left(47\right)\\ B_7\le 2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c.Math..Math.{\sigma }_5.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\frac{2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c}{{\sigma }_5}.Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx},.Math..Math.\mathrm{and}& \left(48\right)\\ B_8\le {\sigma }_6.Math.\alpha .Math..Math.L^4.Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+{\sigma }_7.Math.\alpha .Math..Math.x_a.Math.{\mathrm{cL}}^2.Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}+\left(\frac{\alpha }{{\sigma }_6}+\frac{\alpha .Math..Math.x_a.Math.c}{{\sigma }_7}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2,& \left(49\right)\end{array}$

[0053] All the above σ.sub.3-σ.sub.7 are the positive constants;

[0054] Therefore, according to B.sub.1-B.sub.8, we obtain:

[00011]$\begin{array}{cc}\stackrel{.}{\Delta }\left(t\right)\le -\left(\alpha .Math..Math.{\mathrm{EI}}_b-\frac{\alpha .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b}{{\sigma }_3}-{\sigma }_6.Math.\alpha .Math..Math.L^4\right).Math.{\int }_0^L.Math.{\left[y^{″}\left(x,t\right)\right]}^2.Math..Math.\mathrm{dx}-\left(\alpha .Math..Math.\mathrm{GJ}-\frac{\alpha .Math..Math.\eta .Math..Math.\mathrm{GJ}}{{\sigma }_4}-{\sigma }_7.Math.\alpha .Math..Math.x_a.Math.c.Math..Math.L^2\right).Math.{\int }_0^L.Math.{\left[{\theta }^{\prime }.Math.\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\left(\alpha .Math..Math.m+2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c.Math..Math.{\sigma }_5\right).Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+(\alpha .Math..Math.I_p+\frac{2.Math.\alpha .Math..Math.{\mathrm{mx}}_e.Math.c}{{\sigma }_5}.Math.).Math.{\int }_0^L.Math.{\left[\stackrel{.}{\theta }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+{\sigma }_3.Math.\alpha .Math..Math.\eta .Math..Math.{\mathrm{EI}}_b.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{y}}^{″}\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+{\sigma }_4.Math.\alpha .Math..Math.\eta .Math..Math.\mathrm{GJ}.Math.{\int }_0^L.Math.{\left[{\stackrel{.}{\theta }}^{\prime }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}-\beta .Math..Math.{\mathrm{mx}}_e.Math.c.Math.{\int }_0^L.Math.{\left[\stackrel{.}{y}\left(x,t\right).Math.\stackrel{\mathrm{.Math.}}{\theta }\left(x,t\right)+\stackrel{\mathrm{.Math.}}{y}\left(x,t\right).Math.\stackrel{.}{\theta }\left(x,t\right)\right].Math.}^2.Math.\mathrm{dx}+\alpha .Math..Math.\theta \left(L,t\right)\left[M\left(t\right)+\eta .Math..Math.\stackrel{.}{M}\left(t\right)\right]-\alpha .Math..Math.y\left(L,t\right)\left[F\left(t\right)+\eta .Math..Math.\stackrel{.}{F}\left(t\right)\right]+\left(\frac{\alpha }{{\sigma }_6}+\frac{\alpha .Math..Math.x_a.Math.c}{{\sigma }_7}\right).Math.{\mathrm{LF}}_{b.Math..Math.\max }^2& \left(50\right)\end{array}$