To identify the instantaneous frequencies of time-varying signals, the theoretical relationship between the frequency and rotational angle α in a signal was derived based on the definition of the fractional Fourier transform. Then the fractional Fourier transform was interpreted to be essentially an algorithm combining the ordinary Fourier transform with the dilation and translation window. A general expression of the signal instantaneous frequency in the fractional Fourier domain was thereafter formulated so that the structural instantaneous frequency can be extracted accordingly. The feasibility and reliability of the proposed method were verified with a simulated nonlinear frequency modulation signal and a numerical example of a 3DOF damped time-varying structure system. The results show that, the results of the proposed method are in good agreement with the theoretical values, and the method has a certain degree of anti-noise capability. Subsequently, the proposed method is applicable to the identification of the instantaneous frequencies of time-varying structures.

Structural vibrations of ultra-large structures during on-orbit assembly should be prevented to the maximum extent, given the extreme structural flexibility and low natural frequencies. The assembling process was divided into 4 stages: the grasping stage, the position-attitude adjusting and stabilizing stage, the mounting stage and the crawling stage. For the mounting stage, the dynamics and control were addressed, and a collinear assembly trajectory planning method was proposed to prevent structural vibrations. First, a dynamic model for the on-orbit assembly system including the main structure, the space robot and the assembling structure, was established based on natural coordinate formulation and absolute node coordinate formulation. Second, the requirements of collinear assembly were transformed into a trajectory planning problem of the space robot. The distance from the space robot mass center to the main structure should remain fixed, which is the main idea of the proposed collinear assembly method. Numerical simulation results show that, the proposed assembly method can effectively prevent the transverse motions of the ultra-large structure and reduce the required grasping moment. Finally, the influences of the system parameters on the dynamic responses during the assembly process were studied. The work provides a theoretical basis for the on-orbit assembly of ultra-large spacecraft.

The complex nonlinear coupling in the spatial cooperation process of mobile manipulators, makes it extremely tedious to directly model the spatial cooperative systems with the Lagrange equation or the Newton-Euler method. A dynamic modeling method, combining the Udwadia-Kalaba (U-K) method with the Lagrange equation, was proposed for spatial cooperation of dual-arm mobile manipulators. The load was simplified as a connecting link during modeling. The load center was selected to be disconnected for decomposition, so that the lack of constraint information was avoided between the end joint angle and the end link angle caused by the disconnection of the manipulator end joint; the segmented 2 subsystems were modeled with the Lagrange equation, thus, the dynamic model for the subsystems was obtained. The inherent geometric relationships of the cooperative system were introduced in the form of constraints, and the U-K method was applied to obtain the dynamic model for the cooperative system. The computation for modeling was reduced. The numerical simulation verifies the accuracy of the model.

The nonlinear vibration responses of functionally graded materials (FGMs) shells with different cone angles under external loads were studied. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the 1st-order shear deformation theory, the von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations were simplified into a 1DOF nonlinear vibration differential equation under Volmir’s assumption. Finally, the nonlinear motion equations were solved with the harmonic balance method and the Runge-Kutta method, and the amplitude frequency response characteristic curves of the FGMs conical shells were obtained. The effects of different material distribution functions and different ceramic volume fraction exponents on the amplitude frequency response curves of conical shells were discussed. The bifurcation diagrams of conical shells with different cone angles, as well as time process diagrams and phase diagrams for different excitation amplitudes, were described. The motion characteristics were characterized by Poincaré maps. The results show that, the FGMs conical shells present the nonlinear characteristics of hardening springs. The chaotic motions of the FGMs conical shells are restrained and not prone to motion instability with the increase of the cone angle. The FGMs conical shell present a process from the periodic motion to the multi-periodic motion and then to chaos with the increase of the excitation amplitude.

The sine-cosine method was applied to the wave equation for nonlinear elastic rods, and some new periodic and solitary solutions to the equation were obtained (with material constant n other than 1). The graphs of some solutions were given through the math software. The results are helpful to further research on existence of solitary waves in the nonlinear elastic rods.

The effective bulk modulus and the effective in-plane shear modulus of nano-fiber composites were investigated with the interface model and the interphase model based on the generalized self-consistent method, and the closed-form analytical solutions of the effective bulk modulus and all equations for numerically predicting effective in-plane shear modulus based on the 2 models, were presented. With the interface model, interface effects of the effective bulk modulus and the effective in-plane shear modulus were discussed through numerical examples. Furthermore, the solutions of the interface model were proved to be degenerated ones of those of the interphase model, where the effective bulk modulus can be obtained through analytical degeneration and the numerical results of the effective in-plane shear modulus through numerical degeneration. An example of aluminium containing nano voids shows that, the effective bulk modulus and the effective in-plane shear modulus predicted with the interface model have large deviations from those with the interphase model for small void radii, however, small deviations for larger void radii.

The dual-phase-lag thermoelasticity theory with the memory-dependent differential can perfectly describe the phenomenon of non-Fourier heat conduction, nevertheless, it has not been comprehensively considered: the mechanical responses of materials under the size-dependent effects and the multiphysics coupling effects such as magnetic, thermal and elastic fields. A modified dual-phase-lag thermoelasticity theory with memory-dependent effects and non-local effects was established. Based upon this theory, the magneto-thermoelastic coupling problem of narrow long thin plates subjected to cyclical heat sources was investigated. First, the governing equations for the problem were formulated. Then with the boundary conditions and initial conditions, the solution to the problem was obtained through the Laplace transform and the inverse transform techniques. Finally, the influences of the magnetic field, the phase lag, the time-delay, the kernel function, non-local effect and the time on the dimensionless quantities were investigated respectively. The work provides a powerful reference for the dynamic responses of micro-scale materials.

Based on the Lyapunov stability theory, the matrix analysis method and the linear matrix inequality method, etc, the generalized H₂ control of singularly perturbed uncertain-control time-varying delay systems with control input and disturbance input, was studied. A memory state generalized H₂ controller was designed, and the decision theorem for the specific design method was given. With a new lemma for delay-dependent and delay-independent cases, the relatively less conservative stability criterion was derived. The obtained results were linearized, the selected numerical examples were used to verify the effectiveness and feasibility of the derived conclusions. The results show that, the closed-loop system is asymptotically stable in the whole range from zero to the singular perturbation upper bound, which expands the generalized H₂ stability space and reduces the L₂-L_∞ performance index. The comparison of the stability state parameter index with the related literatures indicate that, the proposed method has certain advantages and less conservatism, and is suitable for standard and non-standard cases.

The event-based state estimation problem was investigated for a class of complex-valued neural networks with mixed delays. Based on the measurement output, a novel event-triggering scheme was introduced to reduce the frequency of updating while ensuring the estimation performance. A waiting time was first employed to avoid the Zeno phenomenon. By means of the Lyapunov direct method and some properties of complex-valued matrices, a sufficient criterion was established to guarantee the globally asymptotic stability for the error system. The weighted parameters and gain matrices were designed with resort to the feasible solution of matrix inequalities. A numerical simulation example illustrates the effectiveness of the proposed method.

The Boussinesq system, as a model to describe many geophysical phenomena, is a zero-order approximation of the coupling between the Navier-Stokes equations and the thermodynamic equations. The multi-dimensional viscous Boussinesq equations were considered. By means of the implicit function theorem, the global well-posedness of the mild solutions was obtained with the small initial data in the scaling invariant spaces.

The projection algorithm is one of the main methods to solve variational inequality problems. At present, the research on projection algorithms usually requires the assumptions that the mapping is monotone and Lipschitz continuous, but in practical problems, these assumptions are often unsatisfied. A new double projection algorithm for solving non-monotone variational inequality problems was proposed with the line search method. Under the assumption that the mapping is uniformly continuous, the sequence generated by the algorithm was proved to strongly converge to the solution of the variational inequality. The numerical experiments illustrate the effectiveness and superiority of the proposed algorithm.