Shuxia Wang

The transient responses include exponential growth terms and the time-domain responses are divergent based on the complex damping constitutive model. This study aims to propose a convergent time-domain direct integration method. Based on the relationship between the displacement and velocity, an improved time-domain complex damping motion equation is obtained. Combined with Newmark-[ ] method, an improved time-domain direct integration method of complex damping constitutive model is proposed for single-degree-of-freedom (SDOF) systems and multi-degree-of-freedom (MDOF) systems. The harmonic vibration responses of SDOF systems show that the proposed method can eliminate the divergent components of the responses within each time step, and the time-domain results are convergent. The random excitation responses of SDOF and MDOF systems show that the calculation results with the proposed method are approximately equal to those with the frequency-domain method for the systems with large natural frequencies. The relative error of peak displacements with the frequency method may be greater than 10% for systems with small natural frequencies. The improved time-domain direct integration method can overcome the divergent limitation and obtain stable time-domain responses. Besides, compared with the frequency-domain method, the proposed method can further consider the influence of the transient responses. The calculation accuracy is improved.

The total mass matrix cannot satisfy the requirement of orthogonality for linearly damped systems with added mass. Traditional modal superposition method will be not be directly used for calculating time-domain responses. In order to calculate the responses, a real domain modal superposition method based on simplified damping matrix and a complex domain modal superposition method based on state space method are constructed, respectively. The numerical examples show that the calculation results of these two methods are approximately equal. Then the dynamic responses of the linearly damped system with different distributions of added mass are analyzed. As the number of storey with added mass decreases, the lateral resistance of the structure increases. The added mass should be prioritized at the bottom storey.

The complex damping model is only applied in frequency-domain. Based on the constitutive relation of viscoelastic materials, a vibration equation is equivalent to the constitutive relation of complex damping theory and the frequency response function satisfying the causality constraint. Compared with the existing equivalent complex damping model, the proposed equivalent damping model can consider more natural frequencies. The calculation formulas of Gauss precise integral method and improvement constant average acceleration method are derived. By the equivalent complex damping, the seismic time-history response of the typical example could carry on the numerical calculation. Compared with the Gauss precise integral calculation results of complex damping vibration equation, some conclusions can be analyzed. The proposed two methods could avoid divergent phenomenon in the time-domain numerical solution of complex damping vibration equation. The time-domain integral method of equivalent complex damping theory is stable and convergent. The Gauss precise integral method has strong equivalence and big computational complexity. The improvement constant average acceleration method has weak equivalence and small computational complexity.

The complex damping model can only be used to calculate the steady-state responses, while the transient responses are divergent. Based on the complex damping model, the Hilbert transform is introduced to establish a hysteretic damping model, eliminating the divergence phenomenon. However, with the increase of the loss factor, the damped natural frequency also increases. To overcome this shortcoming, a frequency-independent damping model is proposed based on the hysteretic damping model. However, traditional time-domain methods are no longer applicable to frequency-independent damping models. Therefore, the transient response and steady-state response are separated, and the assumption of external excitation acceleration is introduced. Time-domain methods-based linear polynomial assumption, quadratic polynomial assumption and trigonometric function assumption are proposed, respectively. Numerical examples show that the time-domain methods based on linear polynomial assumption and quadratic polynomial assumption have high computational efficiency. But these two methods cannot take into account the vibration frequency of external excitation acceleration. Hence, the computational accuracy is low. Compared with them, the time-domain method based on trigonometric function assumption has the lowest computational efficiency and the highest computational accuracy.

Low reflectivity is of great significance to photoelectric devices, optical displays, solar cells, photocatalysis and other fields. In this paper, vanadium oxide is deposited on pattern SiO₂ via atomic layer deposition and then annealed to characterize and analyze the anti-reflection effect. Scanning electron microscope (SEM) images indicate that the as-deposited VO_x film has the advantages of uniformity and controllability. After annealing treatment, the VO₂@pattern SiO₂ has fewer crevices compared with VO₂ on the accompanied planar SiO₂ substrate. Raman results show that there is tiny homogeneous stress in the VO₂ deposited on pattern SiO₂, which dilutes the shrinkage behavior of the crystallization process. The optical reflection spectra indicate that the as-deposited VO_x@pattern SiO₂ has an anti-reflection effect due to the combined mechanism of the trapping effect and the effective medium theory. After annealing treatment, the weighted average reflectance diminished to 1.46% in the visible near-infrared wavelength range of 650–1355 nm, in which the absolute reflectance is less than 2%. Due to the multiple scattering effect caused by the tiny cracks generated through annealing, the anti-reflection effect of VO₂@pattern SiO₂ is superior to that of VO_x@pattern SiO₂. The ultra-low reflection frequency domain amounts to 705 nm, and the lowest absolute reflectance emerges at 1000 nm with an astonishing value of 0.86%. The prepared anti-reflective materials have significant application prospects in the field of intelligent optoelectronic devices due to the controllability of atomic layer deposition (ALD) and phase transition characteristics of VO₂.

The free vibration responses are divergent based on a complex time-domain damping model. The traditional step by step integration method cannot be used to calculate the time-domain responses. Based on the theoretic solution that eliminates the divergent term for the complex damping model, the equivalent viscous damping model is proposed and the corresponding average acceleration method is realized in this paper. The numerical cases show that the calculated results of the equivalent viscous damped system are convergent, which are equal to the ones of the deleted divergent term based on the complex damping model. The correctness of the proposed method is verified, and the computational efficiency is high.

In order to realize the time-domain analysis based on hysteretic damping model, the frequency-independent time-domain damping model of single degree of freedom (SDOF) system is constructed. Based on the assumed relationship of vibration responses, the equivalent frequency-independent time-domain damping model in complex domain and real domain are proposed. The characteristic that the dissipated energy in each cycle is not related to the vibration frequency of external excitation is retained for the two equivalent damping models. Combined with Newmark- β method, the corresponding numerical methods are obtained. The numerical examples show that the free vibration responses are stably convergent based on equivalent damping models. The numerical results of vibration responses of SDOF system due to earthquake wave have high calculation accuracy. Compared with equivalent frequency-independent time-domain damping model in real domain, the computational accuracy of equivalent frequency-independent time-domain damping model in complex domain is higher, and the computational efficiency is lower.