Compared with the displacement-based structural damage assessment method, the coefficient-based method considering structural vibration characteristics has obvious advantages of high assessment efficiency and independent of manual judgment. When structural vibration characteristics are used to assess building damage, commonly used parameters include period (or frequency), vibration mode, damping ratio, frequency response function, curvature mode , modal flexibility, modal strain energy, Ritz vector and residual stress vector. The four parameters of natural vibration period (or frequency), vibration mode, damping ratio and frequency response function can be obtained not only by dynamic characteristic test, but also by analysis of numerical simulation of the finite element model. For the four parameters of curvature mode, modal flexibility, modal strain energy, residual stress vector, they can only be obtained by numerical simulation analysis.

Affected by uncertain factors such as material properties, contact connection and boundary conditions, the established finite element model is difficult to accurately simulate the real situation of the building structure. Therefore, when curvature mode, modal flexibility, modal strain energy, residual stress vector and other parameters are used to assess building damage, the results can easily deviate from the actual situation. . With the development of test instruments and signal analysis tools, the natural vibration period, vibration mode, damping rate and other parameters can only be obtained by analyzing the data of actual measured structural response. In addition, compared with the structural dynamic characteristics such as vibration mode and damping rate, the natural vibration period has the obvious characteristics of easy acquisition and high identification efficiency. Therefore, the natural vibration period is generally chosen as the main parameter to quickly assess the global damage of the structure.

In the process of establishing a structural seismic damage assessment method based on the natural vibration period, it is necessary to focus on two issues. (a) A fundamental period estimation formula should be established that is simplices and clear in physical meaning, and relatively accurate estimation results. (b) A functional relationship should be constructed between the fundamental period and the structural seismic damage. For the above, relevant researchers have conducted a lot of research and obtained certain research results.

Gilles et al.^{1} analyzed the theoretical basis of the seismic action calculation in the equivalent static method of the NBCC standard and investigated the differences in basic shear calculation results for buildings of different heights between the use of the fundamental period formula of standard experience and methodology. The results show that when the fundamental period is calculated by an empirical formula to estimate the results, the shear at the base of the structure is lowered by 3.5 times.

Sofi et al.^{2} focused on analyzing the mechanical principles and main features of various formulas for calculating the fundamental period, and analyzed the impact of masonry-filled walls, concrete or cement block partitions on the fundamental period .

Sangamnerkar and Dubey^{3} analyzed 36 reinforced concrete frame structures with different underlying dimensions and tree spacing structures. And the influencing factors on the fundamental period are researched such as the underlying width, the size of the column section and the stiffness of the structural base. The results show that the growth rate of the structural fundamental period is proportional to the growth of the underlying width.

Young and Adeli^{4} designed 12 eccentric cantilever steel frame structures with different heights, spans, and spatial stiffness distributions, and used ETABS to analyze fundamental periods of different structures. By comparing the ASCE7-10 formula, the Raylei formula and the results of ETABS analysis, the formula for calculating the fundamental recommendation period for different types of eccentric cantilever steel frame structures is given.

Wang et al.^{5} analyzed 414 reinforced concrete structures of high rise, very high rise and composite structures. The main influencing factors of the fundamental period are thoroughly analyzed, and the formula for calculating the fundamental period for the structure of high-rise buildings is adapted. And 15 vibration table test data, 27 pulse test, wind test data and Chinese standardized calculation formula calculation results are used to check the adjusted formula. After the correction, the formula for calculating the fundamental period and the first three-order duty cycle relationship for high-rise and high-rise buildings are given.

Based on 90 fundamental period data of steel plate shear wall structures collected from the literature, Jiang et al.^{6} determined a new calculation formula according to the calculation theory of the dynamic characteristics of the multi-freedom structure. And the research formula is verified by the vibration table tests.

In response to the fundamental period deficiencies of shear wall structures in the Indian seismic code, Mandanka et al.^{seven} selected 23 irregular shear walls considering regular stiffness with different plane dimensions, structural height, shear wall size, etc., and analyzed the fundamental period of these buildings using ETABS software. Based on the numerical simulation analysis data, a new fundamental period estimation formula for irregular stiffness of reinforced concrete shear wall structure is adjusted considering the influencing factors such as height structural width, structural width and moments of inertia.

Elfath and Elhout^{8} applied Egyptian code to design 36 steel framed structures with different structural height, seismic intensity and elastic drift angle, and fundamental period change pattern was analyzed focusing on structural stiffness changes and height distribution. It is believed that the fundamental period of the bending frame structure is closely related to the seismic intensity and the drift angle of the story.

For the coupling relationship between the fundamental period and the structural earthquake damage, the relevant researchers were sought.

Eleftheriadou and Karabinis^{9} statistically analyzed the damage data of 164,135 buildings during the Parnitha 5.9 earthquake in 1999. The relationships between the range of different fundamental periods and the damage rate of buildings corresponding to different damage levels were investigated.

Based on 300,000 nonlinear seismic response time history analysis data, Katsanos and Sextos^{ten} used the elastoplastic response spectrum theory to study the calculation method of the periodic elongation of the building structure under the state of seismic damage. The research results show that the rate of periodic elongation of damaged structures is significantly affected by the period of structural elasticity and the rate of degradation of structural stiffness.

Sarno and Amiri^{11} established a single-degree-of-freedom nonlinear structural system for reinforced concrete structures, taking into account structural factors such as structural ductility coefficient and stiffness decay rate, and used software input OpenSees to account for aftershock main sequence ground motion during nonlinear time historical analysis. The relationship between the period extension rate after structural failure and the distance from the epicenter, the PGA ratio of the main replica, the type of site, the duration, the elastic fundamental period, the ductility coefficient, the Stiffness degradation rate, cumulative damage and other factors have been studied emphatically, and the final formula for estimating period extension rate is given.

The structural failure factor is established according to the structural dynamic equation of Gunawan^{12}. The iterative calculation method of the structural failure factor is constructed using the high-order Runge-Kutta method, and the evaluation of the sensitivity of the structural failure factor was checked using a system single degree of freedom and double degree of freedom.

Gunawan et al.^{13} applied Euler–Bernoulli beam theory to construct a structural damage assessment formula taking the period of structural natural vibration as the dominant factor.

According to the above research literature, it is known that the necessary researches have been conducted in the calculation of structural fundamental period and the assessment of structural damage based on the periodic changes. However, most existing period formulas are based on empirical formulas, and most formulas use the height of the structure or the number of stories to directly estimate the fundamental period of the structure. Although fewer independent variables may increase the convenience of applying the formula, it also sacrifices the accuracy of the formula for calculating the fundamental period of complex and diverse structures, which is unfavorable for structural damage assessment based on periodic changes. At the same time, the results of existing research mainly focus on the rate of extension of the structural period after the complete destruction of the structure. While the relative research results on the interval of period change corresponding to different damage levels, including light damage, moderate damage, severe damage and destruction, are few. In this paper, aiming at the shortcomings of the existing research works, the mechanical analysis of the generalized system with one degree of freedom is carried out using the theory of structural dynamics, and the formula for estimating the fundamental period of the structure based on the displacement is obtained. Using the direct coupling relationship of structural damage, structural displacement response, structural stiffness degradation, and structural periodic change, the structural periodic change factor estimation interval corresponding to the different failure levels is established. Finally, the example of seismic damage is used to verify the research method.