###### Abstract  
Although many families of integration methods have been successfully developed with desired numerical properties, such as second order accuracy, unconditional stability and numerical dissipation, they are generally implicit methods. Thus, an iterative procedure is often involved for each time step in conducting time integration. Many computational efforts will be consumed by implicit methods when compared to explicit methods. In general, the structure-dependent integration methods (SDIMs) are very computationally efficient for solving a general structural dynamic problem. A new family of SDIM is proposed. It exhibits the desired numerical properties of second order accuracy, unconditional stability, explicit formulation and no overshoot. The numerical properties are controlled by a single free parameter. The proposed family method generally has no adverse disadvantage of unusual overshoot in high frequency transient responses that have been found in the currently available implicit integration methods, such as the WBZ-\(\alpha\) method, HHT-\(\alpha\) method and generalized-\(\alpha\) method. Although this family method has unconditional stability for the linear elastic and stiffness softening systems, it becomes conditionally stable for stiffness hardening systems. This can be controlled by a stability amplification factor and its unconditional stability is successfully extended to stiffness hardening systems. The computational efficiency of the proposed method proves that engineers can do the accurate nonlinear analysis very quickly.  
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in high frequency transient responses. These properties preclude them from practical applications. In a recent study, [PERSON] proposed different families of explicit, dissipative algorithms, that generally have uncommon high frequency overshoot behavior [22; 23]. However, the proposed explicit SDIM does not have an unusual overshoot property in both high frequency displacement and velocity responses. The viscous dampers, generally called velocity dependent dampers, have been equipped in many buildings to largely dissipate seismic energy. A velocity dependent damping is considered in an SDOF system numerical example. As a result, the proposed family method (CVM) has good agreement with the average acceleration method (AAM); generally, both methods do not have a displacement and velocity overshoot.  
A novel family of SDIM is proposed in this study. This family of integration method is one-parameter controlled, has an explicit formulation, and is unconditionally stable for linear elastic and nonlinear stiffness softening systems. A numerical analysis was carried out for an SDOF system with bilinear inelastic hysteric behavior, and the results confirm that CVM can be used to solve a highly nonlinear system [33]. A stability amplification factor is included in the coefficients of the displacement difference equation. It helps to improve the stability properties of the proposed method and it has been verified in this paper [28; 29]. The numerical examples are conducted and the results show that a stability amplification factor extended the unconditional stability range of the proposed method to the nonlinear stiffness hardening systems. The overshoot property of the proposed family method for the high frequency responses to non-zero initial conditions is analytically and numerically verified. A recently developed Chang family method is considered to compare the overshoot behavior [30]. As a result, there is no unusual overshoot behavior in the proposed family method for the high frequency responses in both displacement and velocity. Furthermore, the proposed method has unconditional stability for the linear elastic, nonlinear stiffness softening and nonlinear stiffness hardening systems. In addition, this method, which is computationally efficient for solving a new series of mass spring system, proves that engineers can perform fast computation with maximum accuracy.