## 1 Introduction
Generally, the assumption that mass is exactly known or invariant due to possible damage for only updating stiffness in the traditional BMUA is adopted to avoid the coupling effect. However, this assumption may be problematic when a significant change in mass is present, or a structure is experiencing damages in both mass and stiffness. Several works [37] used the Bayesian approach to update both mass and stiffness. Even though the mass is well-estimated, identifying mass and stiffness is challenging when the coupling effect remains. The coupling effect has been recently successfully addressed and achieved acceptable results in identifying mass and stiffness. However, it still requires certain prior information and quantifying parameter uncertainties remains a challenge. [PERSON] et al. [38] proposed a time-domain nonlinear restoring force to identify mass and stiffness; however, an external force is required. [PERSON] and [PERSON] [39] presented a loop substructure identification method for mass and stiffness, while the mass at the sensor's location should be known. [PERSON] and [PERSON] [40] established a time series-based model to identify mass and stiffness features. [PERSON] et al. [41] employed an extended Kalman filter (EKF) to determine the mass-stiffness coupled coefficient using incomplete measured data. Nevertheless, these methods cannot quantify parameter uncertainties. [PERSON] et al. [42] proposed an evolutionary-based model updating approach to simultaneously update structural parameters, mass and stiffness, while only partial uncertainty due to measurement noise was available. In this paper, a novel Bayesian model updating framework is proposed to address the coupling effect for simultaneously updating mass and stiffness, as well as quantifying parameter uncertainties. In the first place, the new eigenequations are reformulated by two sets of measured data acquired from an original system and a stiffness-modified system, which aims to address the coupling effect of mass and stiffness, giving the new prior PDF and new posterior PDF. The new objective functions are then obtained by taking the negative logarithm of the posterior PDF to circumvent complex integrals. Analytical formulations of optimal modal parameters (natural frequency and mode shape) and structural parameters (mass and stiffness) are derived using a linear optimization method; associated uncertainties are also quantified by an inverse of the Hessian matrix of the objective function. Finally, all uncertain parameters are updated in an iterative manner. It is worth mentioning that some successful applications show the feasibility of adding stiffness to the original structure for creating a modified system. The modified system with the attaching of additional structural components has been widely used for stiffness enhancement. For example, the Federal Emergency Management Agency (FEMA) provides guidelines for retrofitting structures by adding additional bracing in the existing structures [43]. Furthermore, curved dampers (CDs) and fluid viscous dampers (FVDs) are installed in building structures to enhance initial stiffness (Figure 1a,b) [44; 45]. In addition, some specially made braces such as the buckling-restrained brace (BRB) allow the structural system to have a better seismic-bearing ability (Figure 1c), as stiffness is improved [46]. Steel diagonal braces added to the existing building frame is a common method to provide adequate strength and stiffness (Figure 1d). The additional springs were attached to the cantilever beam to achieve a stiffness increase in an experimental program (Figure 1e) [47; 48]. Besides, different types of shear connectors in structures have been widely used to increase post-crack strength and initial stiffness, such as steel-concrete joint with bearing plates [49], semi-rigid connections [50], headed studs [51], Perfobond connectors [52], etc.