# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 1 Introduction  
The effect of multiple support excitation on the responses of large-span structures has been studied for decades. Errors that existed in modeling structures have been studied extensively. There are three methods mainly utilized for modeling the structures in multiple support excitation (MSE) problems: Relative motion method [1] (RMM), large mass method [2] (LMM) and absolute displacement method [3] (ADM). RMM is the most widely used method. In RMM, the response of the structure under multiple support excitation is separated into dynamic responses and pseudo-static responses, these tworesponses are solved separately. In the equation of motion (EOM) of RMM, the damping term in the excitation is assumed to be negligible. [PERSON] et al. [1] suggested that the damping term is small relative to the inertial term in the excitation. ADM neglects support damping in its EOM, which will definitely induce errors in the simulation. The LMM is an approximation method, when the large masses approaches infinity and when the modal damping method is used with LMM, the result is seen as 'perfect' in this paper. however, directly using the Rayleigh damping method with LMM would induce damping formulating error. Many researchers have been exploring the mechanism of the error caused by damping formulating methods. [PERSON] [4] found that the mass proportional damping causes the rigid mode to be falsely damped. [PERSON] [5] indicated that the non-stiffness-proportional damping would cause errors in the simulation in the MSE problem. He also compared the non-stiffness-proportional damping to dampers connected to the coordinates. [PERSON] et al. [6] also found the mass proportional damping issue and compared this part of damping to dampers connected to the ground. However, in ADM, even though the stiffness-proportional damping is used, errors still exist. In the MSE problem, the error is not merely caused by the non-stiffness-propositional dampings that exist in the damping matrix. Except for the damping forces of the connection between the structure and the grounds, all the other unbalanced damping forces are artificially induced by the damping formulating scheme. These unbalanced damping forces are defined as virtual damping in this paper, and the error caused by virtual damping is called the virtual damping error. The virtual damping can be compensated in the excitation to complement the false damping in the traditional damping methods. In the MSE problem, the virtual damping error can be aroused by the relative motion of the supports as well as the rigid body modes. [PERSON] et al. [7] studied the error in RMM and ADM and introduced a new method named the massless rigid element method, in this method massless rigid elements needed to be added between the support and the ground. [PERSON] et al. [8] derived the expression of the support damping from the EOM of the converged LMM for the model damping method; with this support damping matrix, the virtual damping error can be eliminated. In this paper, the EOM derived by [PERSON] et al. [8] is expanded to support the Rayleigh damping method. The relationship between this EOM and the RMM is also studied. Another source of error in damping formulation is using simplified damping formulating method such as the Rayleigh damping method instead of the modal damping method. Many damping formulating methods have been developed to approximate real damping of structures. Proportional damping is usually preferred in studying the dynamic characteristics of a structure. The modal damping method is an ideal proportional method. However, it requires the calculation of all the natural frequencies and mode shapes of the structure and is computationally inefficient. The Caughey damping method approaches the modal damping method when the length of the Caughey series approaches infinity. [5] Many researchers have been devoted to optimizing the Caughey damping method [9; 10; 11] or developing new damping formulation methods to better approach the modal damping method [12; 13; 14].