# Error Caused by Damping Formulating in Multiple Support Excitation Problems
## 6 Conclusions  
In this paper, the errors caused by virtual damping error and damping truncation error in the three modeling methods, the large mass method (LMM), the relative motion method (RMM), and the absolute displacement method (ADM), widely used in analyzing multiple support excitation (MSE) problems, are studied.  
The term virtual damping is used to define the wrongly induced damping forces generated between the reference coordinates and the structure in the process of damping matrix formulation. The virtual damping error can be eliminated with the formula proposed in this paper. A new Rayleigh damping formula is proposed for LMM to eliminate the virtual damping error. With modal damping and the new Rayleigh damping formula, an integrated equation of motion (EOM) derived from the converged LMM is proved free from the virtual damping error and equivalent to RMM.  
The damping truncation error is defined as the error caused by using Rayleigh damping instead of modal damping in the modelling of structures. The error in the structural frequency response function caused by the virtual damping error and the damping truncation error in the aforementioned three modeling methods are studied and the formulas measuring the largest error ratio are derived. The mechanism of how the effect of MSE affects the virtual damping error and the damping truncation error is studied by spectral analysis. A simplified mass-spring model, a short-span rigid-frame bridge and a long-span cable-stayed bridge are used for numerical simulation to testify the conclusions obtained in the theoretical derivation.  
The following conclusions are acquired.  
1. RMM is free from the virtual damping error.
2. When modal damping is used, the LMM converged to RMM. Using Rayleigh damping in LMM consists of both virtual damping error and the damping truncation error. When the newly-introduced Rayleigh damping equation is used in LMM, the LMM also converged to RMM and the virtual damping error in LMM is eliminated.
3. The virtual damping error exists in all the structural modes of ADM. In ADM, the error ratio of the virtual damping increase with the frequency.
4. The largest damping truncation errors in LMM, RMM, and ADM appear at the resonance frequency of the mode having the lowest damping ratio.
5. For a particular structure, the damping truncation errors in LMM, RMM and ADM increase with the parameter \(\
u\), which is the quotient of the two frequencies used in formulating the Rayleigh damping matrix.
6. The effect of MSE changes the energy distribution among modes at the same time changes the distribution of error. Compared with the corresponding uniform excitation problem, the effect of MSE can either increase or decrease the virtual damping error and the damping truncation error in the model. The damping truncation error in the MSE problem can be significantly larger than the error in the corresponding uniform excitation problem.  
The bench-mark model of the cable-stayed bridge is available on the website: [[https://datacenterhub.org/resources/257](https://datacenterhub.org/resources/257)]([https://datacenterhub.org/resources/257](https://datacenterhub.org/resources/257)). The data used to support the findings of this study are available from the corresponding author upon request.  
Conceptualization, H.Q. and L.L.; methodology, H.Q. and L.L.; software, H.Q.; validation, H.Q.; formal analysis, H.Q. and L.L.; investigation, H.Q.; resources, L.L.; data curation, H.Q.; writing--original draft preparation, H.Q.; writing--review and editing, L.L.; visualization, H.Q.; supervision, L.L.; project administration, L.L.; funding acquisition, L.L.; All authors have read and agreed to the published version of the manuscript.  
This work was funded in part by the National Key R&D Program of China [No. 2017 YFC0703600] and the National Natural Science Foundation of China [No. 51678116].  
The authors declare no conflict of interest.