Numerical method of time-domain dynamic responses for timber frame structures with viscoelastic damper

Song, Z., He, C., and Wang, S. (2026). "Numerical method of time-domain dynamic responses for timber frame structures with viscoelastic damper," BioResources 21(4), 10269–10275.

Abstract

The seismic performance analysis of timber frame structures with viscoelastic damper is important. Combined with Rayleigh damping model and stiffness proportional damping model, the damping matrix of the timber frame structures with viscoelastic damper is constructed. Based on the fine calculation method, the numerical method of time-domain dynamic responses for timber frame structures with viscoelastic damper is proposed, which can be used to calculate the time-domain dynamic responses. Then the peak top displacement and peak bottom shear force are selected as the indicators, which are used to analyze the arrangement of viscoelastic dampers in wooden frame structures. Numerical results show that compared to viscoelastic dampers at the top and middle, viscoelastic damper at the bottom is the most advantageous for improving the seismic performance of the structure, which can be given priority consideration in design.

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Numerical Method of Time-domain Dynamic Responses for Timber Frame Structures with Viscoelastic Damper

Zhe Song,^a Chongyang He,^b and Shuxia Wang ^c,d,e,*

DOI: 10.15376/biores.21.4.10269-10275

Keywords: Dynamic responses; Timber frame structures; Viscoelastic damper

Contact information: a: Henan Technical College of Construction, Zhengzhou 450064, China; b: Henan Earthquake Agency, Zhengzhou 450018, China; c: School of Materials Science and Engineering, Zhengzhou University of Aeronautics, Zhengzhou 450046, China; d: Lanshen Group Co. Ltd., Nanjing 211500, China; e: School of Materials Science and Engineering, Zhengzhou University, Zhengzhou 450001, China;

* Corresponding author: [email protected]

INTRODUCTION

Timber frame structures have advantages such as light weight and good toughness (Kharrazi et al. 2008; Akan et al. 2021). However, traditional timber structures have disadvantages such as insufficient node stiffness and weak energy dissipation capacity (Matteoni et al. 2024; Pei et al. 2025). The lateral bearing capacity of the pure wood frame structure is insufficient, and the displacement between layers under strong earthquakes is prone to exceed the specified limit (Xu et al. 2025; Gholamizoj et al. 2026).

As an efficient passive energy dissipation component, viscoelastic damping devices can dissipate seismic energy through deformation, providing an effective way for seismic reinforcement of timber structures. Relevant research has become a hot topic in the field of seismic resistance of timber structures. The viscoelastic damping device can effectively improve the seismic performance of timber frame structures, and related research provides theoretical and technical support for seismic reinforcement of timber structures. Nakamura and Matsumura (2022) compared the installation methods of cross-story and inter-story in multi-story timber frames. It was found that dampers can significantly reduce the extension of the basic period and amplitude of the structure and thereby suppress the accumulation of residual deformation. Yi et al. (2024) pointed out that viscoelastic dampers can effectively control the residual deformation of damaged wooden frames on slopes and improve the deformation control ability in weak areas based on the full-scale vibration table tests. However, a simple and effective time-domain calculation method of timber structures with viscoelastic dampers still needs to be further studied.

This paper conducts seismic performance analysis on timber frame structures with viscoelastic dampers. The damping matrix is established based on Rayleigh damping and stiffness-proportional damping models. A refined numerical approach is proposed to calculate the time-domain dynamic responses of the timber structures with viscoelastic dampers. Compared with the common complex modal superposition method based on state space, the calculation process of the proposed method is simpler. The peak top displacement, peak bottom shear force and peak story drift are selected as the indicators, the damper layout effect is quantitatively investigated. The seismic performances of timber frame structures with viscoelastic dampers at the top, middle and bottom are analyzed, respectively.

TIME-DOMAIN METHOD OF DYNAMIC RESPONSES FOR TIMBER FRAME STRUCTURES WITH VISCOELASTIC DAMPER

The time-domain motion equation of the timber frame structure is as follows,

The fine calculation method is used to calculate the numerical results (Zhong and Cai 2000). Thus, the stepwise integration calculation of Eq. (4) can be carried out. The numerical method of time-domain dynamic responses for timber frame structures with viscoelastic damper is realized.

NUMERICAL EXAMPLES

A three-layer shear type timber frame model is established, which is defined as Model A, as shown in Fig. 1. Here the lumped‑mass shear‑layer model is adopted. The mass and stiffness parameters are assumed, which is only for verifying the method. According to BS EN 1995‑1‑1: 2004+A1: 2008, the damping ratio of timber frame structures is 0.02, and the corresponding loss factor of timber frame structures is 0.04.

Fig. 1. Schematic diagram of timber frame structures

Based on Model A, the timber frame models with different viscoelastic dampers are established, respectively, as shown in Fig. 2. The damper of Model B is arranged at the bottom. The damper of Model C is arranged at the middle. The damper of Model D is arranged at the top. The parameters of numerical models are shown in Table 1.

Fig. 2. Schematic diagrams of timber frame structures with different viscoelastic dampers

Table 1. The Parameters of Numerical Models

Taking Model B as an example, El Centro wave and Tianjin wave are selected. The top time-history displacements are calculated by the proposed method and the complex modal superposition method (Chen and Peng 2022), respectively. The corresponding calculation results are shown in Fig. 3. Under El Centro wave, the peak top time-history displacements by the proposed method and the complex modal superposition method are 14.31 mm and 14.25 mm, respectively. Under Tianjin wave, the peak top time-history displacements by the proposed method and the complex modal superposition method are 15.88 mm and 15.86 mm, respectively. The results show that the calculation results of the two methods are consistent. The correctness of the proposed method has been verified. Compared with complex modal superposition method, the calculation process of the proposed method is simpler.

Fig. 3. Top time history displacements due to seismic waves

The undamped natural frequencies of the numerical model are 0.87 Hz, 1.82 Hz and 2.55 Hz, respectively. The proposed method is used to calculate the time-domain responses of the timber frame models due to El Centro wave. The peak top displacement, peak bottom shear force and peak story drift are selected as the indicators for seismic performance analysis. The comparative analysis of different models is shown in Table 2. The results show that compared with Model A, the peak bottom shear force and peak story drift of Model B are the smallest. The viscoelastic damper at the bottom can enhance resistance to seismic forces and improve local stiffness. Therefore, the viscoelastic dampers at the top and middle are not the most advantageous choices. The viscoelastic damper should be priority installed at the bottom of timber frame structures. However, the peak top displacement of Model C is the smallest. The peak top displacement of Model B is greater than the one of Model C. The improvement of the overall lateral stiffness of the timber frame structures with the bottom viscoelastic damper is limited.

Table 2. Comparative Analysis of Different Models

CONCLUSIONS

A damping model was developed for timber frame structures, adopting the Rayleigh damping model. An additional viscoelastic damper adopted stiffness proportional damping model. Then the damping matrix of the timber frame structures with viscoelastic damper was constructed. Based on the fine calculation method, a numerical method of time-domain dynamic responses for timber frame structures with viscoelastic damper is proposed.
The arrangement of viscoelastic dampers in timber frame structures was found to directly affect the seismic performance of the structure. The peak top displacement, peak bottom shear force and peak story drift were selected as the indicators. Compared to viscoelastic dampers at the top and middle, viscoelastic damper at the bottom was found to be the most advantageous for improving the seismic performance of the structure, which can be given priority consideration in design. However, the improvement of the overall lateral stiffness is limited.

ACKNOWLEDGMENTS

This work was supported by the Key Research Project of Henan Higher Education Institutions (Grant No. 25A430032).

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Article submitted: May 16, 2026; Peer review completed: August 1, 2026; Revised version received: August 16, 2026; Accepted: August 17, 2026; Published: September 2, 2026.

DOI: 10.15376/biores.21.4.10269-10275