Abstract
The loss factor of wood material is frequency related, which directly affects the calculation method of dynamic responses for wood structures. In this paper, the relationship between loss factor and damping coefficient was determined based on equal dissipated energy. Combined with the time-domain and frequency-domain methods, a modal superposition method was proposed to calculate the dynamic response of wood structures. Compared with the frequency-domain method, the proposed method can additionally consider the transient vibration responses of wood structures. Compared with the equivalent time-domain method based on constant loss factor, the proposed method can additionally consider the influence of frequency related loss factor. The proposed method should be preferred to calculate dynamic responses of wood structures.
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Numerical Calculation Method of Dynamic Responses for Wood Structures with Frequency-related Damping Parameter
Na Chen *
The loss factor of wood material is frequency related, which directly affects the calculation method of dynamic responses for wood structures. In this paper, the relationship between loss factor and damping coefficient was determined based on equal dissipated energy. Combined with the time-domain and frequency-domain methods, a modal superposition method was proposed to calculate the dynamic response of wood structures. Compared with the frequency-domain method, the proposed method can additionally consider the transient vibration responses of wood structures. Compared with the equivalent time-domain method based on constant loss factor, the proposed method can additionally consider the influence of frequency related loss factor. The proposed method should be preferred to calculate dynamic responses of wood structures.
DOI: 10.15376/biores.20.4.11114-11121
Keywords: Dynamic responses; Numerical method; Wood structure; Frequency related; Damping parameter
Contact information: School of Civil Engineering and Architecture, Henan University of Science and Technology, Luoyang, 471000, China; *Corresponding author: cn20200627@163.com
INTRODUCTION
Wood, as a natural building material, has the advantages of being environmentally friendly and easy to process (Nam et al. 2023). Wood structures have good energy dissipation performance and play an effective role in earthquake resistance and wind resistance (Sun et al. 2024a; Vutukuru et al. 2024; Jensen et al. 2025). However, the damping parameter of wood material is frequency related (Zhang and Zhou 2023). Ouis (2003) analyzed the loss factor of wood with different species, and found a dependence on vibration frequency. Elie et al. (2013) analyzed the loss factors in the low- and the mid-frequency domains based on the subspace method, respectively. The loss factor of wood structures has been found to vary with frequency. It is important to know how to calculate the dynamic responses of wood structures. The traditional time-domain calculation methods cannot be directly used to calculate the dynamic responses of structures with frequency related loss factor (Jiang et al. 2010). Kazemirad et al. (2013) analyzed the dynamic properties of frequency-dependent damping systems with soft materials. Sun et al. (2024b) constructed the complex modal superposition of multi-degree-of-freedom systems with frequency related loss factor. However, the calculation processes of these methods are complex, and these methods are difficult to apply to wood structures.
In this paper, the relationship between loss factor and damping coefficient was determined based on equal dissipated energy. A modal superposition method is proposed to calculate the dynamic response of wood structures with frequency related damping parameter. Moreover, the frequency-domain method based on frequency related loss factor, the equivalent time-domain method based on constant loss factor, and the proposed method are compared and analyzed.
MODAL SUPERPOSITION METHOD OF WOOD STRUCTURES WITH FREQUENCY RELATED DAMPING PARAMETER
Construction of Damping Matrix
The time-domain motion equation of a single-degree-of-freedom system can be expressed according to Nkibeu et al. (2024),
(1)
where m is the mass; c is the damping coefficient; k is the stiffness; and f is the external excitation; x(t) is the displacement; 
is the velocity; and 
is the acceleration.
The displacement response of a single-degree-of-freedom system under harmonic action is:
(2)
where A is the amplitude of the structural displacement response and θ is the vibration frequency of the external excitation harmonic.
Under the influence of harmonics, the energy dissipated per cycle in a single-degree-of-freedom system based on the viscous damping model is (Clough and Penzien 1993):
(3)
Substituting Eq. 2 into Eq. 3:
(4)
In the process of harmonic vibration, the maximum potential energy within one cycle is (Clough and Penzien 1993):
(5)
The loss factor of dissipated energy is (Li et al. 2025):
(6)
The dissipated energy based on the loss factor is:
(7)
Eq. 4 is then set equal to Eq. 7, namely:
(8)
The relationship between loss factor and damping coefficient is shown in Eq. 8. The vibration frequency can be further expanded from the external excitation frequency to the entire frequency domain (Bert 1973), Eq. 8 can be rewritten as,
(9)
where 
is the structural vibration frequency.
However, the wood loss factor is a function related to the vibration frequency (Jiang et al. 2010). The constant loss factor can be replaced with loss factor function (McDaniel et al. 2000). Equation 9 can be further expressed as:
(10)
The expression for the damping matrix can be further obtained from Eq. 10, which is:
(11)
MODAL SUPERPOSITION METHOD
The time-domain motion equation of multi-degree-of-freedom systems for wood structures is,
(12)
where M is mass matrix; K is stiffness matrix; C is damping matrix; x(t) is structural displacement vector; and f is the vector of external excitation.
The modal vector of Eq. 12 is:
(13)
The damping matrix x(t) can be linearly expressed by the complex mode vector, namely:
(14)
Equation 14 is substituted into Eq. 12, which can be decoupled into N single degree of freedom equations. The single degree of freedom equation of the n-th vibration mode can be expressed as:
(15)
(16)
(17)
(18)
(19)
and 
is the natural frequency of the n-th vibration mode.
(20)
where 
is the general solution of homogeneous equation of Eq. 12; 
is the special solution of non-homogeneous equation of Eq. 12.
Eq. 20 is substituted into Eq. 15, namely:
(22)
For Eq. 21, the vibration frequency is approximately equal to the natural frequency. Eq. 21 can be rewritten as:
(23)
To solve Eq. 23, the time is discretized as:
(24)
Equation 24 is substituted into Eq. 23, and the solution of Eq. 23 is obtained as (Wang et al. 2023),
(25)
(26)
In Eq. 22, the loss factor is a function of vibration frequency. The time-domain method cannot be directly applied to solve the Eq. 22. Therefore, the frequency-domain method is considered for solving the Eq. 22. Based on Fourier transform method, the frequency-domain expression of Eq. 22 can be obtained as,
(27)
where 
is the Fourier transform term of ; and 
is Fourier transform term of 
.
By aid of frequency-domain method, 
can be obtained based on discrete frequencies. Then, the frequency-domain solution of Eq. 27 can be obtained. Based on inverse Fourier transform method, is obtained.
Both 
and are substituted into Eqs. 14 and 20, respectively. Then, 
is obtained and the modal superposition method of wood structures with frequency-dependent damping parameter is realized.
NUMERICAL EXAMPLES
The loss factor of wood material is frequency related. In Jiang et al. (2010), the loss factors of wood material at different vibration frequencies can be obtained at 25 °C. The frequency range is 0.1 to 100 Hz. By aid of the least square method, the relationship of loss factor and vibration frequency can be obtained, which is shown in Eq. 28. The comparison is shown in Fig. 1,
(28)
where the coefficient of determination R2 is 0.8986.
Fig. 1. The dimensions of frame structures and the corresponding load conditions
Fig. 2. Schematic diagram of numerical model for the wood structure
The numerical model of wood structure was constructed, which is shown in Fig. 2. The relationship of loss factor and vibration frequency for Eq. 28 was adopted. The natural frequencies of numerical model were 0.5687 Hz, 1.3102 Hz, and 1.9129 Hz.
First, the sine wave was selected as the external excitation. The vibration amplitude was 100 mm/s2 and the vibration frequency was 30 rad/s. The proposed method (frequency-related time domain, FRTD) and the frequency-domain method (FD) (Clough and Penzien 1993) were adopted, respectively. The corresponding time-domain responses are shown in Fig. 3. The results show that after the time reached 120 s, the calculation results of the two methods were approximately equal. The correctness of the proposed method was indirectly proven. Moreover, the traditional frequency-domain method cannot consider the influence of the transient vibration response. Compared with the frequency-domain method, the proposed method can be used to calculate the transient vibration responses.
Fig. 3. Time-domain dynamic responses under harmonic wave
The El Centro earthquake wave was selected as the external excitation, and the acceleration time-history is shown in Fig. 4a. The two time-domain methods can be used to calculate the displacement time-history of the numerical model, and the top displacement time-history is shown in Fig. 4b.
Fig. 4. Time-domain dynamic responses under El Centro wave
One is the equivalent time-domain method (ETD) based on constant loss factor. ETD is the traditional modal superposition method (Clough and Penzien 1993). The constant loss factor is the average value of three modal loss factors, which is 0.015. The other is the proposed method (frequency-related time domain, FRTD). Figure 4b shows that there were local differences in the calculation results between the two methods. Moreover, the peak displacement of ETD was 17.4975 mm, and the peak displacement of FRTD was 18.9338 mm. The relative difference was 8.21%, which is greater than 5%. The reason is that ETD cannot consider the influence of frequency related loss factor. The proposed method based on frequency related loss factor should be preferred to calculate dynamic responses of wood structures.
CONCLUSIONS
- Based on the principle of equal dissipated energy, the relationship between loss factor and damping coefficient was determined. Then, the damping matrix of wood structures with frequency related damping parameter was established. A combination of time-domain and frequency-domain methods was proposed to solve the dynamic response of wood structures.
- Compared with the frequency-domain method, the proposed method can additionally consider the transient vibration responses of wood structures. Compared with the equivalent time-domain method based on constant loss factor, the proposed method can additionally consider the influence of frequency related loss factor. The proposed method based on frequency related loss factor should be preferred to calculate dynamic responses of wood structures.
REFERENCES CITED
Bert, C.W. (1973). “Material damping: An introductory review of mathematical models, measures and experimental techniques,” Journal of Sound and Vibration 29(2), 129-154. DOI: 10.1016/S0022-460X(73)80131-2
Clough, R.W., and Penzien J. (1993). Dynamics of Structures, McGraw-Hill, New York.
Elie, B., Gautier, F., and David, B. (2013). “Estimation of mechanical properties of panels based on modal density and mean mobility measurements,” Mechanical Systems and Signal Processing 40(2), 628-644. DOI: 10.1016/j.ymssp.2013.06.009
He, X., Wang, Y., Li, Y., Wei, Y., Quan, P., and Li, X. (2020). “Dielectric properties of Eucalyptus urophylla under an ultra-wide frequency range,” Science of Advanced Materials 12(8), 1236-1241. DOI: 10.1166/sam.2020.3763
Jiang, J., Lu, J., Zhao, Y., and Wu, Y. (2010). “Influence of frequency on wood viscoelasticity under two types of heating conditions,” Drying Technology 28(6), 823-829. DOI: 10.1080/07373937.2010.485084
Jensen, N. F., Rasmussen, T. V, and Hofmann, L. (2025). “Wood frame walls designed with low water vapour diffusion resistance wind shields,” Buildings 15(5), article 706. DOI: 10.3390/buildings15050706
Kazemirad, S., Heris, H. K. and Mongeau, L. (2013). “Experimental methods for the characterization of the frequency-dependent viscoelastic properties of soft materials,” Journal of the Acoustical Society of America 133(5), 3186-3197. DOI: 10.1121/1.4798668
Li, T., Xu, H.C., Cheng, Y.X., Jiang, Z.H., Cao, X.H., Luo, L.X., Gong, S., and Li, Z. (2025). “Design of multi-interfacial porous CuAlMn/Polymer composites and damping performance analysis with nested three-phase model,” Composites Part A: Applied Science and Manufacturing 198, article 109031. DOI: 10.1016/j.compositesa.2025.109031
McDaniel, J. G., Dupont, P., and Salvino, L. (2000). “A wave approach to estimating frequency-dependent damping under transient loading,” Journal of Sound and Vibration 231(2), 433-449. DOI: 10.1006/jsvi.1999.2723
Nam, H. K., Le, T. S. D., Yang, D. W., Kim, B., Lee, Y. G., Hwang, J. S., Kim, Y. R., Yoon, H., Kim, S. W., and Kim, Y. J. (2023). “Smart wooden home enabled by direct-written laser-induced graphene,” Advanced Materials Technologies 8(9), article 2201952. DOI: 10.1002/admt.202201952
Nkibeu, J. B., Makomra, V., Sali, M., and Jerémie, M. (2024). “Damping properties of some tropical wood species determined by free transverse vibration method,” Wood Material Science & Engineering 16(4), article 680. DOI: 10.1080/17480272.2024.2388213
Ouis, D. (2003). “Effect of structural defects on the strength and damping properties of a solid material,” European Journal of Mechanics A/Solids 22(1), 47-54. DOI: 10.1016/S0997-7538(02)01256-1
Sun, P., Li, K., and Sun, G. (2024a). “Seismic performance analysis of wood-steel frame structures,” BioResources 19(4), 8730-8738. DOI: 10.15376/biores.19.4.8730-8738
Sun, P., Yan, Y., and Yang, H. (2024b). “Complex mode superposition method of non-classically damped systems based on hysteretic damping model with frequency-dependent loss factors,” Journal of Sound and Vibration 19(4), 8730-8738. DOI: 10.1016/j.jsv.2023.118122
Vutukuru, K. S., Erwin, J., and Chowdhury, A. G. (2024). “Full-scale simulation of wind-driven rain and a case study to determine the rain mitigation effect of shutters,” Wind and Structures 38(3), 171-191. DOI: 10.12989/was.2024.38.3.171
Wang, S. X, He, J. J., Fan, J., Sun, P. X.. and Wang, D. W. (2023). “A time-domain method for free vibration responses of an equivalent viscous damped system based on a complex damping model,” Journal of Low Frequency Noise Vibration and Active Control 42(3), 1531-1540. DOI: 10.1177/14613484231157514
Zhang, Z., and Zhou, H. (2023). “Dielectric characteristics of poplar powder under high-frequency electric field,” BioResources 18(3), 5798-5812. DOI: 10.15376/biores.18.3.5798-5812
Article submitted: June 28, 2025; Peer review completed: August 9, 2025; Revised version received: October 3, 2025; Accepted: October 17, 2025; Published: November 2, 2025.
DOI: 10.15376/biores.20.4.11114-11121