Comparative Analysis of Viscous Damping Model and Hysteretic Damping Model

Abstract

A damping model is one of the key factors in dynamic analysis. Viscous damping and hysteretic damping models are commonly used in structural damping models. In this study, transient and steady responses are analyzed for a single degree of freedom system based on the two damping models. The attenuation coefficient and damped natural frequency are important parameters of the transient response. In addition, the vibration amplitude is an important parameter of the steady response. When the relative errors of the parameters for the two damping models are less than 10%, the threshold of the damping ratio is selected as 0.1736 and the threshold of the loss factor is 0.3472. The numerical examples show that the dynamic responses based on the viscous damping model are approximately equal to those based on the hysteretic damping model in small damping cases. With the increase in the damping ratio, the difference between the dynamic responses calculated by the two damping models gradually increases. In large damping cases, the two damping models must be distinguished, and the choice of the damping model depends on the characteristic of dissipate energy.

1. Introduction

A damping model is the theoretical basis for the construction of a structural motion equation, which directly affects the solution of the dynamic response [1,2]. Therefore, a damping model is one of the key factors in dynamic analysis. So far, many damping models have been proposed. Viscous damping and hysteretic damping models are the most commonly used damping models [3,4], which can applied for civil engineering, mechanical engineering, material engineering and automotive engineering. Especially, the loss factors of some damping materials and dampers are between 0.10 and 1.00 [5,6,7]. In the calculated process of engineering dynamic response, the choice of two damping models is difficult. The application range of the two damping models is not easily distinguished.

The viscous damping model assumes that the damping force is related to the velocity of the structural system, which is widely used due to the simplicity of the mathematical calculation [8,9,10]. However, the vibration frequency of external excitation is related to the damping coefficient in the viscous damping model, and the dissipated energy in each cycle is related to the external excitation frequency for the structural steady response. The construction process of the damping matrix is very difficult for large structures, which involves the choice of effective frequencies [11]. These effective frequencies include natural frequencies of mode combinations and the dominant frequencies of external excitations [12,13]. It is difficult to consider all effective frequencies within the interest frequency range [14].

The hysteretic damping model (also called complex damping model [15] and rate-independent damping model [16,17]) is different from the viscous damping model, which assumes that the dissipated energy in each cycle is not related to the external excitation frequency for a structural steady response [18]. However, the complex damping is only applied in the frequency-domain analysis, and the corresponding time-domain response is divergent [19,20]. On the basis of the complex damping model, Inaudi and Kelly [21] constructed a hysteretic damping model by using Hilbert transform relation to overcome the time domain divergence problem of the complex damping model. Based on the assumption of the external excitation relation, Sun et al. [22] further proposed the corresponding time-domain calculation method for the single degree of freedom (SDOF) system and the complex mode superposition method for the multi degree of freedom (MDOF) system. However, the time-domain numerical method based on the hysteretic damping model involves Hilbert transform of displacement, and the traditional time-domain integral method cannot be applied directly. Moreover, the frequency response function of the hysteretic damping model is non-causal, which needs to be further improved [23,24].

Elastic analysis is the basic research of dynamic analysis, and the non-linearly elastoplastic analysis can be equivalent to elastic analysis [25]. In this paper, the comparison of the viscous damping model and hysteretic damping model is analyzed in the elastic stage. The dynamic response consists of transient and steady responses. In Section 2, the difference between transient responses based on the two damping models is discussed. The attenuation coefficient and damped natural frequency are compared. In Section 3, the difference between steady responses based on the two damping model is discussed. The vibration amplitude ratio is compared. The numerical examples with different damping ratio are discussed based on the two damping models in Section 4.

2. Comparative Analysis of Transient Responses Based on Different Damping Models

The viscous damping model and hysteretic damping model are the two most common damping models, and the comparative analysis is very important. The dynamic response consists of transient and steady responses [26]. Therefore, the transient and steady responses of the two damping models were compared and analyzed, respectively.

2.1. The Transient Response of Viscous Damped System

The transient response is equivalent to the free vibration response of systems. By analyzing the characteristics of the free vibration response, the law of transient response can be obtained.

Based on the viscous damping model, the time-domain equation for an SDOF system is expressed as

$$m\ddot{x}(t)+f_{\mathrm{v}}+kx(t)=0$$

(1)

where $m$ is the mass; $k$ is the stiffness; $f_{\mathrm{v}}$ is the viscous damping force; $\ddot{x}(t)$ is the acceleration; $x(t)$ is the displacement.

The damping force of the viscous damping model is proportional to the velocity, and the viscous damping force is expressed as

$$f_{\mathrm{v}}=\frac{2\xi k}{\omega }\dot{x}(t)$$

(2)

where $\xi$ is the damping ratio; $\dot{x}(t)$ is the velocity.

$\omega$ is the natural frequency of the system, which is expressed as

$$\omega =\sqrt{\frac{k}{m}}$$

(3)

The corresponding characteristic equation of Equation (1) is

$${\lambda }^2+2\xi \omega \lambda +{\omega }^2=0$$

(4)

where $\lambda$ is the complex eigenvalue.

By solving Equation (4), it is obtained as

$$\{\begin{array}{l}{\lambda }_1=-\xi \omega +\mathrm{i}\omega \sqrt{1-{\xi }^2}\\ {\lambda }_2=-\xi \omega -\mathrm{i}\omega \sqrt{1-{\xi }^2}\end{array}$$

(5)

where $\mathrm{i}$ is an imaginary unit, $\mathrm{i}=\sqrt{-1}$.

The damped natural frequency is

$${\omega }_{\mathrm{v}}=\omega \sqrt{1-{\xi }^2}$$

(6)

Equation (6) shows that when the viscous damped systems are underdamping state, the damping ratio is less than 1.0. The damped natural frequency is less than the natural frequency.

The attenuation coefficient is

$${\alpha }_{\mathrm{v}}=-\xi \omega$$

(7)

Equation (7) shows that the attenuation coefficient is negative and the stability of free vibration response is guaranteed based on the viscous damping model.

The transient response is expressed as

$$x(t)={\mathrm{e}}^{{\alpha }_{\mathrm{v}}}(A\sin {\omega }_{\mathrm{v}}t+B\cos {\omega }_{\mathrm{v}}t)$$

(8)

where A and B are undetermined coefficients.

2.2. The Transient Response of Hysteretic Damped System

Based on the hysteretic damping model, the time-domain equation for an SDOF system is expressed as [21]

$$m\ddot{x}(t)+f_{\mathrm{h}}+kx(t)=0$$

(9)

The hysteretic damping force is expressed as

$$f_{\mathrm{h}}=-\eta k{x}^{\mathrm{H}}(t)$$

(10)

where $\eta$ is the loss factor; $x^{\mathrm{H}}(t)$ is Hilbert transform of the displacement.

It is important to note that the damping force of the hysteretic damping model can be equivalently evaluated by using the rate-independent damping model [27].

Based on the complex plane method, the general solution of Equation (9) is assumed as

$$x(t)=\psi {\mathrm{e}}^{({\alpha }_{\mathrm{h}}+\mathrm{i}{𝜛}_{\mathrm{h}})t}$$

(11)

where $\psi$ is the amplitude of the vibration response.

Then

$$\{\begin{array}{l}{x}^{\mathrm{H}}(t)=-\mathrm{i}\psi {\mathrm{e}}^{({\alpha }_{\mathrm{h}}+\mathrm{i}{𝜛}_{\mathrm{h}})t},𝜛>0\\ x^{\mathrm{H}}(t)=\mathrm{i}\psi {\mathrm{e}}^{({\alpha }_{\mathrm{h}}+\mathrm{i}{𝜛}_{\mathrm{h}})t},𝜛<0\end{array}$$

(12)

$$\ddot{x}(t)=-{𝜛}_{\mathrm{h}}{}^2\psi {\mathrm{e}}^{({\alpha }_{\mathrm{h}}+\mathrm{i}{𝜛}_{\mathrm{h}})t}$$

(13)

Substituting Equations (11)–(13) into Equation (9), the damped natural frequency and the attenuation coefficient are obtained.

The damped natural frequency is

$${\omega }_{\mathrm{h}}=\frac{\sqrt{2}}{2}\omega \sqrt{\sqrt{1+{\eta }^2}+1}$$

(14)

Equation (14) shows that the hysteretic systems are in an underdamping state and the loss factor is limited. However, the law of the damped natural frequency for the hysteretic damping model is different from the one for the viscous damping model. The damped natural frequency is greater than the natural frequency.

The attenuation coefficient is

$${\alpha }_{\mathrm{h}}=-\frac{\sqrt{2}}{2}\omega \sqrt{\sqrt{1+{\eta }^2}-1}$$

(15)

Equation (15) shows that the attenuation coefficient is negative and the stability of the free vibration response is guaranteed based on the hysteretic damping model.

The transient response is expressed as

$$x(t)={\mathrm{e}}^{{\alpha }_{\mathrm{h}}}(A\sin {\omega }_{\mathrm{h}}t+B\cos {\omega }_{\mathrm{h}}t)$$

(16)

2.3. Comparative Analysis of the Transient Response

The relationship between loss factor and damping ratio is [28,29]

$$\eta =2\xi$$

(17)

In order to compare the transient responses of the viscous damped system and hysteretic damped system, the attenuation coefficient ratio of two damping models is

$${\delta }_{\alpha }=\frac{{\alpha }_{\mathrm{v}}}{{\alpha }_{\mathrm{h}}}=\frac{\xi }{\frac{\sqrt{2}}{2}\sqrt{\sqrt{1+4{\xi }^2}-1}}$$

(18)

Equation (18) shows that the attenuation coefficient of the viscous damping model is greater than that of the hysteretic damping model. Based on Equation (18), the attenuation coefficient ratios of the two damping models are shown in Figure 1. With the increase in damping ratio, the attenuation coefficient ratio increases gradually. When the relative error of the attenuation coefficients for the two damping models is less than 10%, namely, the attenuation coefficient ratio is less than 1.1000, the damping ratio is less than 0.5041. The threshold of the damping ratio was selected as 0.5041 based on the attenuation coefficient ratio.

In order to compare with the transient responses of the viscous damped system and hysteretic damped system, the damped natural frequency ratio of two damping models is

$${\delta }_{𝜛}=\frac{{𝜛}_{\mathrm{v}}}{{𝜛}_{\mathrm{h}}}=\frac{\sqrt{1-{\xi }^2}}{\frac{\sqrt{2}}{2}\sqrt{\sqrt{1+4{\xi }^2}+1}}$$

(19)

Equation (19) shows that the damped natural frequency of the viscous damping model is less than the one of the hysteretic damping model. Based on Equation (19), the damped natural frequency ratios of the two damping models are shown in Figure 2. With the increase in damping ratio, the damped natural frequency ratio decreases gradually. When the relative error of damped natural frequencies for the two damping models is less than 10%, namely, the attenuation coefficient ratio is greater than 0.9000, the damping ratio is less than 0.3308. The threshold of damping ratio was selected as 0.3308 based on the damped natural frequency ratio.

The attenuation coefficient ratio and the damped natural frequency ratio are important indexes of comparative analysis for the transient responses based on viscous damping model and hysteretic damping model. The threshold of the damping ratio was selected as the smaller value between 0.5041 and 0.3308. When the damping ratio is less than 0.3308, the transient responses based on the two damping models are approximately equal.

3. Comparative Analysis of Steady Responses Based on Different Damping Models

The steady response is an important part of the structural dynamic response. Seismic waves can be decomposed into a series of harmonic waves. By analyzing the characteristics of the steady vibration response due to the harmonic wave, the law of steady response due to the earthquake wave can be obtained.

3.1. The Steady Response of Viscous Damped System

Due to the harmonic wave, the time-domain equation for an SDOF system based on viscous damping model is expressed as

$$m\ddot{x}(t)+\frac{2\xi k}{\omega }\dot{x}(t)+kx(t)=-mp\sin \theta t$$

(20)

where $\theta$ is vibration frequency of harmonic wave; $p$ is the vibration amplitude of the harmonic wave.

Due to the harmonic wave, the steady response of an SDOF system is assumed as

$$x(t)=C\sin \theta t+D\cos \theta t$$

(21)

where C and D are undetermined coefficients.

The change cure between damping force and displacement is shown in Figure 3. The area of the ellipse is the dissipated energy. Based on the viscous damping model, the dissipated energy in each cycle is

$$W_{\mathrm{v}}={\displaystyle {\int }_0^{\frac{2\pi }{\theta }}\frac{2\xi k}{\omega }\dot{x}(t)\mathrm{d}x(t)}=\frac{2\pi \xi k\theta }{\omega }(A^2+B^2)$$

(22)

According to Equation (22), the dissipated energy e in each period is related to the excitation frequency based on the viscous damping model.

Substituting Equation (21) into Equation (20), the undetermined coefficients are expressed as

$$\{\begin{array}{l}C=\frac{-({\omega }^2-{\theta }^2)P}{{({\omega }^2-{\theta }^2)}^2+4{\xi }^2{\omega }^2{\theta }^2}\\ D=\frac{2\xi \omega \theta P}{{({\omega }^2-{\theta }^2)}^2+4{\xi }^2{\omega }^2{\theta }^2}\end{array}$$

(23)

The vibration amplitude can be expressed as

$$Q_{\mathrm{v}}=\sqrt{C^2+D^2}=\sqrt{\frac{P}{{({\omega }^2-{\theta }^2)}^2+4{\xi }^2{\omega }^2{\theta }^2}}$$

(24)

3.2. The Steady Response of Hysteretic Damped System

Due to the harmonic wave, the time-domain equation for an SDOF system based on hysteretic damping model is expressed as [21]

$$m\ddot{x}(t)-\eta k{x}^{\mathrm{H}}(t)+kx(t)=-mp\sin \theta t$$

(25)

Due to the harmonic wave, the steady response of an SDOF system is assumed as

$$x(t)=E\sin \theta t+F\cos \theta t$$

(26)

where E and F are undetermined coefficients.

Based on Equation (26), $x^{\mathrm{H}}(t)$ can be obtained as

$$x^{\mathrm{H}}(t)=-E\cos \theta t+F\sin \theta t$$

(27)

The resisting force of the hysteretic damping model includes the elastic force and damping force. The hysteresis loop of the resisting force is shown in Figure 4a, and the hysteresis loop of the damping force is shown in Figure 4b. The area of the ellipse is the dissipated energy in Figure 4b. Based on the hysteretic damping model, the dissipated energy in each cycle is

$$W_{\mathrm{h}}=-{\displaystyle {\int }_0^{\frac{2\pi }{\theta }}\eta k{x}^{\mathrm{H}}(t)\mathrm{d}x(t)}=\pi \eta k{A}^2$$

(28)

According to Equation (27), the dissipated energy in each period is not related to the excitation frequency based on the hysteretic damping model. The characteristic of the hysteretic damping model is different from that of the viscous damping model.

Substituting Equation (26) into Equation (25), the undetermined coefficients are expressed as

$$\{\begin{array}{l}E=\frac{-({\omega }^2-{\theta }^2)P}{{({\omega }^2-{\theta }^2)}^2+{\eta }^2{\omega }^4}\\ F=\frac{\eta {\omega }^{2}P}{{({\omega }^2-{\theta }^2)}^2+{\eta }^2{\omega }^4}\end{array}$$

(29)

The vibration amplitude can be expressed as

$$Q_{\mathrm{h}}=\sqrt{E^2+F^2}=\sqrt{\frac{P}{{({\omega }^2-{\theta }^2)}^2+{\eta }^2{\omega }^4}}$$

(30)

3.3. Comparative Analysis of the Steady Response

In order to compare the steady responses of the viscous damped system and hysteretic damped system, the vibration amplitude ratio of the two damping models is

$${\delta }_Q=\frac{Q_{\mathrm{v}}}{Q_{\mathrm{h}}}=\sqrt{\frac{{({\omega }^2-{\theta }^2)}^2+4{\xi }^2{\omega }^4}{{({\omega }^2-{\theta }^2)}^2+4{\xi }^2{\omega }^2{\theta }^2}}$$

(31)

The frequency ratio is assumed as

$$\beta =\frac{\theta }{\omega }$$

(32)

Equation (30) is rewritten as

$${\delta }_Q=\sqrt{\frac{{(1-{\beta }^2)}^2+4{\xi }^2}{{(1-{\beta }^2)}^2+4{\xi }^2{\beta }^2}}$$

(33)

The maximum and minimum values of the vibration amplitude ratio are expressed as

$$\{\begin{array}{l}{\delta }_{Q,\max }=\sqrt{\frac{1}{1-\xi }}\\ {\delta }_{Q,\min }=\sqrt{\frac{1}{1+\xi }}\end{array}$$

(34)

When the relative error of the maximum and minimum values of the vibration amplitude ratio for the two damping model are less than 10%, Equation (33) is rewritten as

$$\{\begin{array}{l}{\delta }_{Q,\max }\le 1.10\\ {\delta }_{Q,\min }\ge 0.90\end{array}$$

(35)

By solving Equation (34), it is obtained as

$$\{\begin{array}{l}\xi \le 0.1736\\ \xi \le 0.2346\end{array}$$

(36)

According to Equation (35), when the relative error of the maximum and minimum values of the vibration amplitude ratio for the two damping models are less than 10%, the damping ratio is less than 0.1736. The relationships between vibration amplitude ratios and frequency ratios with different damping ratios are shown in Figure 5. The amplitude ratio is an important index of comparative analysis of the steady responses based on viscous damping model and hysteretic damping model. The threshold of the damping ratio was selected as 0.1736. When the damping ratio is less than 0.1736, the steady responses based on the two damping model are approximately equal.

4. Numerical Examples

The numerical examples for free vibration responses were analyzed. Then natural frequency of the numerical model is 14 rad/s. The loss factors of some viscoelastic materials are greater than 0.10 [5], and the equivalent loss factor of some isolation dampers can reach 0.40 [25]. So, the material damping ratios were selected as 0.10, 020, 0.30 and 0.40, respectively. These numerical models are defined as Model A, Model B, Model C and Model D. El Centro wave, Tianjin wave and Chi-Chi wave are used to calculate the dynamic responses. The dominant frequencies and peak ground accelerations of the three earthquake waves are different. The corresponding information is shown in

Table 1. Information of earthquake waves.

Earthquake Name	Time (s)	Peak Ground Acceleration (g)	Dominant Frequency (Rad/s)
El Centro	22	0.0051	14.8796
Tianjin	20	0.0102	7.0563
Chi-Chi	20	0.0204	4.1417

. The ratios of dominant frequency and natural frequency are 1.06, 0.50 and 0.30, respectively. Different ratios can represent different frequency bands of earthquake waves. The law of the calculation results is general.

The time-domain displacements of Model A, Model B, Model C and Model D can be calculated by the numerical calculation methods of viscous damping model (VT) and hysteretic damping model (HT), respectively. The corresponding calculation results are shown in Figure 6, Figure 7 and Figure 8. In order to ensure the calculation accuracy and convergence, the time-domain methods based on the viscous damping model and hysteretic damping model are precise calculation methods [22]. Earthquake waves can be decomposed into a series of harmonic waves in the methods. Figure 6, Figure 7 and Figure 8 show that with the increase in damping ratio, the time-history displacements of the viscous damping model and hysteretic damping model are more different.

The comparative analyses of peak displacements, peak velocities and peak accelerations based on the viscous damping model and hysteretic damping model are shown in

Table 2. Comparative analysis of Model A.

Category	El Centro Wave		Tianjin Wave		Chichi Wave
	VT	HT	VT	HT	VT	HT
Peek displacement (mm)	2.3749	2.3368	1.3369	1.3124	10.6377	10.7779
Relative error (%)	—	1.60	—	1.83	—	1.32
Peek velocity (mm/s)	28.1130	27.3384	19.1281	19.1565	133.0985	133.3709
Relative error (%)	—	2.76	—	0.15	—	0.20
Peek acceleration (mm/s2)	457.7155	449.7389	261.4425	261.8755	2197.0835	2144.0104
Relative error (%)	—	1.74	—	0.17	—	2.42

,

Table 3. Comparative analysis of Model B.

Category	El Centro Wave		Tianjin Wave		Chichi Wave
	VT	HT	VT	HT	VT	HT
Peek displacement (mm)	1.7662	1.6725	1.0725	1.0148	8.0569	7.7329
Relative error (%)	—	5.31	—	5.38	—	4.02
Peek velocity (mm/s)	20.9349	19.9960	14.4211	13.9641	93.1646	94.2103
Relative error (%)	—	4.48	—	3.17	—	1.12
Peek acceleration (mm/s2)	313.6426	331.0309	184.3038	180.2010	1622.9458	1606.6127
Relative error (%)	—	5.54	—	2.23	—	1.01

,

Table 4. Comparative analysis of Model C.

Category	El Centro Wave		Tianjin Wave		Chichi Wave
	VT	HT	VT	HT	VT	HT
Peek displacement (mm)	1.4355	1.3101	0.8906	0.7958	6.8338	5.9648
Relative error (%)	—	8.74	—	10.64	—	12.72
Peek velocity (mm/s)	16.8375	15.6483	11.2117	10.6375	75.9327	72.7688
Relative error (%)	—	7.06	—	5.12	—	4.17
Peek acceleration (mm/s2)	246.3872	261.6063	143.7406	150.9666	1399.0967	1483.3127
Relative error (%)	—	6.17	—	5.03	—	6.02

and

Table 5. Comparative analysis of Model D.

Category	El Centro wave		Tianjin wave		Chichi wave
	VT	HT	VT	HT	VT	HT
Peek displacement (mm)	1.2325	1.0669	0.8261	0.6449	6.0843	4.7873
Relative error (%)	—	13.44	—	21.93	—	21.32
Peek velocity (mm/s)	13.9592	12.8003	8.9614	7.6885	64.2697	61.1640
Relative error (%)	—	8.30	—	14.20	—	4.83
Peek acceleration (mm/s2)	206.7430	220.9314	120.8636	131.8000	1292.1100	1425.8649
Relative error (%)	—	6.86	—	9.05	—	10.35

. With the increase in damping ratio, the difference among the peak displacements, peak velocities and peak accelerations calculated based on viscous damping model and hysteretic damping model gradually increases. In the case of small damping, the dynamic responses calculated by the two damping models are approximately equal. In the case of large damping, the dynamic responses calculated by the two damping models are different and the difference cannot be ignored. Specially, when the damping ratio is 0.30, the relative error of peak displacements based on viscous damping model and hysteretic damping model is greater than 10%. When the damping ratio is 0.40, the relative error of peak velocities and peak accelerations based on viscous damping model and hysteretic damping model is greater than 10%.

Based on the comparative analysis of the transient and steady responses in Section 2 and Section 3. The threshold was selected as the smallest value, and the threshold of damping ratio was selected as 0.1736, conservatively.

5. Conclusions

This paper performed a comparative analysis of the viscous damping and hysteresis damping models. The dynamic response consists of transient and steady responses. So, the transient and steady responses of the two damping models were compared and analyzed, respectively. Some conclusions are as follows.

(1)

The attenuation coefficient and damped natural frequency are important parameters of the transient response. When the relative error of the attenuation coefficients for the two damping models was less than 10%, the damping ratio was less than 0.5041. When the relative error of damped natural frequencies for the two damping models was less than 10%, the damping ratio is less than 0.3308.

(2)

Earthquake waves can be decomposed into a series of harmonic waves. The vibration amplitude of the steady response is an important parameter. When the relative error of the maximum and minimum values of the vibration amplitude ratio for the two damping models was less than 10%, the damping ratio was less than 0.1736.

(3)

The numerical examples show that with the increase in damping ratio, the difference between the dynamic responses calculated by the two damping models gradually increases. Based on the comparative analysis of the transient and steady responses, the threshold of the damping ratio was selected as 0.1736, conservatively. When the damping ratio was less than 0.1736, the dynamic responses of the two damping models were approximately equal.