A time-domain method for free vibration responses of an equivalent viscous damped system based on a complex damping model

Abstract

The free vibration responses are divergent based on a complex time-domain damping model. The traditional step by step integration method cannot be used to calculate the time-domain responses. Based on the theoretic solution that eliminates the divergent term for the complex damping model, the equivalent viscous damping model is proposed and the corresponding average acceleration method is realized in this paper. The numerical cases show that the calculated results of the equivalent viscous damped system are convergent, which are equal to the ones of the deleted divergent term based on the complex damping model. The correctness of the proposed method is verified, and the computational efficiency is high.

Keywords

Free vibration responses complex damping equivalent viscous damping average acceleration method

Introduction

A complex damping model¹ is commonly an internal damping model, which is also called the structural damping model^2,3 and frequency-independent damping model.^4,5 The damping parameter of the complex damping model is the loss factor. The loss factor is the important parameter, which plays a critical role in the transient decay responses.⁶ The loss factor can be obtained by a material damping test. The complex damping model is applied for dynamic analysis of engineering structures.^7,8 The complex damping model is constructed based on the frequency-domain responses due to a harmonic wave. And the dissipated energy in each cycle is not related to the vibration frequency of the harmonic wave. The phenomenon is consistent with the experimental results of most materials.⁹ The frequency-domain responses only include steady-state responses. Compared with the frequency-domain model, the time-domain model can be used to calculate both transient responses and steady-state responses.¹⁰ The complex damping model is extended to the time-domain dynamic analysis. However, the calculated time-domain results are divergent.¹¹ The reason is that the free vibration responses are divergent. Therefore, how to overcome the divergent shortcoming is an important factor of the complex time-domain damping model.

A viscous damping model is another common internal damping model and the corresponding time-domain results are convergent.¹² However, the damping ratio is difficult to obtain. Combined with the advantages of the two damping models, the equivalent viscous damping model is constructed based on the material loss factor. To overcome the divergent phenomenon of the complex damped system, the equivalent viscous damping model is proposed based on the theoretic solution that eliminates the divergent term for the complex damping model. Based on the assumption of the average acceleration method, a time-domain method for free vibration responses of the equivalent viscous damped system is realized, which can be applied for single-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) systems. The calculated results are convergent and have high computational efficiency.

Free vibration responses of the complex damped system

Based on the complex damping model, the time-domain equation for a SDOF system is expressed as¹³

\frac{d^{2} x (t)}{d t^{2}} + i η ω^{2} x (t) + ω^{2} x (t) = 0

(1)

where

η

is the loss factor and

i

is an imaginary unit,

i = \sqrt{- 1}

ω

is the natural frequency

ω = \sqrt{\frac{k}{m}}

(2)

where

m

is the mass and

k

is the stiffness.

It is assumed that

x (t) = x_{1} (t) + i x_{2} (t)

(3)

where

x_{1} (t)

is the displacement and

x_{2} (t)

is the dual item of the displacement.

Substituting equation (3) into equation (1), it is obtained as

{\begin{cases} \frac{d^{2} x_{1} (t)}{d t^{2}} - η ω^{2} x_{2} (t) + ω^{2} x_{1} (t) = 0 \\ \frac{d^{2} x_{2} (t)}{d t^{2}} + η ω^{2} x_{1} (t) + ω^{2} x_{2} (t) = 0 \end{cases}

(4)

The whole time history is discretized, and the time can be expressed as

t = k Δ t (k = 0,1,2, \dots)

(5)

where

Δ t

is the time step.

The assumption of the average acceleration method is¹⁴

\frac{d^{2} x (t)}{d t^{2}} = \frac{1}{2} [\frac{d^{2} x (t_{k})}{d t^{2}} + \frac{d^{2} x (t_{k + 1})}{d t^{2}}] (t_{k} < t < t_{k + 1})

(6)

Equation (6) is rewritten as

{\begin{cases} \frac{d x_{1} (t_{k + 1})}{d t} = \frac{d x_{1} (t_{k})}{d t} + \frac{Δ t}{2} [\frac{d^{2} x_{1} (t_{k})}{d t^{2}} + \frac{d^{2} x_{1} (t_{k + 1})}{d t^{2}}] \\ x_{1} (t_{k + 1}) = x_{1} (t_{k}) + Δ t \frac{d x_{1} (t_{k})}{d t} + \frac{Δ t}{2} [\frac{d^{2} x_{1} (t_{k})}{d t^{2}} + \frac{d^{2} x_{1} (t_{k + 1})}{d t^{2}}] \end{cases}

(7)

{\begin{cases} \frac{d x_{2} (t_{k + 1})}{d t} = \frac{d x_{2} (t_{k})}{d t} + \frac{Δ t}{2} [\frac{d^{2} x_{2} (t_{k})}{d t^{2}} + \frac{d^{2} x_{2} (t_{k + 1})}{d t^{2}}] \\ x_{2} (t_{k + 1}) = x_{2} (t_{k}) + Δ t \frac{d x_{2} (t_{k})}{d t} + \frac{Δ t}{2} [\frac{d^{2} x_{2} (t_{k})}{d t^{2}} + \frac{d^{2} x_{2} (t_{k + 1})}{d t^{2}}] \end{cases}

(8)

Substituting equations (7) and (8) into equation (4) gives

{\begin{cases} (\frac{Δ t}{2} + \frac{ω^{2} Δ t}{2}) [\frac{d^{2} x_{1} (t_{k})}{d t^{2}} + \frac{d^{2} x_{1} (t_{k + 1})}{d t^{2}}] + (1 + ω^{2} Δ t) \frac{d x_{1} (t_{k})}{d t} + ω^{2} x_{1} (t_{k}) \\ - \frac{η ω^{2} Δ t}{2} [\frac{d^{2} x_{2} (t_{k})}{d t^{2}} + \frac{d^{2} x_{2} (t_{k + 1})}{d t^{2}}] - η ω^{2} Δ t \frac{d x_{2} (t_{k})}{d t} - η ω^{2} x_{2} (t_{k}) = 0 \\ (\frac{Δ t}{2} + \frac{ω^{2} Δ t}{2}) [\frac{d^{2} x_{2} (t_{k})}{d t^{2}} + \frac{d^{2} x_{2} (t_{k + 1})}{d t^{2}}] + (1 + ω^{2} Δ t) \frac{d x_{2} (t_{k})}{d t} + ω^{2} x_{2} (t_{k}) \\ + \frac{η ω^{2} Δ t}{2} [\frac{d^{2} x_{1} (t_{k})}{d t^{2}} + \frac{d^{2} x_{1} (t_{k + 1})}{d t^{2}}] + η ω^{2} Δ t \frac{d x_{1} (t_{k})}{d t} + η ω^{2} x_{1} (t_{k}) = 0 \end{cases}

(9)

Solving equation (9), it is obtained as

\frac{d^{2} x_{1} (t_{k + 1})}{d t^{2}} = \frac{- (ω^{2} A + 2 B^{2}) x_{1} (t_{k}) - (ω^{2} B + 2 A B) x_{2} (t_{k})}{A^{2} + B^{2}} - \frac{d^{2} x_{1} (t_{k})}{d t^{2}} - 2 \frac{d x_{1} (t_{k})}{d t}

(10)

\frac{d^{2} x_{2} (t_{k + 1})}{d t^{2}} = \frac{- (ω^{2} A + 2 B^{2}) x_{2} (t_{k}) + (ω^{2} B - 2 A B) x_{1} (t_{k})}{A^{2} + B^{2}} - \frac{d^{2} x_{2} (t_{k})}{d t^{2}} - 2 \frac{d x_{2} (t_{k})}{d t}

(11)

Based on equations (7)–(11), the whole time history of free vibration responses can be obtained. The average acceleration method of complex damped systems is realized.

The numerical model is adopted to calculate the free vibration responses, which is defined as Model A. The natural frequency is 4 rad/s. The initial displacement is 20 cm and the initial velocity is 25 cm/s. Namely, $x_{1} (t_{0})$ is 20 cm and $\frac{d x_{1} (t_{0})}{d t}$ is 25 cm/s. Here, the complex damping model has a shortcoming. The initial values are the real parts of complex vibration responses. The imaginary parts $x_{2} (t_{0})$ and $\frac{d x_{2} (t_{0})}{d t}$ can be directly obtained. It is assumed that $x_{2} (t_{0})$ is 0 cm and $\frac{d x_{2} (t_{0})}{d t}$ is 0 cm/s. The free vibration responses of numerical models with different loss factors are calculated based on the average acceleration method of complex damped systems. The loss factors are 0.05 and 0.10, respectively. The corresponding results are shown in Figure 1.

Figure 1.

The free vibration responses of complex damped system: (a) η is 0.05 and (b) η is 0.10.

Figure 1 shows that the free vibration responses are divergent. The average acceleration method is unconditionally stable and convergent. So the divergent phenomenon is not caused by the calculated method but by the theory of the complex damping model. Besides, with the increase in the loss factor, the divergence becomes more serious. The complex damping model cannot be used to calculate the free vibration responses.

Free vibration responses of the equivalent viscous damped system

Divergent analysis of the complex damped system

The time-domain results of complex damped systems are divergent in Section 2. In the section, by means of the constant-transform method, the analytical solutions of the free vibration equation based on the complex damping model are obtained. Then, the divergent reason of the complex damping model is analyzed.

Based on the complex damping model, the corresponding free vibration equation for a SDOF system is expressed as

\frac{d^{2} x (t)}{d t^{2}} + i η ω^{2} x (t) + ω^{2} x (t) = 0

(12)

The corresponding characteristic equation of equation (12) is expressed as

λ^{2} + i η ω^{2} + ω^{2} = 0

(13)

By solving equation (13), the complex eigenvalues are obtained as

{\begin{cases} λ_{1} = - \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} - 1} + i \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} + 1} \\ λ_{2} = \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} - 1} - i \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} + 1} \end{cases}

(14)

The displacement of a SDOF system is

x_{1} (t) = e^{- α t} (A \sin ϖ t + B \cos ϖ t) + e^{α t} (C \sin ϖ t + D \cos ϖ t)

(15)

{\begin{cases} α = \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} - 1} \\ ϖ = \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} + 1} \end{cases}

(16)

Equation (15) shows that the expression of displacement contains an exponential growth term. It means that small-amplitude acoustic disturbances at any frequency will be exponentially amplified.¹⁵ The growth rate of vibration responses is the important factor.¹⁶ The growth item of free vibration responses is $e^{α t}$ . The corresponding growth rate depends on α. Therefore, the free vibration responses of the complex damped system are divergent.

Equivalent viscous damped system based on the complex damping model

In order to obtain the convergent free vibration responses, the equivalent viscous damping model is constructed. The free vibration equation of the equivalent viscous damped system is expressed as

\frac{d^{2} x (t)}{d t^{2}} + E \frac{d x (t)}{d t} + F x (t) = 0

(17)

The corresponding characteristic equation of equation (17) is expressed as

δ^{2} + E δ + F = 0

(18)

The exponential growth term is eliminated, and equation (15) is rewritten as

x (t) = e^{- α t} (A \sin ϖ t + B \cos ϖ t)

(19)

Based on equation (19), the complex eigenvalues are obtained as

{\begin{cases} δ_{1} = - \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} - 1} + i \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} + 1} \\ δ_{2} = - \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} - 1} - i \frac{\sqrt{2}}{2} ω \sqrt{\sqrt{1 + η^{2}} + 1} \end{cases}

(20)

Substituting equation (20) into equation (18) gives

{\begin{cases} {δ_{1}}^{2} + E δ_{1} + F = 0 \\ {δ_{2}}^{2} + E δ_{2} + F = 0 \end{cases}

(21)

By solving equation (21), the undetermined coefficients are obtained as

{\begin{cases} E = \sqrt{2} ω \sqrt{\sqrt{1 + η^{2}} - 1} \\ F = ω^{2} \sqrt{1 + η^{2}} \end{cases}

(22)

The time-domain equation of motion for the equivalent viscous damping system is

\frac{d^{2} x (t)}{d t^{2}} + \sqrt{2} ω \sqrt{\sqrt{1 + η^{2}} - 1} \frac{d x (t)}{d t} + ω^{2} \sqrt{1 + η^{2}} x (t) = 0

(23)

The average acceleration method is adopted, and the velocity and displacement are expressed as

{\begin{cases} \frac{d x (t_{k + 1})}{d t} = \frac{d x (t_{k})}{d t} + \frac{Δ t}{2} [\frac{d^{2} x (t_{k})}{d t^{2}} + \frac{d^{2} x (t_{k + 1})}{d t^{2}}] \\ x (t_{k + 1}) = x (t_{k}) + Δ t \frac{d x (t_{k})}{d t} + \frac{Δ t}{2} [\frac{d^{2} x (t_{k})}{d t^{2}} + \frac{d^{2} x (t_{k + 1})}{d t^{2}}] \end{cases}

(24)

Substituting equation (24) into equation (23), it is obtained as

\frac{d^{2} x (t_{k + 1})}{d t^{2}} = - \frac{\frac{Δ t}{2} (E + F) \frac{d^{2} x (t_{k})}{d t^{2}} + (E + Δ t F) \frac{d x (t_{k})}{d t} + F x (t_{k})}{1 + \frac{Δ t}{2} E + \frac{Δ t}{2} F}

(25)

Based on equations (24) and (25), the free vibration responses are obtained and the average acceleration method of equivalent viscous damped systems is realized. The calculation process is objective and the divergent term is not manually eliminated. Besides, compared with complex damped systems, the initial values of equivalent viscous damped systems are easily obtained, which do not need the dual item of the initial displacement and initial velocity. The initial values are the real parts of complex vibration responses.

Multiple-degree-of-freedom systems based on the equivalent viscous damping model

Based on the equivalent viscous damping model, the free vibration equation of a MDOF system is expressed as

M \frac{d^{2} x (t)}{d t^{2}} + \frac{\sqrt{2} \sqrt{\sqrt{1 + η^{2}} - 1}}{ω} K \frac{d x (t)}{d t} + \sqrt{1 + η^{2}} K x (t) = 0

(26)

where

M

is the mass matrix;

K

is the stiffness matrix; and

x (t)

is the displacement vector.

The eigenvectors of equation (26) are

Φ = [\begin{array}{c} ϕ_{1} & ϕ_{2} & \dots & ϕ_{N} \end{array}]

(27)

Based on the mode superposition method, the displacement is obtained as

x (t) = \sum_{j = 1}^{N} q_{j} (t) ϕ_{j}

(28)

Substituting equation (28) into equation (26), the equation can be decomposed as

\frac{d^{2} q_{j} (t)}{d t^{2}} + \sqrt{2} \sqrt{\sqrt{1 + η^{2}} - 1} ω_{j} \frac{d q_{j} (t)}{d t} + \sqrt{1 + η^{2}} {ω_{j}}^{2} q_{j} (t) = 0 (j = 1,2, \dots, N)

(29)

Based on the average acceleration method of SDOF systems, $q_{j} (t)$ is obtained from equation (29). Then, $x (t)$ can also be obtained from equation (28).

Numerical examples

Numerical example of SODF systems

In the traditional method, it is assumed that the loss factor is approximately twice the damping ratio.¹⁷ Based on the viscous damping model, the corresponding free vibration equation for a SDOF system is expressed as

\frac{d^{2} x (t)}{d t^{2}} + η ω \frac{d x (t)}{d t} + ω^{2} x (t) = 0

(30)

Based on equation (19), the convergent time-domain method based on the complex damping model (CTC) can be used to calculate the free vibration responses, and the results based on CTC are exact solutions. Based on equations (24) and (25), the average acceleration method based on the equivalent viscous damping model (AEV) can be used to calculate the free vibration responses. Based on equation (30), the average acceleration method based on the traditionally equivalent viscous damping model (TEV) can be used to calculate the free vibration responses. CTC, AEV, and TEV are compared. The mean value of time-domain displacements is selected as the index.

The relative error of mean value–based on AEV is

δ_{A E V} = | \frac{λ_{A E V} - λ_{C T C}}{λ_{C T C}} | \times 100 %

(31)

where

λ_{C T C}

is the mean value of time-domain displacements based on CTC and

λ_{A E V}

is the mean value of time-domain displacements based on AEV.

The relative error of mean value–based on TEV is

δ_{T E V} = | \frac{λ_{T E V} - λ_{C T C}}{λ_{C T C}} | \times 100 %

(32)

where

λ_{T E V}

is the mean value of time-domain displacements based on TEV.

Model A is taken as the numerical model. The natural frequency is 4 rad/s. The initial velocity is 25 cm/s and the initial displacement is 20 cm. The loss factors are 0.10, 0.20, 0.40, and 0.60, respectively. The calculated results of CTC and AEV are shown in Figure 2. The corresponding relative errors of mean values based on AEV and TEV are shown in Table 1.

Figure 2.

The free vibration responses of Model A with different methods: (a) η is 0.10; (b) η is 0.20; (c) η is 0.40; and (d) η is 0.60.

Table 1.

Relative errors of the mean values with different methods for Model A.

η	0.10, %	0.20, %	0.40, %	0.60, %
AEV	0.02	0.01	0.01	0.01
TEV	1.01	2.90	8.59	19.23

The calculated results of AEV are convergent, which is approximately equal to the ones of CTC. And the divergent shortcoming is overcome. With the increases in the loss factor, the convergent rate of time-domain responses is faster. The phenomenon is consistent with physical law. The correctness of AEV is verified. As the loss factor increases, the relative error of TEV increases gradually. When the loss factor is 0.60, the time-domain displacements of TEV are obviously different from the ones of CTC, and the relative error of the mean value of TEV is 19.23%. Different from AEV, TEV cannot be applied in large damping cases. AEV can be applied both in small and large damping cases.

Numerical example of MDOF systems

The 5-DOF model is taken as the numerical example of a MDOF system, which is defined as Model B. The mass and stiffness distribution are shown in Figure 3. The loss factors of Model B are 0.10, 0.20, 0.40, and 0.60, respectively.

Figure 3.

Schematic of Model B.

For Model B, the initial velocity vector is

\frac{d x (t_{0})}{d t} = {(\begin{array}{c} 20 & 13 & 7 & 4 & 0 \end{array})}^{T} c m / s

(33)

The initial displacement vector is

x (t_{0}) = {(\begin{array}{c} 15 & 10 & 6 & 3 & 0 \end{array})}^{T} c m

(34)

CTC and AEV are used to calculate the top displacement of Model B, respectively. The corresponding time-domain results are shown in Figure 4. The calculated results of AEV are convergent, which is approximately equal to the ones of CTC. The law is similar to the one of SDOF systems.

Figure 4.

The top free vibration responses of Model B with different methods: (a) η is 0.10; (b) η is 0.20; (c) η is 0.40; and (d) η is 0.60.

The personal computer (CPU: i5-8300H, RAM: 8.00 GB) is used to calculate the free vibration responses of Model B. The time-history of free vibration responses is 20 s. The corresponding computing time of CTC and AEV is 0.90 s and 0.22 s, as shown in Table 2. Therefore, the computational efficiency of AEV is higher. Besides, CTC needs to delete manually the divergent term, which is not mathematically rigorous. AEV is an objective method without human intervention.

Table 2.

The computing time with different methods (unit: s).

	CTC	AEV
Computing time	0.90	0.22

Discussion and conclusions

The complex damping model cannot be directly applied in time-domain analysis, and the free vibration responses are divergent. Based on the theoretic solution that deletes the divergent term for the complex damping model, the equivalent viscous damping model is proposed in this paper.

Based on the assumption of the average acceleration method, a time-domain method for free vibration responses of the equivalent viscous damped system is realized, which can be applied for SDOF and MDOF systems. The numerical cases show that the calculated results of the equivalent viscous damped system are convergent, which are equal to the ones of the deleted divergent term based on the complex damping model. The correctness of the proposed method is verified. Compared with the approximately equivalent method, the proposed method is precisely equivalent method, which can be applied both in small and large damping cases. Besides, the proposed method is the time-domain integration method and has higher computational efficiency.

The proposed equivalent damping model can provide a basis for the expansion of the complex damping model in the time-domain analysis. In future work, the improved time-domain method will be studied based on the short-time Fourier transform, wavelet transform, etc.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was partially supported by the National Natural Science Foundation of China (Grant No. 51878621), Science and Technology Research Project of Henan Province (Grant No. 222102230010), China Postdoctoral Science Foundation (Grant No. 2022M712905), Natural Science Foundation of Henan Province (Grant No. 222300420316), and Key Research Projects of Henan Higher Education Institutions (Grant No. 22A560005).

ORCID iDs

Shuxia Wang

Panxu Sun

Dongwei Wang

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