1/2分数阶线性随机动力系统的非平稳响应解析解

孔凡，许伊键，韩仁杰，徐军，洪旭; Kong Fan; Xu Yijian; Han Renjie; Xu Jun; Hong Xu

引 en

1/2分数阶线性随机动力系统的非平稳响应解析解^*

孔凡¹
机构：
1. 合肥工业大学土木与水利工程学院,合肥 230009
×
，许伊键¹
机构：
1. 合肥工业大学土木与水利工程学院,合肥 230009
×
，韩仁杰²
机构：
2. 同济大学土木工程学院,上海 200092
×
，徐军³
机构：
3. 湖南大学土木工程学院,长沙 410082
×
，洪旭¹
机构：
1. 合肥工业大学土木与水利工程学院,合肥 230009
×

1. 合肥工业大学土木与水利工程学院,合肥 230009；
2. 同济大学土木工程学院,上海 200092；
3. 湖南大学土木工程学院,长沙 410082；

Analytical Solution for Non-Stationary Response of 1/2-Order Fractional Linear Stochastic Dynamical Systems^*

Kong Fan¹
Affiliation：
1. College of Civil Engineering,Hefei University of Technology,Hefei 230009 ,China
×
，Xu Yijian¹
Affiliation：
1. College of Civil Engineering,Hefei University of Technology,Hefei 230009 ,China
×
，Han Renjie²
Affiliation：
2. College of Civil Engineering,Tongji University,Shanghai 200092 ,China
×
，Xu Jun³
Affiliation：
3. College of Civil Engineering,Hunan University,Changsha 410082 ,China
×
，Hong Xu¹
Affiliation：
1. College of Civil Engineering,Hefei University of Technology,Hefei 230009 ,China
×

1. College of Civil Engineering,Hefei University of Technology,Hefei 230009 ,China；
2. College of Civil Engineering,Tongji University,Shanghai 200092 ,China；
3. College of Civil Engineering,Hunan University,Changsha 410082 ,China；

通讯作者:

韩仁杰,E-mail:hanrenjie@tongji.edu.cn

中图分类号:O321;O324

文献标识码:A

文章编号:1672-6553-2023-21(4)-023-009

DOI:10.6052/1672-6553-2023-040

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目录contents

摘要 Abstract
关键词 Keywords
1 单自由度分数阶系统的响应
1.1 特征向量展开
1.2 均方响应求解
2 数值算例
2.1 白噪声
2.2 调制白噪声
2.3 调制修正金井清谱
3 结论
参考文献

摘要

提出随机激励作用下1/2分数阶线性系统非平稳响应解析解的一种新方法.首先,利用特征向量展开得到1/2分数阶阻尼系统的脉冲响应函数解析表达;之后,基于Laplace变换计算得到响应功率谱密度的解析表达式和系统均方响应.通过白噪声、调制白噪声和调制修正金井清谱三种不同随机激励类型的数值算例,利用与蒙特卡洛模拟所得结果对比证明该方法的准确性和适用性.

Abstract

In this paper, a novel method is proposed to obtain an analytical solution for the non-stationary response of 1/2 order systems under stochastic excitation. The method first uses the eigenvector expansion to obtain the analytical expression of the impulse response function. Next, an analytical solution for the response power spectral density is obtained based on the Laplace transform. Through three illustrative numerical examples, including systems subjected to white noise, modulated white noise, and modulated colored noise with modified Kanai-Tajimi spectrum, the responses of the fractional systems are obtained analytically and compared to the pertinent Monte Carlo estimates to demonstrate the accuracy and applicability of the proposed method.

关键词

分数阶；特征向量展开；拉普拉斯变换；功率谱密度；均方响应

Keywords

fractional order ； eigenvector expansion ； Laplace transform ； power spectral density ； mean squared response

引言
分数阶导数模型在工程和科学问题中应用十分广泛，包括机械、控制、生物学、经济学等领域^[1].在土木工程领域中，分数阶导数常用来描述黏弹性材料的本构关系.20世纪初，Nutting^[2]发现一些材料的应力松弛可被模拟为分数阶幂律的时间函数.之后，Gemant^[3]提出利用分数阶导数表示黏弹性材料的刚度和阻尼特性; Caputo^[4]利用分数算子描述黏弹性土的力学行为; Bagley和Torvik^[5]则给出了分数阶微积分应用的物理解释.
分数阶动力系统虽然具有简洁、准确的优势，但由于方程中存在非整数微分算子，求解动力响应存在较大困难.目前有若干方法用于计算分数阶确定性动力系统响应，包括Laplace变换^[6]、傅里叶变换^[7]、数值模拟方法^[8，9]、特征向量展开法^[10]、平均方法^[11]等，但他们很少关注随机动力系统.对于分数阶随机动力系统，一些研究者通常只关注系统的平稳响应，Spanos和Zeldin^[12]提出了一种随机振动分析的频域方法; 孔凡等^[13]利用统计线性化方法得到随机和谐和联合激励下的分数阶系统的平稳响应; Pinnola^[14]则利用特征向量展开和随机过程的复谱矩表征得到分数阶振子的平稳响应; Huang等^[15]利用单自由度强非线性随机动力系统的随机平均法计算系统的稳态响应并考察稳定性; 马颜颜等^[16]同样利用随机平均法求解得到系统的稳态响应; Artale等^[17]利用残差定理确定了平稳高斯白噪声作用下的分数阶振子的整数阶和分数阶谱矩.另一些研究者在关注系统的平稳响应的同时也关注非平稳响应，如Agrawal^[18]利用特征向量展开并利用时域卷积方法得到具有1/2阶阻尼的随机动力系统解析解; Ye等^[19]利用傅里叶变换技术给出了分数阶动力系统在确定性和随机输入下响应的Duhamel积分型表达式和脉冲响应函数.虽然上述方法能够得到系统响应，但是需要利用双重数值积分，计算时间较长.Cao等^[20]虽然利用Laplace变换和Prony-SS（State-Space model）算法得到了任意有理阶阻尼的随机动力系统的数值解，但它存在两个误差来源:一是通过傅里叶逆变换得到脉冲响应函数时忽视了系统初始条件; 二是Prony-SS法得到脉响函数指数形式时的数值误差.
本文提出了一种计算1/2阶分数阶线性随机动力系统非平稳响应解析解的精确高效方法.主要利用Suarez和Shokooh^[10]提出的特征向量展开法以求解分数阶系统脉冲响应函数，以及基于Laplace变换求解随机动力响应.数值算例采用与Monte Carlo模拟对比的方法验证了所建议方法的精度.
1 单自由度分数阶系统的响应
考虑随机激励作用下具有分数阶导数阻尼的单自由度线性系统

m D^{2} x (t) + c D^{α} x (t) + k x (t) = f (t)

(1)

式中:m，c，k分别为系统质量、分数阶阻尼与刚度系数; f（t）为系统外部随机激励; D表示求导数; α为分数阶导数阶数，根据Riemann-Liouville定义

D_{L}^{α} [x (t)] = \frac{1}{Γ (1 - α)} \frac{d}{d t} \int_{0}^{t} \frac{x (t - τ)}{τ^{α}} d τ, 0 < α < 1

(2)

将上式两边同除以m可得

D^{2} x (t) + 2 η ω_{n}^{2 - α} D^{α} x (t) + ω_{n}^{2} x (t) = f^{*} (t)

(3)

式中: $ω_{n} = \sqrt{k / m}$ 为自振频率; $f^{*} （ t ） = f （ t ） / m$ ; η为阻尼比.
1.1 特征向量展开
当α=1/2时，式（1）所示的系统的脉冲响应函数具有简单的解析表达^[10].为此，将式（3）所示的运动方程写为1/2阶状态空间方程的形式.令

\begin{matrix} z_{1} = D^{3 / 2} x (t), z_{2} = D x (t) \\ z_{3} = D^{1 / 2} x (t), z_{4} = x (t) \end{matrix}

(4)

则有

B D^{1 / 2} {z} - A {z} = F

(5)

式中

(6)

F=[0 0 0 f^*（t）]^T且 $a = 2 η ω_{n}^{3 / 2} ， b = ω_{n}^{2}$ .
为求解式（5），可参照整数阶导数系统状态空间方程的特征向量展开方法，实现状态空间坐标解耦.为此，式（5）对应的特征值问题为

G {φ}_{j} = λ_{j} {φ}_{j}

(7)

式中:

G = B^{- 1} A = [\begin{matrix} 0 & 0 & - a & - b \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}]

(8)

且λ_j为特征值; {φ}_j为对应的特征向量.求解式（7）所示特征值问题即可得到4个特征值以及对应的特征向量，具体求解过程见文献^[10]附录.
引入 ${y} = {[y_{1} ， y_{2} ， y_{3} ， y_{4}]}^{T}$ ，利用

{z} = Φ {y}

(9)

实现对式（5）的解耦，式中Φ=[{φ}₁，{φ}₂，{φ}₃，{φ}₄].将式（9）代入式（5），可得1/2阶

D^{1 / 2} Φ {y} - G Φ {y} = F

(10)

对式（10）左乘{φ}^T_j并利用特征向量的正交特性可得解耦后的1/2阶状态空间方程

D^{1 / 2} y_{j} (t) - λ_{j} y_{j} (t) = {φ}_{4 j} f^{*} (t)

(11)

式中

φ_{4 j} = \frac{1}{\sqrt{4 λ_{j}^{3} + a}}

(12)

为得到系统的位移响应，对式（11）进行Laplace变换得

Y_{j} (s) = \frac{φ_{4 j} F (s) + R_{j}}{\sqrt{s} - λ_{j}}, j = 1,2, 3,4

(13)

且 $R_{j} = {D^{- 1 / 2} y_{j} （ t ）|}_{t = 0}$ .对式（13）进行Laplace逆变换并结合式（4）和式（9）可得系统位移表达式

x (t) = \sum_{j = 1}^{4} φ_{4 j} L^{- 1} [Y_{j} (s)]

(14)

式（3）右端为 $f^{*} （ t ） = δ （ t ） / m$ 时，式（14）即为系统脉冲响应函数.此时，脉冲响应函数为

\begin{matrix} h (t) = \frac{1}{\sqrt{π t}} \sum_{j = 1}^{4} (\frac{1}{m} φ_{4 j}^{2} + φ_{4 j} R_{j}) + \\ \sum_{j = 1}^{4} (\frac{λ_{j}}{m} φ_{4 j}^{2} g_{j} (t) + λ_{j} φ_{4 j} R_{j} g_{j} (t)) \end{matrix}

(15)

式中:

g_{j} (t) = e^{λ_{j}^{2} t} [1 + E r f (λ_{j} \sqrt{t})]

(16)

观察式（15），根据文献^[10]附录所示关系可得

\sum_{j = 1}^{4} φ_{4 j}^{2} = 0, \sum_{j = 1}^{4} λ_{j} φ_{4 j}^{2} = 0, \sum_{j = 1}^{4} λ_{j} φ_{4 j} R_{j} = 0

(17)

利用式（17）关系并考虑零初始条件，式（15）可化为

h (t) = \frac{1}{m} \sum_{j = 1}^{4} \frac{λ_{j} / 2}{2 λ_{j}^{3} + η ω_{n}^{3 / 2}} g_{j} (t)

(18)

1.2 均方响应求解
激励为零均值随机激励时，式（1）所示的系统响应同样为零均值随机过程.考虑随机激励为非平稳随机过程的情况，即

f (t) = a (t) f_{s} (t)

(19)

式中，

a (τ) = \sum_{l = 1}^{N} γ_{l} e^{v l τ}

(20)

为调制函数，f_s（t）为平稳随机过程.根据Duhamel积分

X (t) = \int_{0}^{t} h (t - τ) a (τ) f_{s} (τ) d τ

(21)

求响应均方值，即

\begin{matrix} E [X (t_{1}) X (t_{2})] = \int_{0}^{t 1} \int_{0}^{t 2} h (t_{1} - τ_{1}) h (t_{2} - τ_{2}) \times \\ a (τ_{1}) a (τ_{2}) R_{F} (τ_{2} - τ_{1}) d τ_{1} d τ_{2} \end{matrix}

(22)

式中:E[·]表示数学期望; R_F表示平稳随机过程f_s（t）的自相关函数.
根据Wiener-Khinchin定理，激励的相关函数与功率谱密度具有如下傅里叶变换对关系:

R_{F} (τ) = \int_{- \infty}^{\infty} S_{F} (ω) e^{i ω τ} d ω

(23)

S_{F} (ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} R_{F} (τ) e^{- i ω τ} d τ

(24)

将式（23）代入式（22）可得

\begin{matrix} E [X (t_{1}) X (t_{2})] = \int_{0}^{t_{1}} \int_{0}^{t_{2}} h (t_{1} - τ_{1}) h (t_{2} - τ_{2}) \times \\ a (τ_{1}) a (τ_{2}) [\int_{- \infty}^{\infty} S_{F} (ω) e^{i ω (τ_{2} - τ_{1})} d ω] d τ_{1} d τ_{2} \end{matrix}

(25)

将式（25）改写为

E [X (t_{1}) X (t_{2})] = \int_{- \infty}^{\infty} H (t_{1}, ω) \bar{H} (t_{2}, ω) S_{F} (ω) d ω

(26)

式中^-表示复共轭，且

H (t, ω) = \int_{0}^{t} h (t - τ) a (τ) e^{- i ω τ} d τ

(27)

将式（20）代入式（27）并将求和项移到积分符号外可得

H (t, ω) = \sum_{l = 1}^{N} \int_{0}^{t} h (t - τ) γ_{l} e^{κ l τ} d τ

(28)

式中: $κ_{l} = v_{l} - i ω$ .
令式（26）中t₁=t₂=t，利用激励功率谱密度和响应功率谱密度的关系可得均方响应为

E [X^{2} (t)] = \int_{- \infty}^{\infty} S_{X} (t, ω) d ω

(29)

式中

S_{X} (t; ω) = | H (t, ω) |^{2} S_{F} (ω)

(30)

观察式（29）-式（30）可知欲计算系统均方响应，需先得到H（t，ω）.为此，利用时域卷积对应Laplace域乘积的性质对式（28）进行Laplace变换并考虑到式（18），可得

\begin{matrix} \tilde{H} (s, ω) = \sum_{l = 1}^{N} \tilde{h} (s) \frac{γ_{l}}{s - κ_{l}} = \\ \sum_{l = 1}^{N} \sum_{j = 1}^{4} \frac{A_{j}}{m} \frac{γ_{l}}{s - κ_{l}} (\frac{1}{s - λ_{j}^{2}} + \frac{λ_{j}}{\sqrt{s} (s - λ_{j}^{2})}) \end{matrix}

(31)

式中:

A_{j} = \frac{λ_{j} / 2}{2 λ_{j}^{3} + η ω_{n}^{3 / 2}}

(32)

且上标~表示Laplace域的函数.利用极点留数定理将式（31）展开为多项式形式

\begin{matrix} \tilde{H} (s, ω) = \sum_{l = 1}^{N} \sum_{j = 1}^{4} \frac{A_{j}}{m} \frac{γ_{l}}{λ_{j}^{2} - κ_{l}} (\frac{1}{s - λ_{j}^{2}} - \frac{1}{s - κ}) + \\ \frac{A_{j}}{m} \frac{γ_{l} λ_{j}}{\sqrt{s} (λ_{j}^{2} - κ_{l})} (\frac{1}{s - λ_{j}^{2}} - \frac{1}{s - κ}) \end{matrix}

(33)

将式（33）进行Laplace逆变换并注意到

L^{- 1} (\frac{1}{\sqrt{s}} \frac{1}{s - a}) = a^{- 1 / 2} e^{a t} E r f (\sqrt{a t})

(34)

可得到频响函数

H (t, ω) = \frac{1}{m} \sum_{l = 1}^{N} \sum_{j = 1}^{4} \sum_{r = 1}^{4} A_{j} E_{l j r}

(35)

式中:

E_{l j 1} = \frac{γ_{l}}{λ_{j}^{2} - κ_{l}} e^{λ_{j}^{2} t}

(36)

E_{l j 2} = - \frac{γ_{l}}{λ_{j}^{2} - κ_{l}} e^{κ l t}

(37)

E_{l j 3} = \frac{γ_{l}}{λ_{j}^{2} - κ_{l}} e^{λ_{j}^{2} t} E r f (λ_{j} \sqrt{t})

(38)

E_{l j 4} = - \frac{γ_{l}}{λ_{j}^{2} - κ_{l}} \frac{λ_{j}}{\sqrt{κ_{l}}} e^{κ_{l t}} E r f (\sqrt{κ_{l} t})

(39)

利用式（30）可得响应功率谱密度解析解

S_{X} (t, ω) = \frac{S_{F} (ω)}{m^{2}} \sum_{l = 1}^{N} \sum_{q = 1}^{N} \sum_{j = 1}^{4} \sum_{r = 1}^{4} \sum_{k = 1}^{4} \sum_{n = 1}^{4} A_{j} {\bar{A}}_{k} E_{l j r} {\bar{E}}_{q k n}

(40)

激励项不含调制函数，即式（20）中γ_l=1，v_l=0，且N=1时，式（40）退化为

S_{X} (t, ω) = \frac{S_{F} (ω)}{m^{2}} \sum_{j = 1}^{4} \sum_{r = 1}^{4} \sum_{k = 1}^{4} \sum_{n = 1}^{4} A_{j} {\bar{A}}_{k} E_{j r} {\bar{E}}_{k n}

(41)

得到响应的解析功率谱密度后，利用式（29）和数值积分方法即可得到非平稳均方响应均方值.
2 数值算例
为验证所提方法的正确性，作为演示数值算例，给出随机激励为白噪声、调制白噪声和具有调制修正金井清功率谱的情形.
考虑归一化的1/2阶分数阶系统

D^{2} X (τ) + 2 η D^{1 / 2} X (τ) + X (τ) = \hat{f} (τ)

(42)

式中

τ = ω_{n} t

(43)

X = \frac{x}{σ_{x}}

(44)

Ω = ω / ω_{n}

(45)

\hat{f} (τ) = \frac{f^{*} (τ / ω_{n})}{m σ_{x} ω_{n}^{2}}

(46)

S_{{F \hat{}}^{}} (Ω) = \frac{S_{F} (ω)}{σ_{x}^{2} ω_{n}^{3}}

(47)

σ_{x}^{2} = \int_{- \infty}^{\infty} S_{F} (ω) {|\frac{1}{- m ω^{2} + k + 2 m η ω_{n}^{3 / 2} (i ω)^{1 / 2}}|}^{2} d ω

(48)

2.1 白噪声
考虑激励为白噪声时的情况.归一化后激励功率谱密度由式和式给出，此时系统参数为m=1，k=1.选择η=0.1，截止频率为Ω_u=32π，Δτ=1/32.若未作特殊说明，其他数值算例参数均与上述一致.
图1 白噪声作用下系统响应的均方值
Fig.1 Response mean square value of the fractional system subject to white noise
采用与蒙特卡洛模拟（Monte Carlo Simulation，MCS）结果对比的方式验证建议方法.文章建议方法计算得到的响应均方差与10000条样本的蒙特卡洛模拟得到的结果对比如图1所示.从图中可以得知，无论瞬态还是稳态响应，建议理论方法计算的均方值能很好地符合模拟结果.图2和图3分别给出了由所建议方法计算和广义谐和小波（Generalized Harmonic Wavelet，GHW）估计（10000样本响应）得到的时频域中的响应功率谱密度; 图4则给出了不同时刻下响应的功率谱密度.可以看出，建议方法（PM）能较好地给出功率谱密度在时间起始处的瞬态时频分布，而基于MCS的广义谐和小波估计（HME）则由于边缘效应（End Effect）未能给出合理的瞬态时频分布.此外，由于瞬态响应逐渐耗散且系统阻尼较弱，响应功率谱密度带宽随时间推移逐渐变窄.图5表明了所提方法在不同激励强度作用下的适用性.
图2 白噪声作用下系统响应的功率谱密度
Fig.2 Response PSD of the system subject to white noise. Analytical solution obtained by the proposed method
图3 白噪声作用下由谐和小波估计的响应功率谱密度
Fig.3 Response PSD of the system subjected to white noise. PSD is estimated by harmonic wavelet over 10, 000 sample responses
图4 白噪声作用下不同时刻的响应功率谱密度
Fig.4 Response PSD at different time instants of the system subject to white noise
图5 不同强度白噪声激励下系统响应的均方值
Fig.5 Response mean squared value of the fractional system subject to white noise with different PSD
2.2 调制白噪声
考察激励为调制白噪声的情况.为方便计算且不失一般性，调制函数参数选为γ₁=γ₂=4，v₁=-0.05，v₂=-0.1，如图6所示; 其他参数选取与白噪声情况相同.利用文章所建议方法和MSC所得响应均方值的对比如图7所示.可见，所建议方法能很好地符合MCS结果; 响应峰值到达的时间较调制函数晚.
图6 调制函数
Fig.6 Modulating function
图7 调制白噪声作用下由建议方法得到的系统响应功率谱密度
Fig.7 Response mean square value of the fractional system subject to modulated white noise
图8 调制白噪声激励下响应的均方值
Fig.8 PSD surface of the system subject to modulated white noise. Analytical solution obtained by the proposed method
图9 调制白噪声作用下由谐和小波估计的响应功率谱密度
Fig.9 Response PSD of the system subjected to modulated white noise. PSD is estimated by harmonic wavelet over 10, 000 sample responses
图8和图9分别给出了所建议方法计算和广义谐和小波基于10000条样本响应估计的响应功率谱密度.可见，两种方法得到的结果吻合较好，但边缘效应仍造成了时间开始处响应功率谱密度失实.图10进一步给出了不同时刻处系统响应瞬时功率谱密度对比，其中标志线为MCS结果.由图可见，两种方法得到的结果吻合较好，功率谱密度幅值随时间和频率均呈现先升后降的趋势; 不同时刻处，响应功率谱密度峰值对应的频率位置均处于自振频率附近; 响应功率谱密度带宽随着时间推移而变窄.
图10 调制白噪声作用下不同时刻的响应功率谱密度
Fig.10 Response PSD at different time instants of the system subject to modulated white noise
图11 调制白噪声作用下不同阻尼比的系统响应均方值
Fig.11 Response mean squared value for systems with different damping ratios subject to modulated white noise
研究具分数阶阻尼比不同时，所建议方法在计算响应均方值时的适用性.分别取η=0.05，0.1，0.2，对应的响应均方值如图11所示.由图可见，阻尼比越大，系统响应达到峰值的速度越快.注意到，阻尼越大，系统响应均方值越大，这是由阻尼比不同导致的归一化系统激励大小不同造成的.
2.3 调制修正金井清谱
隔震结构隔震层中采用的橡胶是一种黏弹性材料，它的力学性能可很好地利用分数阶导数模型描述.在地震工程中应用该方法时，需考察激励为调制色噪声的情况.地震工程中常见的金井清（Kanai-Tajimi）谱在低频范围内存在较大误差，因而采用激励功率谱密度为修正金井清谱^[21]的情况，即

S_{F} = \frac{ω_{g}^{2} ω^{2}}{{(ω_{g}^{2} - ω^{2})}^{2} + 4 ζ_{g}^{2} ω_{g}^{2} ω^{2}} S_{0}

(49)

式中:S₀为基岩白噪声的谱密度; ω_g和ζ_g分别是覆土层的特征频率和特征阻尼比.根据鹿磊等^[22]所提参数取法，假设场地为二类二组，设防烈度为7度，取ω_g=15.71rad/s，ζ_g=0.72，S₀=16.05.如式所示的非正归化系统参数为ω_n=15.71rad/s，η=0.1; 调制函数选择的参数为γ₁=γ₂=4，v₁=-0.4，v₂=-0.8.系统模型频率参数的选择直接采用最不利情况，即系统自然频率与场地卓越频率相同的情况.修正金井清谱如图12所示，可见该地震地面运动加速度谱改进了金井清谱零频处幅值不为零的缺点.
图12 修正金井清谱
Fig.12 Modified Kanai-Tajimi spectrum
文章所建议方法计算与MCS估计所得响应均方差对比如图13所示，可见，二者之间几乎完全吻合.图14和图15分别给出了时频域内响应时变功率谱密度的理论和模拟结果，二者在峰值水平、峰值出现时间和频率、上升和下降趋势等方面均定性吻合.此外，图16给出了不同时刻处瞬时功率谱密度的理论和模拟结果对比.与前两例相同，除时间点始处由于小波变换边缘效应吻合不好以外，两种方法给出的结果能在其他时间点吻合较好.此外，瞬时功率谱密度体现出随时间推移而频带变宽的特点.图17给出了阻尼比不同时由所建议方法计算和MCS估计得到的响应均方值对比.可见，阻尼比越大，峰值越小且峰会时间越提前.所建议方法在阻尼比具有不同值时均具体较好的精度.
图13 调制修正金井清谱激励作用下响应的均方值
Fig.13 Response mean square value of the system subject to modulated colored noise with modified KT spectrum
图14 调制修正金井清谱随机激励作用下系统响应的功率谱密度
Fig.14 PSD of response subject to modulated modulated modified Kanai-Tajimi spectrum
图15 调制色噪声（修正金井清谱）作用下由谐和小波估计的响应功率谱密度
Fig.15 Response PSD of the system subjected to modulated colored noise with modified KT spectrum. PSD is estimated by harmonic wavelet over 10, 000 sample responses
图16 调制修正金井清谱作用下不同时刻的响应功率谱密度
Fig.16 Response PSD at different time instants of the system subject to modulated colored noise with modified KT spectrum
图17 调制色噪声（修正金井清谱）作用下不同阻尼比系统响应均方值
Fig.17 Response mean square value for systems with different damping ratios subject to modulated colored noise with modified KT spectrum
3 结论
文章提出了一种计算非平稳随机激励作用下线性分数阶系统非平稳响应功率谱密度和均方值的解析方法.方法的关键之处是利用分数阶动力方程的状态空间特征向量展开法得到1/2分数阶系统脉冲响应函数，并基于Laplace变换计算得到了系统在随机动力系统分别在白噪声、调制白噪声、调制修正金井清谱等不同激励作用下的非平稳响应.考察了不同时刻下响应功率谱密度特征以及阻尼比对随机动力系统响应的影响.与蒙特卡洛模拟的对比证明了文章所建议方法的准确性.
该方法的意义在于，只要知道随机激励的功率谱密度表达式，就能够立即得到分数阶系统响应功率谱密度的精确表达式，通过常规数值积分便可方便地得到系统的非平稳均方响应.因此，其计算高效性可进一步为分数阶动力系统参数优化服务.文章所建议方法关注1/2阶阻尼的线性随机动力系统，对于其他有理数阶导数同样适用; 但此时会由于脉冲响应函数为无穷级数而导致截断误差.该方法有待进一步推广到完全非平稳随机动力激励下单/多自由度非线性和/或任意分类阶动力系统.
参考文献
- [1] SUN H G,YONG Z,BALEANU D,et al.A new collection of real world applications of fractional calculus in science and engineering [J].Communications in Nonlinear Science and Numerical Simulation,2018,64.
- [2] NUTTING P G.A new general law of deformation [J].Journal of the Franklin Institute,1921,191(5):679-685.
- [3] GEMANT A.A method of analyzing experimental results obtained from elasto-viscous bodies [J].Physics,1936,7(8):311-317.
- [4] CAPUTO M.Linear models of dissipation whose Q is almost frequency independent-Ⅱ [J].Geophysical Journal International,1967,13(5):529-539.
- [5] BAGLEY R L,TORVIK P J.On the fractional calculus model of viscoelastic behavior [J].Journal of Rheology,1986,30(1):133-155.
- [6] BAGLEY R L,TORVIK P J.Fractional calculus in the transient analysis of viscoelastically damped structures [J].AIAA Journal,1985,23(6):918-925.
- [7] GAUL L,KLEIN P,KEMPFLE S.Impulse response function of an oscillator with fractional derivative in damping description [J].Mechanics Research Communications,1989,16(5):297-305.
- [8] KOH C G,KELLY J M.Application of fractional derivatives to seismic analysis of base-isolated models [J].Earthquake Engineering & Structural Dynamics,1990,19(2):229-241.
- [9] SHOKOOH A,SUÁREZ L.A comparison of numerical methods applied to a fractional model of damping materials [J].Journal of Vibration and Control,1999,5(3):331-354.
- [10] SUAREZ L E,SHOKOOH A.An eigenvector expansion method for the solution of motion containing fractional derivatives [J].Journal of Applied Mechanics,1997,64(3):629-635.
- [11] WAHI P,CHATTERJEE A.Averaging oscillations with small fractional damping and delayed terms [J].Nonlinear Dynamics,2004,38(1):3-22.
- [12] SPANOS P D,ZELDIN B A.Random vibration of systems with frequency-dependent parameters or fractional derivatives [J].Journal of Engineering Mechanics,1997,123(3):290-292.
- [13] 孔凡,晁盼盼,徐军,等.随机与谐和联合激励下分数阶非线性系统的统计线性化方法[J].振动工程学报,2021,34(4):756-764.KONG F,CAO P P,XU J,et al.Statistical linearization method for fractional order nonlinear systems under random and harmonic excitation [J].Journal of Vibration Engineering,2021,34(4):756-764.(in Chinese)
- [14] PINNOLA F P.Statistical correlation of fractional oscillator response by complex spectral moments and state variable expansion [J].Communications in Nonlinear Science and Numerical Simulation,2016,39:343-359.
- [15] HUANG Z L,JIN X L.Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative [J].Journal of Sound and Vibration,2009,319(3-5):1121-1135.
- [16] 马颜颜,宁丽娟.高斯白噪声激励下分数阶Duffing-Van der Pol系统的稳态响应 [J].动力学与控制学报,2017,15(4):307-313.MA Y Y,NING L J.Steady-state response of fractional-order Duffing-Van der Pol system under Gaussian white noise excitation [J].Journal of Dynamics and Control,2017,15(4):307-313.(in Chinese)
- [17] ARTALE V,NAVARRA G,RICCIARDELLO A,et al.Exact closed-form fractional spectral moments for linear fractional oscillators excited by a white noise [J].ASME Journal of Risk Uncertainty Part B,2017,3(3):030901.
- [18] AGRAWAL O P.An analytical scheme for stochastic dynamic systems containing fractional derivatives [J].Proceedings of the ASME Design Engineering Technical Conferences,Las Vegas,1999:243-249.
- [19] YE K,LI L,TANG J X.Stochastic seismic response of structures with added viscoelastic dampers modeled by fractional derivative [J].Earthquake Engineering and Engineering Vibration,2003,2(1):133-139.
- [20] CAO Q,HU S L J,LI H.Nonstationary response statistics of fractional oscillators to evolutionary stochastic excitation [J].Communications in Nonlinear Science and Numerical Simulation,2021,103:105962.
- [21] 彭凌云,周锡元,李小军.对已有强震地面运动功率谱模型的改进[J].北京工业大学学报,2011,37(3):388-394.PENG L Y,ZHOU X Y,LI X J.Improvement on the existing power spectrum model of strong ground motion [J].Journal of Beijing University of Technology,2011,37(3):388-394.(in Chinese)
- [22] 鹿磊,赵杨.基于中国抗震规范要求的修正金井清谱的参数选择方法[J].建筑结构,2021,51(S1):783-788.LU L,ZHAO Y.Method for parameter selection of modified Kanai-Tajimi spectrum based on Chinese seismic code [J].Building structure,2021,51(S1):783-788.(in Chinese)

基本信息

中图分类号: O321;O324
文献标识码: A
DOI: 10.6052/1672-6553-2023-040
文章编号: 1672-6553-2023-21(4)-023-009

基金信息

国家自然科学基金资助项目(52078399)；
National Natural Science Foundation of China(52078399)；

引用信息

稿件历史

收稿日期: 2022-10-17

参考文献

[1] SUN H G,YONG Z,BALEANU D,et al.A new collection of real world applications of fractional calculus in science and engineering [J].Communications in Nonlinear Science and Numerical Simulation,2018,64.
[2] NUTTING P G.A new general law of deformation [J].Journal of the Franklin Institute,1921,191(5):679-685.
[3] GEMANT A.A method of analyzing experimental results obtained from elasto-viscous bodies [J].Physics,1936,7(8):311-317.
[4] CAPUTO M.Linear models of dissipation whose Q is almost frequency independent-Ⅱ [J].Geophysical Journal International,1967,13(5):529-539.
[5] BAGLEY R L,TORVIK P J.On the fractional calculus model of viscoelastic behavior [J].Journal of Rheology,1986,30(1):133-155.
[6] BAGLEY R L,TORVIK P J.Fractional calculus in the transient analysis of viscoelastically damped structures [J].AIAA Journal,1985,23(6):918-925.
[7] GAUL L,KLEIN P,KEMPFLE S.Impulse response function of an oscillator with fractional derivative in damping description [J].Mechanics Research Communications,1989,16(5):297-305.
[8] KOH C G,KELLY J M.Application of fractional derivatives to seismic analysis of base-isolated models [J].Earthquake Engineering & Structural Dynamics,1990,19(2):229-241.
[9] SHOKOOH A,SUÁREZ L.A comparison of numerical methods applied to a fractional model of damping materials [J].Journal of Vibration and Control,1999,5(3):331-354.
[10] SUAREZ L E,SHOKOOH A.An eigenvector expansion method for the solution of motion containing fractional derivatives [J].Journal of Applied Mechanics,1997,64(3):629-635.
[11] WAHI P,CHATTERJEE A.Averaging oscillations with small fractional damping and delayed terms [J].Nonlinear Dynamics,2004,38(1):3-22.
[12] SPANOS P D,ZELDIN B A.Random vibration of systems with frequency-dependent parameters or fractional derivatives [J].Journal of Engineering Mechanics,1997,123(3):290-292.
[13] 孔凡,晁盼盼,徐军,等.随机与谐和联合激励下分数阶非线性系统的统计线性化方法[J].振动工程学报,2021,34(4):756-764.KONG F,CAO P P,XU J,et al.Statistical linearization method for fractional order nonlinear systems under random and harmonic excitation [J].Journal of Vibration Engineering,2021,34(4):756-764.(in Chinese)
[14] PINNOLA F P.Statistical correlation of fractional oscillator response by complex spectral moments and state variable expansion [J].Communications in Nonlinear Science and Numerical Simulation,2016,39:343-359.
[15] HUANG Z L,JIN X L.Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative [J].Journal of Sound and Vibration,2009,319(3-5):1121-1135.
[16] 马颜颜,宁丽娟.高斯白噪声激励下分数阶Duffing-Van der Pol系统的稳态响应 [J].动力学与控制学报,2017,15(4):307-313.MA Y Y,NING L J.Steady-state response of fractional-order Duffing-Van der Pol system under Gaussian white noise excitation [J].Journal of Dynamics and Control,2017,15(4):307-313.(in Chinese)
[17] ARTALE V,NAVARRA G,RICCIARDELLO A,et al.Exact closed-form fractional spectral moments for linear fractional oscillators excited by a white noise [J].ASME Journal of Risk Uncertainty Part B,2017,3(3):030901.
[18] AGRAWAL O P.An analytical scheme for stochastic dynamic systems containing fractional derivatives [J].Proceedings of the ASME Design Engineering Technical Conferences,Las Vegas,1999:243-249.
[19] YE K,LI L,TANG J X.Stochastic seismic response of structures with added viscoelastic dampers modeled by fractional derivative [J].Earthquake Engineering and Engineering Vibration,2003,2(1):133-139.
[20] CAO Q,HU S L J,LI H.Nonstationary response statistics of fractional oscillators to evolutionary stochastic excitation [J].Communications in Nonlinear Science and Numerical Simulation,2021,103:105962.
[21] 彭凌云,周锡元,李小军.对已有强震地面运动功率谱模型的改进[J].北京工业大学学报,2011,37(3):388-394.PENG L Y,ZHOU X Y,LI X J.Improvement on the existing power spectrum model of strong ground motion [J].Journal of Beijing University of Technology,2011,37(3):388-394.(in Chinese)
[22] 鹿磊,赵杨.基于中国抗震规范要求的修正金井清谱的参数选择方法[J].建筑结构,2021,51(S1):783-788.LU L,ZHAO Y.Method for parameter selection of modified Kanai-Tajimi spectrum based on Chinese seismic code [J].Building structure,2021,51(S1):783-788.(in Chinese)

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1/2分数阶线性随机动力系统的非平稳响应解析解*

Analytical Solution for Non-Stationary Response of 1/2-Order Fractional Linear Stochastic Dynamical Systems*

摘要

Abstract

关键词

Keywords

1 单自由度分数阶系统的响应

(1)

(2)

(3)

1.1 特征向量展开

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

1.2 均方响应求解

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

2 数值算例

(42)

(43)

(44)

(45)

(46)

(47)

(48)

2.1 白噪声

2.2 调制白噪声

2.3 调制修正金井清谱

(49)

3 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

微信公众号二维码

手机版网站二维码

1/2分数阶线性随机动力系统的非平稳响应解析解^*

Analytical Solution for Non-Stationary Response of 1/2-Order Fractional Linear Stochastic Dynamical Systems^*