Sensitivity analysis of the power spectrum density function for non-viscously damped systems subject to stationary stochastic excitations
SHI Junlei1,2,3，DING Zhe1,2,3，ZHANG Lei1,2,3，ZHANG Yan1,2,3
1.Key Laboratory of Metallurgical Equipment and Control Technology of Ministry of Education, Wuhan University of Science and Technology, Wuhan 430081, China;
2.Hubei Key Laboratory of Mechanical Transmission and Manufacturing Engineering, Wuhan University of Science and Technology, Wuhan 430081, China;
3.Precision Manufacturing Institute, Wuhan University of Science and Technology, Wuhan 430081, China
Abstract：Calculating the first-order derivatives of Power Spectrum Density (PSD) function with respect to design variables is a prerequisite for random responses when gradient-based optimization algorithms are adopted. Unlike viscous damping model, which assumes that the damping force is proportional to the velocity, the damping force of non-viscous damping model depend on the past history of motion via convolution integrals over some suitable kernel functions. Therefore, the non-viscous damping model is more accurate to modelling the energy dissipation behaviors of viscoelastic materials. This paper considers the design sensitivity analysis of PSD function for non-viscously damped systems subjected stationary stochastic excitations. The governing equations of the non-viscously damped system under stationary random excitations are transformed into a deterministic harmonic response problem based on Pseudo-Excitation method (PEM). The expressions of the first-order derivatives of the PSD function is derived by direct differentiate method. Three numerical methods, namely complex-mode based first- and second-order approximation method (PEM-FAM, PEM-SAM) and real-mode based iterative method (PEM-IM), are proposed to calculate the sensitivity of the PSD function. The computational accuracy and efficiency of the three methods are compared by two numerical methods. The results indicate that the PEM-IM would be the best candidate to compute the sensitivities of the PSD function of non-viscously damped systems, especially for large-scale problems.
史俊磊1,2,3，丁喆1,2,3，张磊1,2,3，张严1,2,3. 平稳随机激励下非黏滞阻尼系统功率谱密度函数的灵敏度分析[J]. 振动与冲击, 2023, 42(8): 20-27.
SHI Junlei1,2,3，DING Zhe1,2,3，ZHANG Lei1,2,3，ZHANG Yan1,2,3. Sensitivity analysis of the power spectrum density function for non-viscously damped systems subject to stationary stochastic excitations. JOURNAL OF VIBRATION AND SHOCK, 2023, 42(8): 20-27.
[1] 张亚峰，殷红，彭珍瑞，等. 基于位移频响函数灵敏度分析的模型修正[J]. 振动与冲击，2021，40(17)：63-69.
ZHANG Ya-feng，YIN Hong，PENG Zhen-rui，et al. Model updating based on sensitivity analysis of displacement FRF[J]. Journal of Vibration and Shock，2021，40(17)：63-69.
[2] 周俊贤，吕中荣，汪利. 基于灵敏度分析和不同数据融合的损伤识别方法[J]. 振动与冲击，2019，38(18)：236-241.
ZHOU Jun-xian，LV Zhong-rong，WANG Li. Damage identification based on the sensitivity analysis with hybrid data[J]. Journal of Vibration and Shock，2019，38(18)：236-241.
[3] Gomez F，Spencer B F，Carrion J. Topology optimization of buildings subjected to stochastic wind loads[J]. Probabilistic Engineering Mechanics，2021，64：111111.
[4] Su C，Li B，Chen T，et al. Stochastic optimal design of nonlinear viscous dampers for large-scale structures subjected to non-stationary seismic excitations based on dimension- reduced explicit method[J]. Engineering Structures，2018，175：217-230.
[5] 孙攀旭，杨红. 基于等效黏性阻尼模型的非比例阻尼体系反应谱CCQC法[J]. 工程力学，2021，38(10)：160-172.
SUN Pan-xu，YANG Hong. A response spectrum CCQC method for non-proportionally damped systems based on equivalent viscous damping model[J]. Engineering Mechanics，2021，38(10)：160-172.
[6] Mitseas L P，Beer M. Modal decomposition method for response spectrum based analysis of nonlinear and non-classically damped systems[J]. Mechanical Systems and Signal Processing，2019，131：469-485.
[7] 项盼，赵岩，林家浩. 复合随机振动分析的混合PC-PEM方法[J]. 振动与冲击，2015，34(04)：35-39.
XIANG Pan，ZHAO Yan，Lin Jia-hao. Hybrid PC-PEM for complex random vibration analysis [J]. Journal of Vibration and Shock，2015，34(04)：35-39.
[8] Zhu Z，Wang L，Costa P A，et al. An efficient approach for prediction of subway train-induced ground vibrations considering random track une venness[J]. Journal of Sound and Vibration，2019，455：359-379.
[9] Zhang C，Cheng L，Qiu J，et al. Damage detection based on sparse virtual element boundary measurement using metal-core piezoelectric fiber[J]. Structural Health Monitoring-an International Journal，2018，17(1)：15-23.
[10] Zhao X，Wu B，Lai S，et al. A PEM-based topology optimization for structures subjected to stationary random excitations[J]. Engineering Structures，2021，229：111613.
[11] 郑泽栋，陈伟欢，吕中荣，等. 基于功率谱灵敏度分析的结构损伤识别方法[J]. 振动与冲击，2014，33(16)：76-79.
ZHENG Ze-dong，CHEN Wei-huan，LV Zhong-rong，et al. Structural damage identification based on sensitivity analysis of power spectral density[J]. Journal of Vibration and Shock，2014，33(16)：76-79.
[12] Johnson E A，Wojtkiewicz S F. Efficient sensitivity analysis of structures with local modifications. II：transfer functions and spectral densities[J]. Journal of Engineering Mechanics，2014，140(9)：04014068.
[13] Liu Q. Sensitivity and Hessian matrix analysis of power spectral density functions for uniformly modulated evolutionary random seismic responses[J]. Finite Elements in Analysis and Design，2012，48(1)：1370-1375.
[14] Yan W，Wan H，Ren W. Analytical local and global sensitivity of power spectrum density functions for structures subject to stochastic excitation[J]. Computers & Structures，2017，182：325-336.
[15] Ding Z，Shi J，Huang Q，et al. Sensitivity and Hessian matrix analysis of power spectrum density function for non-classically damped systems subject to stationary stochastic excitations[J]. Mechanical Systems and Signal Processing，2021，161：107895.
[16] 沈洪宇，段忠东，李惠. 基于Volterra积分方程Taylor展开的卷积型非粘滞阻尼系统响应的时程分析方法[J]. 应用力学学报，2018，35(02)：261-266.
SHEN Hong-yu，DUAN Zhong-dong，LI Hui. A time-history analysis algorithm based on Taylor expansion of Volterra integral equation for convolutional type non-viscously damped system[J]. Chinese Journal of Applied Mechanics，2018，35(02)：261-266.
[17] 钱向东，李震东，李晨. 混凝土重力坝非黏滞阻尼地震响应分析[J]. 河海大学学报(自然科学版)，2021，49(04)：342-348.
QIAN Xiang-dong，LI Zhen-dong，LI Chen. Non-viscous damped seismic response analysis of concrete gravity dam[J]. Journal of Hohai University (Natural Sciences)，2021，49(04)：342-348.
[18] 李创第，陈明杰，葛新广. 基于Clough-Penzien谱激励的指数型非黏滞阻尼结构随机地震动响应简明封闭解[J]. 应用数学和力学，2021，42(03)：282-291.
LI Chuang-di，CHEN Ming-jie，GE Xin-guang. A simple closed response solution to random ground motion for exponential non-viscous-damping structures based on the Clough-Penzien spectrum excitation[J]. Applied Mathematics and Mechanics，2021，42(03)：282-291.
[19] 史俊磊，张磊，丁喆. 非黏滞阻尼系统频响函数灵敏度分析的改进计算方法[J]. 振动与冲击，2021，40(21)：1-9.
SHI Jun-lei，ZHANG Lei，DING Zhe. An improved method for sensitivity analysis of FRF of a system with non-viscous damping[J]. Journal of Vibration and Shock，2021，40(21)：1-9.
[20] Li L，Hu Y，Wang X. Harmonic response calculation of viscoelastic structures using classical normal modes：An iterative method[J]. Computers & Structures，2014，133：39-50.
[21] Liu Q. Stationary random response of non-viscously damped polymer matrix composite structure systems[J]. Composite Structures，2018，202：1-8.
[22] JIANG D，XU Y，CAO Z，et al. Temperature-dependent thermo-elastic parameter identification for composites using thermal modal data[J]. Advances in Mechanical Engineering，2019，11(10)：1-14.
[23] Wang S，Tian Q，Hu H，et al. Sensitivity analysis of deployable flexible space structures with a large number of design parameters[J]. Nonlinear Dynamics，2021，105(3)：2055-2079.
[24] LI L, HU Y, WANG X, et al. A hybrid expansion method for frequency response functions of non-proportionally damped systems[J]. Mechanical Systems and Signal Processing, 2014, 42(1): 31-41.
[25] Liu H，Zhang W，Gao T. A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations[J]. Structural and Multidisciplinary Optimization，2015，51(6)：1321-1333.
[26] 刘伟，刘大维，陈焕明，等. 基于联合仿真的半主动悬架车辆行驶平顺性研究[J]. 农业机械学报，2009，40(06)：16-22.
LIU Wei，LIU Da-wei，CHEN Huan-ming，et al. Ride comfort of MR-damper semi-active suspension systems based on co-simulations[J]. Transactions of the Chinese Society for Agricultural Machinery，2009，40(06)：16-22.
[27] 桂水荣，陈水生，万水. 路面谱函数对车桥耦合随机振动敏感性分析[J]. 振动.测试与诊断，2018，38(02)：353-359.
GUI Shui-rong，CHEN Shui-sheng，WAN Shui. Sensitivity analysis of vehicle-bridge coupling random vibration based on road roughness spectral function [J]. Journal of Vibration, Measurement & Diagnosis，2018，38(02)：353-359.
