【新文速递】2024年1月6日固体力学SCI期刊最新文章

今日更新：International Journal of Solids and Structures 5 篇，Journal of the Mechanics and Physics of Solids 1 篇，International Journal of Plasticity 1 篇，Thin-Walled Structures 6 篇

International Journal of Solids and Structures

Analytical solution of multilayered structures with Green’s function

Shouming Shang, Jinxin Xiao, Tianlin Jiang, Wen-Hua Zhang

doi:10.1016/j.ijsolstr.2023.112636

具有格林函数的多层结构的解析解

It is well known that for solving the problem of thin multilayered structures in complex working conditions, the calculation accuracy and efficiency of the finite element method are significantly inferior to those of the analytical solution. However, the prediction of road life requires precise and detailed mechanical performance analysis. To satisfy this requirement, this study first utilized the three-dimensional Green’s function adapted to transversely isotropic multilayered structures with arbitrary thicknesses based on the mirroring image principle. Then, the analytical solution of multilayered pavements subjected to multiple loadings without complex integral or inverse transformations was derived. Furthermore, an equivalent sliding interface model was proposed to simulate the imperfect interface or thin interlayer of multilayered pavements. Numerical examples verified the suitability and accuracy of the analytical solution thorough the comparison with the other theory and experimental results. Moreover, the key factors affecting the performance of pavements, such as the anisotropy and thickness of each layer, loading direction, and pavement friction coefficient, were investigated to provide reliable numerical and theoretical bases for the optimal design of pavement structures.

众所周知，在解决复杂工况下的薄多层结构问题时，有限元法的计算精度和效率明显低于解析法。然而，道路寿命的预测需要精确详细的力学性能分析。为了满足这一要求，本研究首先利用了基于镜像原理的三维格林函数，该函数适用于任意厚度的横向各向同性多层结构。然后，在不进行复杂积分或反变换的情况下，推导出了多层路面在多重荷载作用下的解析解。此外，还提出了一个等效滑动界面模型，用于模拟多层路面的不完美界面或薄夹层。通过与其他理论和实验结果的对比，数值实例验证了分析解决方案的适用性和准确性。此外，还研究了影响路面性能的关键因素，如各层的各向异性和厚度、加载方向和路面摩擦系数等，为路面结构的优化设计提供了可靠的数值和理论依据。

Approximate contact force model for viscoelastic materials based on generalized Maxwell model

Suhang Ding, Yiqun Hu, Bin Jian, Yuhang Zhang, Lei Su, Re Xia, Guoming Hu

doi:10.1016/j.ijsolstr.2024.112645

基于广义麦克斯韦模型的粘弹性材料近似接触力模型

This paper proposes a novel approximate model for estimating the normal contact force between a rigid sphere and a viscoelastic half-space based on the generalized Maxwell model. The relaxation function derived from the generalized Maxwell model is applied to the solution to the elastic contact problem based on the elastic-viscoelastic correspondence to obtain an integral expression of pressure for viscoelastic contact. Using the approximate method on the basis of Taylor series, the integral expression is translated into an explicit function of displacement, velocity, time and material properties. A normal contact force model is established by integrating the pressure over contact area. A relaxation test of Nafion® XL is conducted to capture the viscoelastic parameters, and these parameters are employed to simulate the contact behaviors using the finite element method (FEM) to determine the scale factor in the contact force model. In order to verify the validity of the approximate model, the outcomes obtained from the model are compared with those generated by the FEM. A parametric study is conducted to analyze the effects of variations in parameters on normal contact behaviors. The results indicate that the established model is able to predict the contact responses of the viscoelastic materials with complex mechanical characteristics.

本文基于广义麦克斯韦模型，提出了一种估算刚性球体与粘弹性半空间之间法向接触力的新型近似模型。根据弹性-粘弹性对应关系，将广义麦克斯韦模型导出的松弛函数应用于弹性接触问题的解，从而得到粘弹性接触压力的积分表达式。利用基于泰勒级数的近似方法，将积分表达式转化为位移、速度、时间和材料属性的显式函数。通过对接触面积上的压力进行积分，建立了法向接触力模型。对 Nafion® XL 进行松弛测试以获取粘弹性参数，并利用这些参数通过有限元法（FEM）模拟接触行为，以确定接触力模型中的比例因子。为了验证近似模型的有效性，将模型得出的结果与有限元法得出的结果进行了比较。还进行了参数研究，分析参数变化对正常接触行为的影响。结果表明，所建立的模型能够预测具有复杂机械特性的粘弹性材料的接触响应。

A fully coupled electromagnetic-thermo-mechanical model for honeycomb microwave absorbing structure

Han Yan, Bin Fu, Yimeng Shan, Yungang Sun, Xuefeng Yao

doi:10.1016/j.ijsolstr.2024.112646

蜂窝微波吸收结构的全耦合电磁-热力学模型

Honeycomb microwave absorbing structures typically operate under multiple physical fields, resulting in complex electromagnetic-thermo-mechanical coupling phenomena. In the present work, a multi-physics macroscopic model of the absorbing performance for honeycomb microwave absorption structures is proposed by incorporating the concept of load factor and the transmission line theory. Then, the interactive relationship between electromagnetic-thermo-mechanical coupling fields and the absorbing performance of the structure were thoroughly explored. Furthermore, the evolution law of the load factors under both single-field and multi-field coupling is analyzed and verified. The main originality of the proposed model is its capability to predict the evolution of absorbing performance under electromagnetic-thermo-mechanical coupling loading, which provides valuable theoretical guidance for designing advanced honeycomb microwave absorbing structures.

蜂窝微波吸收结构通常在多个物理场下工作，从而产生复杂的电磁-热力学耦合现象。本研究结合负载因子概念和传输线理论，提出了蜂窝微波吸收结构吸收性能的多物理场宏观模型。然后，深入探讨了电磁-热力学耦合场与结构吸波性能之间的交互关系。此外，还分析并验证了单场耦合和多场耦合下负载因子的演化规律。该模型的主要独创性在于能够预测电磁-热力学耦合载荷下的吸波性能演变，为设计先进的蜂窝微波吸波结构提供了宝贵的理论指导。

Magnetoactive asymmetric mechanical metamaterial for tunable elastic cloaking

Quan Zhang, Gengkai Hu, Stephan Rudykh

doi:10.1016/j.ijsolstr.2024.112648

用于可调弹性隐形的磁活性非对称机械超材料

We propose a magnetic field-induced asymmetric mechanical metamaterial for tunable transformation-based elastic cloaking. The metamaterial is designed by integrating hard-Magnetic Active Elastomers (hMAEs) and combining zero-stress collapse modes to produce a unique asymmetric behavior controlled by an external magnetic field. The relationship bridging the microstructure and the desired cloaking performance is presented. This magneto-metamaterial design is applied to achieve tunable static elastic cloaking. The theoretical predictions, together with the numerical tests under various static loads, demonstrate encouraging cloaking performance. The study also highlights the impact of magneto-mechanical coupling and offers the first remotely-controllable hMAE-based cloaking solution, with promising potential in various applications including stress shielding and stealth technologies.

我们提出了一种磁场诱导的非对称机械超材料，用于基于可调变换的弹性隐形。这种超材料是通过整合硬磁活性弹性体（hMAEs）设计的，并结合了零应力塌缩模式，从而产生了一种由外部磁场控制的独特非对称行为。本文介绍了微结构与所需隐形性能之间的关系。这种磁性超材料设计可用于实现可调静态弹性隐形。理论预测和各种静态负载下的数值测试都证明了令人鼓舞的隐形性能。该研究还强调了磁-机械耦合的影响，并提供了首个可远程控制的基于 hMAE 的隐形解决方案，在包括应力屏蔽和隐形技术在内的各种应用中具有广阔的发展前景。

Non-contact reconstitution of the traction distribution using incomplete deformation measurements: Methodology and Experimental Validation

Yue Mei, Dongmei Zhao, Rongyao Kang, Xinyu Wang, Bo Wang, Dawei Song, Li Dong, Hao Jiang, Stephane Avril

doi:10.1016/j.ijsolstr.2024.112650

利用不完全变形测量对牵引力分布进行非接触式重组：方法与实验验证

Traction distribution is often unknown in solid mechanics. However, identifying traction distribution with both high accuracy and spatial resolution can sometimes be critical. In this paper, we consider the frequent situation where incomplete deformation measurements, such as partial surface measurements obtained by digital image correlation, are available. For such situations, we propose to achieve the complete identification of boundary tractions, using an optimization based inverse method. The feasibility and robustness of the proposed approach are firstly assessed with numerical simulations. We show that complete traction distributions can be identified with high accuracy even when there is noise added to the deformation measurements. More specifically, the relative error was less than 3% even with 10% noise. Moreover, the effect of noise depends significantly on the level of incompleteness of the deformation data. We show for instance that when the measurement area is 8.3% of the solid, the relative error can be 5 times larger than when the measurement area is 33.3%. However, even with 8.3% deformation measurement area of the solid, the relative error of the mapped traction distribution can be less than 6%. Besides, the accuracy of the mapped traction distribution can be further improved if the region of the deformation measurement is closer to the traction boundary. We eventually validate the proposed inverse approach using incomplete experimental datasets consisting in tensile tests carried out on two elastic samples. We obtain very promising results, with relative errors below 4% using the measured displacements of only 3 points, which is promising to open potential applications in different fields of engineering.

在固体力学中，牵引力分布往往是未知的。然而，以高精度和高空间分辨率识别牵引力分布有时至关重要。在本文中，我们考虑了经常出现的不完整变形测量的情况，例如通过数字图像相关性获得的部分表面测量。针对这种情况，我们建议使用基于优化的逆方法来实现边界牵引的完整识别。我们首先通过数值模拟评估了所提方法的可行性和稳健性。我们发现，即使在变形测量中添加了噪声，也能高精度地识别出完整的牵引力分布。更具体地说，即使噪声为 10%，相对误差也小于 3%。此外，噪声的影响很大程度上取决于变形数据的不完整程度。例如，我们发现当测量区域为实体的 8.3% 时，相对误差可能比测量区域为 33.3% 时大 5 倍。然而，即使变形测量面积为实体的 8.3%，绘制的牵引力分布图的相对误差也可小于 6%。此外，如果变形测量区域更靠近牵引边界，则绘制牵引力分布图的精度还能进一步提高。最终，我们利用对两个弹性样本进行拉伸测试的不完整实验数据集验证了所提出的逆方法。我们获得了非常有前景的结果，使用仅 3 个点的测量位移，相对误差低于 4%，这为在不同工程领域的潜在应用带来了希望。

Journal of the Mechanics and Physics of Solids

Thermal fluctuations (eventually) unfold nanoscale origami

Matthew Grasinger, Pradeep Sharma

doi:10.1016/j.jmps.2023.105527

热波动（最终）展开纳米级折纸

Origami is a scale invariant paradigm for morphing robotics, deployable structures (e.g. satellites, disaster relief shelters, medical stents), and metamaterials with tunable thermal, mechanical, or electromagnetic properties. There has been a resurgence of interest in using origami principles–along with 2D materials or DNA–to design a wide array of nanoscale devices. In this work, we take cognizance of the fact that small-scale devices are vulnerable to entropic thermal fluctuations and thus a foundational question underlying small-scale origami pertains to its stability, i.e. the origami structure’s propensity to “unfold” due to thermal fluctuations and the rate at which the unfolding will ensue. To properly understand the behavior of these origami-based nanodevices, we must simultaneously consider the geometric mechanics of origami along with the interplay between thermal fluctuations, entropic repulsive forces, van der Waals attraction, and other molecular-scale phenomena. In this work, to elucidate the rich behavior underpinning the evolution of an origami device at the nanoscale, we develop a minimal statistical mechanics model of folded nanoscale sheets. We use the model to investigate (1) the thermodynamic multistability of nanoscale origami structures and (2) the rate at which thermal fluctuations drive its unfolding–that is, its temporal stability. We identify, for the first time, an entropic torque that is a critical driving force for the unfolding process. Both the thermodynamic multistability and temporal stability have a nontrivial dependence on the origami’s bending stiffness, the radii of curvature of its creases, the ambient temperature, its thickness, and its interfacial energy (between folded layers). Specifically, for graphene, we show that there is a critical side length below which it can no longer be folded with stability; similarly, there exists a critical crease diameter, membrane thickness (e.g. for multilayer graphene), and temperature above which a crease cannot be stably folded. To investigate the rate of thermally driven unfolding, we extend Kramers’ escape rate theory to cases where the minima of the energy well occurs at a boundary. Rates of unfolding are found to span from effectively zero to instantaneous, and there is a clear interplay between temperature, geometry, and mechanical properties on the unfolding rate.

折纸是变形机器人、可部署结构（如卫星、救灾避难所、医疗支架）以及具有可调热、机械或电磁特性的超材料的规模不变范例。人们对利用折纸原理以及二维材料或 DNA 设计各种纳米级设备的兴趣再次升温。在这项研究中，我们认识到小尺度器件容易受到熵热波动的影响，因此小尺度折纸的一个基本问题是其稳定性，即折纸结构由于热波动而 "展开 "的倾向以及展开的速度。要正确理解这些基于折纸的纳米器件的行为，我们必须同时考虑折纸的几何力学以及热波动、熵排斥力、范德华吸引力和其他分子尺度现象之间的相互作用。在这项工作中，为了阐明折纸器件在纳米尺度上演化的丰富行为，我们建立了一个折叠纳米级薄片的最小统计力学模型。我们利用该模型研究了：(1) 纳米级折纸结构的热力学多稳定性；(2) 热波动驱动其展开的速率，即其时间稳定性。我们首次发现了一种熵力矩，它是展开过程的关键驱动力。热力学多稳定性和时间稳定性都与折纸的弯曲刚度、折痕曲率半径、环境温度、厚度和界面能（折叠层之间）有着非同小可的关系。具体来说，对于石墨烯，我们发现存在一个临界边长，低于该边长的折纸无法再稳定折叠；同样，存在一个临界折痕直径、膜厚（例如多层石墨烯）和温度，高于该温度的折痕无法稳定折叠。为了研究热驱动解折的速率，我们将克拉默逃逸率理论扩展到能量井的最小值出现在边界的情况。研究发现，展开率从实际为零到瞬时展开，温度、几何形状和机械特性对展开率的影响是显而易见的。

International Journal of Plasticity

A dislocation-based damage-coupled constitutive model for single crystal superalloy: unveiling the effect of secondary orientation on creep life of circular hole

Zixu Guo, Ziyuan Song, Haohao Liu, Daijun Hu, Dawei Huang, Xiaojun Yan, Wentao Yan

doi:10.1016/j.ijplas.2024.103874

基于位错的单晶超合金损伤耦合构造模型：揭示二次取向对圆孔蠕变寿命的影响

Circular holes in single crystal (SX) turbine blades are susceptible to creep failure at high temperatures, due to the stress concentrations near the holes. In this study, a dislocation-based damage-coupled constitutive model for SX superalloys is developed, validated, and applied to unveil the effect of secondary orientation on the creep life of circular holes. The structure-induced geometrically necessary dislocations (GND) density resulting from the macroscopic strain gradient is incorporated. The dissipated energy is employed as the indicator to predict crack nucleation. Initially, the specimens without a hole are employed to calibrate the parameters and validate the model in the absence of structure-induced GND. Creep experiments incorporating the digital image correlation technique are conducted on the specimens with a hole under different secondary orientations, which aims to validate the simulated strain distribution, crack nucleation location, and creep life under the influence of structure-induced GND. The results indicate that the incorporation of structure-induced GND contributes to mitigating the conservatism in the creep life prediction of circular holes. Because, the GND can enhance dislocation hardening near the hole, extending the predicted life and improving the accuracy of life prediction. Meanwhile, both experiments and simulations indicate that the secondary orientation influences the crack nucleation locations at the hole edge. The variation in the creep life at different secondary orientations is attributed to the changes in both the maximum value and uniformity of resolved shear stresses distributed across slip systems. Finally, the model is applied to simulate the creep behavior of a circular hole under various secondary orientations, stresses, and temperatures. An empirical model is developed to rapidly evaluate the creep life of a circular hole in the SX superalloy, which facilitates the engineering application of the proposed model.

由于单晶（SX）涡轮叶片上的圆孔附近存在应力集中，因此在高温下很容易发生蠕变破坏。本研究针对 SX 超合金开发、验证并应用了基于位错的损伤耦合构成模型，以揭示二次取向对圆孔蠕变寿命的影响。宏观应变梯度导致的结构诱发的几何必要位错（GND）密度被纳入其中。耗散能量被用作预测裂纹成核的指标。起初，采用无孔试样来校准参数，并在无结构诱导 GND 的情况下验证模型。结合数字图像相关技术，在有孔的试样上进行了不同次要方向的蠕变实验，旨在验证结构诱导 GND 影响下的模拟应变分布、裂纹成核位置和蠕变寿命。结果表明，加入结构诱导 GND 有助于减轻圆孔蠕变寿命预测的保守性。因为 GND 可以增强孔附近的位错硬化，延长预测寿命，提高寿命预测的准确性。同时，实验和模拟都表明，二次取向会影响孔边缘的裂纹成核位置。不同次要方向上蠕变寿命的变化归因于分布在滑移系统上的解析剪应力的最大值和均匀性的变化。最后，应用该模型模拟了圆孔在不同次生方向、应力和温度下的蠕变行为。建立的经验模型可快速评估 SX 超级合金圆孔的蠕变寿命，从而促进了所提模型的工程应用。

Thin-Walled Structures

Post-fire behaviour and resistances of square recycled aggregate concrete-filled stainless steel tube stub columns

Ziyi Wang, Yukai Zhong, Ke Jiang, Meini Su, Ou Zhao

doi:10.1016/j.tws.2024.111564

方形再生骨料混凝土填充不锈钢管桩柱的火灾后行为和抗性

Experimental and numerical studies on the cross-section compressive behaviour and residual resistances of square recycled aggregate concrete-filled stainless steel tube (RACFSST) stub columns after exposure to fire are reported in this paper. An experimental programme was firstly carried out on twelve stub column specimens with three recycled coarse aggregate replacement ratios (0%, 35% and 70%) after exposure to the ISO-834 standard fire for 0 min (i.e. at ambient temperature), 15 min, 30 min and 45 min. The test results, including load–end shortening curves, failure loads and failure modes, were presented, with the initial compressive stiffness and confinement effect analysed. The experimental programme was followed by a numerical modelling programme, where thermal and mechanical finite element models were developed and validated against the test results and afterwards used to conduct parametric studies to generate additional numerical data over a wide range of cross-section dimensions. Based on the test and numerical data, the relevant design rules for square natural aggregate concrete-filled carbon steel tube stub columns at ambient temperature, as specified in the European code, Australia/New Zealand standard and American specification, were evaluated, using post-fire material properties, for their applicability to square RACFSST stub columns after exposure to fire. The evaluation results generally revealed that the European code and Australian/New Zealand standard led to a good level of design accuracy, while the American specification resulted in slightly conservative post-fire cross-section compression resistance predictions.

本文报告了对方形再生骨料混凝土填充不锈钢管（RACFSST）短柱在遭受火灾后的截面抗压行为和残余阻力的实验和数值研究。首先，对 12 个具有三种再生粗骨料替代率（0%、35% 和 70%）的短柱试样进行了试验，这些试样在 ISO-834 标准火灾中暴露了 0 分钟（即在环境温度下）、15 分钟、30 分钟和 45 分钟。试验结果包括荷载端缩短曲线、破坏荷载和破坏模式，并分析了初始压缩刚度和约束效应。实验计划之后是数值建模计划，根据测试结果开发和验证了热和机械有限元模型，之后用于开展参数研究，以生成更多横截面尺寸范围的数值数据。根据测试和数值数据，对欧洲规范、澳大利亚/新西兰标准和美国规范中规定的环境温度下方形天然骨料混凝土填充碳钢管桩柱的相关设计规则进行了评估，使用了火灾后材料特性，以确定其是否适用于火灾后的方形 RACFSST 桩柱。评估结果普遍表明，欧洲规范和澳大利亚/新西兰标准的设计精确度较高，而美国规范对火灾后截面抗压性的预测略显保守。

Compression performance of 3D-printed thermoplastic auxetic structures

Pan He, Siwen Wang, Miaomiao Zhang, Lin Sang, Liyong Tong, Wenbin Hou

doi:10.1016/j.tws.2024.111558

三维打印热塑性辅助结构的压缩性能

The influence of lattice structural design on the mechanical property has been abundantly investigated, however, the effect of material on the load-carrying performance and energy absorption was seldom studied. In the current work, the auxetic structures fabricated using four materials including polyamide (PA), carbon fiber (CF) reinforced PA (PA/CF), polyether ether ketone (PEEK) and PEEK/CF were 3D-printed, and the compression performance and deformation behavior were compared and explored. The results showed that the deformation mode of auxetic structure were similar, which was not dependent on the material type. However, the compressive strength and modulus of the auxetic structure showed strong dependence on the material properties. Besides, PA and PA/CF auxetic structures displayed better rebound behavior after compression, which was perhaps attributed to the good resilience of PA resin. Therefore, mathematical equations were utilized to predict the compressive property based on the material parameters, showing a good agreement. On the other hand, the simulation of compression process was established and conducted, which confirmed the deformation process as experimental presented. The rebound behavior was further predicted by a loading-releasing cycle with different compression strains. Accordingly, the compression parameters showed great relations with material parameters but little effect on the deformation behavior.

格子结构设计对力学性能的影响已得到大量研究，但材料对承载性能和能量吸收的影响却鲜有研究。本研究采用聚酰胺（PA）、碳纤维（CF）增强 PA（PA/CF）、聚醚醚酮（PEEK）和 PEEK/CF 四种材料制作了 3D 打印辅助结构，并对其压缩性能和变形行为进行了比较和探讨。结果表明，辅助结构的变形模式相似，与材料类型无关。然而，辅助结构的抗压强度和模量与材料特性有很大关系。此外，PA 和 PA/CF 辅助结构在压缩后显示出更好的回弹特性，这可能与 PA 树脂的良好回弹性有关。因此，根据材料参数利用数学方程预测了压缩性能，结果显示两者吻合良好。另一方面，建立并进行了压缩过程模拟，证实了与实验结果一致的变形过程。通过不同压缩应变的加载-释放循环，进一步预测了回弹行为。因此，压缩参数与材料参数关系密切，但对变形行为影响不大。

Real-time hybrid model test to replicate high-rise building resonant vibration under wind loads

Zhou Huimeng, Shao Xiaoyun, Zhang Jianwen, Yao Hongcan, Liu Yanhui, Tan pin, Chen Yangyang, Xu Li, Zhang Yin, Gong Wei

doi:10.1016/j.tws.2024.111559

在风荷载作用下复 制高层建筑共振的实时混合模型试验

In May 2021, the 72-story Shenzhen Saige building experienced abnormal vibration that was strongly felt on many floors and triggered social panic. The tower was therefore closed for more than 2 months resulting in huge economic losses. A preliminary study consisting of field resonant excitation tests and a series of numerical simulations were carried out. It was concluded that the wind loads provoked the higher modes of the building vibration. Specifically, the vibration of the mast, which is located at the building top, induced this abnormal vibration. To validate this conclusion, real-time hybrid model (RTHM) tests was developed to reproduce this building vibration incident. This paper presents the details of the validation RTHM tests including testing design and result discussions. Structural vibration parameters obtained from the field tests were used in the numerical substructure building model, and the experimental substructures were the two scaled down mast models (the cantilever beam section of the masts). During RTHM test, the restoring force of the experimental substructure due to real wind loads induced by an air fan was measured and used in the numerical simulation to compute interface motions. A shaking table was then used to impose the interface motion back to the bottom of the mast model to reproduce the abnormal vibration incident. The demonstrated ability of the developed RTHM testing method to reproduce the resonant phenomenon of the wind-induced tower vibration provides an alternative experimental method to study vibration responses of high-rise buildings in future.

2021 年 5 月，72 层高的深圳赛格大厦发生异常振动，许多楼层都有强烈震感，引发了社会恐慌。赛格大厦因此关闭了两个多月，造成了巨大的经济损失。初步研究包括现场共振激振试验和一系列数值模拟。研究得出的结论是，风荷载引发了建筑物的高频振动模式。具体来说，位于建筑物顶部的桅杆的振动引发了这种异常振动。为了验证这一结论，开发了实时混合模型（RTHM）测试来重现这一建筑振动事件。本文介绍了验证 RTHM 试验的细节，包括试验设计和结果讨论。从现场测试中获得的结构振动参数被用于下层建筑的数值模型，而实验下层建筑则是两个按比例缩小的桅杆模型（桅杆的悬臂梁部分）。在 RTHM 试验期间，测量了实验下部结构在风扇引起的实际风荷载作用下的恢复力，并将其用于数值模拟，以计算界面运动。然后使用振动台将界面运动施加回桅杆模型底部，以重现异常振动事件。所开发的 RTHM 测试方法能够再现风引起的塔楼振动的共振现象，为今后研究高层建筑的振动响应提供了另一种实验方法。