Applied Mathematical Modelling最新文献_第7页

Control and experiment of bistable nonlinear dynamics in a non-smooth hydraulic rock drill model 非光滑液压凿岩机模型双稳态非线性动力学控制与实验

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2025-12-24 DOI: 10.1016/j.apm.2025.116722

Siyuan Chang , Wei Ma , Min Ye , Joseph Páez Chávez , Yelin Li , Yuchuan Ma , Jiale Zhang

This paper addresses the bistable motion that occurs in hydraulic rock drills during drilling by applying a nonsmooth-system-based bistability control method to the drill's hard-impact system. This method enables the transition from undesired attractors to desired attractors via continuous pathways without affecting the original state of the system. First, a multi-degree-of-freedom dynamic model of the rock drill is established based on the dry friction rock theory, examining both viscous and non-viscous movements. An analysis of unidirectional bifurcation with respect to impact frequency reveals that viscous motion can easily induce chaos in the system. When ω = 3, the system is in the optimal motion state p₁q₁r₁, and the rate of penetration (ROP) reaches its maximum value. In the high-frequency range, the system exhibits poor stability, showing chaotic behavior and bistability. Bidirectional bifurcation analysis at ω = 9, conducted with different initial values, reveals two bistable attractors, p₀q₁r₁ and p₁q₀r₂. The global attraction domain of the p₁q₀r₂ attractor accounts for up to 90.6%. Subsequently, linear and nonlinear control methods are employed to facilitate transitions between these two attractor regions. Additionally, a pseudo-arclength continuation method is adopted for path tracking to determine the parameter domains a and b suitable for the proposed control strategy. Numerical simulations show that controlling parameters such as impact force amplitude, offset, or collision spacing can effectively alter the attractor states. Finally, the validity of the model is verified experimentally. The hydraulic rock drill exhibits bistable motion, which is attributed to unstable stress waves generated by the impact of drill tools on heterogeneous rock during high-frequency drilling. Furthermore, improper matching among the impact force, propulsion force, and buffer force causes jumps in hydraulic pressure frequency at the moment of direction reversal, which consequently alters the initial working parameters of the rock drill.

本文将基于非光滑系统的双稳控制方法应用于水力凿岩机的硬冲击系统，解决了凿岩机在钻井过程中出现的双稳运动问题。该方法能够在不影响系统原始状态的情况下，通过连续路径从不需要的吸引子过渡到需要的吸引子。首先，基于干摩擦岩石理论，建立了凿岩机多自由度动力学模型，同时考虑了凿岩机的粘性运动和非粘性运动；单向分岔与冲击频率的关系分析表明，粘性运动容易引起系统混沌。当ω = 3时，系统处于最佳运动状态p1q1r1，钻速（ROP）达到最大值。在高频范围内，系统稳定性较差，表现为混沌和双稳态。在ω = 9处用不同的初始值进行双向分岔分析，得到两个双稳态吸引子p0q1r1和p1q0r2。p1q0r2吸引子的全局吸引域占比高达90.6%。随后，采用线性和非线性控制方法来促进这两个吸引区域之间的转换。此外，采用伪弧长延拓法进行路径跟踪，确定适合所提控制策略的参数域a和b。数值模拟表明，控制冲击力振幅、偏移量或碰撞间距等参数可以有效地改变吸引子状态。最后，通过实验验证了模型的有效性。水力岩石钻机表现为双稳态运动，这是由于高频钻井过程中钻头对非均质岩石的冲击产生的不稳定应力波所致。此外，由于冲击力、推进力和缓冲力的匹配不当，导致换向时刻液压频率出现跳变，从而改变了凿岩机的初始工作参数。

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引用次数: 0

A double-level dimension-reduction alternating interval inverse method incorporating parameter correlation 结合参数相关性的双层降维交替区间反演方法

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2025-12-31 DOI: 10.1016/j.apm.2025.116729

Jiachang Tang , Yong Lei , Taolin Zhang , Chengji Mi , Qishui Yao , Lixiong Cao

A double-level dimension-reduction alternating interval inverse method incorporating parameter correlation is proposed to solve the uncertain nonlinear inverse problem. This method is employed to address the issue of concurrent output uncertainty and model uncertainty, as well as correlation among model parameters. First, an uncertain inverse problem is modeled where both the output and the model exhibit uncertainty, and the model parameters are correlated. Second, a multidimensional parallel interval model is utilized to characterize the domain of uncertainty in correlation parameters, and an affine coordinate transformation is introduced to transform the uncertainty domain of correlated parameters into a standardized parameter domain. Third, a double-level dimension-reduction alternating interval inverse method is proposed, where the first level of dimension-reduction converts the uncertain function response into the sum of the input uncertain function response and the model's uncertain function response, and the second level dimension-reduction utilizes interval dimension-reduction analysis to construct a deterministic optimization model. By employing outer optimization and inner interval computation, the alternating solution for the interval midpoint and radius is achieved. Finally, the effectiveness and efficiency of this method are demonstrated through three examples.

针对不确定非线性逆问题，提出了一种结合参数相关性的双层降维交替区间逆方法。该方法解决了并发输出的不确定性和模型的不确定性以及模型参数之间的相关性问题。首先，建立了一个不确定反问题，其中输出和模型都具有不确定性，并且模型参数是相关的。其次，利用多维并行区间模型对相关参数的不确定域进行表征，并引入仿射坐标变换将相关参数的不确定域转化为标准化参数域；第三，提出了一种双层降维交替区间逆方法，其中第一级降维将不确定函数响应转换为输入不确定函数响应与模型不确定函数响应之和，第二级降维利用区间降维分析构建确定性优化模型。通过外部优化和内部区间计算，实现了区间中点和半径的交替求解。最后通过三个算例验证了该方法的有效性和高效性。

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引用次数: 0

Nonlinear dynamics of an ice plate overlaying deep water under supercritical compression 超临界压缩下覆盖在深水上的冰板的非线性动力学

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2025-12-24 DOI: 10.1016/j.apm.2025.116714

Y.A. Stepanyants , N.M. Zubarev

We investigate a supercritical regime of a compressed ice plate floating on infinitely deep water. Our derived governing equations describe wave amplitude dynamics when compression exceeds a critical threshold in 2D and 3D cases. Results show that small deviations in the ice plate initially grow exponentially, leading to either complete structural failure (at high supercriticality) or oscillations around equilibrium states. We provide quantitative estimates of wave amplitudes, oscillation periods in stable regimes, and instability growth rates using realistic ice plate parameters. This research offers valuable insights for ice-working researchers and engineers regarding hazardous compression conditions and the relationship between compression supercriticality and resulting ice dynamics.

研究了漂浮在无限深水上的压缩冰板的超临界状态。我们推导的控制方程描述了在二维和三维情况下压缩超过临界阈值时的振幅动态。结果表明，冰板中的小偏差最初呈指数增长，导致结构完全破坏（在高超临界下）或在平衡状态周围振荡。我们提供了定量估计的波动幅度，振荡周期在稳定的制度，和不稳定增长率使用现实的冰板参数。这项研究为冰工研究人员和工程师提供了关于危险压缩条件和压缩超临界与冰动力学之间关系的宝贵见解。

引用次数: 0

Analysis of rectangular stiffened plates using a finite difference-embedded neural network approach 矩形加筋板的有限差分嵌入神经网络分析

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2025-12-23 DOI: 10.1016/j.apm.2025.116721

Zhong-Min Huang , Lin-Xin Peng

This paper presents a finite difference–embedded neural network method for analyzing the mechanical responses of stiffened plates under bending and free vibration, based on the Kirchhoff plate and Euler–Bernoulli beam theories. Displacement transformation equations are first derived from the continuity conditions at the plate–stiffener interface. The plate’s displacement field is then approximated using an artificial neural network, with the domain discretized by points that simultaneously serve as both integration and finite difference points. The coordinates of these points are taken as network inputs. High-order derivatives required for strain energy evaluation are directly obtained from the network outputs through convolution operations formulated on the finite difference principle. Incorporating the displacement transformation equations, the total potential energy of the plate–stiffener system is expressed as a loss function and minimized via the Adam optimizer during network training. Numerical analyses of stiffened plates under various boundary conditions and stiffener heights, validated through comparisons with finite element results and published data, demonstrate the accuracy and reliability of the proposed method.

基于Kirchhoff板理论和Euler-Bernoulli梁理论，提出了一种有限差分嵌入神经网络方法，用于分析加筋板在弯曲和自由振动下的力学响应。首先从板-加劲界面的连续性条件出发，推导了位移变换方程。然后用人工神经网络逼近板的位移场，并用同时作为积分点和有限差分点的点对区域进行离散。这些点的坐标作为网络输入。应变能评估所需的高阶导数是通过有限差分原理的卷积运算直接从网络输出中获得的。结合位移变换方程，将板-加劲系统的总势能表示为损失函数，并在网络训练中通过Adam优化器最小化。对不同边界条件和加劲板高度下的加筋板进行了数值分析，并与有限元结果和已发表的数据进行了比较，验证了所提方法的准确性和可靠性。

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引用次数: 0

Emergence of cooperation and commitment in optional Prisoner’s Dilemma 选择性囚徒困境中合作与承诺的出现

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2025-11-13 DOI: 10.1016/j.apm.2025.116603

Zhao Song, The Anh Han

Commitment is a well-established mechanism for fostering cooperation in human society and multi-agent systems, while most research on commitment-based cooperation assumes compulsory participation. This overlooks many real-world scenarios where individuals are free to abstain from an interaction. We address this gap by proposing a two-stage evolutionary game model that integrates commitment with the optional Prisoner’s Dilemma. In our model, players first decide whether to commit and then choose to cooperate, defect, or exit. We find that optional participation boosts commitment acceptance but fails to foster cooperation, leading instead to widespread exit behaviour. To address this, we compare two institutional incentives: a strict rule rewarding only committed cooperators, and a flexible rule rewarding any committed non-defector. Results show that the strict rule is better for promoting cooperation, as the flexible rule creates a loophole for opportunistic exits. However, the flexible rule can yield higher social welfare when exit options are highly beneficial. Our study highlights the need for well-designed incentives in voluntary settings to effectively promote both cooperation and social welfare.

承诺是人类社会和多主体系统中促进合作的一种完善机制，而大多数基于承诺的合作研究都假设了强制性参与。这忽略了许多现实世界的场景，在这些场景中，个人可以自由地避免互动。我们通过提出一个两阶段进化博弈模型来解决这一问题，该模型将承诺与可选的囚徒困境相结合。在我们的模型中，参与者首先决定是否承诺，然后选择合作、叛变或退出。我们发现，选择性参与提高了承诺接受度，但未能促进合作，反而导致了广泛的退出行为。为了解决这个问题，我们比较了两种制度激励：严格的规则只奖励承诺的合作者，而灵活的规则奖励任何承诺的非叛逃者。结果表明，严格的规则更有利于促进合作，而灵活的规则则为机会主义退出创造了漏洞。然而，当退出选择非常有利时，灵活规则可以产生更高的社会福利。我们的研究强调，需要在自愿环境中设计良好的激励措施，以有效促进合作和社会福利。

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引用次数: 0

Impact of the WHO's 90-70-90 Strategy on HPV-related cervical cancer control: A mathematical model evaluation in China 世界卫生组织90-70-90策略对hpv相关宫颈癌控制的影响：中国的数学模型评估

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2026-01-01 DOI: 10.1016/j.apm.2025.116735

Hua Liu , Chunya Liu , Yumei Wei , Qibin Zhang , Jingyan Ma

In August 2020, the World Health Assembly approved the Global Strategy to eliminate cervical cancer, marking the first time that numerous countries committed to eliminating a form of cancer. China introduced the HPV vaccine in 2016 and has made significant advancements in both prevention and treatment strategies. However, due to the relatively late introduction of the vaccine, the burden of cervical cancer in China continues to rise. In light of this, we develop a compartmental model to assess the impact of the WHO's 90–70–90 strategy, along with adult catch-up vaccination, on the control of HPV-induced cervical cancer in China. We analyze the basic properties of the model and provide proofs of the local and global asymptotic stability of the equilibrium points. Additionally, a sensitivity analysis is performed, and we use the MCMC algorithm to fit the number of new cervical cancer cases and cumulative deaths in China from 1990 to 2021. The estimated basic reproduction number before and after the introduction of the HPV vaccine in China is 1.2133 (95% CI: 1.1593–1.2673) and 0.8317 (95% CI: 0.8042–0.8592), respectively. The sensitivity analysis reveals that screening, as a non-pharmaceutical intervention, plays a crucial role in controlling the spread of the disease. We apply the 90–70–90 strategy to predict the future number of new cervical cancer cases in China. The results indicate that prioritizing the 70–90 target combination is the most cost-effective approach and can achieve the goal of zero new cervical cancer cases by 2063 (within the range of 2052–2066). Finally, an optimal control model is developed to explore the best implementation strategies for HPV vaccination and screening under various plausible scenarios.

2020年8月，世界卫生大会批准了《消除宫颈癌全球战略》，标志着众多国家首次承诺消除一种癌症。中国于2016年推出了HPV疫苗，并在预防和治疗战略方面取得了重大进展。然而，由于疫苗引进相对较晚，中国宫颈癌的负担继续上升。鉴于此，我们开发了一个分区模型来评估世界卫生组织的90-70-90战略以及成人补种疫苗对中国hpv诱导宫颈癌控制的影响。我们分析了模型的基本性质，并给出了平衡点的局部和全局渐近稳定性的证明。此外，我们还进行了敏感性分析，并使用MCMC算法拟合1990年至2021年中国宫颈癌新发病例数和累积死亡人数。中国引入HPV疫苗前后的估计基本繁殖数分别为1.2133 （95% CI: 1.1593-1.2673）和0.8317 （95% CI: 0.8042-0.8592）。敏感性分析表明，筛查作为一种非药物干预手段，在控制疾病传播方面发挥着至关重要的作用。我们应用90-70-90策略预测未来中国宫颈癌新发病例数。结果表明，优先考虑70-90目标组合是最具成本效益的方法，可实现到2063年（2052-2066年范围内）零宫颈癌新发病例的目标。最后，建立了一个最优控制模型，以探索在各种可能的情况下HPV疫苗接种和筛查的最佳实施策略。

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引用次数: 0

Dynamic modelling and vortex dynamics in elastic-joint caudal fin propulsor for efficient swimming 高效游泳弹性关节尾鳍推进器的动力学建模和涡动力学

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2026-01-05 DOI: 10.1016/j.apm.2025.116737

Yiming Lu , Haicheng Zhang , Xubin Sun , Daolin Xu

Marine animals achieve remarkable swimming performance through precisely coordinated interactions among different body segments. Current bio-inspired swimmers generally fail to account for the distinct mechanical functions of various anatomical structures and often overlook the critical nonlinear fluid-structure interaction mechanisms. To bridge this gap, an elastic-joint rigid fin model is developed that conceptualizes propulsion as three synergistic phases: active muscular actuation, elastic peduncle energy transmission, and vortex-generating caudal fin motion. This model uniquely enables the quantitative dissection of individual component contributions to overall performance. To address the fluid-structure interaction problem in swimming, a Nonlinear Vortex Sheet Method (NVSM) is proposed. The method seamlessly integrates a vortex sheet model based flow reconstruction, a coupled system modeling for structural motion and vortex tracking, and the Broyden method based synchronized solving, achieving robust and efficient performance. Rigorously validated against computational fluid dynamics, the NVSM accurately captures the key physics of dynamic response and vortex shedding with significantly reduced computational cost.

Comparative analyses demonstrate a threefold enhancement in peak propulsion efficiency at optimal frequencies over rigid counterparts. A detailed examination of the fluid-structure interaction reveals the propulsion mechanisms. These are identified as hydrodynamic phase synchronization, suppression of energy-dissipating vortex structures, and superior fluid energy utilization efficiency. Furthermore, this study establishes a quantitative mapping between structural stiffness and key performance metrics, providing actionable design guidelines for next-generation biomimetic propulsion systems.

海洋动物通过不同身体部位之间精确协调的相互作用，获得卓越的游泳性能。目前的仿生游泳者普遍未能考虑到各种解剖结构的独特力学功能，往往忽视了关键的非线性流固相互作用机制。为了弥补这一差距，研究人员开发了一种弹性关节刚性鳍模型，该模型将推进概念定义为三个协同阶段：主动肌肉驱动、弹性肢节能量传输和尾鳍运动产生涡流。该模型独特地能够定量分析单个组件对整体性能的贡献。针对游泳过程中流固耦合问题，提出了一种非线性涡片法（NVSM）。该方法将基于旋涡片模型的流动重建、结构运动与旋涡跟踪的耦合系统建模和基于Broyden方法的同步求解无缝集成，实现了鲁棒性和高效率。NVSM经过了严格的计算流体动力学验证，能够准确地捕捉到动态响应和涡流脱落的关键物理特性，大大降低了计算成本。对比分析表明，在最佳频率下，峰值推进效率比刚性频率提高了三倍。对流固相互作用的详细研究揭示了推进机制。这些特性被确定为流体动力相位同步，抑制能量耗散涡结构，以及优越的流体能量利用效率。此外，该研究建立了结构刚度和关键性能指标之间的定量映射，为下一代仿生推进系统提供了可操作的设计指南。

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引用次数: 0

Neural networks for solving elliptic problems: Analysis and adaptivities 求解椭圆问题的神经网络：分析与适应性

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2025-12-23 DOI: 10.1016/j.apm.2025.116716

Jiao Li , Haocheng Mei , Qinfeng Ou , Hongqiao Wang , Jinyong Ying

The application of deep learning methods in solving partial differential equations is increasingly recognized for its significance. In this paper, we consider physics-informed neural networks to solve elliptic problems with or without interface conditions. By leveraging a gradient-involved loss function and employing a shallow neural network, a unified convergence analysis is presented and verified, showing the approximation has a convergence rate depending on the width of shallow networks and the number of sampling points. Meanwhile, to further improve solutions’ accuracies, two different types of adaptive methods are designed and analyzed. In particular, for the adaptive method in enlarging the hypothesis space, where radial basis functions serve as the enriched functions, new improved error estimates are presented and numerical tests on different types of elliptic problems (including Darcy problems) are used to demonstrate its effectiveness and its advantage over four other different adaptive methods.

深度学习方法在求解偏微分方程中的应用越来越受到人们的重视。在本文中，我们考虑物理通知神经网络来解决有或没有界面条件的椭圆问题。通过利用梯度损失函数和浅层神经网络，提出并验证了统一的收敛分析，表明近似具有取决于浅层网络宽度和采样点数量的收敛率。同时，为了进一步提高解的精度，设计并分析了两种不同类型的自适应方法。特别是对于以径向基函数作为丰富函数的扩大假设空间的自适应方法，提出了新的改进误差估计方法，并对不同类型的椭圆问题（包括Darcy问题）进行了数值试验，证明了其有效性及其相对于其他四种不同自适应方法的优势。

引用次数: 0

The evolution of fairness induced by marginal loss in ultimatum games within the networked population 网络群体中最后通牒博弈中边际损失诱导的公平演化

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2026-01-14 DOI: 10.1016/j.apm.2026.116753

Haojie Xu , Changwei Huang , Chengyi Xia

Fairness stands as a foundational principle in social and economic interactions, whereas in classical ultimatum game models it is generally grounded in linear, lossless payoffs that neglect the real-world costs of resource transfers. In this study, we extend the spatial ultimatum game by introducing a nonlinear loss function with two interpretable parameters: the loss intensity coefficient and the marginal loss sensitivity, which together capture diminishing marginal losses. Comprehensive numerical simulations reveal a pronounced optimization pattern: peak fairness appears when the loss intensity is large and the marginal loss sensitivity is moderate, a regime that simultaneously elevates average proposals and acceptance thresholds while damping long-term fluctuations. Mechanistically, the marginal loss effect imposes evolutionary pressure on unfair strategies, raising acceptance thresholds and compelling proposers to offer more equitable splits. Notably, the location of this high-fairness region remains robust across diverse network topologies, underscoring the generality of the marginal loss mechanism in fostering fairness. These findings contribute insights into the role of loss effects in promoting fairness, with potential applications in economic modeling, multi-agent systems, and the design of platforms that optimize social resource allocation.

公平是社会和经济互动的基本原则，而在经典的最后通牒博弈模型中，它通常建立在线性、无损失的回报基础上，忽略了资源转移的现实成本。本文通过引入具有两个可解释参数的非线性损失函数来扩展空间最后通牒博弈：损失强度系数和边际损失灵敏度，它们共同捕获边际损失递减。综合数值模拟揭示了一个明显的优化模式：峰值公平性出现在损失强度大而边际损失敏感性适中的情况下，这种制度同时提高了平均建议和接受阈值，同时抑制了长期波动。从机制上讲，边际损失效应对不公平策略施加了进化压力，提高了接受门槛，迫使提议者提供更公平的分配。值得注意的是，这种高公平区域的位置在不同的网络拓扑结构中保持稳健，强调了边际损失机制在促进公平方面的普遍性。这些发现有助于深入了解损失效应在促进公平中的作用，并在经济建模、多智能体系统和优化社会资源配置的平台设计中具有潜在的应用前景。

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引用次数: 0

Thermoelastic damping of thin circular plate with arbitrary boundary considering three-dimensional heat conduction 考虑三维热传导的任意边界圆薄板热弹性阻尼

IF 4.4 2区工程技术 Q1 ENGINEERING, MULTIDISCIPLINARY

Applied Mathematical Modelling

Pub Date : 2026-07-01 Epub Date: 2025-12-24 DOI: 10.1016/j.apm.2025.116723

Yukun Qin, Zhijing Wu, Shurui Wen, Fengming Li

The development of accurate thermoelastic damping (TED) models constitutes a critical prerequisite for predicting the quality factor of microelectromechanical resonators. This study aims to develop a universal and accurate TED model to overcome the limitations of existing approaches in predicting the TED for non-axisymmetric modes. Based on the three-dimensional (3D) heat conduction theory, a 3D TED model for the thin circular plate subject to arbitrary boundary constraints is proposed using the thermal energy method. The equation of motion is derived according to the Hamilton’s principle and Rayleigh-Ritz method. The correctness is validated by the agreement of the model’s results with the reference data and those from the finite element simulations, respectively. The effects of different boundary spring types, boundary conditions, geometric parameters, and modal orders on the TED are investigated. It is found that the stiffness of linear boundary springs has a greater impact on the TED of circular plates than that of rotational boundary springs. As the modal order increases, the peak values of TED progressively move toward the direction of small thickness-to-radius ratios. Furthermore, the RMS error of the 3D TED model is <1.84%. Compared with the finite element method, the model demonstrates higher computational efficiency, reducing computation time by >97%.

建立准确的热弹性阻尼（TED）模型是预测微机电谐振器质量因子的重要前提。本研究旨在建立一个通用的、准确的TED模型，以克服现有方法在预测非轴对称模TED时的局限性。基于三维热传导理论，利用热能法建立了受任意边界约束的薄板三维TED模型。根据哈密顿原理和瑞利-里兹方法推导了运动方程。通过与参考数据和有限元模拟结果的比较，验证了模型的正确性。研究了不同边界弹簧类型、边界条件、几何参数和模态阶数对边界弹簧的影响。研究发现，线性边界弹簧的刚度对圆板TED的影响大于旋转边界弹簧的刚度。随着模态阶数的增加，TED的峰值逐渐向小厚度半径比方向移动。三维TED模型的均方根误差为1.84%。与有限元方法相比，该模型具有更高的计算效率，计算时间缩短了97%。

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引用次数: 0