Applied Mathematics and Mechanics was founded in 1980 by CHIEN Wei-zang, a celebrated Chinese scientist in mechanics and mathematics. The current editor in chief is Professor LU Tianjian from Nanjing University of Aeronautics and Astronautics. The Journal was a quarterly in the beginning, a bimonthly the next year, and then a monthly ever since 1985. It carries original research papers on mechanics, mathematical methods in mechanics and interdisciplinary mechanics based on artificial intelligence mathematics. It also strengthens attention to mechanical issues in interdisciplinary fields such as mechanics and information networks, system control, life sciences, ecological sciences, new energy, and new materials, making due contributions to promoting the development of new productive forces.
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2026, Volume 47, Issue 4
publish date：April 01 2026
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2026, 47(4): 391-403.
doi: 10.21656/1000-0887.470101
Abstract:
Presently, scientific research is becoming increasingly active, interdisciplinary integration is deepening, and artificial intelligence is rapidly entering knowledge production, reconsidering what truly constitutes highquality research has become a matter of immediate practical relevance. This paper argues that scientific research should uphold four essential criteria: significance, necessity, originality, and feasibility. Significance asks whether a study is worth pursuing; necessity asks why it must be undertaken and why now; originality concerns what substantive advance it makes; and feasibility asks whether its outcomes can truly stand and be translated under scientific, technical, engineering, economic, and practical constraints. These four criteria are not isolated labels, but form a logical chain from problem formulation to the establishment of credible results. Several tendencies in current research practice deserve particular caution: replacing judgment on scientific problems with journal labels and impact factors, reversing research logic by chasing international trends and toptier journals, packaging marginal variations as originality, and presenting ideas that lack object constraints, manufacturability, cost awareness, durability assessment, and application compatibility as “frontier breakthroughs.” In the age of artificial intelligence, big data, and automated research, results can be generated faster, appearances of completeness can be produced more easily, and ideas can be packaged more persuasively; reiterating these four criteria is therefore not a conservative reaction against new tools, but a basic safeguard against the deviation of scientific judgment by formal abundance. For Applied Mathematics and Mechanics, reaffirming the four criteria also means clarifying the journal’s orientation: mechanics should remain the point of application, engineering goals the central guide, and applied mathematics a source of essential methodological support.
Presently, scientific research is becoming increasingly active, interdisciplinary integration is deepening, and artificial intelligence is rapidly entering knowledge production, reconsidering what truly constitutes highquality research has become a matter of immediate practical relevance. This paper argues that scientific research should uphold four essential criteria: significance, necessity, originality, and feasibility. Significance asks whether a study is worth pursuing; necessity asks why it must be undertaken and why now; originality concerns what substantive advance it makes; and feasibility asks whether its outcomes can truly stand and be translated under scientific, technical, engineering, economic, and practical constraints. These four criteria are not isolated labels, but form a logical chain from problem formulation to the establishment of credible results. Several tendencies in current research practice deserve particular caution: replacing judgment on scientific problems with journal labels and impact factors, reversing research logic by chasing international trends and toptier journals, packaging marginal variations as originality, and presenting ideas that lack object constraints, manufacturability, cost awareness, durability assessment, and application compatibility as “frontier breakthroughs.” In the age of artificial intelligence, big data, and automated research, results can be generated faster, appearances of completeness can be produced more easily, and ideas can be packaged more persuasively; reiterating these four criteria is therefore not a conservative reaction against new tools, but a basic safeguard against the deviation of scientific judgment by formal abundance. For Applied Mathematics and Mechanics, reaffirming the four criteria also means clarifying the journal’s orientation: mechanics should remain the point of application, engineering goals the central guide, and applied mathematics a source of essential methodological support.
2026, 47(4): 404-414.
doi: 10.21656/1000-0887.460151
Abstract:
In cold regions, the formation of ice covers over water bodies during winter is a common phenomenon. The continuous growth of ice covers significantly impacts human activities, making it practically important to understand and predict ice growth behavior for the prevention of ice-related hazards. Ice cover growth is influenced by multiple factors, and the underlying mechanisms have not yet been fully elucidated. To investigate the complexity of ice cover growth, a finite element computational model was established, and the equivalent heat capacity method was employed to numerically simulate the ice growth process. The accuracy of the proposed model and method was validated through comparison with experimental data. A comparative analysis was conducted between numerical results considering and neglecting natural convection. Furthermore, both the proposed method and the freezing degree-day method were applied to estimate the ice thickness at a specific cross section of the Songhua River. The root mean square errors of the 2 methods were provided, further confirming the effectiveness of the equivalent heat capacity method in real river environments. The results demonstrate that, the established computational model and the numerical approach can effectively represent physical processes such as heat transfer and fluid motion, and handle water-ice phase transition problems. This study provides an effective method for simulating ice cover growth under multi-physics coupling effects.
In cold regions, the formation of ice covers over water bodies during winter is a common phenomenon. The continuous growth of ice covers significantly impacts human activities, making it practically important to understand and predict ice growth behavior for the prevention of ice-related hazards. Ice cover growth is influenced by multiple factors, and the underlying mechanisms have not yet been fully elucidated. To investigate the complexity of ice cover growth, a finite element computational model was established, and the equivalent heat capacity method was employed to numerically simulate the ice growth process. The accuracy of the proposed model and method was validated through comparison with experimental data. A comparative analysis was conducted between numerical results considering and neglecting natural convection. Furthermore, both the proposed method and the freezing degree-day method were applied to estimate the ice thickness at a specific cross section of the Songhua River. The root mean square errors of the 2 methods were provided, further confirming the effectiveness of the equivalent heat capacity method in real river environments. The results demonstrate that, the established computational model and the numerical approach can effectively represent physical processes such as heat transfer and fluid motion, and handle water-ice phase transition problems. This study provides an effective method for simulating ice cover growth under multi-physics coupling effects.
2026, 47(4): 415-425.
doi: 10.21656/1000-0887.460048
Abstract:
The nonlocal continuum theory effectively integrates microscopic structural features and macroscopic mechanical responses by introducing a cross-scale correlation mechanism, providing a new theoretical paradigm for solving multi-scale mechanical problems. However, due to the incorporation of long-range interaction integral terms in the constitutive relationship, its control equations exhibit the characteristics of high-order partial integro-differential equations, and significantly increase the computational complexity. A novel finite element-differential quadrature coupling algorithm (FE-DQ) was established and applied to the study of the free vibration characteristics of graphene equivalent nanoplatelets. Based on the parameterized calculation, the influential mechanisms of key variables such as characteristic sizes and non-local parameters on the non-local effects of buckling loads were revealed through systematic parametric studies. The results show that, the scale effect of buckling loads exhibits a significant nonlinear attenuation characteristic and is positively correlated with the size of the non-local parameter. With the gradual increase of the structural size, the non-local effect on the free vibration frequency will gradually weaken; conversely, with the continuous increase of the non-local parameter value or the rise of the vibration modal order, the non-local effect on the free vibration frequency will intensify significantly. The research provides a reference for the study of structural dynamic characteristics on a nanoscale in related fields.
The nonlocal continuum theory effectively integrates microscopic structural features and macroscopic mechanical responses by introducing a cross-scale correlation mechanism, providing a new theoretical paradigm for solving multi-scale mechanical problems. However, due to the incorporation of long-range interaction integral terms in the constitutive relationship, its control equations exhibit the characteristics of high-order partial integro-differential equations, and significantly increase the computational complexity. A novel finite element-differential quadrature coupling algorithm (FE-DQ) was established and applied to the study of the free vibration characteristics of graphene equivalent nanoplatelets. Based on the parameterized calculation, the influential mechanisms of key variables such as characteristic sizes and non-local parameters on the non-local effects of buckling loads were revealed through systematic parametric studies. The results show that, the scale effect of buckling loads exhibits a significant nonlinear attenuation characteristic and is positively correlated with the size of the non-local parameter. With the gradual increase of the structural size, the non-local effect on the free vibration frequency will gradually weaken; conversely, with the continuous increase of the non-local parameter value or the rise of the vibration modal order, the non-local effect on the free vibration frequency will intensify significantly. The research provides a reference for the study of structural dynamic characteristics on a nanoscale in related fields.
2026, 47(4): 426-439.
doi: 10.21656/1000-0887.450303
Abstract:
For leader-less multi-agent systems, the problem of distributed formation optimization in predefined time under unknown disturbances was studied, and the global cost function composed of local strongly convex functions for all agents was minimized. A class of formation optimization algorithms based on the sliding mode control was proposed to realize the formation control of multi-agent systems within the predefined time. The algorithm was divided into 3 parts: firstly, the integrated sliding mode control strategy was used to guide each agent to approach the sliding mode surface in the predefined time, and the external interference was effectively suppressed; then, the design protocol control was employed to guide each agent state to the minimum point of its local cost function; finally, the leaderless formation was realized for all agents to reach the minimum point of the global cost function. The algorithm does not require agents to share the gradients and Hessian matrix information of neighbors, thus saving the information exchange cost, and can deal with highly nonlinear multi-valued strongly convex cost functions. Several examples of numerical experiments demonstrate the effectiveness and reliability of the design control protocol algorithm.
For leader-less multi-agent systems, the problem of distributed formation optimization in predefined time under unknown disturbances was studied, and the global cost function composed of local strongly convex functions for all agents was minimized. A class of formation optimization algorithms based on the sliding mode control was proposed to realize the formation control of multi-agent systems within the predefined time. The algorithm was divided into 3 parts: firstly, the integrated sliding mode control strategy was used to guide each agent to approach the sliding mode surface in the predefined time, and the external interference was effectively suppressed; then, the design protocol control was employed to guide each agent state to the minimum point of its local cost function; finally, the leaderless formation was realized for all agents to reach the minimum point of the global cost function. The algorithm does not require agents to share the gradients and Hessian matrix information of neighbors, thus saving the information exchange cost, and can deal with highly nonlinear multi-valued strongly convex cost functions. Several examples of numerical experiments demonstrate the effectiveness and reliability of the design control protocol algorithm.
2026, 47(4): 440-453.
doi: 10.21656/1000-0887.460023
Abstract:
The clearance in the cage pocket of cylindrical roller bearings influences the kinematic characteristics such as slip and collision between the rollers and the cage, as well as the overall vibration of the bearing. the limitation of traditional dynamic models solely considering viscous drag effects of lubricant on rollers for cylindrical roller bearings, was addressed. Instead, the lubricant was described as a timevarying friction coefficient related to the contact force between the rollers and raceways, along with flow resistances and resistance torques on the cage. Additionally, nonlinear spring and damping elements were employed to simulate the collision contacts between the rollers and the cage, highlighting the impacts of the pocket clearance. The accuracy of the proposed model was validated, and the influences of the cage pocket clearance on the kinematic characteristics of the rollercage speed, slip, and collision, as well as the vibration characteristics of the bearing, were investigated. Simulation results indicate that, as the cage pocket clearance increases within the range of 0.1 mm to 0.7 mm, the cage slip rate rises, leading to more severe speed fluctuations and compromising cage stability. Simultaneously, the roller’s spin slip rate decreases, which reduces the collision frequency between the roller and the front and rear ends of the cage pocket, while the collision force increases. Furthermore, the overall vibration of the bearing intensifies with the increasing pocket clearance. The study can provide valuable insights for the design and failure analysis of cylindrical roller bearings.
The clearance in the cage pocket of cylindrical roller bearings influences the kinematic characteristics such as slip and collision between the rollers and the cage, as well as the overall vibration of the bearing. the limitation of traditional dynamic models solely considering viscous drag effects of lubricant on rollers for cylindrical roller bearings, was addressed. Instead, the lubricant was described as a timevarying friction coefficient related to the contact force between the rollers and raceways, along with flow resistances and resistance torques on the cage. Additionally, nonlinear spring and damping elements were employed to simulate the collision contacts between the rollers and the cage, highlighting the impacts of the pocket clearance. The accuracy of the proposed model was validated, and the influences of the cage pocket clearance on the kinematic characteristics of the rollercage speed, slip, and collision, as well as the vibration characteristics of the bearing, were investigated. Simulation results indicate that, as the cage pocket clearance increases within the range of 0.1 mm to 0.7 mm, the cage slip rate rises, leading to more severe speed fluctuations and compromising cage stability. Simultaneously, the roller’s spin slip rate decreases, which reduces the collision frequency between the roller and the front and rear ends of the cage pocket, while the collision force increases. Furthermore, the overall vibration of the bearing intensifies with the increasing pocket clearance. The study can provide valuable insights for the design and failure analysis of cylindrical roller bearings.
2026, 47(4): 454-467.
doi: 10.21656/1000-0887.460004
Abstract:
Viscoelastic materials are extensively utilized in civil engineering, aviation, and other domains owing to their superior vibration damping characteristics. The nonlinear stiffness Zener model was employed to replace the energy transfer elements in the conventional nonlinear energy trap, thereby making a novel viscoelastic energy trap device. Furthermore, the bifurcation behavior of the model under simple harmonic excitation was investigated. Initially, the nonlinear dynamic control equation for the coupled main structureenergy trap system was formulated based on the nonlinear stiffness Zener model. Subsequently, the slowvarying system’s equation under the 1∶1 resonance condition was analytical derived with the complex variable averaging method. On this basis, the effects of key parameters on the bifurcation behavior of the system's viscoelastic energy trap under slowchanging conditions were elucidated. Through integration of the numerical simulation approach and in view of the vibration reduction efficiency and energy transfer efficiency of the main structure as evaluation indicators, the vibration suppression effectiveness of the viscoelastic energy trap across different bifurcation regions was further explored. The results indicate that, the newly developed viscoelastic energy trap can effectively modulate saddlenode bifurcations and Hopf bifurcations through parameter adjustments, thereby significantly enhancing the system’s vibration damping efficiency and energy transfer efficiency while effectively restraining the displacement response of the main structure. This research provides a robust theoretical foundation for the engineering design and parameter optimization of the innovative viscoelastic energy trap.
Viscoelastic materials are extensively utilized in civil engineering, aviation, and other domains owing to their superior vibration damping characteristics. The nonlinear stiffness Zener model was employed to replace the energy transfer elements in the conventional nonlinear energy trap, thereby making a novel viscoelastic energy trap device. Furthermore, the bifurcation behavior of the model under simple harmonic excitation was investigated. Initially, the nonlinear dynamic control equation for the coupled main structureenergy trap system was formulated based on the nonlinear stiffness Zener model. Subsequently, the slowvarying system’s equation under the 1∶1 resonance condition was analytical derived with the complex variable averaging method. On this basis, the effects of key parameters on the bifurcation behavior of the system's viscoelastic energy trap under slowchanging conditions were elucidated. Through integration of the numerical simulation approach and in view of the vibration reduction efficiency and energy transfer efficiency of the main structure as evaluation indicators, the vibration suppression effectiveness of the viscoelastic energy trap across different bifurcation regions was further explored. The results indicate that, the newly developed viscoelastic energy trap can effectively modulate saddlenode bifurcations and Hopf bifurcations through parameter adjustments, thereby significantly enhancing the system’s vibration damping efficiency and energy transfer efficiency while effectively restraining the displacement response of the main structure. This research provides a robust theoretical foundation for the engineering design and parameter optimization of the innovative viscoelastic energy trap.
2026, 47(4): 468-486.
doi: 10.21656/1000-0887.450309
Abstract:
The stability of stochastic reaction-diffusion systems under boundary sampling control was discussed. With the fully accessible system state, a boundary sampled-data controller was proposed, and a piecewise discontinuous Lyapunov function related to the sampling interval was constructed. For stochastic reaction-diffusion systems, sufficient conditions for mean-square exponential stability, both in nominal and robust settings, were obtained with the Wirtinger inequality for spatial integrals and isomorphic discrete transformations, in the form of matrix inequalities. With the not fully accessible system state, an observer-based boundary sampled-data control strategy was proposed, and results for the mean-square exponential stability and the robust mean-square exponential stability of the system were obtained, respectively. Finally, the feasibility of the proposed methods was demonstrated through 3 numerical examples.
The stability of stochastic reaction-diffusion systems under boundary sampling control was discussed. With the fully accessible system state, a boundary sampled-data controller was proposed, and a piecewise discontinuous Lyapunov function related to the sampling interval was constructed. For stochastic reaction-diffusion systems, sufficient conditions for mean-square exponential stability, both in nominal and robust settings, were obtained with the Wirtinger inequality for spatial integrals and isomorphic discrete transformations, in the form of matrix inequalities. With the not fully accessible system state, an observer-based boundary sampled-data control strategy was proposed, and results for the mean-square exponential stability and the robust mean-square exponential stability of the system were obtained, respectively. Finally, the feasibility of the proposed methods was demonstrated through 3 numerical examples.
2026, 47(4): 487-495.
doi: 10.21656/1000-0887.460071
Abstract:
In response to the instability problem caused by the coupling of strong nonlinear factors such as time-varying mesh stiffness and backlash in high-speed heavy-duty gear systems, an external periodic excitation control strategy was introduced, to establish and numerically solve a dynamic model for single-degree-of-freedom gear transmission systems. With the cell mapping method, the effects of control parameters on the erosion and bifurcation transition process of the system safety-attraction basin, as well as the evolution law of the attraction domain proportion p, were quantitatively analyzed. Based on the Floquet multiplier analysis, a quantitative mapping relationship between the doubling coefficient, the excitation amplitude, and the bifurcation threshold, was established. Combined with the system phase diagram and the Poincaré mapping diagram, the stable control mechanism realized through reconstruction of the phase space topology under external excitation, was revealed, and the control mechanism of key control parameters on the global stability transition of the system was quantitatively elucidated. The results shows that, the low-frequency excitation can easily induce high period attractors, leading to motion boundary instability; the high frequency excitation can trigger safety-attraction basin erosion and bifurcation, where the P3S attractor is stable and the P2S attractor undergoes inverse doubling bifurcation to transition to the P1S single period safe orbit; and the reverse excitation amplitude will destroy the system stability, while increasing the forward excitation amplitude can accelerate the stabilization process, ultimately achieving full coverage of the P1S attraction domain. The research provides a theoretical support for vibration suppression, parameter optimization, and safety design of gear transmission systems.
In response to the instability problem caused by the coupling of strong nonlinear factors such as time-varying mesh stiffness and backlash in high-speed heavy-duty gear systems, an external periodic excitation control strategy was introduced, to establish and numerically solve a dynamic model for single-degree-of-freedom gear transmission systems. With the cell mapping method, the effects of control parameters on the erosion and bifurcation transition process of the system safety-attraction basin, as well as the evolution law of the attraction domain proportion p, were quantitatively analyzed. Based on the Floquet multiplier analysis, a quantitative mapping relationship between the doubling coefficient, the excitation amplitude, and the bifurcation threshold, was established. Combined with the system phase diagram and the Poincaré mapping diagram, the stable control mechanism realized through reconstruction of the phase space topology under external excitation, was revealed, and the control mechanism of key control parameters on the global stability transition of the system was quantitatively elucidated. The results shows that, the low-frequency excitation can easily induce high period attractors, leading to motion boundary instability; the high frequency excitation can trigger safety-attraction basin erosion and bifurcation, where the P3S attractor is stable and the P2S attractor undergoes inverse doubling bifurcation to transition to the P1S single period safe orbit; and the reverse excitation amplitude will destroy the system stability, while increasing the forward excitation amplitude can accelerate the stabilization process, ultimately achieving full coverage of the P1S attraction domain. The research provides a theoretical support for vibration suppression, parameter optimization, and safety design of gear transmission systems.
2026, 47(4): 496-504.
doi: 10.21656/1000-0887.460228
Abstract:
Two typical definite integrals are calculated with the method of modern potential theory by virtue of complex analysis. In the context of calculus, the integrals in consideration are difficult to calculate and are usually directly applied as established results in fracture and contact problems. However, they may be expressed explicitly by means of the residue theorem, the infinite arc lemma and the infinitesimal arc lemma in the context of complex analysis. This work perfects the theoretical fundamentals of 3D fracture and contact problems, and further demonstrates the intrinsic advantages of complex analysis in calculation of definite integrals.
Two typical definite integrals are calculated with the method of modern potential theory by virtue of complex analysis. In the context of calculus, the integrals in consideration are difficult to calculate and are usually directly applied as established results in fracture and contact problems. However, they may be expressed explicitly by means of the residue theorem, the infinite arc lemma and the infinitesimal arc lemma in the context of complex analysis. This work perfects the theoretical fundamentals of 3D fracture and contact problems, and further demonstrates the intrinsic advantages of complex analysis in calculation of definite integrals.
2026, 47(4): 505-515.
doi: 10.21656/1000-0887.450288
Abstract:
With a class of weighted difference formulas and explicit Euler methods to discretize diffusion terms and the 1st-order temporal derivative, respectively, a new type of 2-level, explicit and monotonic finite difference methods is established for 2D Fisher-KPP equations. As α，p，θ and the grid step size satisfy some constraining conditions, their numerical solutions can inherit the properties of the exact solutions, such as positivity preservation, boundedness and monotonicity. Furthermore, the maximum norm error estimate is obtained, rigorously. Numerical experiments illustrate that the numerical results agree well with the theoretical findings.
With a class of weighted difference formulas and explicit Euler methods to discretize diffusion terms and the 1st-order temporal derivative, respectively, a new type of 2-level, explicit and monotonic finite difference methods is established for 2D Fisher-KPP equations. As α，p，θ and the grid step size satisfy some constraining conditions, their numerical solutions can inherit the properties of the exact solutions, such as positivity preservation, boundedness and monotonicity. Furthermore, the maximum norm error estimate is obtained, rigorously. Numerical experiments illustrate that the numerical results agree well with the theoretical findings.
2026, 47(4): 516-528.
doi: 10.21656/1000-0887.460032
Abstract:
The number of determining modes was estimated for the incompressible non-Newtonian micropolar fluid with infinite delay in 2D bounded domains. The results show that, the asymptotic behavior of any weak solution to the non-Newtonian micropolar fluid equations with infinite delay depends completely on the asymptotic behavior of their 1st finite number of Fourier modes.
The number of determining modes was estimated for the incompressible non-Newtonian micropolar fluid with infinite delay in 2D bounded domains. The results show that, the asymptotic behavior of any weak solution to the non-Newtonian micropolar fluid equations with infinite delay depends completely on the asymptotic behavior of their 1st finite number of Fourier modes.
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Founded in 1980 ( monthly )
Supervisor：Chongqing Jiaotong University
Founder：CHIEN Weizang
Editor-in-Chief：LU Tianjian
ISSN 1000-0887
CN 50-1060/O3
