Carton stack stability evaluation system based on dynamic deformation monitoring

Abstract

The paper box stack stability evaluation system based on dynamic deformation monitoring discloses a kind of measurement and calculation technical field, the system includes: sensor part, dynamic stability analysis part and stability evaluation part;The sensor part includes multiple sensor units, and each sensor unit collects the real-time displacement and real-time shear deformation of detection point position;Dynamic stability analysis part is used to construct the deformation characteristic tensor of stacked paper box;Calculate strain energy density field and construct dynamic stability function;The interlayer coupling effect in stacking process is analyzed, and the interlayer coupling tensor is calculated;The stability evaluation part is used to carry out nonlinear correction operation on nonlinear response spectrum function, and the stability value of stacked paper box is calculated after obtaining nonlinear correction value.The present application provides reliable support for stacking design optimization and safety control in logistics transportation and warehouse management, effectively reduces instability risk and improves transportation efficiency.

Description

Technical Field

[0001] This disclosure relates to, but is not limited to, the fields of measurement and calculation technology, and in particular to a carton stacking stability evaluation system based on dynamic deformation monitoring. Background Technology

[0002] In the fields of logistics, transportation, and warehousing management, cardboard box stacking is a common packaging and storage method, playing a vital role in improving space utilization, reducing transportation costs, and ensuring cargo safety. However, the stability of the cardboard box stacking structure directly affects the safety of cargo transportation and the efficiency of storage management. Existing technologies have yielded numerous studies and engineering practices addressing the stability issues of cardboard box stacking, covering multiple areas such as stacking design, material optimization, mechanical analysis, and dynamic monitoring; however, many problems still remain to be solved.

[0003] Traditional cardboard box stacking design relies primarily on rules of thumb and static analysis models. For example, in practical applications, workers design stacking structures based on the size, weight, and number of layers of the boxes, combined with experience, and use simple stability assessment methods (such as top pressure tests) to evaluate the load-bearing capacity of the stacked structure. While this empirical method has some practicality, it lacks systematicity and scientific rigor, and cannot accurately predict the risk of dynamic instability under complex stacking conditions. Furthermore, due to the complexity of factors such as the elastic properties of cardboard materials, dimensional errors, and contact surface friction, traditional static evaluation methods struggle to encompass the dynamic characteristics of the actual stacking environment, such as vibrations, impacts, and center-of-gravity shifts during transportation.

[0004] In recent years, mechanical analysis methods, especially numerical simulation techniques based on finite element analysis, have been widely used in the study of cardboard box stacking stability. By establishing a mechanical model of the stacked structure and simulating stress distribution and deformation modes under different stacking conditions, researchers can conduct a more in-depth analysis of the stability of cardboard box stacks. For example, some studies calculate the maximum load-bearing capacity and instability modes of the stacked structure based on the material properties of the cardboard box (such as elastic modulus and Poisson's ratio), combined with stacking loads and gravity conditions. This method has good results in static stacking environments, but it still has significant limitations under dynamic conditions (such as vibration and impact during transportation). In dynamic environments, the deformation and stress distribution of the stacked structure are time-varying, and traditional static analysis methods cannot effectively capture the impact of dynamic changes on stacking stability.

[0005] In addition, some progress has been made in recent years in the modification of cardboard box materials and the optimization of their structures. For example, high-strength corrugated boxes have been developed, multi-layer materials have been used to improve the compression resistance of the boxes, and the edge structure of the boxes has been optimized to reduce the risk of shear instability. However, these material and structural improvements mainly target the mechanical properties of individual cardboard boxes and fail to fully consider the complexity of interlayer interactions in stacked structures. The stability of stacked structures depends not only on the mechanical properties of individual cardboard boxes but is also significantly affected by factors such as interlayer friction, mass distribution, contact area, and deformation coupling between adjacent layers. Existing material optimization and structural design methods often fail to systematically consider these complex interlayer interactions, resulting in significant uncertainties in the stacking stability in practical applications. Summary of the Invention

[0006] This disclosure provides a cardboard box stacking stability evaluation system based on dynamic deformation monitoring. This system automates the entire process from deformation monitoring to stability value calculation, offering real-time performance, accuracy, and adaptability. Compared to existing technologies, this invention can more comprehensively capture the nonlinear characteristics and instability trends of stacking systems in dynamic environments, providing reliable support for stacking design optimization and safety control in logistics transportation and warehousing management, effectively reducing instability risks and improving transportation efficiency.

[0007] To solve the above problems, the technical solution of the present invention is implemented as follows:

[0008] A cardboard box stacking stability evaluation system based on dynamic deformation monitoring is provided. The system comprises a sensor section, a dynamic stability analysis section, and a stability evaluation section. The sensor section includes multiple sensor units, each positioned at a pre-selected detection point within the stacked cardboard boxes. Each sensor unit collects real-time displacement and real-time shear deformation data at the detection point. The dynamic stability analysis section constructs a deformation characteristic tensor of the stacked cardboard boxes based on the real-time displacement and shear deformation data at each detection point. Based on the deformation characteristic tensor, it calculates the strain energy density field and constructs a dynamic stability function. The system then evaluates the stability of the cardboard boxes based on the deformation characteristic tensor, strain energy density field, and... The dynamic stability function analyzes the interlayer coupling effect during the stacking process and calculates the interlayer coupling tensor. The stability evaluation part combines the strain energy density field, interlayer coupling tensor, deformation characteristic tensor, and dynamic stability function to perform nonlinear response spectrum analysis, obtain the nonlinear response spectrum function, and perform nonlinear correction calculation on the nonlinear response spectrum function to obtain the nonlinear correction value. Then, the stability value of the stacked cartons is calculated and compared with the preset stability threshold. If it is lower than the set stability threshold, the stacked cartons are judged to be unstable and an early warning is issued. In the stacked cartons, all cartons are made of the same material.

[0009] Furthermore, the sensor unit includes a laser displacement sensor and a strain gauge; the sampling frequency of the laser displacement sensor and the strain gauge is at least 100Hz.

[0010] Furthermore, the deformation characteristic tensor D is represented using the following formula:

[0011]

[0012] Where n is the number of detection points; Δx i (t) represents the displacement component along the X-axis at time t for the i-th detection point; Δy i (t) represents the displacement component along the Y-axis at time t for the i-th detection point; Δz i (t) represents the displacement component along the Z-axis at time t for the i-th detection point; γ ixy (t) represents the shear deformation of the i-th detection point in the X-axis direction toward the Y-axis at time t; γ ixz (t) represents the shear deformation of the i-th detection point in the X-axis direction toward the Z-axis at time t; γ iyx (t) represents the shear deformation of the i-th detection point in the Y-axis direction towards the X-axis at time t; γ iyz (t) represents the shear deformation of the i-th detection point in the Y-axis direction toward the Z-axis at time t; γ izx (t) represents the shear deformation of the i-th detection point in the Z-axis direction towards the X-axis at time t; γ izy (t) represents the shear deformation of the i-th detection point in the Z-axis direction toward the Y-axis at time t; r i L represents the position vector of the i-th detection point; c The characteristic length is defined as one-third of the average height of all cartons in a stack; ||·|| is the modulus operation of the vector.

[0013] Furthermore, the strain energy density field E is calculated based on the deformation characteristic tensor using the following formula:

[0014]

[0015] Where C is the elastic property tensor of the material of the stacked cardboard boxes; : represents the double dot product operation between tensors; α is the weighting factor, used to adjust the influence of deformation gradient on strain energy density, and is a set value; V represents the spatial region where all detection points are located; dV is the spatial integration variable; This is the gradient operator.

[0016] Furthermore, the dynamic stability function S is constructed using the following formula:

[0017]

[0018] Where dτ is the time integration variable; E c The critical energy density; ||·|| F For F-norm operations; D max This represents the maximum allowable deformation of all cartons in a stack.

[0019] Furthermore, when analyzing the interlayer coupling effect during the stacking process, the stacked cartons are divided into multiple layers from top to bottom, with multiple detection points in each layer. The interlayer coupling effect during the stacking process is analyzed using the following formula, based on the deformation characteristic tensor, strain energy density field, and dynamic stability function, to calculate the interlayer coupling tensor C:

[0020]

[0021] Where μ is the coefficient of friction of the cardboard box surface; N is the number of layers of the stacked cardboard boxes; D k Let D be the deformation feature tensor of the k-th layer; k+1 The deformation feature tensor of the (k+1)th layer; A k，k+1 M represents the contact area between the k-th stacked cardboard box and the (k+1)-th stacked cardboard box; k M represents the mass of the k-th layer of stacked cartons; k+1 Let the mass of the (k+1)th layer of stacked cartons be denoted as 'k'. For Hadama accumulation.

[0022] Furthermore, by combining the strain energy density field, interlayer coupling tensor, deformation characteristic tensor, and dynamic stability function using the following formula, nonlinear response spectrum analysis is performed to obtain the nonlinear response spectrum function R:

[0023]

[0024] in, This is the Fourier transform function.

[0025] Furthermore, the nonlinear response spectrum function is nonlinearly corrected using the following formula:

[0026]

[0027] Where G is the nonlinear correction value; det|·| is the matrix determinant operator.

[0028] Furthermore, the stability value Λ of the stacked cartons is calculated using the following formula:

[0029]

[0030] Where tr(·) is the trace operation of the matrix; ||·||2 is the L2 norm.

[0031] The cardboard box stacking stability evaluation system based on dynamic deformation monitoring of this invention has the following beneficial effects: First, this invention achieves real-time, high-precision acquisition of the deformation characteristics of stacked cardboard boxes, which lays a solid foundation for stability analysis. By using laser displacement sensors and strain gauges, this invention can monitor the three-dimensional displacement and shear deformation of stacked cardboard boxes at each detection point. This sensor combination design not only has the advantage of non-contact measurement but also captures the complex dynamic deformation behavior during cardboard box stacking, especially under dynamic environments such as vibration, impact, and center of gravity shift during transportation, providing high-frequency and accurate data input. Compared with existing technologies that rely on single-point monitoring or static acquisition, the sensor module of this invention significantly improves the ability to capture the deformation characteristics of the stacking system, ensuring that the system can comprehensively and in real-time acquire the mechanical properties of the stacked structure. Second, this invention provides a mathematical modeling basis for the mechanical behavior of complex stacking systems by constructing deformation characteristic tensors, strain energy density fields, interlayer coupling tensors, and dynamic stability functions. These mechanical models organically combine multidimensional factors that are difficult to fully cover in existing technologies. For example, by using the deformation characteristic tensor, this invention can quantify the dynamic deformation characteristics of each detection point in the stacked system and rationally allocate the influence of different detection points on the overall deformation through a spatial attenuation function. The strain energy density field describes the internal stress state of the stacked system from the perspective of energy distribution, revealing the potential risks of local instability regions. The introduction of the interlayer coupling tensor captures the complex relationships of interlayer interactions in the stacked system, such as the interlayer slip tendency caused by friction, mass differences, and contact area. The dynamic stability function further reveals the dynamic characteristics of the stacked system through frequency domain analysis, especially when the external disturbance frequency is coupled with the system's inherent vibration mode, enabling accurate assessment of instability risk. The innovative combination of these models allows this invention to not only reflect the static stability of the stacked system but also comprehensively characterize its evolution process in a dynamic environment, providing a comprehensive and reliable data foundation for subsequent stability evaluation. Attached Figure Description

[0032] Figure 1 This is a schematic diagram of the system structure of a cardboard box stacking stability evaluation system based on dynamic deformation monitoring, provided in an embodiment of the present invention. Detailed Implementation

[0033] To make the technical problems, technical solutions, and beneficial effects to be solved by this disclosure clearer and more understandable, the disclosure will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of this disclosure and are not intended to limit this disclosure.

[0034] Example 1, Reference Figure 1 A cardboard box stacking stability evaluation system based on dynamic deformation monitoring is described. The system comprises a sensor section, a dynamic stability analysis section, and a stability evaluation section. The sensor section includes multiple sensor units, each positioned at a pre-selected detection point within the stacked cardboard boxes. Each sensor unit collects real-time displacement and shear deformation data at the detection point. The dynamic stability analysis section constructs a deformation characteristic tensor of the stacked cardboard boxes based on the real-time displacement and shear deformation data at each detection point. Based on the deformation characteristic tensor, it calculates the strain energy density field and constructs a dynamic stability function. The system then evaluates the stability of the cardboard boxes based on the deformation characteristic tensor, strain energy density field, and... The dynamic stability function analyzes the interlayer coupling effect during the stacking process and calculates the interlayer coupling tensor. The stability evaluation part combines the strain energy density field, interlayer coupling tensor, deformation characteristic tensor, and dynamic stability function to perform nonlinear response spectrum analysis, obtain the nonlinear response spectrum function, and perform nonlinear correction calculation on the nonlinear response spectrum function to obtain the nonlinear correction value. Then, the stability value of the stacked cartons is calculated and compared with the preset stability threshold. If it is lower than the set stability threshold, the stacked cartons are judged to be unstable and an early warning is issued. In the stacked cartons, all cartons are made of the same material.

[0035] Specifically, in the stability evaluation of cardboard box stacking, the role of sensors is to capture displacement and shear deformation data at key detection points in the stacked cardboard boxes in real time, thus providing basic data support for subsequent dynamic analysis and stability evaluation. To achieve this goal, sensor units are strategically placed at detection points in the cardboard box stack that represent the overall mechanical state. These detection points are typically selected in the four corners, edges, and middle of the cardboard box—areas with weak mechanical properties—to comprehensively capture the deformation behavior of the cardboard box under external forces. The working principle of the sensors is based on the mechanical measurement of deformation. Specifically, each sensor unit can sense the displacement changes and shear deformation at the detection point. These parameters reflect the combined effects of compressive stress, tensile stress, and shear stress experienced by the cardboard box during stacking. The sensor units transmit these deformation parameters to the system's analysis module in the form of real-time signals using high-precision measurement technology. To achieve efficient monitoring, the sensors employ a combination of strain gauge and displacement measurement methods. Strain gauge measurement utilizes the change in resistance of the material under stress to detect deformation; the strain gauges embedded inside the sensor can sense minute stretching or compression on the surface of the cardboard box in real time. Displacement measurement, on the other hand, is based on optical or laser technology. By capturing the spatial displacement of a specific point, it can accurately calculate the geometric changes of a cardboard box under mechanical action.

[0036] In dynamic deformation monitoring, another key technical characteristic of the sensor component is its dynamic response capability. The deformation that may occur during the stacking of cartons is rapid and complex, requiring sensors to capture data at extremely high sampling frequencies to avoid missing crucial deformation information. To this end, the sensor unit employs high-frequency sampling technology, enabling multiple data acquisitions within milliseconds, while also featuring anti-interference signal processing to ensure the accuracy and stability of the collected data. Furthermore, the sensor utilizes a distributed architecture to achieve multi-point synchronous monitoring. Each sensor at each detection point can operate independently, transmitting data to a central processing module for centralized analysis. This architecture effectively covers the entire carton stacking area, forming a comprehensive deformation monitoring network. Regarding data transmission, the sensor unit typically connects to the central processing module wirelessly or via wired connections. Wireless connections employ low-latency, high-bandwidth communication protocols, enabling efficient transmission of large-scale data in dynamic environments. Especially in dynamic environments such as logistics transportation, wireless sensor systems can adapt to positional changes during carton stacking. Wired connections, on the other hand, are more suitable for static scenarios, offering stronger data transmission stability and meeting the needs of long-term, high-volume data monitoring. In addition, the sensor has built-in data caching and preprocessing functions, which can perform preliminary processing on the acquired signals, such as filtering and noise removal, thereby improving transmission efficiency and the accuracy of subsequent calculations.

[0037] Shear deformation is a key parameter in stack stability analysis, reflecting changes in interlayer slip and internal stress distribution. To measure shear deformation, the sensor combines optical interferometry with contact measurement technology, ensuring measurement accuracy while avoiding error accumulation that might occur with non-contact techniques. Simultaneously, the sensor can separate effective signals from dynamic environments using sophisticated signal demodulation algorithms. For example, when a cardboard box is subjected to external impact, the sensor captures transient deformation signals, demodulates them to extract the true deformation trend, and removes interference signals. This dynamic demodulation technology significantly improves monitoring accuracy and robustness, enabling the system to operate stably in complex scenarios. To meet practical application requirements, the sensor also needs strong environmental adaptability. Since the monitoring scenario of stacked cardboard boxes may involve various complex factors such as temperature, humidity, and vibration, the sensor unit adopts an anti-environmental interference design. For example, the sensor surface is covered with a waterproof and dustproof coating, and the internal structure is designed to resist vibration, ensuring accurate data acquisition even in harsh environments. In addition, the power consumption optimization design of the sensor is also particularly important, especially in wireless sensing scenarios. Low power consumption can extend the sensor's operating time, thereby reducing maintenance costs and improving the overall reliability of the system.

[0038] The dynamic stability analysis requires real-time displacement and shear deformation data provided by the sensors as input to construct a deformation characteristic tensor reflecting the internal mechanical state of the stacked cartons. The deformation characteristic tensor is a three-dimensional matrix-based mechanical description tool used to represent the geometric deformation occurring at each detection point of the cartons during stacking. This tensor includes not only normal strain (i.e., the amount of compression or tension of the cartons along a specific direction under external force) but also shear strain (reflecting interlayer slip and relative displacement). By spatially interpolating the deformation data at each detection point, the deformation characteristic tensor can comprehensively characterize the local deformation characteristics and overall geometric deformation pattern of the entire stacking system. The significance of this step lies in transforming discrete deformation data into a continuous mechanical description, thus providing a unified computational basis for subsequent analysis. Based on the deformation characteristic tensor, the system further calculates the strain energy density field of the stacked cartons. The strain energy density field is a key parameter in mechanical analysis, reflecting the distribution of elastic strain energy stored in material elements under external force. For stacked cartons, the strain energy density field can reveal areas of stress concentration and potential instability points during stacking. The calculation process integrates the deformation characteristic tensor with material mechanical properties, such as elastic modulus and Poisson's ratio, to obtain the energy density value distributed in three-dimensional space. The spatial gradient of the strain energy density field can further reflect the propagation path and intensity changes of stress in the stacked system, which is of great significance for identifying weak points and interlayer interactions in stacked structures. The construction of the strain energy density field not only enhances the analytical depth of the system but also lays the foundation for establishing a dynamic stability function.

[0039] The dynamic stability function is the core analytical tool of this module, used to quantify the overall stability state of the stacked system by integrating the deformation characteristic tensor, strain energy density field, and dynamic changes in the time domain. The construction of this function requires consideration of multiple physical factors, including the dynamic loading of external forces, the temporal evolution of deformation, and interlayer coupling. By mathematizing these complex physical processes, the dynamic stability function can capture the nonlinear behavior of the stacked system over time, thereby revealing the dynamic mechanism of its transition from equilibrium to instability. In particular, the time-domain rate of change of the dynamic stability function reflects the transient response characteristics of the stacked system, i.e., the recovery ability and instability tendency of the cardboard box stacking structure under external disturbances. This time-domain analysis method significantly improves the system's adaptability to complex dynamic environments, enabling not only the evaluation of the current stability state but also the prediction of potential instability risks. In dynamic stability analysis, interlayer coupling effects are a crucial factor that cannot be ignored. The interlayer contact and friction behavior of the stacked cardboard boxes determine the mechanical properties of the overall stacked system, and these behaviors typically exhibit nonlinear and highly coupled characteristics. The dynamic stability analysis section quantitatively describes the interaction strength and relative motion modes between layers by calculating the interlayer coupling tensor. The interlayer coupling tensor not only reveals the cross-layer propagation laws of stress and deformation during stacking but also provides necessary supplementary information for the dynamic stability function. For example, when a layer is subjected to an external force, the interlayer coupling tensor can predict how this disturbance will propagate to adjacent layers and the overall system's response amplitude. By introducing the interlayer coupling effect into the dynamic analysis framework, the system can more comprehensively capture the complex mechanical behavior of the stacked system. Furthermore, the dynamic stability analysis section utilizes numerical simulation technology to refine the dynamic characteristics of the stacked system. This process combines finite element analysis and multibody dynamics simulation. By performing high-precision modeling of the structure and material properties of the stacked cartons, the system can simulate the mechanical behavior in actual stacking environments. For example, when the stacked system is affected by external factors such as gravity, vibration, or impact, dynamic simulation can predict the deformation modes and stress distribution changes of the cartons, thereby verifying the accuracy of the dynamic stability function. Simulation technology not only enhances the system's analytical capabilities but also provides important references for optimizing stacking design and material selection.

[0040] During the stability evaluation process, the system first obtains multidimensional parameters describing the key mechanical properties of the cardboard box stacking process based on the output of the dynamic stability analysis module. These parameters include not only the spatially distributed strain energy density field and deformation characteristic tensor, but also the coupling tensor reflecting the interaction between stacked layers and the time-varying characteristics of the dynamic stability function. Through these parameters, the system can comprehensively capture the stress distribution, deformation modes, and response behavior of the stacked system under static and dynamic states, providing a sufficient data foundation for stability evaluation. Since the cardboard box stacking system exhibits significant nonlinear characteristics, stability evaluation requires the use of nonlinear analysis methods to fully consider the changes in mechanical behavior under complex conditions. Therefore, the system introduces nonlinear response spectrum analysis technology to perform comprehensive analysis of the dynamic stability function in both the frequency and time domains, thereby revealing the dynamic characteristics and instability trends of the stacked system under external disturbances. Nonlinear response spectrum analysis is the core tool of the stability evaluation section. Its principle lies in transforming the time-varying characteristics of the dynamic stability function into an amplitude response curve in the frequency domain through mathematical methods such as Fourier transform. This method can effectively analyze the stability performance of the stacked system in different frequency ranges, especially identifying key vibration modes and frequencies that may lead to instability. Specifically, the system uses the input of the dynamic stability function to generate a nonlinear response spectrum function, reflecting the response intensity of the stacked system under various frequency perturbations. Nonlinear response spectrum analysis can not only identify the inherent vibration modes of the stacked system but also reveal the coupling relationship between external perturbations and the system's dynamic characteristics, such as the interaction effect between vibration frequency and interlayer coupling tensor. This in-depth frequency domain analysis enables the system to accurately predict the dynamic instability point of the stacked system, providing a reliable foundation for calculating stability values.

[0041] After obtaining the nonlinear response spectrum function, the system further performs nonlinear correction calculations on it. The core of this step lies in dynamically adjusting the response spectrum function by combining the interlayer coupling tensor and the strain energy density field to eliminate the influence of non-ideal conditions in the stacked system (such as changes in contact surface friction and material heterogeneity) on the stability evaluation. The nonlinear correction value is a dynamically adjusted parameter that optimizes the response spectrum analysis results based on the actual mechanical characteristics during the stacking process, ensuring higher accuracy and applicability in the final stability value calculation. This nonlinear correction mechanism not only enhances the system's adaptability to complex stacking environments but also makes the stability evaluation results more robust and reliable. After completing the nonlinear correction, the system calculates the stability value of the stacked cartons by combining all mechanical parameters. The stability value is a comprehensive quantitative index used to measure the overall stability state of the stacked system. Its calculation process considers not only the geometric deformation mode reflected by the deformation characteristic tensor but also the spatial distribution of the strain energy density field and its dynamic interaction characteristics with the interlayer coupling tensor. Therefore, the stability value can comprehensively reflect the mechanical state of the stacked system, including its compressive strength, shear strength, and resistance to dynamic disturbances. This quantitative indicator allows the system to clearly determine whether the stacked system meets design requirements. Finally, the system compares the calculated stability value with a preset stability threshold. The stability threshold, set based on specific application scenarios and material properties, represents the minimum stability requirement used to identify the instability risk of the stacked system. When the stability value falls below this threshold, the system determines that the stacked system is at risk of instability and issues a timely warning signal. This warning mechanism is a crucial function of the stability evaluation section, providing users with timely feedback to take appropriate intervention measures and prevent safety issues caused by stacked system instability. The system can also visualize the stability evaluation results, displaying the instability risk areas and key instability modes of the stacked system, helping users further optimize stacked design and operational procedures.

[0042] Example 2: The sensor unit includes a laser displacement sensor and a strain gauge; the sampling frequency of the laser displacement sensor and the strain gauge is at least 100Hz.

[0043] Specifically, the main function of the laser displacement sensor is to measure the three-dimensional spatial displacement changes of the detection points on the cardboard box. The laser displacement sensor emits a high-precision laser beam and calculates the displacement changes of the measured point using optical triangulation or laser interferometry principles. The advantages of this measurement method lie in its non-contact nature and extremely high accuracy, enabling it to capture minute displacements of the detection points in real time during dynamic stacking. For example, when stacked cardboard boxes are subjected to external forces, the laser displacement sensor can accurately measure the local displacement of the cardboard box surface, providing high-precision displacement data for constructing the deformation feature tensor. Furthermore, due to the extremely high response speed of the laser displacement sensor, with a sampling frequency of at least 100Hz, it can ensure the capture of rapid deformation events (such as transient deformation caused by external impact during stacking). This high sampling rate greatly enhances the system's responsiveness to dynamic characteristic changes, ensuring that the system can capture critical deformation information.

[0044] Strain gauges, another core measurement component, are primarily used to detect strain changes in cardboard materials under stress. Strain gauges measure normal and shear strain by detecting minute changes in the material's resistance under stress. This contact-based measurement method directly senses the internal deformation characteristics of the material, providing highly sensitive strain measurement data, especially in mechanically weak areas such as the edges or corners of the cardboard. Strain gauges and laser displacement sensors complement each other: the laser displacement sensor captures macroscopic displacement changes, while the strain gauge focuses on the microscopic strain characteristics of the localized areas of the cardboard. Combined, the system not only comprehensively covers the geometric deformation of the stacked system but also provides in-depth analysis of the stress distribution within the cardboard material. A sampling frequency of at least 100Hz means that 100 sets of data can be collected per second, reflecting the dynamic changes of the stacked system in real time. This high-frequency sampling is particularly important in dynamic stacking environments, such as during logistics transportation, where cardboard boxes may be subjected to continuous vibration, impact, or other disturbances; high-frequency sampling can capture the detailed characteristics of these transient deformations. Furthermore, the high sampling frequency provides ample data for subsequent dynamic analysis, making the construction of dynamic stability functions and nonlinear response spectrum analysis more accurate. Especially in stacking processes with significant nonlinear characteristics, a high sampling rate can ensure that the system captures key mechanical change nodes, thereby providing more reliable data support for stability evaluation.

[0045] Example 3: The deformation characteristic tensor D is represented by the following formula:

[0046]

[0047] Where n is the number of detection points; Δx i (t) represents the displacement component along the X-axis at time t for the i-th detection point; Δy i (t) represents the displacement component along the Y-axis at time t for the i-th detection point; Δz i(t) represents the displacement component along the Z-axis at time t for the i-th detection point; γ ixy (t) represents the shear deformation of the i-th detection point in the X-axis direction toward the Y-axis at time t; γ ixz (t) represents the shear deformation of the i-th detection point in the X-axis direction toward the Z-axis at time t; γ iyx (t) represents the shear deformation of the i-th detection point in the Y-axis direction towards the X-axis at time t; γ iyz (t) represents the shear deformation of the i-th detection point in the Y-axis direction toward the Z-axis at time t; γ izx (t) represents the shear deformation of the i-th detection point in the Z-axis direction towards the X-axis at time t; γ izy (t) represents the shear deformation of the i-th detection point in the Z-axis direction toward the Y-axis at time t; r i L represents the position vector of the i-th detection point; c The characteristic length is defined as one-third of the average height of all cartons in a stack; ||·|| is the modulus operation of the vector.

[0048] Specifically, the deformation characteristic tensor D is derived from the displacement components and shear deformation at the detection point. Displacement components (such as Δx) i (t), Δy i (t), Δz i (t) is used to represent the geometric displacement of each detection point in three-dimensional space during the stacking process. These displacements reflect the macroscopic motion state of the stacked cartons under force. For example, when the stacked cartons are compressed or stretched, the displacement components of the detection points can intuitively reflect the motion trend in various directions. Shear deformation (such as γ) ixy (t), γ ixz (t) etc. describes the relative displacement slip occurring in different directions, such as interlayer shear slip or bending deformation. By combining displacement and shear deformation, the deformation characteristics of each detection point can be represented by a 3×3 matrix, which fully describes the state of that point in three-dimensional mechanical space. However, the deformation characteristics in the stacked system are not determined by a single detection point, but are a comprehensive reflection of all detection points. Therefore, in order to describe the global deformation of the entire stacked system, the formula superimposes the tensor matrices of all detection points. This superposition is not a simple linear addition, but incorporates the spatial weights of the detection point positions. Specifically, through This exponential decay function assigns a weight to the contribution of each detection point, which increases with the distance of the detection point from the center of the stacked system. i || Increases and decreases. Here, r i L represents the relative position vector of the detection point. cIt is the characteristic length, used to define the spatial range of attenuation.

[0049] The exponential decay function simulates the spatial distribution of mechanical forces in a stacked system. Physically, the mechanical properties of a stacked system typically exhibit local concentration; that is, regions closer to the center contribute more to the overall deformation, while regions farther from the center contribute less. For example, the central region of a stacked cardboard box usually bears the greatest pressure and deformation, so the detection points in these regions should contribute more to the global tensor. Through the exponential decay function, the weights of these detection points are significantly increased, while the influence of detection points farther from the center on the overall deformation is relatively weakened due to their smaller weights. Feature length L c The introduction of the tensor provides control over the attenuation range, allowing the formula to adapt to stacking systems of different sizes and shapes. Temporal dynamics are another important feature of the formula. All deformation components in the formula are time-varying (t), indicating that the deformation characteristic tensor D is a dynamically changing quantity over time. In dynamic environments (such as logistics transportation, vibration, or impact), the deformation characteristics of stacked cartons change continuously over time. By calculating the deformation characteristic tensor in real time, the system can capture the transient behavior of deformation during stacking, such as shear slip or local instability tendencies at a detection point under impact. This temporal dynamics ensures that the deformation characteristic tensor accurately reflects the mechanical evolution of the stacking system, providing timely and accurate data support for subsequent stability analysis. The deformation characteristic tensor also possesses important mathematical properties. Through the matrix form of the tensor, the formula realizes a multidimensional representation of the deformation of the stacking system. The core principle of this representation method is to express the mechanical properties in three-dimensional space in matrix form, enabling it to describe not only displacement or deformation in a single direction but also the mutual influence between different directions. For example, shear deformation reflects coupling effects in different directions, which are particularly important in stack stability analysis because interlayer slip or rotational instability are often the main causes of stack failure. These complex mechanical properties are systematically captured and quantified using the deformation characteristic tensor, providing comprehensive information for subsequent analysis.

[0050] Example 4: The strain energy density field E is calculated based on the deformation characteristic tensor using the following formula:

[0051]

[0052] Where C is the elastic property tensor of the material of the stacked cardboard boxes; : represents the double dot product operation between tensors; α is the weighting factor, used to adjust the influence of deformation gradient on strain energy density, and is a set value; V represents the spatial region where all detection points are located; dV is the spatial integration variable; This is the gradient operator.

[0053] Specifically, the first term of the formula This represents the energy stored during the elastic deformation of stacked cardboard boxes. This part primarily relies on the deformation characteristic tensor D, which describes the geometric deformation characteristics of each detection point under stress. The deformation characteristic tensor is a 3×3 symmetric matrix, where the diagonal elements represent normal strain (tension or compression) in each direction, and the off-diagonal elements represent shear deformation. By coupling with the material property tensor C, the formula can calculate the elastic energy corresponding to the deformation, taking into account the elastic properties of the cardboard material. The material property tensor C is a higher-order tensor used to describe the elastic response characteristics of the cardboard material in different stress directions, such as the elastic modulus and Poisson's ratio. This coupling relationship indicates that different material properties directly affect the magnitude of the deformation energy storage. For example, materials with a larger elastic modulus have higher deformation energy storage, exhibiting stronger resistance to deformation. The physical meaning of the double dot product operation D:C:D lies in quantifying the interaction between the deformation characteristic tensor and the material property tensor in the form of energy. This higher-order tensor operation mechanism is similar to an inner product, effectively combining the components of the deformation characteristic and the anisotropy of the material properties. This operation transforms complex multidimensional mechanical relationships into scalar form, resulting in an energy expression that is easy to understand and calculate. Furthermore, the pre-coefficients... This ensures the symmetry of energy calculations, meaning that the corresponding stored energy remains consistent regardless of whether the deformation is caused by loading or unloading. This symmetry is a fundamental requirement for energy calculations in mechanics of materials. The second term of the formula... This is a description of the energy of the deformation gradient. The deformation gradient is the spatial rate of change of the deformation feature tensor, and its physical meaning lies in describing the deformation difference between detection points. For example, stacked cardboard boxes may experience significant shear slip or bending deformation in local areas, and these non-uniform deformations lead to the accumulation of additional strain energy. The formula uses the gradient operator... Extract the local rate of change of deformation using a double dot product. These rates of change are linked to the energy relationship. The square of the deformation gradient reflects the degree of deformation inhomogeneity; when the gradient is large, the energy storage in local regions increases significantly. The weighting factor α is an important adjustment parameter for the deformation gradient term. It is used to adjust the weight of the deformation gradient's influence on the total strain energy density field. Specifically, when the deformation gradient of the stacked system changes significantly, such as at the edges or contact regions, a larger α value can highlight the contribution of these regions to the energy distribution. Conversely, when the gradient change is small, a lower α value can avoid the gradient term excessively affecting the overall energy calculation. This adjustability allows the formula to flexibly adapt to different stacking scenarios and material properties. For example, for flexible materials, the energy contribution of the deformation gradient may be relatively large, while for rigid materials, the elastic energy storage term may dominate. The volume integral ∫ in the formula... V(·)dV is used to extend the local energy distribution to the entire spatial range V of the stacked system. The physical meaning of the integration operation is to summarize the local energy contributions of all detection points in the stacked system, thereby forming a global expression for strain energy density. This process can capture the whole picture of the internal mechanical state of the stacked system. For example, the integration result may show that the strain energy density is higher in the central region of the stack, reflecting that these regions are subjected to greater compressive and shear stresses. In the edge regions, the strain energy density may gradually decrease, indicating that the mechanical effects in these regions are weaker. Through volume integration, the formula organically combines local and global mechanical properties, providing a complete mechanical analysis perspective for the system.

[0054] Example 5: Construct the dynamic stability function S using the following formula:

[0055]

[0056] Where dτ is the time integration variable; E c The critical energy density; ||·|| F For F-norm operations; D max This represents the maximum allowable deformation of all cartons in a stack.

[0057] Specifically, the first part of the formula This is the core term describing the dynamic response strength of the deformation tensor. The deformation characteristic tensor D captures the deformation characteristics of stacked cartons, and its second time derivative represents the acceleration of the deformation component in the stacked system, that is, the degree of drastic change in deformation over time. The double dot product operation couples the components of the deformation acceleration tensor to generate a scalar value used to measure the dynamic response strength of the entire stacked system. Physically, this term reflects the deformation sensitivity of the stacked system under dynamic loads such as vibration and impact. A larger acceleration indicates that the system's deformation changes rapidly and drastically, potentially leading to a higher risk of instability. For example, during logistics transportation, external vibrations may trigger transient responses in the stacked system; the formula captures this behavior and quantifies its impact through the acceleration term. Part Two This is an energy density weighting factor used to reflect the impact of the current strain energy density state on dynamic stability. Strain energy density E is an indicator of the elastic energy stored in a material during deformation, while Ecritical is the critical energy density, defining the energy boundary between stability and instability in a stacked system. By normalizing E and introducing an exponential decay function, the formula can nonlinearly adjust the contribution of energy density to stability. When E is much smaller than Ecritical... c When E approaches or exceeds E, the value of the exponential factor is close to 1, indicating that the current energy state has little impact on stability; when E approaches or exceeds E, the value of the exponential factor is close to 1, indicating that the current energy state has little impact on stability. cWhen the index factor rapidly approaches zero, it reflects that the system is in a high-energy state, and the risk of instability increases significantly. This factor not only characterizes the internal stress state of the stacked system, but also emphasizes the important role of critical energy density in stability assessment through nonlinear mapping. For example, when local energy accumulation approaches the critical value due to compression or shear during the stacking process of cardboard boxes, the possibility of instability of the system will increase rapidly.

[0058] Part Three This is a deformation constraint term used to quantify the contribution of the geometric deformation state of a stacked system to its stability. Here, ∥D∥ F is the F-norm of the deformation tensor, representing the overall amplitude of the deformation characteristic tensor, i.e., the global deformation of the system. The F-norm is a scalar value obtained by summing the squares of all components of the tensor and taking the square root, used to measure the overall deformation intensity. D... max This is the maximum allowable deformation of the stacked system, typically determined by the material's yield strength or design requirements. The physical meaning of this limiting factor is that when the deformation ||D||... F Approaching or exceeding D max At that time, the risk of system instability increases rapidly. Through The design allows the formula to dynamically adjust the influence of deformation on stability. When ||D|| F Much smaller than D max When the factor value is close to 1, it indicates that the deformation has little impact on the current stability; when ||D|| F Approaching D max At this point, the factor value rapidly decreases to zero, reflecting that the system is approaching the instability boundary. Finally, the time integral... The combined results of the above three parts are accumulated over a given time range to generate the dynamic stability function S. The integral variable τ represents the time step. Through cumulative calculation, the formula can not only capture the instantaneous dynamic stability state, but also reflect the stability evolution trend of the stacked system over the entire time period. The introduction of time integration is particularly important for handling stacked systems in dynamic environments. For example, during logistics transportation, the stacked system may be affected by multiple vibrations or impacts, and its stability will gradually change over time. Through time integration, the formula can accurately capture these dynamic changes, transforming transient characteristics into global evaluation results.

[0059] Example 6: When analyzing the interlayer coupling effect during the stacking process, the stacked cartons are divided into multiple layers from top to bottom, with multiple detection points in each layer. The interlayer coupling effect during the stacking process is analyzed using the following formula based on the deformation characteristic tensor, strain energy density field, and dynamic stability function, and the interlayer coupling tensor C is calculated:

[0060]

[0061] Where μ is the coefficient of friction of the cardboard box surface; N is the number of layers of the stacked cardboard boxes; D k Let D be the deformation feature tensor of the k-th layer; k+1 The deformation feature tensor of the (k+1)th layer; A k，k+1 M represents the contact area between the k-th stacked cardboard box and the (k+1)-th stacked cardboard box; k M represents the mass of the k-th layer of stacked cartons; k+1 Let the mass of the (k+1)th layer of stacked cartons be denoted as 'k'. For Hadama accumulation.

[0062] Specifically, the first part of the formula It is the driving force behind the entire tensor computation. Gradient operator. The rates of change of the dynamic stability function S and the strain energy density field E in space were extracted to reflect the non-uniformity of deformation and energy distribution in stacked cardboard boxes. The dynamic stability function S represents the evolution of the overall stability of the stacked system over time and space, while the strain energy density field E describes the energy storage at different locations within the stacked cardboard boxes. Gradient calculations, by capturing the local changes in these variables, reveal the interaction between deformation and energy. For example, when the dynamic stability of a layer changes significantly, its deformation triggers a local redistribution of energy, and these changes are propagated to adjacent layers through gradient coupling. (Hadamard product) The two gradient fields are then multiplied component-by-component to precisely quantify the interaction between deformation and energy change. Specifically, the Hadamard product is an element-level operation that distributes the coupling effect of the two into each spatial component, thus forming a detailed coupled tensor field. This part of the design directly captures the core dynamic mechanism of interlayer deformation and energy interaction, providing a foundation for subsequent tensor calculations. In the formula... This paper describes the effects of interlayer mass distribution differences and frictional effects on coupling. In a stacked carton system, the weight differences between different layers of cartons can lead to uneven contact pressure distribution, thus affecting the interlayer mechanical transmission characteristics. Mass difference | M k -M k+1 | is a key indicator for measuring the degree of uneven contact between two layers. When the mass difference is large, the lower layer of the carton bears greater pressure, which enhances the mechanical coupling at the contact surface. Furthermore, the coefficient of friction μ characterizes the magnitude of the sliding resistance at the contact surface. Friction is an important limiting factor for interlayer mechanical coupling. When the coefficient of friction is large, interlayer slippage is suppressed, and the coupling effect is enhanced; when the coefficient of friction is small, the tendency for interlayer slippage increases, and the coupling effect weakens. By accumulating the mass difference of N layers of carton, the formula can systematically capture the overall influence of the mass distribution of the entire stacking system on the coupling effect.

[0063] The exponential decay term in the formula This further characterizes the dynamic regulation of coupling effect by deformation differences. Here, D... k and D k+1 Let be the deformation feature tensors of the k-th and (k+1)-th layers, representing the overall deformation state of each layer. The norm difference between the two deformation tensors is parallel to D. k -D k+1 || represents the deformation difference between two layers. When the deformation states of the two layers are similar, the coupling effect is strong; however, when the deformation difference is large, the interlayer coupling effect is significantly weakened. Normalization factor ||D|| F This is the global norm of the deformation feature tensor, used to standardize deformation differences, making it applicable to different stacking scenarios. The design of the exponential decay function introduces a nonlinear effect: when the deformation difference is small, the exponent value is close to 1, indicating enhanced interlayer coupling; while when the deformation difference is large, the exponent value rapidly approaches 0, indicating that the coupling effect essentially disappears. This mechanism physically reflects the importance of contact surface deformation matching for mechanical transmission; for example, when slippage or instability tendencies increase, interlayer contact becomes unstable. (Contact area term) The interlayer coupling model was further improved through geometric constraints. Contact area A k，k+1 This represents the actual contact area between layer k and layer (k+1), a crucial factor affecting interlayer mechanical transmission. A larger contact area results in more uniform force transmission and enhanced interlayer coupling; conversely, a smaller contact area leads to unstable mechanical transmission and weakened coupling. The formula introduces the reciprocal of the contact area. The text emphasizes the weakening effect of insufficient contact area on coupling strength. For example, in uneven stacking or when there is localized slippage on the surface of the carton, the reduction in contact area directly leads to a weakening of the interlayer mechanical connection.

[0064] Example 7: Using the following formula, combined with the strain energy density field, interlayer coupling tensor, deformation characteristic tensor, and dynamic stability function, nonlinear response spectrum analysis is performed to obtain the nonlinear response spectrum function R:

[0065]

[0066] in, This is the Fourier transform function.

[0067] Specifically, the first part of the formula is the effect of the dynamic stability function S. The dynamic stability function S itself is a time function of the stability state of the stacked system, describing the dynamic response trend of the stacked system under external forces. The formula further captures the influence of the interlayer coupling tensor C on stability by combining S with an exponential decay factor. (Exponential decay factor) The physical significance lies in quantifying the nonlinear relationship between interlayer coupling and deformation and energy density gradients. The norm ∥C∥ of the interlayer coupling tensor C represents the interlayer coupling strength of the entire stacked system, while... This is the Hadamard product norm of the gradient field of the dynamic stability function and the strain energy density field, reflecting the driving force of deformation and energy change on the coupling effect. By normalizing ∥C∥ to the norm of the gradient field, the formula effectively captures the relative relationship between interlayer coupling and deformation driving force. When the interlayer coupling is strong and the driving force is weak, the exponent value is small, indicating high system stability; while when the coupling is weak or the driving force is too large, the exponent value decreases rapidly, reflecting the tendency of the stacked system to become unstable. The Fourier transform of the formula... The dynamic stability function S and its modulation factor are transformed in the frequency domain. The Fourier transform is an important tool in dynamic analysis, revealing the dynamic response characteristics of a stacked system at different frequencies by converting a time-domain signal into a frequency-domain signal. Specifically, the Fourier transform can identify the natural vibration frequencies of the stacked system and the response intensity at a specific external force frequency. The dynamic behavior of nonlinear systems often contains multiple frequency components, and the Fourier transform can decompose these components, aiding in the analysis of the system's dynamic stability. For example, a significant amplification of certain frequency components may indicate a risk of instability in the stacked system at the corresponding frequencies. The formula extracts frequency domain information through the Fourier transform, providing a deeper analysis of nonlinear dynamic stability. The formula also includes an energy weighting factor. This is used to adjust the contribution of the strain energy density field E to the nonlinear response spectrum. The strain energy density E is a quantitative indicator of the elastic energy stored within the system, and E... c This is the critical energy density of the stacked system. Through normalization, the formula incorporates the ratio of the current energy state to the critical energy density into the response spectrum analysis. When E approaches Ecritical, the energy weighting factor increases, indicating that the influence of internal energy storage on the dynamic response is enhanced; conversely, when E is much smaller than Ecritical... c When the energy weighting factor decreases, it indicates that the system is in a lower energy state and contributes less to stability. This design reflects the central role of the energy state in the nonlinear response of a stacked system. (The last part of the formula...) This further captures the influence of interlayer friction and mass difference on the response spectrum. Here, the friction coefficient μ reflects the sliding resistance of the contact surface, while the mass difference |M k -M k+1 The | indicates the disturbance of mechanical transmission caused by interlayer mass inhomogeneity. By accumulating these factors layer by layer, the formula effectively describes the weakening effect of interlayer inhomogeneity on the dynamic response of the entire stacked system. For example, when the mass difference between some layers is too large or the friction coefficient is low, the product term in the formula will significantly reduce the value of the nonlinear response spectrum function, reflecting the instability of the interlayer contact.

[0068] Example 8: Nonlinear correction is performed on the nonlinear response spectrum function using the following formula:

[0069]

[0070] Where G is the nonlinear correction value; det|·| is the matrix determinant operator.

[0071] Specifically, the core structure of the formula is the second-order partial derivative matrix, which uses the second-order partial derivatives of R and C as elements to reflect the nonlinear response characteristics of the stacked system in the x, y, z spaces. These partial derivatives describe the dynamic changes and interactions of the system in different spatial dimensions. For example, the second derivative of the nonlinear response spectral function R... This represents the acceleration or curvature of the system in the x-direction; the larger the value, the more sensitive the system is to perturbations in that direction. Similarly, the partial derivative of the interlayer coupling tensor C... This captures the coupling changes between the x and y directions; these cross derivatives reflect the complex dynamic interactions between layers. The determinant calculation integrates these second-order rates of change into a single scalar value, reflecting the overall nonlinear characteristics of the system. The magnitude of the determinant can be interpreted as an index of the system's nonlinear strength; a large value indicates significant nonlinear behavior in multidimensional space, potentially leading to a higher risk of dynamic instability. The exponential decay factor of the strain energy density field E plays a crucial role in the formula. This factor is expressed as... Relating the current energy density E to the critical energy density E c The ratio is used to introduce the calculation of the nonlinear correction value. The strain energy density E is a scalar field of elastic potential energy stored in a material under deformation, while the critical energy density Ec c This is the reference value used in system design to define the instability boundary. When E is less than E... c When E approaches or exceeds E0, the exponential factor approaches 1, indicating that the current energy state has little impact on the nonlinear behavior of the system. However, when E approaches or exceeds E0... c When the exponential factor rapidly decays to near zero, it indicates that the system has entered a high-energy state and its dynamic nonlinear behavior is significantly enhanced. This design allows the formula to dynamically adjust the nonlinear correction weights in high-energy regions of the system. For example, in regions with high interlayer contact pressure, such as stack edges or areas with high interlayer contact pressure, local energy density increases are often observed. In these cases, the exponential factor can significantly reduce the contribution of these regions to the overall correction value, thus reflecting the potential risk of local dynamic instability.

[0072] The generation of the nonlinear correction value G is not merely a simple superposition of R and C, but a deep analysis of their interaction. The response spectrum function R describes the dynamic characteristics of the stacked system in the frequency domain, and its spatial variation reflects the system's response amplitude to perturbations at different frequencies. The interlayer coupling tensor C captures the mechanical interactions between the layers in the stacked system, including friction effects, differences in mass distribution, and the influence of contact area. By incorporating the higher-order partial derivatives of C into the correction matrix, the formula effectively quantifies the influence of interlayer interactions on the dynamic response. For example, when the coupling strength between certain layers is low (e.g., insufficient contact area or low friction coefficient), the partial derivative values ​​decrease significantly, thereby reducing the determinant value and reflecting potential weaknesses in these regions. The introduction of the exponential decay factor further demonstrates the moderating effect of energy state on nonlinear behavior. The dynamic behavior of a stacked system is often closely related to its energy distribution. In low-energy states, the dynamic response of the system tends to be linear, and interactions in all directions are relatively stable. However, in high-energy states, the accumulation of deformation and stress may trigger complex nonlinear behaviors, such as slip, local instability, or even overall collapse. Through the exponential factor... The formula dynamically captures the energy-driven characteristics of a system transitioning from stability to instability, thus allowing for reasonable adjustment of the correction value G. The formula's design, through a combination of determinants and exponential factors, systematizes the dynamic response characteristics of the stacked system into a scalar correction value G that is easy to analyze and interpret. The second derivative matrix in the determinant provides the foundation for capturing spatial nonlinearity, while the exponential factor further refines the physical meaning of the correction value through adjustments to the energy state. The resulting G reflects the stability state of the stacked system under complex dynamic environments, especially its nonlinear characteristics under high-energy or strongly coupled conditions. For example, in logistics transportation, cardboard box stacking systems may be subjected to frequent vibrations or impacts. By calculating the value of G, it is possible to identify which regions or directions in the stacked system have a higher risk of instability and provide guidance for optimizing the stacking structure.

[0073] Example 9: Calculate the stability value Λ of stacked cartons using the following formula:

[0074]

[0075] Where tr(·) is the trace operation of the matrix; ||·||2 is the L2 norm.

[0076] Specifically, the numerator of the formula combines the deformation characteristic tensor D, the interlayer coupling tensor C, and the spatial distribution characteristics of the stacking system. The deformation characteristic tensor D is the key description of the deformation of stacked cartons under external force. TThe trace operation of D quantifies the deformation intensity and direction at each detection point. The matrix trace operation reflects the combined change of the deformation tensor in all directions, describing the deformation distribution characteristics of stacked cartons in space. This term is achieved through the spatial integral ∫ V dV summarizes the deformation characteristics throughout the entire space of the stacked system, forming a comprehensive expression of deformation energy. The F-norm of the interlayer coupling tensor C is ∥C∥. F This quantifies the mechanical coupling strength between different layers, especially during the dynamic changes of cardboard box stacking, where the influence of interlayer friction, mass distribution differences, and contact area on overall stability cannot be ignored. By combining these two aspects, the formula can comprehensively consider the deformation and interlayer interaction of the stacked cardboard box system from a mechanical perspective, thereby evaluating the stability of the stacking system. Secondly, the formula adjusts the influence of the nonlinear correction value G on the stability value through max(|G|, 0.5). The nonlinear correction value G is calculated based on the behavior of stacked cardboard boxes in complex dynamic environments, reflecting the nonlinear response strength of the stacking system when subjected to external disturbances (such as vibration, impact, or external force loading). By comparing the calculated result of the correction value with 0.5, the formula can provide stable calculation results when G is small, preventing calculation instability caused by excessively strong or weak nonlinear correction. In this way, the influence of nonlinear response on the stability value is reasonably adjusted, ensuring that the stability value remains within a physically reasonable range under different dynamic conditions.

[0077] Next, the exponential decay factor in the formula This is a correction to the strain energy density field E, reflecting the stability of the stacked system under different energy states. The strain energy density E represents the energy stored in the stacked cartons under stress, while E... c This is a preset critical energy density, marking the energy boundary from stability to instability of the system. When the energy state of the stacked system approaches or exceeds E... cWhen the exponential factor decays rapidly, it indicates that the system's stability weakens and the risk of instability increases. Conversely, when the stacked system is in a lower energy state, the exponential factor approaches 1, indicating that the system is more stable in this state. In this way, the formula can dynamically adjust the stability assessment of the stacked system, ensuring that the influence in the high energy density region is appropriately amplified, while maintaining a smaller influence in the low energy state. In addition, the formula also introduces the L2 norms ∥R∥2 and ∥S∥2 of the nonlinear response spectrum function R and the dynamic stability function S to measure the dynamic response strength and stability of the stacked system under different frequency conditions. The L2 norm ∥·∥2 is a measure of the total energy or overall size of the function, and ∥R∥2 and ∥S∥2 reflect the overall response strength of the nonlinear response spectrum and the dynamic stability function. By correcting the ratio of these two, the formula can identify the stability performance of the stacked system under specific dynamic conditions, especially when stacked cartons are prone to instability under high-frequency dynamic disturbances, accurately reflecting this trend. Finally, the formula... The influence of interlayer deformation differences is further introduced. This term captures the contribution of deformation differences between adjacent layers to the stability of the stacked system. Deformation difference ∥ D k -D k+1 ∥ F This is the F-norm difference in deformation between adjacent layers, reflecting the degree of deformation variation between different layers. When the deformation difference between adjacent layers is large, the coupling effect weakens, and the stability of the stacked system decreases accordingly. Through this term, the formula can accurately incorporate the interlayer deformation difference into the stability assessment, reflecting the stability of different layers of the stacked carton system under dynamic loading.

[0078] The preferred embodiments of this disclosure have been described above with reference to the accompanying drawings, but this does not limit the scope of the claims of this disclosure. Any modifications, equivalent substitutions, and improvements made by those skilled in the art without departing from the scope and substance of this disclosure shall be within the scope of the claims of this disclosure.

Claims

1. A cardboard box stacking stability evaluation system based on dynamic deformation monitoring, characterized in that, The system comprises a sensor section, a dynamic stability analysis section, and a stability evaluation section. The sensor section includes multiple sensor units, each positioned at a pre-selected detection point within the stacked cartons. Each sensor unit collects real-time displacement and shear deformation data at the detection point. The dynamic stability analysis section constructs a deformation characteristic tensor of the stacked cartons based on the real-time displacement and shear deformation of each detection point. It calculates the strain energy density field and constructs a dynamic stability function based on the deformation characteristic tensor, strain energy density field, and dynamic stability function. It analyzes the interlayer coupling effect during stacking based on the deformation characteristic tensor, strain energy density field, and dynamic stability function, calculating the interlayer coupling tensor. The stability evaluation section combines the strain energy density field, interlayer coupling tensor, deformation characteristic tensor, and dynamic stability function to perform nonlinear response spectrum analysis, obtaining a nonlinear response spectrum function. It then performs nonlinear correction operations on the nonlinear response spectrum function to obtain a nonlinear correction value, calculates the stability value of the stacked cartons, and compares the calculated stability value with a preset stability threshold. If the value is lower than the set stability threshold, the stacked cartons are deemed unstable, and an early warning is issued. All cartons in the stack are made of the same material.

2. The cardboard box stacking stability evaluation system based on dynamic deformation monitoring as described in claim 1, characterized in that, The sensor unit includes a laser displacement sensor and a strain gauge; the sampling frequency of the laser displacement sensor and the strain gauge is at least 100 Hz.

3. The cardboard box stacking stability evaluation system based on dynamic deformation monitoring as described in claim 2, characterized in that, The deformation characteristic tensor D is represented by the following formula: Where n is the number of detection points; Δx i (t) represents the displacement component along the X-axis at time t for the i-th detection point; Δy i (t) represents the displacement component along the Y-axis at time t for the i-th detection point; Δz i (t) represents the displacement component along the Z-axis at time t for the i-th detection point; γ ixy (t) represents the shear deformation of the i-th detection point in the X-axis direction toward the Y-axis at time t; γ ixz (t) represents the shear deformation of the i-th detection point in the X-axis direction toward the Z-axis at time t; γ iyx (t) represents the shear deformation of the i-th detection point in the Y-axis direction towards the X-axis at time t; γ iyz (t) represents the shear deformation of the i-th detection point in the Y-axis direction toward the Z-axis at time t; γ izx (t) represents the shear deformation of the i-th detection point in the Z-axis direction towards the X-axis at time t; γ izy (t) represents the shear deformation of the i-th detection point in the Z-axis direction toward the Y-axis at time t; r i L represents the position vector of the i-th detection point; c The characteristic length is defined as one-third of the average height of all cartons in a stack; ||·|| is the modulus operation of the vector.

4. The cardboard box stacking stability evaluation system based on dynamic deformation monitoring as described in claim 3, characterized in that, The strain energy density field E can be calculated using the following formula based on the deformation characteristic tensor: Where C is the elastic property tensor of the material of the stacked cardboard boxes; : represents the double dot product operation between tensors; α is the weighting factor, used to adjust the influence of deformation gradient on strain energy density, and is a set value; V represents the spatial region where all detection points are located; dV is the spatial integration variable; This is the gradient operator.

5. The cardboard box stacking stability evaluation system based on dynamic deformation monitoring as described in claim 4, characterized in that, The dynamic stability function S is constructed using the following formula: Where dτ is the time integration variable; E c The critical energy density; ||·|| F For F-norm operations; D max This represents the maximum allowable deformation of all cartons in a stack.

6. The cardboard box stacking stability evaluation system based on dynamic deformation monitoring as described in claim 5, characterized in that, When analyzing the interlayer coupling effect during the stacking process, the stacked cartons are divided into multiple layers from top to bottom, with multiple detection points in each layer. The interlayer coupling effect during the stacking process is analyzed using the following formula, based on the deformation characteristic tensor, strain energy density field, and dynamic stability function, to calculate the interlayer coupling tensor C: Where μ is the coefficient of friction of the cardboard box surface; N is the number of layers of the stacked cardboard boxes; D k Let D be the deformation feature tensor of the k-th layer; k+1 The deformation feature tensor of the (k+1)th layer; A k，k+1 M represents the contact area between the k-th stacked cardboard box and the (k+1)-th stacked cardboard box; k M represents the mass of the k-th layer of stacked cartons; k+1 Let the mass of the (k+1)th layer of stacked cartons be denoted as 'k'. For Hadama accumulation.

7. The cardboard box stacking stability evaluation system based on dynamic deformation monitoring as described in claim 6, characterized in that, Using the following formula, combined with the strain energy density field, interlayer coupling tensor, deformation characteristic tensor, and dynamic stability function, nonlinear response spectrum analysis is performed to obtain the nonlinear response spectrum function R: in, This is the Fourier transform function.

8. The cardboard box stacking stability evaluation system based on dynamic deformation monitoring as described in claim 6, characterized in that, The nonlinear response spectrum function is nonlinearly corrected using the following formula: Where G is the nonlinear correction value; det|·| is the matrix determinant operator.

9. The cardboard box stacking stability evaluation system based on dynamic deformation monitoring as described in claim 8, characterized in that, The stability value Λ of stacked cardboard boxes is calculated using the following formula: Where tr(·) is the trace operation of the matrix; ||·||2 is the L2 norm.

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