An investigation into the effect of cross-ply on energy storage and vibration characteristics of carbon fiber lattice sandwich structure bionic prosthetic foot

Made a pioneering attempt to use the lattice sandwich structure in prosthetic foot design and pioneered the study for the lay-up design of the prosthetic foot. An innovative carbon fiber bionic prosthetic foot was designed using a sandwich structure. The effect of cross-ply on the prosthetic foot’s energy storage properties and vibration characteristics was investigated using the lattice sandwich structure prosthetic foot. The bionic prosthetic foot’s finite element model was constructed under normal working conditions according to international standards. The results indicate that the storage of strain energy increases with an increase in cross-ply under heel-strict working conditions. Under the toe-off condition, the strain energy distribution increases with the increase in cross-ply. The cross-ply number influences the mode of displacement of the bionic foot. The natural frequencies of the bionic foot increase with the increase in the cross-ply.


Introduction
Due to the anisotropic mechanical properties of fiber-reinforced composites, changes in layup design can cause significant responses in energy absorption [1]. The stiffness is directly related to the energy storage characteristics of the foot and ankle prosthesis, and both energy storage characteristics and vibration characteristics are important design criteria for foot and ankle prostheses.
Sandwich structure composites have been applied in aerospace, automobiles, and other fields with higher specific stiffness and strength than carbon fiber laminate structure composites. In this article, we attempt to design a prosthetic foot using sandwich composites and investigate the effect of sandwich structural panel layup design on the energy storage and vibration characteristics of the prosthetic foot.
Cross-layered structures were applied to the layup design of composites. In recent years, the lay-up design of carbon fiber-reinforced composites has been studied a lot.
The Bouligand structure, which is prevalent in fish scales and arthropod cuticles, is a fibrous laminate in which the direction of the fibers gradually changes across the thickness [2]. Recently, researchers have become more interested in complex three-dimensional fracture processes (crack twisting) as a potential source of toughness. In their investigation of the bending of antisymmetric cross-ply laminates, Rodrigues et al. [3] selected five of the most well-liked high-order shear deformation theories (HSDTs) that have been reported in the literature. The Radial Point Interpolation Method was utilized to research the bending of antisymmetric cross-ply laminates using HSDTs. Different laminates were analyzed using the radial point interpolation method and HSDTs. The accuracy and robustness of the numerical technique are demonstrated by comparisons between meshless solutions and analytical and finite element method solutions that are published in the literature. Sasikumar et al. [4] created an asymmetrical laminate by combining plies of varying thicknesses. The suggested unsymmetrical hybrid laminate's damage sequence was compared to that of the thin-ply baseline using C-scan inspection on impacted and quasi-statically indented specimens. The expected delamination area was lowered by the hybrid laminate's bottom intermediate plies, which also delayed and reduced fiber damage. To determine the lay-up of composite laminates with several plies, Fedon et al. [5] presented a brand-new deterministic optimizer. A method that employs beam search to explore the design space encourages quick convergence to optimum or nearly optimal solutions. It is demonstrated that the suggested optimizer can quickly recover symmetric laminate layups with up to 300 plies. Ply drop sequence (PDS) is a concept that Yang et al. [6] proposed for building composite laminate constructions with many sections. For the PDS-based blending optimization, a genetic algorithm (GA) with specific operators and codification is used, which ensures that the design is thoroughly blended within the GA rounds. To show the adaptability and potential of the suggested strategy, it is applied to a benchmark issue with 18 panels called the horseshoe problem. An innovative method for the vibration and damping analysis of arbitrary curved n-layered sandwich beams is presented by Arikoglu and Ozturk [7]. A parametric mid-section curve and a frequency-dependent viscoelastic core in a spiral-shaped sandwich beam are explored. The analysis of the core thickness and subtended angle's impact on vibration and damping behavior is done in great detail. Xu et al. [8] provided a governing equation for forecasting the free vibration of the graded corrugated lattice core sandwich beam using the continuous homogeneous theory. To solve the governing equation for various kinds of boundary conditions, the Rayleigh-Ritz technique is used. The theoretical predictions are verified by numerical simulations and experimental findings, and the natural frequencies are determined. The effects of the face sheet thickness, core height, beam length, and graded parameters were investigated on the natural frequencies of the graded corrugated lattice core sandwich beams. The vibrational behavior of smart orthotropic cross-ply laminated stepped beams is studied by Fazeli et al. [9]. To determine the natural frequencies and forced vibrational response of a smart carbon fiber/poly ether-ether-ketone cantilever beam, experimental research was conducted to test the correctness of the analytical approach. Comparisons show how effectively and accurately the offered closed-form technique can forecast the free and forced vibrational behavior of smart-stepped laminated composite beams.
The vibration characteristics and mechanical properties of cross-layered structures have been studied in depth in theory and experiments, but relatively little research has been done on the mechanical and vibration characteristics of cross-layers under specific operating conditions. Although sandwich structure composite materials have been applied in many fields, prosthetic foot manufacturing materials are mainly laminated. The following literature discusses the stiffness research methods of lattice sandwich structures.
Zhang et al. [10] suggested a brand-new pyramid lattice sandwich construction with active variable stiffness, The structure exhibits good active and self-adjusting vibration reduction. The dynamics equations for the entire structure are developed using the assumed modes approach, Hamilton's principle, and supersonic piston theory. Xiao et al. [11] suggested using multiscale topology optimization as a design strategy for graded lattice sandwich structures (GLSSs). The distribution optimization of lattice cells in the core layer and the thickness optimization of two solid face sheets are both implemented based on the Kriging metamodel. To demonstrate the benefits of the suggested strategy, numerical examples of compliance and natural frequency optimization of GLSSs are given. The findings of bending tests show that GLSSs have bigger natural frequencies and are stiffer than uniform lattice sandwich constructions. According to Ghannadpour et al. [12], who introduced 10 strut-based topologies of the latticecore sandwich beams, the octet-truss structure resisted far larger deflection, star and diamond structures had higher stiffness, and they are the best option in the sandwiches under compressive and bending loads. It was clear that Grid had a superior reaction under compressive loads while having the poorest response to three-point bending. A novel heterogeneous glass sponge lattice structure with a circle-/ grid-like appearance (GSLS) was created by He et al. [13]. The FEA simulation demonstrates that the heterogeneous unit cell in the GSLS may improve strut connection, distribute stress, and display a distinct fracture process of cell-after-cell and layer-by-layer. According to an approach presented by Bai et al. [14], new types of curving lattice structures may be created using a circular or elliptical arc in the lattice struts. Its mechanical qualities were significantly improved by changing the original stress distribution under the loads and efficiently relieving the stress concentration at the struts' nodes.
The fundamental theory of lattice structure, design and preparation techniques, and stiffness and damage modes of lattice sandwich structures are all covered in earlier research on this topic. Lattice sandwich technology has never been used in a prosthetic foot. The lattice sandwich structure is employed to offer substantial stiffness for research into the bionic performance of prostheses because it possesses the properties of energy absorption cushioning, low relative density, high specific stiffness, and specific strength.
Stiffness is an important design index of the prosthetic foot. Adamczyk et al. [15] developed and assessed a technique for measuring the angular stiffness of the prosthetic foot using a straightforward linear compression test by calculating the angular stiffness as a function of tower angle, normal force, and center of pressure concerning the rate of change of linear displacement. In an experimental foot prosthesis, Adamczyk et al. [16] adjusted the stiffness of the forefoot and the hindfoot. They then calculated the results of these computations, including the prosthesis' energy return, center of mass mechanics, ground reaction forces, and joint mechanics, and determined how sensitive these variables were to component stiffness. To determine the impact of foot stiffness on kinematics, kinetics, muscle activity, prosthetic energy storage return, and mechanical efficiency during amputee walking, Fey et al. [17] carried out a thorough biomechanical investigation. Furthermore, it identifies the impact of changed prosthetic foot stiffness on muscle and foot function using forward dynamics models of amputee walking [18]. Five experimental feet were utilized in the investigation to assess the impact of prosthetic foot forefoot flexibility on oxygen costs [19]. Flexibility has a big impact on the preference ranking. The number of flexural hinges in the forefoot regions of the experiment was changed to affect flexibility. According to the study's findings, solid-ankle prosthetic foot models with excessively flexible forefoot parts may have a "drop-off" impact in late stance and when switching loads between the prosthetic and contralateral limbs [20].
Bionic methods based on gait analysis, anthropometric data, exercise oxygen consumption, and mechanical performance tests are mostly used in the structural design of the prosthetic foot.
Hamzah and Ghannadpour [21] presented a brandnew carbon fiber-epoxy composite ankle prosthesis concept. The design has a good energy return reaction and a smooth roll-over shape. Jang et al. [22] created a flexible keel for an energy-storing prosthesis while creating a cost function for the keel's performance assessment. The contribution of each element to the cost function was explored. The ideal flexible keel was made with better energy storage capacity and is appropriate for more vigorous prosthesis walking. Alleva et al. [23] constructed a dimensionless kinematic model of the lower limb to determine the angle of the lower limb during gait. following the creation of a prototype prosthesis with a mechanical design. The ankle joint angle during stride resembled that of a healthy person, according to early tests of the prototype performed by itself in the lab.
Previous studies have shown that stiffness and energy storage characteristics are important indicators affecting the performance of prosthetic foot. However, the effects of cross-ply on the energy storage characteristics and dynamic character of prostheses foot are unclear. This article presents an innovative lattice sandwich structure bionic prosthetic foot, introducing a pyramid lattice sandwich structure to provide walking stiffness and determine its walking characteristics according to ISO standards. Finally, the effect of the layered design of the lattice sandwich structure on the energy storage characteristics and vibration performance of the prosthetic foot is discussed.

Methods
Energy storage characteristics and vibration characteristics are the key indicators of the prosthetic foot. In this article, we study the effect of cross-ply on the energy storage characteristics and vibration characteristics of prosthetic foot. When the structural system is deformed elastically under the action of external forces, energy is stored in the structure in the form of strain energy, and the larger the strain energy is, the better the energy storage characteristics of the structure. Therefore, this article adopts strain energy as an index to study the energy storage characteristics.
An innovative carbon fiber bionic prosthetic foot was designed using a sandwich structure. As the sandwich structure's character of high specific stiffness, the bionic prosthetic foot has sufficient structural stiffness. The large stiffness of the lattice structure is utilized to translate the energy to the cross-ply surface. Hence, to obtain the effect of the surface cross-ply on the energy store of the structure. The structural stiffness was analyzed using theory.
ISO10328 is a specified test method for examining the mechanical properties of the bionic foot. The standard specifies two typical working conditions, which are used to represent the extreme force conditions of a human walking with a bionic foot. Therefore, the mechanical properties of the bionic foot are simulated using the two conditions specified in ISO10328.
The bionic prosthetic foot's finite element model was constructed under normal working conditions according to ISO10328 standards. The typical conditions of the bionic foot during human walking with the bionic foot are calculated, and the results include the strain energy under the two conditions. The effect of cross-layering on the energy characteristics is then obtained by varying the cross-ply design.
The numerical model used for the study was validated by tests cited in the literature to demonstrate the reliability of the analytical data.
The vibration characteristics are the intrinsic frequency and the vibration mode. The intrinsic frequency is a direct indicator of the dynamic characteristics of the system, and the vibration mode corresponding to each order of the intrinsic frequency indicates the form of the system response under the applied excitation, which is a guideline for measuring the damage mode of the system.
Based on the validation of the numerical model, the free and constrained modes of the bionic foot were analyzed using the numerical model to obtain the inherent frequencies and vibration modes of the bionic foot. Different crosslayering schemes are simulated separately to study the effect of cross-ply on the vibration characteristics.

Bionic design
Sandwich structure prosthetic ankle-foot was designed according to anthropometry, with a total length (L) of 260 mm, and a width of 6 mm based on anthropometric data [24]. As shown in Figure 1, the prostheses model consists of an elastic energy storage ankle and a bionic foot. The ankle is designed as a large deformation flexible double-leaf spring to reduce impact and store the energy of heel strike. The bionic foot is composed of a heel, arch, and forefoot. The heel and forefoot are concave to provide smooth rolling, and the arch is convex to store middle stand energy. A pyramid lattice sandwich structure is designed for the foot, which is composed of upper and lower plates and a lattice sandwich, as shown in Figure 1. Figure 1(a) is the axonometric perspective view of the bionic foot, and Figure 1(b) is the lateral view of the foot.

Calculation of bionic prosthesis strain energy
The structure is made simpler as a cantilever model and a straightforward curved beam model by applying the homogenization procedure. The letters c and s stand for curved and straight (cantilever) beams, respectively. Axial, shear, and bending moment forces combine to produce a vertical deflection at the foot's point of contact with the ground. When the radius-to-thickness is more than 10, however, the effects of the axial stress and shear are minimal [25].
To determine the beam's deflection in this investigation, the strain energy that was exclusively attributable to bending was employed: where F is the ground reaction force of the composite foot, and the total strain energy U of the bionic prosthesis foot is expressed as follows: the strain energy of the straight part, U C is expressed as follows:

Numerical model
Numerical models were developed to analyze the effect of cross-ply on energy and dynamic properties. The strain energy of typical working conditions is used to represent the energy characteristics of the prosthesis. Free and constrained modes are used to analyze the vibration characteristics of the prosthesis. The stiffness under compressive loading is used to verify the accuracy of the model. Using the geometric model of the bionic prosthesis created in the previous section. The bionic prosthesis FE model was divided into HyperMesh using the solid mesh, then import the model into Abaqus. Using a "tie" connector to join the panels and cores of the sandwich structure, and the cell type is 8-node hexahedron elements C3D8R. Local coordinate systems were established to set up composite lay-ups. The composite materials were laid using discrete coordinate systems in the upper and lower panels of the sandwich structure, with the material stacking direction along the panel thickness and the 0degree fiber direction of the composite materials along the length of the prosthetic foot (Figure 1(a), L). Each core bar is laid separately using a local l-cylinder coordinate system with material stacking direction along the radius direction of the bar and 0-degree fiber direction along the length direction (Figure 1(a), L c ) of the core bar. The composite layups of all core bars were along 0 degree fiber direction; cross layup variations were set up in the upper and lower panels of the prosthetic sandwich structure. FE models were established for 0 cross-layers, two 45 ±°cross-layers, four cross-layers of 45 ±°, six crosslayers of 45 ±°, and eight cross-layers of 45 ±°, respectively. The models of each layup design were simulated under heel strike and toe-off according to the ISO10328 standard, free mode and constrained mode were also calculated. A total of 20 sets of simulations were performed. The results were used to analyze the effect of cross-layers on energy and dynamic characteristics.
The material of the simulation model was set according to the material parameters in the literature [26] using a highstrength carbon/epoxy prepreg (T 300), whose material properties are E11 132 GPa To replicate the testing conditions outlined in the ISO10328:2016 standard for lower-limb prosthetics, a rigid plate was placed at the appropriate angle on the toe and heel of the foot model. This setup ensured compliance with the requirements and test methods specified for the foot assembly. Elements on the sole blade and the rigidly specified plate were characterized as being in free frictional sliding contact. The FE model and a schematic diagram of walking with a prosthetic foot for these two working conditions are shown in Figure 2. The fixed support is at the ankle of the foot.
The stiffness of the foot is analyzed in Abaqus/ explicit. The finite element software ABAQUS is used to calculate the natural frequencies of the bionic prosthesis. A linear perturbation analysis step is created, and a frequency extraction procedure is carried out using the Lanczos solver. The first nine modes were obtained.

Model verification
The accuracy of the simulation model was verified using the carbon fiber lattice sandwich structure foot and ankle prosthesis vertical compression test in the literature [26]. In the literature, YT 2017 06 -autoclave was used to prepare carbon fiber pyramidal sandwich prosthesis. The prosthesis was vertically loaded on MTS E45 testing machine according to ASTMD5961 standard, and the acoustic emission signal during the test was collected according to the nondestructive testing standard (Physical Acoustuc Co.PAC PCI-2, Micro-II Digital AE System). The load-displacement curves of the prosthesis were obtained during the test. The carbon fiber material used in the literature, the sandwich structure parameters, and the loading condition are consistent with this article. Therefore, the simulation reliability of this article is verified by using the results of the literature ( Table 1).
The accuracy of the model can be demonstrated by comparing the reaction force corresponding to the maximum displacement of the simulation model with the test data in the literature, when the displacement is 35 mm, the simulated reaction force of the bionic foot is 644.356 N, and the prosthetic test force data in the literature is 571.420 N. The magnitude of the two sets of data is the same, indicating that the results calculated using the finite element model are accurate.

Static analysis
A numerical simulation was conducted to analyze the static characteristics of the bionic prosthesis. The study compared the effects of different configurations, specifically 2, 4, 6, and 8 layers of ±45°cross-ply. The strain energy distribution in the upper and lower sheet regions of the prosthesis lattice sandwich structure is more concentrated; we choose to show the strain distribution in this region; and the location and perspective of both sheets in the geometric system of the prosthesis are shown in Figure 3. The specific strain values of other regions are compared in Figure 4. The strain distribution of the bionic foot is studied under the toe-off conditions, as shown in Figure 3. Under this condition, the strain of the bionic foot is mainly distributed at the heel, where the strain distribution on the upper plate is more than that on the lower plate, and the strain is the most concentrated at the contact point of the face and core of the upper and lower plates.
The toe-off condition strain distribution in the upper and lower panels of the sandwich structure of the prosthesis foot is shown below: The strain distribution is linear in the non-cross-ply bionic foot and gradually changes to a planar distribution in the 2-8 ply bionic foot.
The strain distribution in the up sheet of the toe-off condition is better than in the down sheet. 2,4,6 Cross-ply has the best strain energy stored in both heel-strict and toe-off conditions.
The general concept of energy storage and release in the prosthetic foot is that the strain energy is stored in the structural element, and the strain energy stored in the ankle, the up sheet, the down sheet, and the core in the heel-strike condition are shown in Figure 4.
The strain on energy storage increases with the crossply increase in heel-strict conditions. Most strain energy is stored in the ankle of the prosthesis foot under the heel-strict condition, and the lattice sandwich core stores the least energy in the heel-strict condition. The lower sheet store strain energy increases with the increase in cross-ply numbers.
The storage of strain energy is almost the same as the increase in cross-ply in toe-off conditions, and the up sheet stores most strain energy in toe-off conditions. The core structure stores the least strain energy in both heel-strict and toe-off conditions.

Mode analysis
Mode analysis is performed to measure the dynamic characteristics of different cross-ply foot systems. Three freemode shapes are obtained. The mode shapes of free mode are shown in Figure 5. The first mode is bending vibration, the third mode is second-order bending, the fifth mode is third-order bending vibration, and the sixth mode is bending. The second-order model is the first-order plane water ripple vibration, and the fourth-order mode is the second-order plane water ripple vibration. The seventh mode corresponds to the third-order water ripple combined torsional mode. The eighth and ninth modes are torsional.
The dangerous mode shapes of free mode appear in the first, third, and fifth mode shapes.
The constrained modal vibration mode of each ply model is the same. The constrained modal vibration mode diagram of the 0 cross-ply prosthesis foot is as follows: The constrained mode shapes are shown in Figure 6. The first mode shapes are pure bending, the second mode shapes are torsion, the third mode shapes are bending, the fourth mode shapes are bending, the fifth mode shapes are bending, the sixth, seventh, and eighth mode shapes are bending torsion combination, and the ninth mode shapes are bending.
The first, third, fourth, and ninth modes are the dangerous modes of constrained mode.
The free mode and construction mode maximum displacements of the first nine modes for all cross-ply numbers of the prosthesis foot are shown in Figure 7.
The free mode displacement of the bionic foot without cross-ply is the smallest in the first mode displacement, the largest in the ninth mode displacement, the first to third mode displacement gradually increases, the third to fifth mode displacement gradually decreases, the fifth to seventh mode displacement slowly increases, and the seventh to ninth free-mode displacement sharply increases.
For the free-mode displacement of the bionic foot with cross-ply, the second-mode displacement is the smallest and the eighth mode displacement is the largest.
Among the constrained modal displacements of the bionic foot without cross-ply, the fifth modal displacement is the smallest, the second modal displacement is the second, and the eighth constrained modal displacement is the largest. The first to second modal displacements decrease, the second to third modal displacements increase, the third to fifth modal displacements decrease, the fifth to eighth modal displacements increase, and the eighth to ninth modal displacements decrease.
In the constrained mode with the cross-ply bionic foot, the third and fifth-order modes have the smallest displacement, the seventh-order mode displacement is the largest, the first to second descending mode displacement decreases, the second to third-order constraint mode displacement increases, the third to fifth-order constrained mode displacement decreases, the fifthto seventh-order constrained mode displacement increases, and the seventh-to ninth-order constrained mode displacement decreases.
The peak value of free-mode displacement appears in the seventh order, and the peak value of constrainedmode displacement is in the eighth order. The cross-ply has little effect on the first three modes and has an obvious effect on the high-order mode displacement. The cross-ply has little effect on the fourth-and fifthorder displacement of free mode.
The fifth-order displacement of the free mode is the smallest. The first nine modal displacements of the bionic foot without cross-ply are different from those of the bionic foot with cross-ply. The displacements of the second-, third-, and eighth-order free modes differ greatly, and the displacements of the first-, eighth-, and ninth-order constrained modes differ significantly from those of other constrained modes.
The free mode and construction mode frequencies of the first nine modes for all cross-ply numbers of the prosthesis foot are shown in Figure 8.
In the free modal frequencies of the bionic foot of each layer, the frequencies from the first mode to the fourth mode increase slowly, the frequencies from the fourth mode to the sixth mode increase sharply, the frequencies of the sixth and seventh modes are similar, and the frequencies from the seventh to the ninth modes increase gradually.
The free mode frequencies of the bionic foot without cross-ply and the bionic foot with cross-ply have little difference in the first four steps, and the difference is Figure 5: The first nine mode shapes of non-cross-ply prosthesis foot: 1-first mode shape, 2-second mode shape, 3-third mode shape, ... , 9-ninth mode shape.
obvious from the fifth to the eighth step. The changing trend of the mode frequencies of the bionic foot with cross-ply and the bionic foot without cross-ply is different from the eighth to the ninth step.
For the bionic foot with cross-ply, the first four free mode frequencies are similar, and the difference in natural frequencies increases from the fifth mode. The free mode frequencies of two cross-ply bionic foot and four cross-ply bionic foot are highly overlapped. With the increase in the number of cross-ply, the natural frequency of free mode increases.
The overall trend of the constrained modal frequency of each bionic foot is as follows: the frequency increases gently from the first-order constrained modal to the thirdorder constrained modal, sharply from the third-order to the fifth-order modal, slowly from the fifthto the seventh- Figure 6: The first nine modes of constrained mode shapes of non-cross ply prosthesis foot: 1-first mode shape, 2-second mode shape, 3-third mode shape, ... , 9-ninth mode shape. order modal, and sharply from the seventh-to the ninthorder constrained modal.
Compared with the bionic foot without a cross-ply, the constrained modal frequencies of the bionic foot with a cross-ply are higher than those of the bionic foot. There is little difference in the first three constrained modal frequencies of the bionic foot with various cross layers, but with the increase in modal order, the gap between the modal frequencies of the bionic foot with various layers increases.
Except for the sixth mode, the frequencies of the constrained modes of the bionic foot with a two-layer crossply and the bionic foot with a four-layer cross-ply are similar. The constrained modal frequency of the bionic foot increases with the increase in the number of attached cross plies. The frequency of the first three constrained modes increases slowly, and that of the fourth to ninth constrained modes increases with the increase in the number of cross plies.
With the increase in cross-ply, the natural frequency will increase, and the influence in orders 5-9 is greater than that in orders 1-4. There is no significant difference between two layers and four layers. The first three natural frequencies increase slowly, and the higher modes increase sharply. The natural frequencies of order 7 are similar.
The influence of cross-ply on the first three modes of constrained mode is less than that on the fourth to ninth modes. The difference in constrained mode frequencies between the second and fourth layers is small, and 4-6-8 increases gradually. The second and third modes have similar vibration frequencies. 5-to 7-order frequencies increase slowly, and other modal frequencies increase sharply with the increase in order.  This study has two limitations. The first limitation is that we cannot find enough peer-reviewed studies on the subject. Therefore, we used approximate test data from previous studies to verify the simulation model. The second limitation is cost constraints. Our funds are not enough to support us to design a special testing system to complete the testing of foot and ankle prosthetics, which somewhat weakens the convincing of the results. We will provide further data in subsequent studies.

Conclusions
A lattice sandwich bionic prosthetic foot is designed and simulated according to the ISO 10328 standard. The influence of cross-ply on the energy storage characteristics of the prosthetic foot is analyzed. The free modes and constrained modes of each cross-ply bionic prosthetic foot structure are calculated, respectively. The effects of cross-ply on modal shapes and natural frequencies are analyzed, leading to the identification of dangerous working conditions for the lattice sandwich bionic foot.
Overall, our findings show that the storage of strain energy increases with the rise of cross-ply under heelstrict working conditions, while the strain energy distribution increases with the increase in cross-ply under toe-off conditions. We also discovered that the dangerous modes of free mode are the first, third, and fifth modes, while the dangerous modes of constrained mode are the first, second, fifth, and ninth modes. Furthermore, we found that the natural frequencies of both free mode and constrained mode of the bionic foot increase with the increase in cross-ply, and that cross-ply and non-cross-ply have  different effects on the free mode and constrained mode displacement of the foot. The method we employed in this study can be applied to the optimization of ply for composite structures and to the study of the energy storage and vibration properties of prosthetic foot. Our results provide a valuable contribution to the development of prosthetic devices and offer insight into the potential for further research in this field.