Buffering and energy-absorbing characteristics of disc spring composite monomer under impact dynamic load

Abstract

In deep coal mines, roadway supports endure intense impact ground pressure from rock bursts, severely challenging the load-bearing capacity of underground support equipment. Yet, research on the energy buffering mechanisms of disc springs under such loads remains scarce, impeding the development of optimized impact-resistant support structures for underground environments. This study aims to design a novel modular disc spring-type buffering and energy-absorbing device installed on hydraulic support top beams to mitigate impact damage (e.g., column fracture, cylinder explosion) caused by rock bursts. A hybrid methodology integrating physical drop hammer impact tests (validated using pressure sensors) and dynamic simulations was employed. An ADAMS dynamic simulation model was constructed to compare performance discrepancies between flexible and rigid Disc Spring Composite Monomers under variable loads ranging from 0 to 7500 kg. The flexible Disc Spring Composite Monomer reduced peak support reaction by 10% versus rigid counterparts, exhibited higher rebound height and longer buffering time, and effectively suppressed displacement mutation; its load-deformation relationship followed a sub-linear growth trend, showing high sensitivity at low loads and stiffness-driven saturation at high loads. Flexible Disc Spring Composite Monomers demonstrate superior energy absorption, peak load suppression, and stability against repeated impacts, providing a new technical pathway for impact-resistant roadway support design.

Introduction

In recent years, as coal resource mining has gradually moved to deep areas, the mine working environment has become increasingly complex, and dynamic disasters such as rock bursts occur frequently, which pose a great threat to the safe production of mines. The occurrence of rock bursts usually leads to serious damage to roadways, casualties, and even local earthquakes and surface damage. Therefore, how to effectively prevent and deal with impact loads has become an important research topic in current coal mine engineering^1,2,3,4,5. Previous studies have mainly focused on the prediction, monitoring and early warning of rock bursts, while the research on roadway support technology has lagged behind. As the last line of defense against impact loads, the design and research of roadway support are crucial to the safety of mines^6,7,8.

In the technology of dealing with impact loads, metal sandwich structures are widely used in various protective devices and buffer structures because of their excellent impact resistance. For example, Wu Yongzheng et al.⁹ proposed the ‘pressure relief-support–protection’ collaborative prevention and control technology, which combines hydraulic fracturing pressure relief, four - high bolt support and composite energy-absorbing structure protection, effectively reducing the impact energy and improving the stability of surrounding rocks. Dai Lin et al.¹⁰ designed a vertical impact test bench with a combined power source of a hydraulic pump and an accumulator, and realized high - precision synchronous braking through fuzzy PID control, which solved the problem of secondary impact under heavy loads and provided an effective control strategy for improving the impact protection performance. In addition, Liu Chengzhi et al.¹¹ optimized the dynamic response of the active protection system under wire impact based on multi-body dynamics simulation technology, and improved the energy dispersion and impact resistance of the system. Although the above studies have made significant progress, there is still much room for improvement in the design and optimization of roadway protection support structures, especially in the research combining new materials and structural forms.

As a material with excellent elasticity and energy-absorbing performance, disc springs have gradually attracted attention in impact protection applications. Disc spring structures can effectively absorb and disperse impact energy through their unique geometric shape, and have been applied in shock absorption devices, impact energy-absorbing devices and other fields¹². However, the current research on disc springs in impact protection is still in the initial stage, especially the energy - absorbing mechanism and mechanical properties of disc springs in low-speed impact environments have not been fully studied^13,14,15,16. The research on the application of disc springs as impact protection materials in complex environments such as coal mine roadways is relatively scarce, and the optimization of key parameters such as geometric size design and connection methods still needs in-depth exploration^{17,18,19,20,21,22,23}.

Based on the existing academic achievements, this study innovatively carries out experimental research by adjusting the load parameters of the load plate, successfully designs a disc spring impact protection structure suitable for coal mine roadways, and empirically analyzes its protection efficiency through drop hammer impact tests to explore its application value in the field of coal mine roadway impact protection^24,25. Specifically, the research team accurately simulates impact load conditions of different intensities by systematically changing the mass of the load plate, so as to obtain the mechanical response data of disc springs in dynamic impact environments. In the research process, the experimental data are compared with the dynamic model built on the ADAMS platform to effectively verify the reliability and accuracy of the simulation model. Based on the above research results, the bearing performance and energy-absorbing characteristics of the disc spring structure under various load conditions are further analyzed. Finally, a new disc spring-type buffering and energy-absorbing device integrated with disc spring composite monomers is proposed, which provides a new technical path and theoretical reference for the innovation and development of coal mine roadway support technology.

Model construction

Mathematical model

As shown in Fig. 1, this study proposes and designs a new disc spring-type buffering and energy- absorbing device integrated with disc spring composite monomers, which can be installed on the top beam of the support to effectively improve the energy absorption and structural protection capabilities under impact loads.

The disc spring composite monomer is composed of eight-disc springs, including four large disc springs and four small disc springs, as shown in Fig. 2. The large disc spring has an outer diameter D=125 mm, an inner diameter d=64 mm, a thickness t of 8 mm (7.5 mm in special working conditions), a single piece free height H₀ = 10.6 mm, and a deformation h₀ = 2.6 mm in the flattened state without a supporting surface; the small disc spring has an outer diameter D=80 mm, an inner diameter d=41 mm, a thickness t=5 mm, a single piece free height H₀ = 6.7 mm, and a flattened deformation h₀ = 1.7 mm without a supporting surface. According to relevant standards, the material elastic modulus E=206,000 N/mm² and Poisson’s ratioµ = 0.3.

Fig. 1

Overall structure of a new type of buffering energy absorbing device.

Fig. 2

Disc spring composite monomer.

The load - bearing capacity F of a single disc spring is calculated as:

$$F=\frac{{4E}}{{1 - {\mu ^2}}} \cdot \frac{{{t^4}}}{{{K_1}{D^2}}} \cdot K_{2}^{2} \cdot \frac{f}{t}\left[ {K_{2}^{2}\left( {\frac{{{h_0}}}{t} - \frac{f}{t}} \right)\left( {\frac{{{h_0}}}{t} - \frac{f}{{2t}}} \right)+1} \right]$$

(1)

In the Eq., K₁ represents the geometric coefficient related to the diameter ratio of the disc spring; K₂ represents the correction coefficient considering the influence of the supporting surface, which is only applicable to disc springs with a supporting surface; h₀ represents the calculated value of the deformation of the disc spring without a supporting surface when flattened.

$${F_c}=\frac{{4E}}{{1 - {\mu ^2}}} \cdot \frac{{{h_0}{t^3}}}{{{K_1}{D^2}}} \cdot K_{2}^{2}$$

(2)

In the Eq. (2), F_c represents the calculated value of the disc spring load when flattened;

$${K_1}=\frac{1}{\pi } \cdot \frac{{{{[(C - 1)/C]}^2}}}{{(C+1)/(C - 1) - 2/\ln C}}$$

(3)

$$C=\frac{D}{d}$$

(4)

$${K_2}=\sqrt { - \frac{{{C_1}}}{2}+\sqrt {{{\left( {\frac{{{C_1}}}{2}} \right)}^2}+{C_2}} }$$

(5)

$${C_1}=\frac{{{{(t^{\prime}/t)}^2}}}{{[(1/4) \cdot ({H_0}/t) - t^{\prime}/t+3/4][(5/8) \cdot ({H_0}/t) - t^{\prime}/t+3/8]}}$$

(6)

$${C_2}=\frac{{{C_1}}}{{{{(t^{\prime}/t)}^3}}}\left[ {\frac{5}{{32}}{{\left( {\frac{{{H_0}}}{t} - 1} \right)}^2}+1} \right]$$

(7)

For disc springs without a supporting surface, the value of K₂ is set to 1; for disc springs with a supporting surface, the value of K₂ needs to be calculated according to Eq. (7). In the calculation process of Eq. (1) and Eq. (2), the corrected thickness is used $t{\prime }$ to replace the original thickness t, and ${h_0}^{\prime }={H_0}^{\prime } - t^{\prime}$ is used to replace h to ensure that the calculation results are in line with the actual working conditions.

The disc spring composite monomer adopts a hierarchical combination design: four large disc springs are connected in series in an opposed combination way, and four small disc springs are also connected in series in an opposed combination way. Under ideal working conditions where friction effects are neglected, the mechanical performance parameters of the opposed combined disc springs composed of i disc springs of the same specification can be calculated and analyzed based on the following theoretical model.

Where f_z represents the deformation of the combined disc spring; F_z represents the load of the combined disc spring; H_z represents the free height of the combined disc spring; H₀ represents the free height of a single disc spring.

$${F_z}=F$$

(8)

$${f_z}=i \cdot f$$

(9)

$${H_z}=i \cdot {H_0}$$

(10)

ADAMS dynamics simulation model

Due to the dual limitations of material structural characteristics and sensor measurement accuracy, it is difficult to obtain detailed data on the interaction force between disc springs and subtle deformations during the experiment. In view of this, this paper uses ADAMS View2020 software to carry out the simulation analysis of the impact test bench and build the corresponding simulation test model, as shown in Fig. 3.

In terms of constraint conditions, the supporting force contact is defined between the disc spring top plate and the disc spring; the cylindrical constraint is adopted between the buffer column and the telescopic rod to simulate the axial sliding and rotation characteristics; the contact force constraint is set between the initial support force top plate and the buffer top plate to ensure the effective transmission of force. In addition, the buffer column, the bottom plate and the sensor bottom plate are fixed on the simulation ground to realize the stable constraint of the boundary conditions. To ensure the cooperative working performance of the disc spring group, all disc springs are rigidly connected through fixed pairs. The initial support force top plate is placed above the disc spring group to bear the impact load generated by the self - weight and falling of the load plate; the sensor base is fixed by bolt anchoring, and all components are nested in four columns to finally complete the construction of the ADAMS simulation test bench for the disc spring - type buffering and energy - absorbing device, providing a digital analysis platform for the in-depth study of disc spring performance.

The disc spring is modeled with a steel flexible material. To balance the calculation efficiency and dynamic response accuracy, the number of modes is set to 6, and the deformation coefficient is 1.0 to intuitively present the dynamic deformation characteristics of the disc spring. In the construction of the model connection relationship, fixed constraints are adopted between the bottom plate and the sensor bottom plate, and between the base and the simulation ground, which equivalently simulate the rigid connection effect of high - strength bolts in the actual working condition.

For the contact behavior between the components of the model, except for the fixed connection part, the other components adopt universal contact constraints. The contact force model selects the normal contact force suitable for impact simulation: the contact stiffness coefficient is set to 1.0e + 05, the force index is 2.2, and the contact damping parameter is 10; at the same time, to simplify the calculation and focus on the normal mechanical response, the tangential friction is ignored. To realistically simulate the buckling and crushing phenomena of the disc spring composite monomer during impact and avoid non-physical penetration, the maximum allowable penetration depth is limited to 1.0. The simulation environment settings adhere to international standards, with the gravitational acceleration G set to 9.80665 N/kg and its direction along the positive Z-axis, ensuring that the simulation conditions are consistent with the actual physical environment. The normal-contact stiffness and damping are justified/calibrated using the representative measured peak-impact case Impact 1d in Table 3, and then kept unchanged for all other cases.

Fig. 3

ADAMS impact test platform model of disc spring composite monomer.

As shown in Fig. 4, the simulation specimen consists of a falling top plate and a disc spring composite monomer as the core structural system. The falling top plate is a square rigid component with dimensions of 800 mm × 800 mm; the disc spring composite monomer has an initial height of 340 mm and adopts a hierarchical series design: each group of disc springs includes four large disc springs and four small disc springs, which are assembled in series through opposed combination respectively. The large and small disc spring groups are rigidly connected by an annular connector with an outer diameter of 90 mm and an inner diameter of 41 mm, and are vertically supported by secondary and tertiary telescopic rods. This design ensures that the force direction of the disc spring composite monomer is strictly consistent with the direction of gravity during the impact process, effectively avoiding the risk of disintegration of the disc spring group caused by lateral forces, and ensuring the structural integrity and the reliability of mechanical response. Considering the elastic characteristics of the disc spring, it is subjected to flexibilization processing in ADAMS software.

In terms of parameter setting, the mass of the falling top plate is fixed at 20 kg, the mass of the initial support force top plate is set at 10 kg, and the mass of the load plate is taken as the only variable. To verify the reliability of the simulation model, the load plate mass of 4500 kg is selected for typical working condition simulation. Considering the balance between the model scale and the calculation accuracy, the simulation workspace is set to 1200 × 1200 area, divided by a square grid of 50 × 50 mm to meet the needs of refined calculation.

Fig. 4

Dynamic model of disc spring composite monomer.

Material model

According to GB/T1222–2016 ‘Spring Steel Part 1: Hot - rolled Steel’²⁶ and GB/T1298–2008 ‘Technical Conditions for Spring Steel’²⁷, 60Si2MnA spring steel is selected as the material for preparing disc springs. 60Si2MnA is a medium - carbon silicon - manganese alloy spring steel, which has excellent elastic performance, high strength and good fatigue resistance. It is widely used in the manufacture of elastic components under heavy load and high stress conditions, especially suitable for elastic structures with concentrated load such as disc springs.

As an elastic component with highly concentrated stress, disc springs have high requirements for material performance, requiring high strength, high elastic modulus and excellent fatigue life. As shown in Table 1, on the basis of mature heat treatment process, 60Si2MnA steel can obtain stable mechanical performance. It has good matching of strength and toughness, strong elastic recovery ability, and moderate material cost, which is suitable for mass industrial production. This steel grade has been widely verified in practical engineering. The selection of 60Si2MnA can effectively ensure the long - term stability and safety of disc springs under alternating loads and impact loads, and meet the comprehensive performance requirements of high - performance mechanical elastic components.

Table 1 Material parameters of 60Si2Mn spring steel.

Experimental process

Comparison and verification of experiment and simulation

As shown in Fig. 5, the impact test rig consists of an electromagnet release unit, a falling top plate, a disc spring composite buffer monomer, and force sensors. The 30 kg top plate is held and released by the electromagnet to achieve free fall and impact on the disc spring composite monomer, reproducing the impact–deformation process under practical conditions²⁸. The specimen is mounted on a 220 mm × 220 mm impact base, and four pressure/load sensors are installed at the four corners for distributed force measurement. The sensor model is QLMH-2802-10T, and all four channels are synchronously recorded by a high-speed data-acquisition system at 1500 Hz using identical settings for all tests. The initial drop height ${h}_{1}$is set using the vertical scale of the test bench and verified before each release by aligning the top-plate reference mark with the scale reading. Although the drop-height variability was not statistically logged in the current campaign, the same procedure and reference points are used for repeated drops at a given nominal height to minimize setup variation. This distributed sensing layout mitigates errors associated with eccentric loading or local contact imperfections. The sensors are wired to the acquisition system and carefully leveled to ensure full contact with the disc spring assembly and stable load transfer through the guiding structure.

The test bench integrates a load plate, a buffer top plate, multi - specification disc spring assemblies (including standard specifications such as D80_d41 and D125_d64), high - precision pressure sensors and their safety installation chassis. This modular design can simultaneously capture the real - time deformation of the disc spring under impact load and measure the energy absorption data, providing reliable test data support for the study of disc spring mechanical performance and structural optimization, and effectively simulating the working state of the disc spring under high - speed impact load in engineering practice.

Fig. 5

Layout of pressure sensors.

To further study the influence of the load plate on the disc spring composite buffer monomer, a load plate is added on it. Taking the working condition with a load plate mass of 4500 kg as the research object, the simulation time is set to 1 s and the number of simulation steps is 400 to analyze the support reaction between the initial support force top plate and the buffer top plate. The support reaction curve is shown in Fig. 6.

Fig. 6

Post treatment curve of reaction force under 4500 kg load plate support.

The test bench gradually enters a stable oscillation state about 0.5 s after the impact. Subsequently, the oscillation of the disc spring group begins to subside, with the stable value around 44,338.7464 N. When the disc spring group reaches a stable state, the buffer top plate supports the weights of the initial - support - force top plate, the load - bearing plate, and the top plate. According to G = 9.80665 N/kg, and based on F = mg, the theoretical value of the support reaction force should be 44,326.058 N. Thus, it can be concluded that the error between the simulation data and the actual data is approximately 0.03%. This indicates that the constructed ADAMS model can accurately reflect the actual constraint conditions, and the experimental conclusions derived from this model have high credibility and practical application value.

Considering the contingency of a single experiment, to further verify the reliability and integrity of the model, study the deformation mode and energy absorption characteristics of the disc spring composite monomer under impact load, 6 sets of simulation experiments are designed for comparative study. The subsequent experiments strictly follow the consistency principle. On the premise of keeping the simulation time, steps and other parameters and experimental steps unchanged, the energy absorption effect of the disc spring is studied by changing the weight of the load plate. The specific numbers and experimental parameters of the specimens are shown in Table 2.

Table 2 Shock experiment control group.

Dynamic process of simulation experiment

To ensure the integrity of the simulation procedure and the credibility of the results, the total simulation time is set to 5 s with 400 simulation steps, while all other conditions remain unchanged. In this study, the only variable among the test cases is the mass of the load-bearing plate (Table 2). The impact condition is defined by the initial falling height h₁ = 1125mm (Stage I free fall) and gravitational acceleration g. The top plate is released from rest (v = 0); under free fall, the theoretical velocity at first contact can be expressed as v₀ = $\sqrt{2gh_{1}}$, and the corresponding input impact energy is E₀ = mgh₁, consistent with the energy conversion described in Eqs. (13)–(16). Unless otherwise stated, tangential friction is ignored and only the gravitational acceleration is considered, acting vertically downward along the positive direction of the global $Z$-axis. The boundary conditions and constraints follow the model setup in Sect.  2.2, where base-related components are fixed and contact/constraint pairs are applied to ensure effective load transmission.

As shown in Fig. 7, taking the load - bearing plate mass of 7500 kg and the falling top plate mass of 20 kg as an example, according to the phenomena of the impact test, the entire impact process is divided into five stages for systematic analysis. In this paper, the buffering performance is consistently evaluated using the quantities reported throughout the manuscript, including the peak support reaction, rebound height, stabilization (buffering) time, and disc spring deformation.

Fig. 7

Dynamic process simulation in ADAMS software.

Fig. 8

Schematic diagram of deformation of ADAMS flexible disc spring assembly.

The deformation characteristics and stress distribution of the flexible disc spring composite under impact loading are presented in Fig. 8. Based on the predefined stress nephogram color scale (blue to red gradient representing increasing stress levels), subfigures (a)-(d) clearly illustrate: (1) distinct stress evolution patterns among individual disc springs within the composite; (2) non-uniform stress distribution between the edge and center regions of each disc spring; (3) significant stress concentration zones. Simulation results reveal that the flexible disc spring exhibits complex multi-directional deformation modes under impact, including axial compression coupled with radial expansion/contraction and angular torsion. Compared with rigid-body simulations, the flexible-body model adopted in this study more accurately captures the actual mechanical behaviors, providing a more precise theoretical basis for topology optimization, fatigue life prediction, and reliability assessment of disc spring assemblies.

Stage I (Free-fall stage, Fig. 7a and c)

At the start of the experiment (t = 0 s), the electromagnet is de-energized, and the top plate loses its constraint, starting to move in free fall in the gravitational field. According to the kinematic equation., its falling speed increases linearly with time, and the gravitational potential energy (Eq. (11)) is gradually converted into kinetic energy (Eq. (12)). Until t = 0.3569 s, the top plate makes the first contact with the load - bearing plate. At the moment of contact, the disc spring group starts to compress, and the kinetic energy is converted into elastic potential energy through nonlinear elastic deformation (Eq. (13)). As the compression amount $\delta$ of the disc spring increases, the speed of the top plate decays with damping. By t = 0.5168 s, the top plate reaches the lowest displacement point, at this time the speed drops to zero, and the elastic potential energy of the disc spring reaches the maximum value. The energy conservation relationship in this stage is shown in Eq. (14), where h₁ is the initial falling height, and $\varDelta{E}_{d1}$ represents the initial energy dissipation at the contact interface. The instant when the speed of the top plate is zero is the critical state between Stage I and Stage II.

$${E_p}{\text{ }}={\text{ }}mgh$$

(11)

$${E_k}=\frac{1}{2}m{v^2}$$

(12)

$${E_s}=\int_{0}^{\delta } k (\delta )d\delta$$

(13)

$$mg{h_1}+\frac{1}{2}mv_{1}^{2}={E_{s1}}+\Delta {E_{d1}}$$

(14)

Stage II (First rebound stage, Fig. 7d and e)

At t = 0.5168 s, the elastic restoring force of the disc spring drives the top plate to move upward, and the elastic potential energy is gradually converted into kinetic energy during the unloading process of the disc spring. Although the simulation model ignores air resistance and friction, in the actual impact process, local vibrations at the contact interface of the disc spring and material internal friction will cause part of the kinetic energy to be converted into vibrational energy and thermal energy. There is internal friction in the actual energy conversion process, resulting in the rebound kinetic energy ${E}_{k2}$ being less than the stored elastic potential energy ${E}_{s1}$. At t = 0.6697 s, the kinetic energy of the top plate is completely converted into gravitational potential energy (Eq. (15)), and the first rebound process ends here. Experimental observation shows that the first rebound height is as shown in Eq. (16), where ${\eta}_{1}\left(0<{\eta}_{1}<1\right)$ is the efficiency factor of the first rebound, which is used to reflect the loss degree of the system’s mechanical energy.

$${E_{k2}}=mg{h_2}$$

(15)

$${h_2}={\eta _1}{h_1}$$

(16)

Stage III (Secondary falling stage, Fig. 7f and g)

At t = 0.6697 s, the top plate’s velocity at the highest rebound point returns to zero, and then it enters the secondary free-falling stage. This stage follows the assumption of an ideal conservative system, where only reversible conversion between gravitational potential energy and kinetic energy occurs. The motion equation. satisfies Eq. (17), and the velocity increases quadratically with the increase of falling displacement. At t = 0.707 s, the top plate impacts the load-bearing plate for the second time, and the instantaneous kinetic energy at this moment is as shown in Eq. (18). Among them, part of the energy is converted into the elastic potential energy ${E}_{s2}$ from the secondary compression of the disc spring, and the remaining energy excites the high-frequency vibration mode of the disc spring group. Compared with the first compression, the nonlinear stiffness characteristic of the disc spring during the secondary compression leads to the compression amount ${\delta _{max2}}=\beta {\delta _{max1}}$ (where $\beta$is the stiffness degradation coefficient), which reflects the stiffness attenuation effect of the material under cyclic loading. The critical state between Stage III and Stage IV is the moment when the top plate reaches the lowest point at t = 707 s.

$$h(t)={h_2} - \frac{1}{2}g{(t - {t_3})^2}$$

(17)

$${E_{k3}}=mg{h_2}$$

(18)

Stage IV (Secondary rebound stage, Fig. 7h and i)

After t = 0.707 s, the disc spring starts to unload. Its elastic potential energy ${E}_{s2}$ is not only converted into the upward kinetic energy of the top plate, but also needs to overcome the hysteretic resistance at the contact interface and the internal friction loss of the material. During the rebound process, the object is subjected to inertial force, damping force and spring restoring force. According to Newton’s second law, Eq. (19) can be obtained. In the Eq. (19), c represents the equivalent viscous damping coefficient, which is used to describe the energy dissipation mechanism. At t = 0.7407 s, the kinetic energy of the top plate is completely converted into gravitational potential energy again, and the secondary rebound process ends. Experimental data show that the secondary rebound height is as shown in Eq. (20), and ${\eta}_{2}<{\eta}_{1}$, which indicates that energy dissipation has a cumulative effect, resulting in a gradual decrease in rebound efficiency. The deformation curve of the disc spring in this stage shows significant hysteretic characteristics, and the area of the hysteresis loop corresponds to the energy loss during a single rebound.

$$m\ddot {z}+c\dot {z}+kz=0$$

(19)

$${h_3}={\eta _2}{h_2}$$

(20)

Stage V (Attenuation and stabilization stage, Fig. 7h and i)

After t = 0.7407 s, the top plate enters the third impact process, and by this time, the system energy has been significantly attenuated. At t = 0.7533 s during the third impact, the remaining kinetic energy ${E}_{k5}$ is mainly used to excite the disc spring to produce high-frequency micro-vibrations. Subsequently, the system enters the attenuation vibration stage. During the periodic compression-rebound process of the disc spring, energy is continuously dissipated through the vibrational friction between the disc spring groups, and the system eventually enters a stable state over time.

Results and discussion

Comparative analysis of experimental and simulation results

To verify the reliability of the disc spring group impact–rebound simulation model, physical experiments were conducted. The impact and rebound process of a steel solid plate on the disc spring group was recorded through process state diagrams, and pressure sensors were evenly distributed at the four corners of the disc spring group’s bearing surface to collect impact force data under multiple working conditions. Simultaneously, an ADAMS model was established to simulate the mechanical response under the same working conditions, as shown in Fig. 9. It should be noted that Table 3 compares the peak support reactions during impact (dynamic peak forces), rather than post-impact quasi-static reactions. According to Table 3, the experimental and ADAMS-simulated peak impact forces show very close agreement, with a maximum relative error of about 0.50%, and highly consistent response trends. Among the records in Table 3, Impact 1d (Total = 0.087 t) exhibits the maximum measured peak reaction and is therefore selected as the representative calibration baseline (most critical case). The ADAMS normal-contact stiffness and damping are justified/calibrated by matching the peak force of this case (852.6 N experimentally vs. 860.4156 N in ADAMS), after which the same contact parameters are kept unchanged for the remaining cases; therefore, the errors reported in Table 3 reflect model predictability rather than case-by-case parameter tuning. Overall, the validated ADAMS model provides a reliable basis for the subsequent parametric and comparative analyses presented in this study.

Fig. 9

Comparison of experimental and simulation experiment records.

Table 3 Experimental data analysis of pressure sensors.

In the baseline ADAMS setup, tangential friction/hysteresis is neglected and energy dissipation is mainly represented by normal contact damping. To assess whether this simplification affects the buffering performance, a sensitivity comparison is conducted by enabling friction only in the guide-column sliding pair while keeping all other settings unchanged. Coulomb friction is adopted with a static coefficient of 0.3 and a dynamic coefficient of 0.1, and a smooth transition is applied using the stiction transition velocity (100) and friction transition velocity (1000) in ADAMS.

Fig. 10

Sensitivity of support reaction to guide-column friction (with/without friction).

Figure 10 compares the support reaction histories for the frictionless and friction-enabled cases. In the figure, Avg is the stabilized-stage mean support reaction for the frictionless case, and Avg2 is the corresponding mean value for the friction-enabled case. The results show Avg = 44338.7464 N and Avg2 = 44111.1485 N. The difference is 227.5979 N, corresponding to a relative change of about 0.51%. This small deviation indicates that, under the investigated configuration and friction settings, guide-column friction has a negligible influence on the stabilized support reaction level and does not change the overall response trend. Accordingly, neglecting friction/hysteresis is acceptable for the subsequent peak-response-based dynamic analysis.

Comparative analysis of support reactions under different load plates

A comparative analysis of the experimental data in Table 4 reveals that as the mass of the load plate increases, the support reaction between the initial support force top plate and the buffer top plate shows a significant linear growth trend, increasing from 195.15 N to 73,815.72 N. Further verification by comparing the simulation results with theoretical calculations shows that the theoretical values are in good agreement with the experimental values under all working conditions, with errors ranging from 0.03% to 0.50%, where the maximum error is 0.50% and the minimum error is only 0.03%.

Table 4 Reaction error of lower support of different load plates.

As shown in Fig. 11, the data indicate a positive correlation between the support reaction and the load, which is consistent with the mechanical analysis expectations of the model. The extremely low error percentage (all less than 0.50%) confirms the high accuracy of the model.

Fig. 11

Comparison and error analysis of experimental and theoretical values of reaction forces.

Error Analysis: The main source of error may be attributed to the constraint relationships between various structural components in the ADAMS model. Based on the analysis of issues during model debugging, it can be concluded that different setting points of the constraint pairs for the same component can have a significant impact on the experimental data. In this study, the constraint positions closest to the actual situation were selected through debugging and verification, but minor errors may still exist.

Comparative analysis of disc spring materials under different load plates

Based on the ADAMS multi-body dynamics simulation platform, this study constructed flexible and rigid disc spring composite monomer models respectively to systematically investigate the differences in the response of support reactions with the initial support force top plate under the same dynamic load conditions (simulation duration: 0 ~ 5s). The experiment adopted the single-variable control method, parameterizing only the mass of the load plate and constructing multiple groups of experimental control groups with mass gradients to conduct impact simulation experiments. The specific working condition parameters and simulation results are as follows.

As shown in Fig. 12, the comparative analysis of the top plate displacement curves (with the flexible disc spring composite monomer model on the left and the rigid disc spring composite monomer model on the right) indicates that: in the initial stage, the top plates are all in a free-falling state, and their displacement increases linearly and rapidly with time. After the top plate comes into contact with the disc spring composite monomer, significant differences in dynamic responses appear between the two models. The rigid disc spring composite monomer model, due to its inability to deform, cannot buffer and absorb impact energy, resulting in direct transmission of the impact load and a relatively low rebound height. In contrast, the flexible disc spring composite monomer model achieves efficient absorption of impact energy through elastic deformation at the moment of contact, effectively alleviating the impact on the top plate. When the external force disappears, the stored elastic potential energy is released and converted into kinetic energy for the flexible body to rebound, resulting in a higher rebound height. Therefore, the process of energy absorption and release by the flexible body is relatively slow, and its rebound is more sustained compared to the rigid body, extending the buffering time. This allows the system more time to adapt to the impact, reducing the acceleration change rate caused by the impact and alleviating the stress on related components.

Fig. 12

Analysis of displacement curves of top plate under different load conditions.

Taking the experimental results with a load plate mass of 4500 kg in Fig. 13(a) as an example, in the initial stage of impact, the rigid disc spring composite monomer model, due to the lack of material deformation capacity and energy dissipation mechanism, experiences a sharp rise in support reaction to a peak of 133,060 N, then exhibits significant rebound characteristics at 0.25 s, and only shows a relatively stable trend at approximately 0.9 s. In contrast, the flexible disc spring composite monomer model, relying on the synergistic effect of multiple physical mechanisms such as elastic deformation energy storage, damping dissipation, and stress wave diffusion, reduces the peak support reaction to 120,180 N, which is 10% lower than that of the rigid body. Moreover, the smoothness of the support reaction curve is significantly improved, stabilizing at approximately 1.8 s, effectively suppressing the rebound amplitude.

Fig. 13

Reaction force variation curve of flexible/rigid disc spring assembly. (a) Load plate mass M1 = 4500 kg, (b) Load plate mass M2 = 6000 kg.

When the load plate mass increases to 6000 kg, as shown in Fig. 13(b), the peak support reaction of the rigid disc spring composite monomer model further increases, and the secondary impact effect caused by the impact persists until 1.0 s. Although the stabilization time of the flexible disc spring composite monomer model is extended to 2.0 s, the fluctuation amplitude of the support reaction is much gentler than that of the rigid body. According to the momentum theorem (Eq. (21)), the extremely short action time leads to a significant increase in the peak reaction force. Although the rigid disc spring can achieve rapid stabilization, this response characteristic comes at the expense of structural safety. The flexible body significantly reduces the instantaneous impact force by extending the action time, effectively protecting structural integrity and avoiding structural damage caused by concentrated stress.

$${F_{{\text{peak}}}}=\frac{{m\Delta v}}{{\Delta t}}$$

(21)

Comparative analysis of disc spring deformation under different load plates

Statistical analysis of the data in Table 5 shows that the disc spring group deformation increases monotonically with the mass of the load plate. Here, “Initial Height” and “Stabilized Height” in Table 5 denote the axial point-to-point distance extracted from the ADAMS model along the global $Z$-axis, where gravity is defined to act vertically downward along the positive $Z$-direction. Specifically, the initial value ${Z}_{0}$is measured at $t=0$before impact, and the stabilized value ${Z}_{s}$is measured after the impact response decays to a steady state; the disc spring axial deformation is calculated consistently as ${\Delta}Z={Z}_{s}-{Z}_{0}$for all cases. Under this definition, ${\Delta}Z$increases from 0.0752 mm at a load plate mass of 1500 kg to 0.2927 mm at 7500 kg. The sub-millimeter magnitude is consistent with the relatively high axial stiffness of the disc spring assembly within the investigated load range, and the exported distance units were verified to be millimeters (mm) using the same reference points for all cases. Therefore, the results indicate that, within 0–7500 kg, the disc spring group remains in the elastic deformation regime without plastic deformation or structural failure, confirming the working reliability and structural stability of the disc spring group in this load range.

Based on the nonlinear regression analysis results of the experimental data, a trend line Eq. (22) was fitted, yielding:

$$y=(1.83894 \times {10^{ - 4}}) \cdot {w^{0.8264}}$$

(22)

where the coefficient represents the reference deformation per unit load; the exponent indicates that the deformation has a sublinear growth relationship with the load mass, i.e., the growth rate of deformation gradually slows down as the load increases. This phenomenon may be caused by the gradual increase in the stiffness value of the system under high loads, resulting in material nonlinearity or contact effect nonlinearity.

Table 5 Disc spring deformation variables under different load plates.

The goodness-of-fit test of the model shows that the coefficient of determination R² = 0.99962 indicates that the power function model can explain 99.96% of the variation in deformation. ReducedChi - sqr = 3.69179 × 10^− 6, and the extremely small residual sum of squares indicates high fitting accuracy, verifying the reliability and validity of this functional relationship.

As shown in the load-deformation curve in Figs. 14, in the low-load region (0–3000 kg), the deformation rate exhibits a significant rapid growth trend; while in the high-load range (3000–7500 kg), the increment of the deformation rate is significantly lower than that in the low-load stage. This phenomenon is highly consistent with the power function relationship. By performing linear fitting on the low-load and high-load segments respectively, it is found that both segments are accurately adapted to the piecewise function Eq. (23). This result indicates that under low-load conditions, the effective stiffness of the system is relatively low, similar to a spring in an incompletely compressed state, resulting in a high sensitivity of deformation to load changes.

$$\left\{ {\begin{array}{*{20}{c}} {y=(4.62667 \times {{10}^{ - 5}})x+0.0026} \\ {y=(3.40067 \times {{10}^{ - 5}})x+0.03849} \end{array}} \right.$$

(23)

Further confirmation of the above conclusion can be obtained through specific data comparison: when the deformation increases from 4500 to 6000, the increase is 0.0514 mm, which is slightly lower than the increase of 0.0523 mm when the deformation increases from 3000 to 4500. This phenomenon can be attributed to multiple factors: first, under high loads, the thin-walled structure of the disc spring may undergo slight buckling, leading to a decrease in effective stiffness; second, the deformation of the disc spring group during impact destroys the uniformity of force transmission. In the high-load state, as the spring is gradually compacted or affected by structural constraints, the stiffness gradually increases, resulting in a significant decrease in the growth rate of deformation. When the spring reaches a fully compacted state, its mechanical properties tend to be consistent with those of the rigid disc spring, fully demonstrating the simulation characteristics of the rigid disc spring.

Fig. 14

Fitting curve of deformation of flexible disc spring assembly.

This phenomenon reveals the sensitivity of the system in the initial loading stage, providing a key basis for optimization design: through pre-tightening structures, material selection, and nonlinear modeling, more stable deformation responses can be achieved over the entire load range.

Conclusions

This study systematically conducted load mass gradient experiments and combined comparative analysis of disc spring composite monomers of different materials to deeply explore their application characteristics in impact dynamics simulation, clarifying the significant advantages of the flexible disc spring composite monomer model compared to the traditional rigid model.

(1) In terms of energy absorption mechanism, the flexible disc spring composite monomer model, relying on the dual effects of elastic energy storage and damping dissipation, has much higher energy absorption efficiency than the rigid structure. Dynamic analysis shows that compared with the rigid body, the flexible disc spring composite monomer model can reduce the peak support reaction by 10%, effectively alleviating the instantaneous impact load on the structure, which is of great significance for prolonging the service life of the structure. Experimental and simulation results fully confirm that the use of flexible disc spring composite monomers as buffering devices has significantly better impact dynamics performance than rigid bodies.

(2) The study further reveals the sensitivity of the system to the initial loading stage, which points out the direction for optimization design. Through reasonable design of pre-tightening structures, precise material selection, and adoption of nonlinear modeling methods, stable and controllable deformation responses over the entire load range can be achieved. The study found that there is a significant correlation between the load mass and the peak force: below 4500 kg, the two show a highly linear relationship (${R}^{2}=0.99$); beyond 4500 kg, due to the buckling effect of the disc spring, the growth of the peak force tends to be nonlinearly saturated. Experimental data show that a load mass of 4500 kg is the optimal configuration, where the system has the highest energy absorption efficiency, which can not only effectively reduce the peak force but also achieve a fast and stable response.

In engineering applications, the research findings provide a quantitative basis for the impact protection design of roadway support systems. Through in-depth analysis of the mechanical properties of the disc spring assembly, it is recommended to prioritize the use of flexible disc spring composite monomers as buffer elements in roadway support scenarios. These elements can effectively absorb the impact energy generated by surrounding rock deformation and reduce the dynamic load on the support structure.