Contact Response Analysis of Vertical Impact between Elastic Sphere and Elastic Half Space

At present, the contact problem between the particle and the plane plate is generally equivalent to the rigid sphere impacting the elastic half space or the elastic sphere impacting the rigid surface. However, in the actual contact process, there will be no rigid body, and both contact and contacted object will deform and absorb energy. ,e research results obtained from the equivalent of the contact material to the rigid body are less accurate. In order to obtain the accurate mechanical relation and contact response, we took the research of impact between particles and the metal plate as a breakthrough in which the particle is equivalent to an elastic sphere and the metal plate is equivalent to an elastic half space and established the theory of vertical impact contact between elastic sphere and elastic half space by the Hertz contact theory. ,rough the dynamic simulation of an elastic sphere which has similar properties with rock impacting target in elastic half space in LS-DYNA, the correctness of the established theory and the feasibility of the contact process simulated by LS-DYNA are verified. Based on the established theory and 3D simulation, we studied the influence law of material parameters on the contact response and analyzed the differences of the collision vibration signals caused by the different contact objects. From the above research results, we obtain that the theoretical model is more accurate to predict the maximum contact force and contact displacement in this paper than traditional Hertz theory. And the sphere radius and both contact objects’ elastic modulus have larger influence on the contact response than sphere density, while the Poisson’s ratio has the smallest influence on the contact response results. Different material properties will cause the different contact response. ,e conclusions of this paper provide a theoretical calculation method for contact and a 3D simulation method for elastic half space and provide theoretical guidance for the differences analysis of the vibration signal.


Introduction
In engineering applications, impact and contact problems exist everywhere.e impact of the particles on metal plate can be seen in many fields such as in the top coal mining process which is accompanied by coal or gangue impacting on the metal of the cover beam or tail beam of the caving coal hydraulic powered support, multibody systems, vehicle impact etc. e contact problem of particles impacting on the metal plate has always been a classical problem of the contact mechanism in engineering practice.At the same time, the contact between the particles and the metal plate is a complicated process, and it involves many aspects, such as the short duration of the impacts, rapid increase of transient stress, local large deformation of particle, elastoplastic properties of particle and metal plates, metal plate's flexural deflection, and other issues.At present, most theoretical research studies on the contact between particles and planes are based on the Hertz contact theory [1][2][3][4][5][6], and the particle is equivalent to spherical particles [7].Cermik, Rossikhin, and Xie et al. studied the impact vibration response of an elastic sphere on the rigid surface with flexible balls or inflatable thin-walled balls striking the rigid target surface vertically or obliquely [8][9][10][11][12][13][14].
e target surface was equivalent to a rigid surface, ignoring the elastic deformation of the object being impacted.Wang et al. [15], Wang et al. [16], Willert et al. [17], Mougin et al. [18], and Jäger [19] studied the impact contact response of the rigid sphere impacting elastic plane or elastic sphere, respectively.e active plane was used as the rigid surface, and the target plane was used as the elastic half space.ey took the deformation and energy absorption of the target object into account, but they ignored the deformation and energy absorbing of the active contact object during the contact process.Wu et al. [20], Liu et al. [21], Vu-Quoc et al. [22,23], Wang et al. [24], and Peng et al. [25] simulated and analyzed the contact process through simulation methods.Pham, Chen, Liu, and Cheng et al. investigated the effects of contact stiffness on the impact behaviour of RC beams, fracture and fragmentation responses of laminated glass under impact, ballistic performance of monolithic and multilayered steel targets penetrated by EFP, and low-velocity impact performance of scarf-repaired composite laminates [26][27][28][29].
e contact object and the target object were simultaneously equivalent to elastic materials in the simulation setup and meshed separately, and the contact response when the material under elastic-plastic deformation or the damage and destruction of the material is studied.
Based on the above research conclusions and deficiencies, in order to reveal the derivation mechanism of impact, contact response, and impact vibration law in the process of the sphere impacting the metal plate and correctly analyze the differences in vibration signals, we equated the particle to the elastic sphere and made the metal plate equivalent to the elastic half space target plate.And theoretical model of vertical impact contact between the elastic sphere and target plate in elastic half space by the Hertz contact theory is established.Meanwhile, we combined the theory with simulation by LS-DYNA to research the impact contact response between the elastic sphere and the metal plate in elastic half space and explored the influence law of material parameters on contact response.Based on the above research results, we further analyze the differences of the collision vibration signals caused by the different contact objects.
e remainder of the paper is organized as follows: Section 2 establishes the theoretical model of vertical impact contact between the elastic sphere and target plate in elastic half space.Section 3 introduces the method of constructing the simulation model of the vertical elastic impact between the elastic sphere and half space target plate and compares the analysis result of theory with simulation.Section 4 analyzes the effect of material parameters on contact response.Section 5 performs the differences analysis of vibration signals.Section 6 shows some related work and our conclusions.

Contact Mathematical Model of Elastic
Sphere Vertically Impacting Elastic Half Space e contact theory of elastic ball impacting rigid surface was proposed by Hertz long before.As is shown in Figure 1, Hertz theory [1][2][3][4][5][6]30] provides the following expressions for the contact load under the condition of statics: As is shown in Figure 2, the contact force and contact deformation of the rigid sphere contacting with elastic half space under static conditions meet the following relationship [31]: (2) According to the law of energy conservation, in the critical end time of compression stage, From Equations ( 3) and ( 4), the maximum contact deformation and maximum contact force under two different contact conditions are obtained in the following form: where and ] 2 are the elastic modulus and Poisson's ratio of the sphere (elastic or rigid) and surface (rigid surface or elastic half space), respectively, a is the contact radius, and δ i (i � 1, 2) is the contact deformation.
For the problem of the elastic sphere impacting elastic half space, as is shown in Figure 3, the functional relationship of the system in the critical end time of compression stage can be expressed as Solutions are given by 2 Shock and Vibration δ � where δ it (i � 1, 2) is the contact deformation of the elastic sphere and elastic half space, respectively, and δ is the displacement of the centre of the sphere.
For free fall impact of the elastic sphere on the metal plate, such as coal and gangue impacting the metal plate, when the rock particles such as coal or gangue is dropped from the height H in the initial vertical velocity v ⊥0 � 0 m/s, we obtain from the law of energy conservation Select a rock material, take the material property in elastic area as the material property of the elastic sphere, and take the metal plate as target plate in elastic half space.e material parameters are shown in Table 1.
e change curves of contact displacement and contact force with different impact velocity v are obtained from Equations ( 5)- (8) and Equations ( 12)-( 13), as is shown in Figures 4 and 5.
Under the ideal elastic contact condition, with the increase of impact velocity, the displacement and contact force of the rock sphere increase gradually.At the same impact speed, the displacement of centre of sphere (DCS) under condition of the elastic sphere impacting the elastic half space is biggest, the DCS under condition of the elastic sphere impacting the rigid surface is smaller, and the DCS under condition of the rigid sphere impacting the elastic half space is smallest.On the contrary, at the same impact speed, the maximum contact force under condition of rigid sphere impacting the elastic half space is biggest, the maximum contact force under condition of the elastic sphere impacting the rigid surface is smaller, and the maximum contact force under condition of the elastic sphere impacting the elastic half space is smallest.Combining Figures 4 and 5, the change curves of maximum contact force-contact displacement (MCF-CD) are obtained as shown in Figure 6.According to Equation (16), the changing curves of maximum contact force-rockfall height (MCF-RH) and contact displacementrockfall height (CD-RH) are obtained as shown in Figure 7.
From Figure 6, with the increase of the DCS, the contact force increases exponentially.At the same DCS, the maximum contact force under condition of the rigid sphere impacting the elastic half space is biggest, the maximum contact force under condition of the elastic sphere impacting the rigid surface is smaller, and the maximum contact force under condition of the elastic sphere impacting the elastic half space is smallest.Combining Figures 4 and 5 we obtain that, the DCS calculated by the elastic sphere impacting elastic half space contact theory proposed in this paper is greater than that of the traditional theory, and the contact force is less than that of the traditional theory.Because the traditional Hertz contact theory only considers one of the contact body or the contacted body as the flexible body, the predicted contact force is too large, and the accuracy is lower when predicting the contact force by traditional theory.From Figure 7, DCS and contact force increase exponentially in the power of 2/5 and 3/5 of rockfall height, respectively.

Numerical Simulation of Vertical Impact
Contact between Viscoelastic Sphere and Target Plate in Elastic Half Space

e Simulation Model.
In this paper, we mainly studied the vertical impact contact response between the elastic sphere and target plate in elastic half space and used the finite element software LS-DYNA to establish the dynamic model of the sphere impacting the metal plates.It can be Shock and Vibration seen from the Figure 8, the dynamic simulation of the single sphere with di erent properties impacting the metal plates vertically at di erent speeds was conducted.In the process of modeling, the target plate was set as the thin metal plate with homogeneous continuous medium, and the sphere models were set as the viscoelastic constitutive models with the properties of rock (coal or gangue).
In the modeling process, other factors such as air resistance were ignored.To simulate Earth's gravity, the acceleration of the sphere was de ned as 9.8 m/s 2 .In order to ignore the in uence of the shape factor, the size R of the spheres of di erent attributes were all taken as 25 mm, and the simulation of 5 groups of di erent impact velocities such as 2 m/s, 4 m/s, 6 m/s, 8 m/s, and 10 m/s for di erent properties of spheres was performed.In order to prevent the sphere from overlapping with the mesh of metal plate, the gap between the end point of the sphere and the upper endface of the metal plate was de ned to be 0.000326 m, and the velocity increment of the sphere over this gap was less than 0.01 m/s, that is, less than 1/200 of the de ned speed.
erefore, the e ect of the gap can be negligible.To simulate the target plate in elastic half space, the boundary constraint was applied to the lower end of the metal plate after meshing the metal plate, and the constraint type was de ned as full   At the same time, in order to improve the quality of the simulation, hexahedral mesh is used to construct the model.e grid size of the sphere was defined as 0.002 m, and the metal plate was divided into 9 parts for meshing.e mesh of the contact areas between the sphere and the metal plate was refined.e models contain 86080 elements and 99431 nodes.Impact contact of the sphere on the metal plate is a complicated nonlinear contact.Owing to this, we set the contact type as Automatic_ Surface_ To_Surface.e surface of the sphere was set as the contact surface, and the surface of the metal plate was the target surface.e stiffness of the simulation was defined according to the contact stiffness calculated by the Hertz theory, and damping was applied during the contact process.We set the material type of the plate as elastic, and the basic material properties of the plate are similar to the 45 # steel.e material properties of spheres and plate are shown in Table 2 (1 # and 2 # elastic spheres are the elastic spheres which have the similar properties to different coal, respectively, and 3 # and 4 # elastic spheres are the elastic spheres which have the similar properties to different gangue, respectively.).We set the solution time as 0.001 s, each step lasts 5 × 10 −7 s.To ensure the reliability of the simulation results, the hourglass energy should be less than 5% of total energy.

Comparative Analysis of eory and Simulation Results.
When the sphere with different properties impacts on the metal plates, the theoretical and simulated values of impact contact response were obtained under different impact velocities, respectively.Based on this, we got the fitting curves of the simulated values and theoretical values of the DCS, contact force, maximum contact pressure, maximum internal energy of the sphere, and the maximum internal energy of metal plate when the 1 # sphere particle impacted on the metal plate, as shown in Figure 9. e linear correlation coefficient R 2 between the simulated values and theoretical values of the contact response of each group of materials is shown in Table 3.
From Figure 9 and Table 3, the correlation coefficient R 2 of four materials for the DCS and contact force between the simulated values and theoretical values is higher than 0.96, which shows the quite higher correlation between the simulation values and theoretical values of the DCS and contact force.e correlation coefficient R 2 for maximum internal energy of the sphere is larger than 0.85, and R 2 for maximum internal energy of the metal plate is no less than 0.84, which shows the higher correlation between the simulation values and theoretical values of the maximum internal energy of the sphere and maximum internal energy of the metal plate.Because of the large uncertainty of contact pressure, the correlation coefficient R 2 for the contact pressure is just ranging from 0.7483 to 0.85486, and there is a positive correlation between the simulated contact pressure and theoretical contact pressure.In summary, simulation results simulated by LS-DYNA have the same changing trends with the theoretical results, and the linear correlations are significant.
Figure 10 shows the variation curves of the contact force and the DCS during the initial impact and the theoretical curves of the maximum contact force and the DCS when the four types of spheres impact on the metal plate with different velocities.From the Figure 10, we can see that when the impact between the sphere and metal plate occurs, the  Table 2: Parameters of contact model of the elastic sphere and metal plate.Shock and Vibration contact force of the sphere and DCS gradually grows with the increase of time, and then gradually decreases after reaching the peak.e reason is that when the initial kinetic energy of the sphere is converted into the energy absorbed and consumed by the sphere and the metal plate, the velocity of the sphere drops to zero, and the system enters the rebound recovery phase from the compression phase.Due to the randomness and uncertainty of the vibration, the contact force curves between the spheres and the metal plate in each group show dense vibration waves instead of smooth curves.erefore, the method of FFT lter was used to process the simulated contact force.For di erent material spheres at di erent speeds, the theoretical values of the maximum contact force are very close with the simulation values of contact force after FFT lter, the DCS presents a smooth curve, and the simulation values of maximum DCS under various working conditions are also very close to the theoretical values.
us, the maximum contact force and   Shock and Vibration maximum displacement obtained by the proposed theory of the elastic sphere impacting the elastic half space in this paper are basically consistent with the corresponding results obtained by simulation.Figure 11 shows the curves of the contact force which changes with the DCS when the four di erent material spheres impact the metal plate.e thick solid black lines are the theoretical curves of the compression stage, the rest are the simulation curves, and the contact force of simulation has been processed by the FFT lter.As can be seen from the gure, when the spheres with di erent properties impact the metal plate at di erent velocities, the contact forcedisplacement curves are all closed hysteresis loops, which is consistent with the properties revealed by the energy model considering energy absorption [32][33][34][35][36][37][38][39][40][41].e reason for the appearing of energy hysteresis loops is the energy dissipation of the system damping, and the shapes of the contact force-displacement curves are similar to that of the experimental and simulation results [42][43][44][45][46].When the sphere impacts the metal plate at ve di erent velocities, the contact force-displacement simulation curves of the four materials uctuate around the theoretical curves during the compression stage (before the DCS with di erent materials reaching its maximum values), which shows a good agreement between theoretical curves and simulation curves.e reasons for the uctuations are as follows: (1) the randomness and uncertainty of the vibration in the process of impact contact; (2) the vibration recovery of the compressed material caused by the elasticity of the sphere and the metal plate; and (3) the calculation error of simulation software.
is paper establishes a theoretical model of an elastic sphere impacting the elastic half space.
e simulation model of the viscoelastic sphere which had similar properties

Shock and Vibration
with coal-gangue elastic impacting target plate in elastic half space vertically is also established.By the comparison of theory and simulation, it can be obtained that the simulation results of impact contact response have the same changing trends with the theoretical results.And the theoretical maximum contact force of the elastic sphere and theoretical maximum DCS is basically consistent with the corresponding simulation results.
e theoretical models are established in this paper only considering the contact response of the sphere compression stage.e contact forcedisplacement simulation curves in the compression phase are in good agreement with the theoretical curves.Based on the above conclusions, the correctness of the theoretical models and the feasibility of simulation methods in the compression stage can be verified.

The Influence of Material Properties and Sphere Radius on the Impact Contact Response
When the elastic sphere impacts the elastic half space, the properties of the elastic sphere and the metal plate and the size of the elastic sphere have a great influence on the impact contact response of the two objects.In order to research the difference of the impact contact response caused by the change of parameters, set the variation intervals of the material properties of the elastic sphere and the metal plate (the change of properties is within the limits of different rock's properties) and set the variation interval of radius of the elastic sphere, as shown in Table 4.
From Equations ( 12)-( 13), we obtain that Hence, the DCS changes linearly with the radius R of the sphere, and the contact force is proportional to R 2 , and from Table 4, δ/R � 0.013373106060040, P/R 2 � 2.161252014548062 × 10 6 .
e R 1 has much larger influence on DCS and maximum contact force than ρ 1 .
According to the material parameters in Table 4 and Equations ( 17)- (18), the influence curves of the material parameters (ρ 1 , E 1 , ] 1 ) of the elastic sphere on the contact force and DCS can be obtained as shown in Figure 12.Meanwhile, the change curves of the contact force and DCS with the parameters of the metal plate material (E 2 , ] 2 ) can be also obtained in Figure 13.
It can be seen from Figure 12, with the increase of the elastic modulus E 1 , the DCS presents an exponential decrease tendency, the rate of decrease gradually reduces, and the contact force increases at a decreasing rate.e E 1 has larger influence on DCS and maximum contact force than ρ 1 .With the increase of the Poisson's ratio ] 1 , the DCS gradually decreases, the contact force increases gradually, and the rate of change gradually increases.
e (1 − ] 2 1 ) changes very small due to the little variations of ] 1 , and the change of (1 − ] 2 1 ) 2/5 is smaller than (1 − ] 2 1 ).However, the DCS and the contact force are approximately proportional to , respectively, which leads the smallest influence of ] 1 on contact response.According to Equations ( 17)-( 18) and Figure 12, the influence relationship of parameters on the DCS and the maximum contact force is (the sensitivity of DCS and the maximum contact force to the change of the parameter is defined as Figure 13 presents that with the increase of the elastic modulus E 2 of the metal plate, the DCS shows a sharp decrease and then slows down, the maximum contact force increases quickly then slows down, and the amplitude of variation decreases gradually.With the increase of the Poisson's ratio ] 2 , the DCS gradually decreases, the maximum contact force gradually increases, and the amplitude of variation gradually increases.Comparing the rangeability of the DCS and the maximum caused by the change of the metal plate's elastic modulus E 2 and by the Poisson's ratio ] 2 , the rangeability of contact response caused by ] 2 is extremely small.erefore, the influence of the metal plate parameters changes the DCS, and the maximum contact force is Ω(E 2 ) » Ω(] 2 ).
Elastic modulus E and the Poisson's ratio] of the sphere and the metal plate both affect the results of the impact contact response.To compare the influence range of the response results caused by E 1 , E 2 and] 1 , ] 2 , define the elastic modulus E and the Poisson's ratio] of the sphere and the metal plate with the same value and range of variation, respectively, as shown in Table 5.
Combining Equations ( 12)-( 13), the response surface diagram of the DCS and the maximum contact force with the elastic modulus E and Poisson's ratio] can be obtained, as shown in Figure 14.
From Figure 14, it can be seen that with the increase of E 1 and E 2 , the DCS and the maximum contact force decrease gradually, with the increase of ] 1 and ] 2 , the DCS and the maximum contact force also gradually decrease.As shown in the figure, when E 1 , E 2 or ] 1 , ] 2 take the same value and change range, the resulting trend which belongs to the DCS and the maximum contact force is exactly the same, and the reason is that the relationship between the DCS/the maximum contact force and E 1 is similar to the relationship between the DCS/the maximum contact force and E 2 , and the relationship between the DCS/maximum contact force and ] 1 is similar to the relationship between the DCS/the maximum contact force and ] 2 , as shown in Equations ( 17)- (18).erefore, if However, as for the working condition where the elastic sphere which had similar properties with rock impacting the metal plate vertically, because of the limit of intrinsic properties of the material, the E 2 is much larger than E 1 (usually at least an order of magnitude) and the values and ranges of variation of E 1 and E 2 are di erent.In order to study the in uence of the sphere with similar properties with coal and gangue on the response of the metal plate under the two working conditions with elastic modulus E 1 and E 2 , the values of E 1 and E 2 in Table 4 were taken in combination with Equations ( 12)-( 13), and the response surface diagrams of the DCS and the maximum contact force with changes in the E 1 and E 2 , respectively, are obtained, as shown in Figure 15.
In the range of values, with the increase of E 1 and E 2 , the DCS presents a decreasing trend, and the maximum contact force increases.However, changing amplitude of the DCS and the maximum contact force caused by the change of the

Shock and Vibration
E 2 is much greater than that of the E 1 .In that case, the sensitivity of the DCS and the maximum contact force to the change of the parameters is Ω(E 2 ) > Ω(E 1 ).Meanwhile, the elastic modulus E 2 is much larger than E 1 , and ] 1 is close to ] 2 , which results in From Equations ( 17)-( 18), the in uence of ] 1 on the contact response is larger than the in uence of ] 1 on the contact response.erefore,

Case Study: Difference Analysis of Impact Contact Response of Different Materials
For the four kinds of elastic spheres with similar properties to rock as shown in Table 2, impacting the same metal plate at a speed of 10 m/s, respectively, the simulation curves of contact force-time and contact force-DCS are obtained as shown in Figures 16 and 17.
When the elastic sphere impacts the same metal plate at the same speed, the maximum contact force P 3# » P 4# > P 2# > P 1# and the DCS δ 4# > δ 1# > δ 2# » δ 3# .Comparing the parameters of the 4 kinds of elastic spheres according to Table 2, we know therefore, the e ect of parameters R and Poisson's ratio µ can be ignored.rough the theoretical analysis of the e ect of the variation of the parameters on the response results, we can get that when the elastic spheres impact the metal plate, the relationship of the sensitivity Ω of the DCS and the maximum contact force to the changing parameters is Ω(R) » Ω(E 1 ) > Ω(ρ 1 ) > Ω(] 1 ), and the DCS will decrease and the maximum contact force will gradually increase with the increasing of elastic modulus E 1 of the elastic sphere, and the DCS and the maximum contact force are all increasing with the density ρ 1 of the elastic sphere, and the maximum contact force P 3# » P 4# > P 2# > P 1# and the DCS δ 4# > δ 1# > δ 2# » δ 3# can be obtained, which are same as the conclusions obtained in Figures 16 and 17.
In the four kinds of elastic spheres, 1 # and 2 # are elastic spheres similar to coal, and 3 # and 4 # are elastic spheres similar to gangue.From Figures 16 and 17, 1 # and 2 # elastic spheres have the obvious di erence to 3 # and 4 # elastic spheres in the maximum contact force and DCS.erefore, coal and gangue will cause di erent contact responses when impacting the metal plates in the top coal mining process, which will cause the di erent vibration signal of the tail beam of the hydraulic support.And the di erences of impact and contact response of tail beam vibration caused by coal and gangue can be used as the basis to identify the coal-gangue.

Conclusion
Based on Hertz contact theory, a theoretical model of elastic sphere impacting elastic half space is established in this paper.And the nite element software LS-DYNA is used to establish a dynamic model of viscoelastic sphere impacting elastic half space, and the simulation analysis is conducted.
rough the combinative and comparative analysis of theory and simulation, we obtain the following conclusions: (1) In this paper, the contact response calculated by the theoretical model of compression stage proposed in this paper is consistent with the simulation results of the changing trend, and the maximum value and the process curve in compression stage verify the correctness of the theoretical model established in this paper and the feasibility of the simulation method in the compression stage.
(2) e theoretical model of elastic sphere impacting elastic half space proposed in this paper is more accurate to predict the maximum contact force and contact displacement.e maximum contact force calculated by the traditional contact theory is too large.(3) e DCS and contact force are all increasing with the density ρ 1 of the elastic sphere.With the increase of elastic modulus E 1 , the DCS decreases gradually and the contact force increases.With the increase of Poisson's ratio μ 1 , the DCS decreases gradually and the contact force increases gradually, and the decreasing velocity of the DCS and the increasing velocity of the contact force are all increasing.With the increase of elastic modulus E 2 , the DCS decreases rapidly rst and then decreases slowly and the contact force increases rapidly rst and then increases slowly.With the increase of Poisson's ratio e conclusions of this study will provide a theoretical basis for the accurate calculation of the contact response results, provide methods for simulation study of the impact behaviour, and provide theoretical guidance for the differences analysis of the vibration signal.

Figure 8 :
Figure 8: Simulation contact model of the elastic sphere impacting the elastic half space.
Figure9: e relationship between the simulated values and theoretical values (take 1 # material as an example).

vDCS
= 2m/s simulation force v = 4m/s simulation force v = 8m/s simulation force v = 6m/s simulation force v = 10m/s simulation force v = 2m/s force after FFT filter v = 4m/s force after FFT filter v = 6m/s force after FFT filter v = 8m/s force after FFT filter v = 10m/s force after FFT filter v = 2m/s simulation variation v = 4m/s simulation variation v = 8m/s simulation variation v = 6m/s simulation variation v = 10m/s simulation variation ball v = 2m/s simulation force v = 4m/s simulation force v = 8m/s simulation force v = 6m/s simulation force v = 10m/s simulation force v = 2 m/s force after FFT filter v = 4 m/s force after FFT filter v = 6 m/s force after FFT filter v = 8 m/s force after FFT filter v = 10m/s force after FFT filter v = 2m/s simulation variation v = 4m/s simulation variation v = 8m/s simulation variation v = 6m/s simulation variation v = 10m/s simulation variation 3

v = 2 m
/s simulation results v = 4 m/s simulation results v = 8 m/s simulation results v = 6 m/s simulation results v = 10 m/s simulation results 1of centre of sphere (m) Displacement of centre of sphere (m) Displacement of centre of sphere (m) Displacement of centre of sphere (m) Contact force (N) Contact force (N) Contact force (N) Contact force (N) 1 # rock ball v = 2 m/s simulation results v = 4 m/s simulation results v = 8 m/s simulation results v = 6 m/s simulation results v = 10 m/s simulation results 2 # rock ball v = 2 m/s simulation results v = 4 m/s simulation results v = 8 m/s simulation results v = 6 m/s simulation results v = 10 m/s simulation results 3 # rock ball v = 2 m/s simulation results v = 4 m/s simulation results v = 8 m/s simulation results v = 6 m/s simulation results v = 10 m/s simulation results 4 # rock ball
the sensitivity of the DCS and the maximum contact force to the 10 Shock and Vibration change of parameters is Ω(E 1 )

3. 5 × 10 - 4
Displacement of centre of sphere (m) Displacement of centre of sphere a er zooming in (m) 3.5 × 10 -4 Displacement of centre of sphere when Poisson's ratio ν 2 changes a er zooming in Elastic modulus E

2 (E 2 (Figure 14 :
Figure 14: E ect of contactants' parameters on DCS and contact force in the contact process of elastic sphere impacting the elastic plate (when the elastic modulus and Poisson's ratio of the sphere are same with the plate).

EFigure 15 :
Figure 15: E ect of E 1 and E 2 on DCS and contact force in the contact process of elastic sphere which has similar properties with rock impacting the metal plate.

Figure 16 :Figure 17 :
Figure 16: Simulation curves of contact force-time at 10 m/s

Table 1 :
Parameters of contact model of the elastic sphere and metal plate.

Table 3 :
e linear correlation coe cient R 2 between the simulated values and theoretical values.

Table 4 :
Parameters of contact model of the elastic sphere and metal plate.Figure 12: E ect of sphere parameters on DCS and contact force.

Table 5 :
Parameters of contact model of the elastic sphere and metal plate with same properties.
Figure 13: E ect of metal plate parameters on DCS and contact force.