Force Analysis for the Impact Between a Rod and Granular Material Memduh Arsalan1 , Hamid Ghaednia1 , Dan B. Marghitu1 , and Dorian Cojocaru2(B) 1 Department of Mechanical Engineering, Auburn University, Auburn, USA 2 Department of Mechatronics and Robotics, University of Craiova, Craiova, Romania cojocaru@robotics.ucv.ro Abstract. The simulation of the normal and oblique impact of a rigid rod with a granular material has been studied. This model solved the inconsistency of the dynamic force at the beginning of the impact. Two diﬀerent impact angles have been analyzed: the normal impact and the impact with an angle θ = 45◦ . The penetration depth, the normal velocity of the tip during impact and the normal contact force during the impact have been analyzed. For the normal impact the dynamic force reaches a maximum during the impact, while the static and the normal forces show continuous increase. For the oblique impact the dynamic and the total forces show a maximum during the impact, while the static force is increasing throughout the impact. Keywords: Impact 1 · Rigid rod · Granular material Introduction Granular materials are not considered as solid, liquid or gas because of having a variety of properties that distinct them from other materials [1]. The penetration depth of the rigid body goes into the granular material, the velocity during penetration, the eﬀects of densities of the rigid body and the granular material, the eﬀect of packing state of the granular materials, and the proper force law during the penetration have been studied by many researchers [2]. The primary goal of the granular impact studies was to determine the contact forces [3]. The dynamics between the rigid bodies and the granular media is complicated and necessary to be studied further. A simple phenomenological model for the equation of motion of penetrating rigid bodies for granular impact cases that follows the generalized Poncelet law was proposed by [4–7]. Ciamarra et al. [8] studied the time of the rigid bodies motion during the penetration. In their study, they demonstrate that the deceleration of solids during penetration is constant and proportional to the impact velocity, that the stopping time is independent of impact velocity, and that the drag force on the rigid body has ﬂuctuations and it is proportional to the impact velocity. c Springer International Publishing AG 2018 C. Ferraresi and G. Quaglia (eds.), Advances in Service and Industrial Robotics, Mechanisms and Machine Science 49, DOI 10.1007/978-3-319-61276-8 10 86 M. Arsalan et al. They proposed for the force on the rigid body at the impact with a granular material a normal, Fn , and a tangential, Fs , component: Fn = − [k δ + mr γn |vn | θ(vn )] n̂, (1) Fs = min [mr γs |vs | μ |Fn |] ŝ, (2) where vn and vs are the normal and the tangential components of velocity, δ is an overlapped length, θ is Heavyside function, γn and γs are viscoelastic constants in normal and tangential directions respectively, and μ is the static friction coeﬃcient. Tsimring and Volfson [7] proposed a generalized Poncelet law. They addi1/3 , with d0 the minimum depth from tionally proposed for the depth d = d20 H zero impact velocity, and H the total height which includes the free fall height and the penetration depth. Katsuragi and Durian showed that the interaction between rigid body and the granular material has an inertial drag force that is velocity dependent and a frictional force that is depth dependent [4]. Paecheco et al., studied the depth dependence friction for the dynamics of the impact with a granular medium [2]. Ciamarra et al. and Katsuragi and Durian [4,8] proposed that the ﬁnal penetration depth is proportional to the 1/3 power of the total height at low impact cases while Walsh et al., showed that it is proportional to 1/4 power of the total height at high impact [9]. Ambroso et al. [10] studied the penetration depth, d, at low velocities. Their study examined the penetration of wooden spheres into a loose noncohesive packing of glass beads in 3D at low velocities that is lower than 5 (m/s). They also analyzed penetration depth for frictional drag in the shallow regime and found a formula for the contact force. Goldman and Umbanhowar in their study varied the diameter of the impacting sphere, R and the density, ρs , to show the eﬀects of R and ρs on the depth [5]. They also proposed a new equation for the drag force that is valid in the steady collision regime which is before the granules start to collapse onto the impacting object. Tiwari et al. [11] stated that determining the drag force for low impact velocities is diﬃcult, so they proposed that drag force is linearly proportional with depth. They also proposed that the velocity dependence drag constant h(z) takes two diﬀerent values that are corresponding to preﬂuidized and ﬂuidized force network of the granular media. Li et al. [12] systematically investigated how the properties of the medium (volume fraction) inﬂuence the performance of a walking robot on granular media. They conclude that small changes in properties of the material produce important changes in speed of motion. Zhang et al. [13] studied the mechanic of movement for a bio-inspired legged robot. Taken as model the movement of desert animal on sandy granular media, the authors performed experiments regarding the dependences between the performances of the displacement and both the mechanics of legged locomotion and the substrates properties. Force Analysis for the Impact Between a Rod and Granular Material 87 In order to support systematic research regarding the control of legged robot walking on granular media, Qian and Goldman [14] designed an experimental system (SCATTER). The system facilitates the possibility of varying a number of locomotions parameters and studies the eﬀect when diﬀerent legged robots walk on diﬀerent terrains. In order to accomplish the robotic tasks involving a mechanical interaction between the robot and parts of the working space, with or without rebound, it is compulsory to assure a control strategy based on the impact model. Designing a robotic system for classical ﬁeld of robotics application, industrial or nonindustrial ones, it is very important to have a coherent model for the mechanical impact between the robots links and diﬀerent kinds of materials composing the working space. If we take into consideration special applications (e.g. rescue applications, prosthesis, mobile walking robots, robots working in dangerous environments etc.) the needs for such models increase dramatically. The control strategies for robotic applications with mechanical impacts start from a number of initial conditions (e.g. angles, velocities) and estimate the dynamic post-impact behaviour of the system. 2 Force Analysis For the granular impact cases, the important interaction between the granular materials and the rigid body during penetration is the resistance force [4,15]. The resistance force consists of two forces that are a depth-dependent frictional force which is static resistance force, Fs (d), and a velocity-dependent frictional force which is dynamic resistance force, Fv (v): FR (d, v) = Fs (d) + Fv (v). (3) The dynamic frictional force is similar to the drag force used in [4]. Although the granular materials are not liquid, the granular materials begin to act liquid behavior during penetrating of the rigid body into the materials while the grains loose their stationary state in contacts [16]. The granular studies which are not eﬀected by static resistance force with using dilute granular material [17–19] and at low speed impact [4,7] show the quadratic drag force model is more convenient and the dynamic frictional force is modeled as v (4) Fd = − ηd ρg Ar v · v, |v| where ηd is the drag coeﬃcient obtained from experiments and Ar is the immersed area of the rigid body [20]. Using Eq. (4) in an impact problem will cause an inconsistency. At the initiation of the collision this equation for the dynamic force will yield into a nonzero total force. To solve this problem we propose the following modiﬁcation on Eq. (4): Fd = − v ηd ρg Ar v · v (1 − e−3 dy /dc ), |v| (5) 88 M. Arsalan et al. where dc is the diameter of the object and dy is the displacement in the granular matter of the tip of the cylinder. Now this equation will results in zero force at the initiation of impact and is solving the inconsistency. The static frictional force is based on the reorganization of the grains and the grain-grain friction [16]. So, gravitation, packing state of the granular material, properties of both the granular materials and the rigid body such as size, and shape. Albert et al. [21] performed their experiments at low speeds and formed the static frictional force as Fs = η g ρ H 2 dc . The horizontal component of the static resistance force at any angle is Fsh = − v ηh g ρg dc d2y , |v| (6) where ηh is a constant which depend on the surface fraction, morphology, and packing state of the granular [21], and dy is the immersed depth of the rigid body. Hill, Yeung, and Koehler [22] suggested an empirical equation for the vertical component of the static resistance force as Fsv v ηv =− |v| dy L λ g ρg V, (7) where ηv and λ are empirical constants [22], and V is the immersed volume. 3 Numerical Results The impact of a rigid rod with a granular material has been numerically analyzed based on the presented equations. The nonlinear equation of motion of the rod has been solved numerically using the Runge-Kutta integration method with the following values for the impact of a rod: length of the rod, L = 0.1524 (m), diameter of the rod dc = 0.00635 (m), density of the rod ρc = 7700 (kg/m3 ), density of the granular ρg = 2500 (kg/m3 ), ηd = 6.5, ηh = 8, ηv = 22, and λ = 1.1. The rod is released vertically without an angular velocity and the initial linear velocity is set to vi = 1.53 (m/s). The numerical simulation stops when the velocity of the tip of the rod reaches to zero. Figure 1 shows the penetration depth during the collision. The maximum penetration is d = 0.031 (m) and the penetration time is t = 0.032 (s) It can be seen that the penetration rate is decreasing throughout the collision and gets to zero at the end of the impact. Figure 2 shows the normal velocity of the tip of the rod for the same impact as shown in Fig. 1. The velocity starts with vi = 1.53 (m/s) and decreases to zero at t = 0.032 (s). The slope of the velocity is zero at t = 0 (s), which is the result of Eq. (5). Figure 3 shows the static and dynamic and the total normal force. The dashed line shows the dynamic normal force, which starts at zero with a large increasing rate. The dynamic force reaches a maximum at t = 0.00495 (s) and starts Force Analysis for the Impact Between a Rod and Granular Material 89 Fig. 1. Penetration depth, dy , for a normal impact with initial velocity vi = 1.53 (m/s). Fig. 2. Normal velocity of the tip, vT , for a normal impact with initial velocity vi = 1.53 (m/s). 90 M. Arsalan et al. Fig. 3. Static (Fs ), dynamic (Fd ) and total force (FR ) for a normal impact with initial the velocity vi = 1.53 (m/s). decreasing after this point until it is zero at the end of the collision. The static force (dotted line) shows an increasing trend throughout the impact. Although the total normal force (solid line) shows an increasing trend during the collision, the slope of the total force decreases when the dynamic force starts decreasing and then increases again. The reason that the contact force is negative is because the positive direction is considered downward into the granular material. For the same numerical values the oblique impact of a same rod with the same initial velocity, and an initial angle, θ = 45◦ has been analyzed for the penetration depth, normal velocity, and the contact forces. Figure 4 shows the penetration time during the impact for initial velocity, vi = 1.53 (m/s), and initial angle, θ = 45◦ . The penetration depth is continuously increasing with a decreasing slope. The maximum penetration depth is d = 0.0145 (m) at the end of impact, t = 0.018 (s). Figure 5 shows the normal velocity of the tip of the rod during the impact for the initial angle, θ = 45◦ and the initial velocity, vi = 1.53 (m/s). The slope at the beginning is zero and the velocity decreases to zero at the end of the impact. There is a diﬀerence between this case and the normal impact shown in Fig. 2. For the normal impact the slope of the velocity was continuously increasing while for the oblique impact at the end of the impact the velocity slope is decreasing. The static, dynamic, and the total normal contact force during the impact is shown in Fig. 6. Same as for the normal impact case shown in Fig. 3 the dynamic force starts at zero and reaches a maximum at t = 0.005 (s) and decreases until the end of the impact. The static force starts with zero and increases continuously until the end of impact. The total force is also diﬀerent from the normal case Force Analysis for the Impact Between a Rod and Granular Material 91 Fig. 4. Penetration depth, dy , for an oblique impact with initial velocity vi = 1.53 (m/s) and an initial angle θ = 45◦ . Fig. 5. Normal velocity of the tip, vT , for an oblique impact with initial velocity vi = 1.53 (m/s) and initial angle θ = 45◦ . 92 M. Arsalan et al. Fig. 6. Static, dynamic, and total force for an oblique impact with initial velocity vi = 1.53 (m/s) and initial angle θ = 45◦ . shown in Fig. 3. For this case the dynamic force is more dominant and causes the total force to reach a maximum at t = 0.007 (s) and decreases until the end of the impact. 4 Conclusions The impact of a rod with granular material has been analyzed. The impact force has been modiﬁed to solve the inconsistency at the beginning of the impact, which is the results of a nonzero dynamic force at zero penetration. An exponential step function has been added to the dynamic force. The normal and oblique impact of a rod have been studied. Dynamic force shows a maximum during the impact in both the cases, while the static force is increasing throughout the impact. The total force however, acts diﬀerently for normal and oblique impact. For the normal impact, the static force has the larger eﬀect and the total force increases throughout the impact, while for the oblique impact the dynamic force has a larger eﬀect and the total force reaches a maximum during the impact. This study was performed in order to support the design process for robotic applications involving mechanical impact. The results are useful both for the design of mechanical parts of the robotic system and for the implementation of an adequate control strategy. The theoretical study will be continued by a practical validation. An experimental environment was designed and implemented: an enclosure with transparent walls and containing balls of plastic material, a system for high speed image Force Analysis for the Impact Between a Rod and Granular Material 93 acquisition and processing, a number of rods, and a system with controlled initial conditions for dropping of the rods. The results obtained here using the proposed theoretical model will be compared with the practical behaviour. References 1. Jaeger HM, Nagel SR (1992) Physics of the granular state. Science 255(5051):1523 2. 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