Shock and Vibration

Volume 2015 (2015), Article ID 129728, 10 pages

http://dx.doi.org/10.1155/2015/129728

## Experimental and Simulation Analysis for the Impact of a Two-Link Chain with Granular Matter

Mechanical Engineering Department, Auburn University, Auburn, AL 36849, USA

Received 30 October 2014; Revised 22 May 2015; Accepted 2 June 2015

Academic Editor: Sundararajan Natarajan

Copyright © 2015 Eliza A. Banu and Dan B. Marghitu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The resistance force of the granular matter is modeled as a linear superposition of a static (quadratic depth-dependent) resistance force and a dynamic (quadratic velocity-dependent) frictional force. The impact is defined from the moment the end point of the system comes in contact with the granular matter surface until the vertical linear velocity of the end point is zero. The variables of interest are the final depth at the end of the penetration phase and the stopping time. The results for a two-link kinematic chain with two points of contact were compared to the results obtained by applying the resistance force formulation developed to corresponding CAD simulation models. The results revealed that the final displacement increases with initial velocity, while the stopping time decreases. The sensitivity to the initial velocity was studied and an improvement to the resistance force formulated as a result. A series of expressions are proposed for the resistance force coefficients.

#### 1. Introduction

Impact with granular matter has been of great interest to researchers for decades, mainly because although the material is a conglomeration of discrete solids, it behaves as a fluid until a solid-like behavior becomes established. As a result, a resistance force that would account for these characteristics during impact has been difficult to model. The dynamics associated with the motion and interaction of rigid bodies with Newtonian fluids is a classical problem in fluid mechanics. Granular materials are collections of polydisperse grains and exhibit complex behaviors. This means that the rheology of the medium is comparable to that of a solid under a critical shear stress, although it performs like a fluid above it. Granular materials have been studied for potential applications in multiple areas. In earth science, for example, problems such as avalanches involve the flow of granular matter [1, 2] and a better understanding of their behavior would greatly facilitate those seeking to model the effects of earthquakes [3], meteorite impact cratering, and low-speed impact cratering [4–6]. Industrial processes such as mixing, stirring, and drilling would also benefit [7, 8]. To appreciate the larger perspective that is involved in these applications, the impact of an object with granular matter is being explored through experiment, simulation, and theory. However, because of the solid-fluid-like behavior of the granular matter, a great deal remains to be explored, especially from a theoretical perspective. The focus of much of the current research on the impact of a rigid body on granular matter is on describing the force experienced due to the reorganization of the grains that are opposing the motion of the intruder. A good understanding of these forces can aid in the design of tools and the design and control of robots designed to maneuver in granular environments. Due to the lack of a universal accepted theoretical model, experiments have generally sought to demonstrate the variation of the drag force in terms of a number of different parameters: impact velocity [9], the diameter of the penetrating object [1, 9], the packing volume [10], and the shape [11] of the penetrating object, for example, cylinders [7, 9, 12–14] or spheres [3, 15, 16]. Others have adopted a slightly different approach by performing their analysis using discrete element methods (DEM) for the same purpose [1, 17, 18].

The penetration of granular matter at high speeds and low speeds has been studied in recent years [5, 6, 16, 19, 20]. In particular, studies of impact cratering have looked at parameters such as the depth and size of the crater, as well as how the form of the crater is affected by the initial impact conditions of the impacting rigid body. To study the motion of a sphere through granular matter resistance force models that treated the problem as the sum of a velocity-dependent force and a static resistance force which is depth-dependent were created [5]. For example, Ambroso et al. [19] suggested that the stopping time of a projectile in granular matter depends on the geometry and density of the rigid body and the initial impact velocity.

Lee and Marghitu [21] extended the work of Katsuragi and Durian [4] to provide the first mathematical model for an oblique impact. They demonstrated that for an oblique impact of a compound pendulum the stopping time in the granular matter increases with decreasing initial impact velocity using a force model that consists of a linear superposition of the static resistance force and drag force.

The focus of this study is on the oblique impact of rigid bodies. Research shows that the study of the impact of rigid bodies with granular matter is potentially of great significance. As yet, to the best of our knowledge there has been no comprehensive universally accepted expression for the penetration resistive force of a rigid projectile into a granular medium. The most important aspect of this analysis is therefore to develop an analytical expression for the resistant force that is valid for low-speed impact both vertically and at an angle. Researchers have analyzed the impact of spherical and cylindrical shaped rigid bodies through experiments and numerical simulation. It was concluded that the drag force [16] should include a quadratic velocity-dependent term [4, 5] and depend on the depth of penetration [2, 7, 9, 11, 22, 23]. Research has also shown that the resistant force and depth of penetration for plunging rigid bodies into granular matter depend on other parameters as well. The resistant force during impact scales with impact velocity [16], diameter of the impacting object [2, 9, 24], and packing fraction [2, 10, 25]. Only a few studies have been reported for oblique angle impact [21] but have shown that the impact angle has an influence on the value of the resistant force [1, 13, 18, 26, 27].

The first objective of this paper is to examine the characteristics of the resistance force presented above and to develop a model that takes into account the granular properties of the medium. Until now, the coefficients employed in these expressions have been determined experimentally. Instead a series of mathematical expressions will be proposed and then validated by comparing the simulation results for depth and stopping time for double impact points at different angles with the experimental results.

A second purpose is to validate the use of a CAD software package in simulating the impact with granular matter by using an equation model that will reduce computational time in comparison to other numerical models and discrete element methods. The results of this study will be of a particular value in robotic design, prosthetics, and biomechanics.

This study is of great importance and a set of mathematical expressions for estimating the values of the scaling that should be used is proposed for each component of the resistance force. The force model presented here shows that the two components are not independent of each other and should thus not be treated separately. The depth depends on impact velocity; therefore the static force will be influenced by the dynamic force.

#### 2. Mathematical Model

The first mathematical model for the impact of an object with granular matter at a specific angle was proposed by Lee and Marghitu [21] based on the force model proposed by Tsimring and Volfson [5] and verified experimentally for a sphere by Katsuragi and Durian [4]. The resistance force proposed takes into account both the rate-dependent force (dynamic frictional force) and the displacement-dependent force (static resistance force):The dynamic force is due to the velocity, , of the intruder in the granular matter, which is made possible by the fluid-like characteristics of the material, up to the point at which the granular material begins to act like a solid surrounding the object, stopping its motion. Hence, the expression for this can take the same form as that for the drag force: . However, it has been shown that the rate-dependent force is proportional to the velocity squared [9, 25]: , where is the density of the medium, is the reference area, and is the drag coefficient. The dynamic force used in this work isThe force is opposing the motion of the penetrating body; therefore, the force expression includes the term . The term is a constant and is the reference area.

The static force, , in (1) represents the force that appears due to the hydrostatic pressure [10] and is based primarily on the reorganization of the particles of matter and grain friction. Thus, this force depends on the material properties, the packing of the matter [2, 17, 25], gravitation, and size and shape of the penetrating object. This will tend to lead to the stress on one side to increasing with depth according to the scaling factor in one direction and according to a power law for depth in another direction.

The horizontal static force increases with the granular pressure [9, 15, 24, 28] that is acting on the surface of the body in the horizontal direction. From experiments conducted [9] at very low speeds of mm/s, the yield force is of the form , where is a constant that depends on the media properties, is the gravitation acceleration, is the immersed dimension of the intruder, and is the diameter of the object. The direction in which the force acts will be opposite to the normal velocity of the center of mass of the immersed volume, , and the immersed length of the rigid body is given by the displacement in the granular matter of the tip of the cylinder, . Therefore the generalized horizontal static force isThe vertical component of the static force represents the resistant force of the granular medium to the immersion of the intruder in the vertical direction. Experiments conducted for vertical impact [3, 15, 23] revealed a nonlinear dependency of the force on the depth of penetration and attempted to formulate a power law to describe this force. The power law form was proven to arise due to the dispersed quality of the grain size of the granular matter found in nature, so that the force chain structure between the grains and the grain intruder exhibits a homogenous distribution. Thus, the expression for the vertical static force iswhere is the length of the rigid body and is the vertical term of the velocity of the center of mass of the immersed volume, .

For a two-link kinematic chain with two contact points, (2), (3), and (4) can be applied for each contact point. The dynamic friction force and static resistance force have the same point of application. The point of application is considered to be the centroid of the immersed volume. For a Cartesian reference frame the terms of the resistance force becomewhere is the lateral dimension of the link, is the diameter, and is the density of the granular medium. Because of the proven dependency of the resistance force of the packing fraction of the granular matter, is the density of the medium and the packing fraction is factored into the value of . The velocity is the linear velocity vector of the centroid of the immersed volume, with being the horizontal component of the velocity and the vertical component of the velocity of the centroid .

The velocity of point , as shown in Figure 1, can be expressed in terms of the velocity of the end point , , angle of impact, , and angular velocity of the bar, :where and are the position vectors of points and :ThenSubstituting (11) into (8) and with , the expression for becomesTherefore the expressions for and to be used in (5), (6), and (7) are