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978-3-319-61276-8 10

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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.
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