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A study on an up-milling rock crushing tool operation of an underwater tracked vehicle Article  in  Proceedings of the Institution of Mechanical Engineers Part M Journal of Engineering for the Maritime Environment · November 2017 DOI: 10.1177/1475090217735934

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Original Article

A study on an up-milling rock crushing tool operation of an underwater tracked vehicle

Proc IMechE Part M: J Engineering for the Maritime Environment 1–18 Ó IMechE 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1475090217735934 journals.sagepub.com/home/pim

Mai The Vu1, Hyeung-Sik Choi2, Dae Hyeong Ji2, Sang-Ki Jeong2 and Joon-Young Kim3

Abstract In this article, we develop the analysis of a new underwater tracked vehicle with rock crushing tool, working under the water. To design the capacity of the underwater tracked vehicle actuator and the rock crushing tool actuator, we analyze the interaction forces and torque between the rock and the rock crushing tool. Since experiments on the underwater tracked vehicle with a rock crushing tool are very difficult, costly, time-consuming, we first perform a mathematical modeling of the underwater tracked vehicle with the rock crushing tool. We analyze the mechanics of the underwater tracked vehicle system that is affected by the forces and moments of the underwater rock crushing, where the force and torque on the rock crushing tool are based on the analysis of the mechanics of an individual cutter tool. We derive a mathematical expression for the forces and moments of the combined system on the underwater tracked vehicle and the rock crushing tool for rock crushing. For this, we study the parameters that affect the mechanics of the underwater tracked vehicle system with the rock crushing tool. To apply the rock crushing tool to underwater rock excavation, we also study the hydrostatic effects to the combined underwater tracked vehicle system with the rock crushing tool. To design the capacity of the actuator of the developing underwater tracked vehicle and the rock crushing tool, we analyze the required tractive or down thrust forces, and the torque to the rotor carriage caused by the cutting system. In addition, we analyze the energy and the power for the rock crushing tool actuator related to the tool characteristics. To support the validity of the analyses, we use the derived equations to perform a number of numerical simulations.

Keywords Rock crushing, underwater tracked vehicle, tractive thrust, down thrust, simulation

Date received: 25 July 2016; accepted: 11 August 2017

Introduction The application of excavation machines for hard rock excavation in both civil and mining engineering fields has increased significantly in recent years. In particular with the recent development of versatile machines capable of effectively coping with different rock conditions, the mechanical excavation industry is destined to play a much bigger role in future construction projects. For the seafloor miner, tracked vehicles are preferred compared to wheeled or legged vehicles due to the larger contact area of tracks with the ground providing better floatation and larger traction forces, which are required for the extremely cohesive soft deep-seafloor soil. In this reason, the interaction between ground and off-road vehicles, such as agricultural tractors, has been an important field of study.1 In order to investigate the performance of tracked vehicles, a number of studies

have been carried out. Rubinstein and Hitron2 developed a three-dimensional (3D) multi-body simulation model for simulating the dynamic behavior of tracked off-road vehicles using the LMS-DADS simulation program and used user-defined force elements to describe 1

Department of Convergence Study on the Ocean Science and Technology, School of Ocean Science and Technology, Korea Maritime and Ocean University, Busan, Republic of Korea 2 Division of Mechanical and Energy Systems Engineering, Korea Maritime and Ocean University, Busan, Republic of Korea 3 Division of Marine Equipment Engineering, Korea Maritime and Ocean University, Busan, Republic of Korea Corresponding author: Hyeung-Sik Choi, Division of Mechanical and Energy Systems Engineering, Korea Maritime and Ocean University, Yeongdo-ku, Busan 606-791, Republic of Korea. Email: [email protected]

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the interaction between each track link and the ground. Solis and Longoria3 described the integration of a realistic and efficient track–terrain interaction model with a multi-body dynamics model of a robotic tracked vehicle, and comparisons between simulated results and those obtained from field testing with a remotely operated unmanned tracked vehicle. Hong et al.4 developed a simplified transient 3D dynamic analysis method for tracked vehicles crawling on extremely soft cohesive soil. Morgan and Cathie5 discussed aspects of terramechanics and mobility that are applicable to the operation of tracked trenchers on very soft clays. Rock crushing (RC) machines are able to perform vertical and horizontal cuts for quarrying natural stone mine. The design of cutting tools and setting parameters of cutting operations requires knowledge about the cutting process. Cutting force is one of the main factors characterizing a cutting process. Theoretical evaluation of the cutting force is not an easy task. The mechanical interaction between the cutter tool and rock has been studied by numerous researchers over the years. The primary motivation behind these research efforts is twofold: on one hand, the need to improve the efficiency of mechanical excavation of rocks, and on the other hand, the possibility of deducing material properties from the action of a tool pressed against the surface of a rock. To maximize the benefits of mechanical excavators to any operation, performance of these machines under specific conditions must be understood. For this purpose, several investigators have formulated and applied a diversity of criteria for determining the efficiency of mechanical techniques of excavation. The most commonly cited criteria are bit cutting force, power consumed, machine cutting rate, and specific energy of the excavation process. Simple analytical models, like those developed by Nishimatsu,6 can provide a very approximate estimation of cutting forces only. Numerical methods based on continuum models, like finite element methods, have serious problems in modeling discontinuities of the material occurring during rock cutting.7 As described by Inyang,8,9 a distinction needs to be made between the bit force needed to penetrate rock to the depth of cut, and the force needed to cut the rock along the cutting profile once the bit has penetrated the rock to the depth of cut (the cutting force). Specific energy is one of the comprehensive criteria for evaluating the efficiency of excavation processes. It is inversely proportional to the efficiency of operation and is defined as the ratio of expended energy to the volume of material excavated. Bailey and Dean10 have shown that when the same tool is used in fragmenting different rocks, the specific energy is a useful basis for comparison. However, laboratory-specific energy and field-specific energy of excavation in the same rock can vary by as much as 80%, as confirmed by the results of tests performed by Rabia.11 Although there are many trenching machines which have been manufactured by companies around the

world, information about the methods to develop such machines is very limited because of proprietary rights held by the companies. Moreover, the previous studies still have some problems such as how to implement the dynamic analysis of the total deep ocean mining system, there is no progress in the cutter rock interaction understanding and modeling, and the effects are not fully studied. Sometimes important factors are neglected, since dynamic interaction between the cutter tools and the rock mass is uncertain, complex, and difficult. In industrial sectors, companies have gained numerous experiences from designing and producing such machines during gradual development and evolution of successive generations of the machines. In rapid development of trenching machine for specific operating conditions with certain performance characteristics requirement, an analytical approach should be developed that can cover all required important features and yet practical. Direct experiments are time-consuming and costly and strict scientific approach from the first principles such as theoretical rock mechanics has numerous difficulties such as finding detailed measured material properties, failure criteria, or selecting which fracture theories should be used. Regarding those considerations, a modeling method for the RC tool and further analysis of the interaction effects of underwater RC tool to underwater tracked vehicle (UTV) is fully studied and analyzed in this article. Our article deals with the forces and power levels in cutting machines having a disk or drum that rotates about an axis perpendicular to the direction of advance. The forces on individual cutter tools are related to their position on the RC tool and to characteristics of tool layout, tool speed, RC tool size, machine advance speed, and RC tool torque. Integration leads to expressions for force components acting on the RC tool axis, taking into account tool characteristics, cutting depth of the RC tool, and RC tool torque. These provide estimates of tractive thrust and thrust normal to the primary free surface. For self-propelled machines, this leads to considerations of traction, normal reaction, weight, and balance. We analyze specific energy consumption and relate it to machine characteristics and strength of the material being cut. We also analyze power requirements for the ejection of cuttings and treat the hydrodynamic resistance of underwater cuttings. Finally, we conduct a number of simulations to generate physical values for the design of the system.

General specifications of the combined UTV with the RC tool Underwater RC tool description Figure 1 shows the proposed system, which consists of a RC tool connected to a UTV, which is lowered to the seabed by a crane. An RC tool is a machine that uses a rotary cutting unit equipped with cutter tools (bits) to excavate in rock for both mining and civil engineering

Vu et al.

Figure 1. An RC tool12 on the UTV. UTV: underwater tracked vehicle; RC: rock crushing.

purposes. The cutting unit consists of a disc (or drum), and a number of drag cutter tools spaced along its length and wound around the disc. To excavate in rock, the RC tool is pushed into the material, and as it rotates while the UTV moves forward, the cutter tools cut through the rock material. The thrust is primarily in the direction of the rock surface advance. The rotary action of the RC tool generates drag that is tangent to the cutter. Thus, the cutter tools rip into the rock surface and gouge off fragments.

3 a certain amount of thrust and torque provided by the mechanical cutting system. Cutter tools are classified into two general types: rolling cutters and drag bits. Cutter tools transmit the energy of the machine to the rock so that it can be fragmented. The geometry and wear characteristics of the cutter tool therefore have a significant effect on the energy transferred to the rock, and the attainable rate of penetration. The two main types of drag cutters used in the mining industry employ radial and conical bits. Radial cutters are limited to the excavation of the softest and least abrasive materials. Continuous miners, long-wall shearers, and borer miners are typical mechanical excavators, where radial cutters are used to cut softer material, such as coal, trona, and salt. Figure 2(a) shows the examples of different radial cutters. Figure 2(b) shows the second type of drag cutters that use conical bits. Compared to radial bits, these are typically used in continuous miners and long-wall shearers to cut harder rock and are also used in roadheaders. Conical bits are more durable than radial bits and have a self-sharpening property, which is an advantage for longer cutter life compared to radial bits. Conical bits can be used to excavate higher strength rocks if the rock mass is significantly weakened by the presence of joints, fractures, bedding, or foliation.

Assumptions To simplify the challenge of modeling the RC tool, the followings are assumed as:

Cutter type A crucial aspect of any mechanical RC tool is the cutter tool, which performs the actual rock penetration under



Cutting mode: up-milling RC tool mode, of which Figure 3 shows the configuration.

Figure 2. Types of cutter bits: (a) radial cutters (Sandvik) and (b) conical cutters (Kennametal).

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Proc IMechE Part M: J Engineering for the Maritime Environment 00(0)

Figure 3. Cutting modes for transverse rotation devices: (a) up-milling mode, (b) climb-milling mode, and (c) slot-milling mode (lateral view of RC tool).











The resultant force on a single cutter tool fluctuates with time in response to the formation of discrete chips in the brittle material. In this article, the tool force does not vary systematically with tool position in a homogeneous material. The traverse velocity U of the UTV, when compared with the RC tool tip speed ut, is small and the absolute tool velocity u is given to a sufficient degree of accuracy by the tangential velocity arising from rotation alone, ut that is, u’ut = Rv = 2pRf. The ratio of tool force components K is assumed to be invariant with the chipping depth of the cutter tool. The tool forces are directly proportional to the chipping depth.

2.

3.

Terminology Some of the terminologies used in this article are given below, while Figure 4 shows all the parameters used in this article. 1.

Cutting modes: Figure 3 illustrates the three main cutting modes. In up-milling, the RC tool is sunk into the work to a depth less than the diameter, the axis of rotation is parallel to the primary free surface, and the direction of rotation is such that the cutter tools move upward on the leading side

4.

5. 6.

of the RC tool. In climb milling, the RC tool is sunk into the work to a depth less than the diameter, the axis of rotation is parallel to the primary free surface, and the direction of rotation is such that the cutter tools move downward on the leading side of the RC tool. In the slot-milling mode, the RC tool cuts across its complete semicircumference, and the axis of rotation is normal to the primary free surface. In this article, we just consider and analyze the interaction of a rock crushing machine with the rock in climb-milling operation mode. Cutting depth d is the depth to which the RC tool is set into the work, measured normal to the primary free surface. Chipping depth ‘ is the depth of penetration of the individual cutter tool into the work, measured in a radial direction. For a given machine speed, it varies continuously through the working sweep of the cutter tool. Effective tool length h is the maximum length of the individual cutter tool that can safely penetrate the work, measured in a radial direction. RC tool radius R is the radius of the RC tool measured to the effective tool tips. Tracking cutters are cutter tools that sweep along a common path in a diametral plane. The number of the tracking cutters in a complete revolution of the RC tool is designated by n.

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5

16.

17.

18.

Figure 4. Definition of symbols (lateral view of RC tool).

7.

8.

9.

10.

11.

12.

13.

14.

15.

Circumferential tooth spacing S is the circumferential distance between the tracking cutters, measured at the tooth-tip radius, that is, S = 2pR=n. Lateral tooth spacing s is the distance between adjacent rings of the tracking cutters, measured along a generator of the RC tool. Angular position of the RC tool is defined by the angle u between a radius running through the point of interest and a radius running normal to the traverse direction. Angular velocity of the RC tool v will often be expressed as a frequency f in this discussion, that is, f is the number of revolutions of the RC tool per unit time (rev/sec). Traverse rotation devices rotate about an axis that is perpendicular to the direction of advance. It is the motion of the entire RC tool relative to the material being cut, that is, the motion of the center of the RC tool. The traverse speed U is the linear speed of the RC tool center relative to the work and is directed parallel to the primary free surface. The absolute tool speed u is given by the velocity of the tool tip relative to the work, which in turn is given by the time derivative of the tool trajectory through the work. The velocity component derived from drum rotation alone, ut, is of special interest. ut is the tangential velocity of the bit tip relative to the center of the RC tool. Tool forces or cutting forces are the forces developed by the individual cutter tool during the cutting process. The resultant force on a cutter tool can be resolved into tangential and radial components fu and fR with respect to the RC tool on which they are mounted. The ratio of tool force components K is fR =fu , that is, the ratio of the radial component to the tangential component. This is the tangent of the angle between the resultant cutting force and the tangential direction. In practical terms, it gives a measure of the sharpness of the cutter tool, with high values of K indicating blunt or worn tools, also varies considerably with tool

19.

20.

21.

22.

design and with the state of wear, but it is not very sensitive to variations of chipping depth ‘.13 RC tool cutting torque T is the net torque developed by the rotary RC tool when it applies tangential cutting forces to the cutter tool tips at constant speed. Net overall cutting force Ft is the tangential force given by Ft = T=R, where R is the radius of the RC tool to the cutter tool tips. Axle forces on the RC tool are the forces developed on the axle by the cutting process. The resultant axle force at any instant is given by a summation of the vector cutting forces on all the active cutter tools. The resultant axle force can be resolved into three components that are (1) parallel to the direction of advance H, (2) normal to the primary free surface V, and (3) radial to the RC tool axis. The component radial to the RC tool axis should be zero for a symmetrical RC tool that is dynamically balanced about the midsection. The component parallel to the direction of advance H, which is often horizontal, equals the sum of the components of fu and fR resolved in that direction. The component normal to the primary free surface V, which is often vertical, equals the sum of the components of fu and fR resolved in the same normal direction. Tractive thrust of a machine FTT is the force parallel to the traverse direction in order to overcome cutting resistance. This equals the axle force component H. When the rotary RC tool is mounted on a UTV, the tractive thrust is the net forward force developed by the wheels or crawler tracks, that is, the ‘‘drawbar pull.’’ Down thrust of a machine FDT, which may be positive or negative, is the force perpendicular to the traverse direction that is required to maintain the RC tool at the required operating depth. This is equal to the axle force component V. When the rotary RC tool is mounted on a UTV, the available down thrust is limited by the weight and balance of the machine. Machine power P can be partitioned broadly as the RC tool power Pr, thrust power PH, and power loss PL. The RC tool power is the power consumed by the RC tool for cutting. The thrust power is the net power used to overcome cutting resistance in the direction of advance. The power loss is the power that does not contribute directly to the cutting process. Power density of a RC tool is a term used here to denote the RC tool power per unit area of cutting surface, that is, power density Q is PR =Rum B, where um = cos1 ½1  (d=R) and B is RC tool width. Since um is normally an operating variable, an arbitrary definition of normal power density is taken as a basis for comparison of machines; making um = p=2, nominal power density is 2PR =pRB.

6 23.

Proc IMechE Part M: J Engineering for the Maritime Environment 00(0) Specific energy of a cutting machine is the energy consumed per unit volume of material removed. Alternatively, it is the power consumption divided by the volumetric removal rate. The overall specific energy for a complete machine is based on the total machine power.

Analysis of force and moment of the RC tool Mellor13 published a series of reports covering the mechanics of various terrestrial cutting and boring machines that work on land such as transverse rotation machines, axial rotation machines, and continuous belt machines. In this article, Mellor’s analytical model is adopted and improved to design and to analyze an underwater RC machine developed at Korea Institute of Ocean Science and Technology (KIOST). In this article, some steps of the analysis process for the RC machine are presented, which can be used as reference for designing an RC machine. The cutting process is highly nonlinear and complex. Moreover, the cutting process in marine environment makes the system more complex. Because the system includes the hydrodynamic calculation and the shapes of typical rock materials are very irregular, the corresponding computation is quite complicated. In this reason, it is necessary to consider the main influencing factors to simplify the system. Proper application of the RC machines to any mining or engineering application depends on the detailed understanding of the parameters described below.

Analysis of cutting forces The first aim of this article is to examine the forces on individual cutter tools when they are mounted on a RC tool and then to determine how an assembly of the cutter tools affects the moments and the forces for the RC tool as the entire of the RC tool. Forces on individual cutter tools. Each cutter tool on a RC tool develops a cutting force that is determined mainly by the cutter tool geometry, rock properties, and operating conditions, and in particular, the chipping depth. Figure 4 shows that the cutting force can be resolved into radial and tangential components fR and fu . For a particular type of cutter tool working in a given isotropic material, both the tangential cutting force fu and the radial cutting force fR increase as the chipping depth ‘ increases.13 In an ideal condition, fu and fR are directly proportional to the chipping depth ‘ when the chipping depth ‘ is very small with the condition of ‘5w (3D cutting). However, experimental data show that the general pattern of the behavior of fu and fR increases nonlinearly with ‘ in two-dimensional (2D) cutting (w  ‘). The rate of increase is dropping off as the chipping depth ‘ increases according to irregular parabolic relation; this is reflected in the more

sophisticated 2D cutting theories. Since analysis of the tool forces on rock-cutting machines is too complicated, the less affecting complexities and the unknowns are neglected. A simple approximation would make fR or fu proportional to the chipping depth; as a general approximation, they can be expressed as  b ‘ fu = ku ð1Þ r  a ‘ ð2Þ fR = kR r From equations (1) and (2), we have the ratio between fR and fu as follows   fR kR ‘ ab = ð3Þ fu ku r where kR and ku are proportionality constants with dimensions of force (embodying the effects of the cutter tool geometry and the rock properties), r is the radius of curvature of the cutter tool tip, and a and b are dimensionless exponents. Some characteristics of a and b can be deduced from the data compiled in Mellor.13 These data show force components either proportional to the chipping depth ‘, or approximately proportional to some fractional power of the chipping depth ‘. They also show fR =fu decreasing slowly with increase in the chipping depth ‘, from a value that is approximately equal to unity when the chipping depth ‘ is small, that is, fR =fu ’1 when ‘4r. Hence, it might be reasoned that a41 a4b a’b

b41

To simplify the tool force variations during one RC tool rotation, assume that a = b = 1, which means that cutting forces are directly proportional to chipping depth. In the foregoing analysis, equations (1) and (2) are only approximate empirical relations, and in many cases, the experimental data from cutting tests can be represented adequately by linear relations of the form   ‘ fR = AR + kR ð4Þ r   ‘ fu = Au + ku ð5Þ r where AR and Au are proportionality constants with dimensions of force, representing tool force components as ‘ tends to zero. In some circumstances, that is, narrow tools or tools cutting deeply in material, the constants AR and Au are small, and it is sufficient to assume direct proportionality   ‘ fR = kR ð6Þ r

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  ‘ fu = ku r

ð7Þ

The chipping depth ‘ is a function of the rotational frequency of the RC tool f, the traverse velocity U, the number of tracking cutters n, and the angular position u ‘=

U sin u fn

ð8Þ

Thus, for a given set of operating conditions, the chipping depth of each individual cutter tool increases in proportion to sin u. With a typical up-milling the RC tool, the variation is from practically zero at the point of entry, up to a maximum that occurs at the point of exit when umax \ ( p2 ), and at u ¼ p2 when umax > p2 . Substituting equation (8) into equations (6) and (7) yields fR =

kR U sin u r fn

ð9Þ

fu =

ku U sin u r fn

ð10Þ

In addition, we can easily deduce the ratio between fR and fu as follows fR kR = =K fu ku

ð11Þ

Since fR and fu are proportional, it is only necessary to treat one component; we select fu , as it relates directly to the torque of the RC tool. In general, with n cutter tools spaced uniformly around the RC tool, there are n cutter tools passes through the work for each revolution, and each cutter tool experiences a tangential force of ku U sin u sin u = f fu = n r nf

um 2p=n

where the angular position is given as follows    d 1 um = cos 1  R

n cos1 ½1  (d=R) 2p

fu0 =

fu 2p=mn

ð16Þ

where fu = f ( sin u=n) in which the expression of f* is defined as follows f =

ð12Þ

ð13Þ

ku U r fn

ð17Þ

uðm

Ft =

fu0 du

ð18Þ

0

From equation (16), we can easily deduce the expression of the net overall cutting force Ft as follows mn Ft = 2p

uðm

fu du

ð19Þ

0

Substituting equation of fu into equation (19) yields ð14Þ

where d is the cutting depth and R is RC tool radius. Substituting equation (14) into equation (13), we have Na =

Net overall cutting force and tool force. The sum of the tangential tool forces at any given time gives a net overall cutting force Ft that has to be overcome by the applied torque of the RC tool under constant speed conditions. When there are only a few cutter tools on the RC tool (n is small), Ft can be obtained by calculating the values of fu for each of the cutters tool in the work and summing them, and plotting the results against the angular position to obtain variation of Ft with position or time. However, when there are many cutter tools in the work together, and Ft does not vary significantly with position or time, an integral expression for Ft is more convenient. Suppose that the cutter tools on a wide, rigid drum RC tool are disposed in m rings across the width of the RC tool, and that there are n uniformly spaced cutter tools in each ring. If the m rings are staggered systematically with respect to neighboring rings, then a side view of the RC tool would show mn cutter tools uniformly spaced around the perimeter. With cutter tools spaced around the RC tool, each cutter tool accounts for an angular interval of 2p=mn, thus, the tangential cutting force per unit angle, fu0 , is

The net overall cutting force is thus

where f = (fu )u = p=2, n = 1 , that is, the maximum tangential force at u = p=2 when there is only one cutter tool. At any given time, the number of cutter tools in the work is the integer given by Na =

The total tangential force acting on the perimeter of the RC tool at any given time is the sum of the individual tangential tool forces.

ð15Þ

mf Ft = 2p

uðm

sin udu

ð20Þ

0

Then, again substituting equation (17) into equation (20), and taking the integration, we can define as follows

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Proc IMechE Part M: J Engineering for the Maritime Environment 00(0) Ft =

mku U d r uta

ð21Þ

where uta = 2pRf is the tangential velocity of the RC tool, d is the cutting depth, and r is a constant in this context. In principle, the RC tool torque T could be estimated from a laboratory test that uses a single cutter tool to define the relationship between fu and ‘. Alternatively, the mean force for a single cutter tool at a given angular position could be estimated from field measurements of torque T or power Pr on the RC tool, recalling that T = Ft R =

Pr 2pf

ð22Þ

So, we can easily obtain the relationship between Ft and fu using equation (22), as follows fu =

‘u T ‘u P r = md(U=uta )R mdU

ð23Þ

It is also instructive to express fu as fu =

2pR sin u Ft nmd

ð24Þ

In addition, from equation (22), we can rearrange equation (24) as follows fu =

2p sin uT Pr sin u = nmd nmdf

fR0 = ð25Þ

For d/R 4 1, the maximum tool forces occur at the maximum value of u. At this maximum, we have ( sin u)=d = ½(2R=d)  11=2 =R, and equation (25) can be defined as 2pT ½(2R=d)  11=2 ( fu )max = mn R

of mobile machines, the available tractive thrust from the UTV can set the limit of performance for an upmilling RC tool. In the following discussion, H will be referred to dimensionless terms in the form of H/Ft, and to give this clearer meaning, we assume that Ft is the maximum value that can be developed when the RC tool is operating at the maximum torque. The down thrust force V, which is perpendicular to the direction of travel and to the work surface, determines the down thrust needed to maintain a given cutting depth d. On mobile machines, hydraulic actuators often provide this down thrust; an upper limit to positive down thrust is set by the weight and balance of the whole machine. If the force V exceeds the thrust capability of the actuators or the available reaction, then cutting depth d or forward speed U will have to change, in order to limit V. As with H, we will discuss the force V in dimensionless terms as V/Ft. When the RC tool has many cutter tools acting at the same time, as described above, each tool force can be divided by an angular interval of 2p=mn, in order to give force per unit angle, fu0 and fR0 . At any angular position u, the unit force in the tangential direction is shown in equation (16), while the force in the radial direction is expressed as

ð26Þ

fu0 = f

ð27Þ

where ( fu )max is the maximum value of time-averaged tangential tool force. This is an important result, in that it gives a very practical method for estimating the maximum tool force. Tractive thrust and down thrust forces on the RC tool axis. Under normal circumstances, a symmetrical RC tool on a transverse rotation machine has no net side force acting parallel to the axis. The resultant force that acts normal to the rotation axis is, in general, inclined to the travel direction at the finite angle; it is convenient to resolve that force into components H and V, which are parallel and normal to the travel direction, respectively. The tractive thrust force H, which is parallel to the travel direction, determines the tractive thrust needed to feed the RC tool into the rock material. In the case

m sin u 2p

ð29Þ

Similarly, substituting equation (9) into equation (28) and using equation (17) yields fR0 =

Pr ½(2R=d)  11=2 nmf R

ð28Þ

Then, substituting equation (10) into equation (16) and using equation (17) yields

or ( fu )max =

fR 2p=mn

kR m sin u f 2p ku

ð30Þ

On each angular increment of the cutting perimeter (du), there are the radial and the tangential forces, fR0 and fu0 , respectively. These forces can be resolved parallel to, and normal to, the travel direction. For an up-milling RC tool, resolution gives  

Parallel to travel direction: fR0 sin u du + 0 fu cos u du; Normal to travel direction: fR0 cos udu fu0 sin udu.

We obtain the forces on the axis of the up-milling RC tool, H and V, by summing or integrating the resolved incremental components uðm

H=



 fR0 sin u + f u0 cos u du

ð31Þ

0

Substituting equations (29) and (30) into equation (31) and then integrating, we finally obtain

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"   1=2  # F t kR R kR d 2R d um  1 H= 1 + 2 2 ku d ku R d R

ð32Þ

Similar to calculating the tractive thrust H, we can define the down thrust V as follows uðm

V=

ð f 0R cos u  f0 u sin uÞdu

ð33Þ

0

From equations (29) and (30), then integrating equation (33), we also finally have "  1=2 #    Ft kR d R d 2R 1 V= 2  um + 1  R d R d 2 ku ð34Þ 1

where um = cos ½1  (d=R) and kR =ku are constants, consistent with the assumptions of equations (9)–(11). H is positive when thrust applied by the machine is in the direction of travel, and V is positive when thrust applied by the machine is downward into the work. As mentioned earlier, the ratio of the radial component to the tangential component K is the tangent of the angle between the resultant cutting force and the tangential direction. The consideration of the effect of K is important for any mechanical RC systems because it affects machine performance directly. The following simulation is given in order to illustrate the application of the concepts and equations. This is based on an actual engineering problem but is simplified to concentrate on the key points. Herein, H and V are expressed as dimensionless terms in the form of H/Ft and V/Ft and, for clarity, Ft is defined as the maximum value that can be developed when the rotor exerts maximum torque or is developing maximum power. The simulation results of equations (32) and (34) are shown graphically in Figures 5 and 6. Figure 5 shows the influence of the ratio K = kR =ku more directly and plots H/Ft and V/Ft against d/R for two different values of K = kR =ku in the up-milling mode of operation. Suppose a machine is fitted with sharp new cutting teeth: fR = fu = kR =ku = K = 1. The RC tool is up-milling and is operated so as to utilize the full power available to the RC tool. With the RC tool just touching the work surface, that is, with d close to zero, the resistance to forward motion H/Ft would be close to 1, and the required down thrust V/Ft would also be close to 1 (with K = 1) and would be close to 2 (with K = 2). Setting the RC tool deeper into the work would produce an increase in H/Ft (as might intuitively be expected) and a decrease in V/Ft (which may not be obvious). The increase in H/Ft is not very great; a maximum value of H/Ft = 1.37 is reached when the RC tool is set to a depth equal to about 50% of the effective radius, that is, at d/R = 0.5. If d is further increased, the value of H/Ft then falls off again. As d is increased, the vertical thrust V/Ft falls off very significantly and at

Figure 5. Variation of the axle force components H and V with the cutting depth d, assuming constant torque.

d/R = 0.5, V/Ft = 0.13. If d is increased even more, V continues to decrease, dropping to zero at d/R = 0.64, and then becoming negative at greater depths. This means that for d/R . 0.64, the RC tool is pulling itself down into the work, and in order to maintain a fixed cutting depth, the thrust actuators have to hold it back. In principle, the depth at which V changes from positive to negative is an indication of the value of K for the cutter tools. Returning to consideration of the same machine in the up-milling mode, assume that the cutter tool wear has increased the value of K (K = fR = fu ) to 2. At d = 0, H/Ft would start at 1.0, as it did with the unworn teeth; but as d increases, H/Ft increases markedly, reaching at d/R = 0.55 a value of 2, and attaining at d/R = 0.9 a maximum value of almost 2.1. The down thrust V/Ft would start off from a value of 2, that is, twice as big as the value for the unworn teeth. As d increases, V/Ft decreases, falling to a value of V/Ft = 1 at d/R = 0.425, and not changing from positive to negative until reaching a depth of d/R = 1.18. Figure 6 shows the effect of K, which reflects the cutter tool geometry, in particular the geometric changes by wear and gives H/Ft and V/Ft as functions of K for three different values of d/R. For shallow cutting with d/R = 0.15 (Figure 6(a)), the effect of K on H is not dramatic. In up-milling, as K increases from 1 to 2, H/Ft increases from 1.28 to 1.64. There is a much stronger effect on V. In up-milling, as K changes from 1 to 2, V/Ft increases from 0.57 to 1.5. For deep cutting with d/R = 0.75 (Figure 6(b)), as K changes, both H and V change by approximately the same amount. In upmilling, an increase in K from 1 to 2 increases H/Ft from 1.34 to 2.06 and V/Ft from 20.9 to + 0.53. For slot milling with d/R = 2.0 (Figure 6(c)), K has no effect on V/Ft, but H/Ft increases markedly. Also, a change of K from 1 to 2 makes H/Ft increase from 0.78 to 1.57

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Figure 6. Variation of the axle force components H and V with K, assuming constant torque and proportional tool force components (a) shallow cutting d/R = 0.15, (b) deep cutting d/R = 0.75 and (c) slot milling d/R = 2.

Resistance forces Other factors to be considered in the design of a RC tool are the resistance forces generated by the rock material, and the working environment when the machine operates underwater, since these affect the performance of the entire system.

water causes additional resistance and the power loss. When analyzing or designing for underwater work, it is necessary to have at least an approximate estimate of the magnitudes of these effects. This leads to an estimate of the hydrodynamic resistance FW for each cutter tool as Fw =

Frictional resistance in rock cutting. The force is also needed to overcome the friction between cutters and the confining work face. This is not significant when radial accelerations are low, of the same order as the gravitational acceleration, but for small high-speed RC tools, it could be significant. Ignoring gravity, and assuming that the cuttings scrape over the work face as a coherent mass, the force needed to overcome the friction FF is _ t FF = mrm vu

ð35Þ

where m is the rock-to-rock friction coefficient, rm is the in situ density of the work material, ut is the tangential velocity of cutter tip, and v_ is the volumetric rate of cutting as shown in equation (49).

1 1 CD Aru2 = CD hct wct ru2 2 2

ð36Þ

where CD is a drag coefficient of order unity, A is the frontal area of the tool and its mount, r is the fluid density, and u is the tool speed. We have taken here a pair of ‘‘effective’’ values for the height of tool hct and width of tool wct, such that hctwct = A. Following this procedure, we can write the mean shear stress t w that is induced by hydrodynamic drag as   1 hct t w = CD ð37Þ ru2 2 S where S is the distance between tracking cutters.

Gravity and buoyancy forces Hydrodynamic resistance during underwater cutting. When a RC tool is operating under the water, churning of the

Gravity and buoyancy forces are also termed restoring forces. In this UTV system, we assume that the center

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of the gravity force and the center of the buoyancy force are at the same point by positioning of buoyancy material. Thus, the two forces can be written as W = mg

ð38Þ

BT = pgVr

ð39Þ

where r is the fluid density, g is the acceleration due to the gravity, m is the total mass of complete system, and Vr is the volume of the RC tool.

CG: center of gravity.

Analysis of the traction of the UTV For a given type of the vehicle running on a given type of ground, the forward tractive thrust is usually provided by the net traction of the wheels or the crawler tracks. This net traction, which excludes the motion resistance of the track system, is known in the field of the vehicle technology as the drawbar pull Dp. It gives a measure of the vehicle’s reserve capacity to pull, push, or climb slopes. A dimensionless ‘‘drawbar coefficient,’’ CT is commonly defined as the drawbar pull divided by the vehicle weight CT =

DP W

ð40Þ

On moderately firm ground surfaces (including dry snow), CT for track-laying vehicles is typically in the range 0.3–0.8. In the case of the UTV that carries a rotation RC tool, the normal force between the running gear and the ground depends on the vehicle gross weight W, the down thrust force V, and the buoyancy force of the overall system BT. In the simple case, the drawbar pull Dp is Dp = CT (W  (V + BT ))

ð41Þ

As discussed earlier, V can be either positive or negative, but follows the convention that V is positive when the machine has to thrust downward into the work. The amount of power represented by the thrust power PH (= UH) and the losses in the running gear (including internal and external motion resistance) is usually quite small so that there is no great difficulty in supplying adequate power to the tracks. Thus, the drawbar pull is limited by the tractive efficiency of the running gear, which is expressed by CT; in order to traverse a RC tool, Dp must be equal to, or greater than, H H4CT (W  (V + BT ))

Figure 7. Moments affecting the balance of a combined UTV system with the RC tool.

ð42Þ

For an up-milling RC tool mode, this leads to the condition       Ft R K d 1 um W5 1  2 þK CT R CT 2 d 1=2  #   d 2R K  1 1 1 R d CT ð43Þ

where K = (kR =ku ).

Analysis of the moment of the RC tool (weight and balance) We neglect the additional hydrodynamic inertia forces, resulting from the added masses normally considered for accelerating submerged bodies and also do not take into account the effect of the underwater cable in this article. Figure 7 shows the main external forces acting on the RC tool and moment arms, where H, V, Br, Wr, Bv, Wv, and Dp are the tractive thrust, the down thrust, the buoyancy force of the RC tool, the weight of the RC tool, the buoyancy force of the UTV, the weight of the UTV, and the drawbar pull, respectively. With an RC tool on a UTV, it is important that the RC tool be mounted in such a way that the pitching moment developed by the axle force stays within acceptable limits. The RC tool is often mounted in such a way that forces on it have appreciable moments about points on the supporting system. Both the deadweight of the RC tool and the cutting force have moments that need to be accounted for in the design of the carriage system and the manipulating mechanism. For illustrative purposes, we will take moments about the center of area of the UTV. An ordinary, unmodified tractor is likely to have its weight Wv distributed over the running gear in such a way that the center of the gravity is more or less directly above the center of the area of the track system. The static balance may be designed to make the machine slightly nose heavy, to compensate for the small moment developed by pulling or pushing. A rotary RC tool attached to the front or the rear of such a tractor immediately disturbs the static balance, and there are further complications when the RC tool begins to operate. Figure 7 gives the simple diagram of the forces and the moments. For the net pitching moment to be zero, the condition is (Wv  Bv )X1 + (Wr  Br  V)X2  HX3 = 0

ð44Þ

When V is positive and greater than Wr, there is clearly an advantage to having the distance X2 as short

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as possible. When V is negative (the RC tool pulling itself into the work) or less than Wr, it may be more convenient to eliminate its moment by having reaction shoes or rollers that restrain the RC tool from further penetration. The moment represented by the third term of equation (44) is small if X2 is small, and the moment created by H is not likely to cause much difficulty, since the tractors are designed to accommodate such a moment. The moment arm X2 should obviously be kept as short as possible, if only for structural reasons. The effect of the positive V is partly offset by the RC tool weight Wr, and with the negative V, the RC tool can be fitted with auxiliary running gear (depth limiters) to provide local reaction against the surface.

Analysis of the energy and power of the RC tool Energy and power considerations are very important because they directly affect machine performance.

Machine power The energy considerations for cutting machines can be discussed conveniently in terms of the power consumed in the various parts of the system. The power for any component or subsystem can often be obtained from the product of the force and the velocity. For a single cutter tool at any part of its working stroke, the power Pc is essentially the product of the tangential force component fu and the tangential velocity of cutter tip ut Pc = fu ut

ð46Þ

The thrust power PH that is needed to traverse the RC tool through the work is PH = HU

ð47Þ

where H is the tractive thrust and U is the traverse speed.

Specific power The specific energy of a cutting machine is defined here as the energy required to cut a unit volume of the material. The overall specific energy for a complete machine EST is based on the total power output of the machine PT EsT =

PT v_

ð50Þ

PT = 2pfT + UH + PL

ð51Þ

In this context, PL is the power that does not contribute directly to the cutting process. The processspecific energy Es for cutting is based on the actual power used for cutting or excavating, excluding PL Es =

PT  PL Pr + PH = v_ v_

ð52Þ

From equation (52), we obtain Es =

2pfT + UH UBd

ð53Þ

Power density The term power density is used here to denote RC tool power per unit area of cutting surface. For a transverse rotation RC tool of radius R, width B, and power Pr, the power density Q is Q=

Pr Rum B

ð54Þ

where um = cos1 ½1  (d=R). Since Q varies with um , that is, with d/R, it is also convenient to define a nominal power density for some fixed value of um . For typical machines that operate with d/R \ 2, the nominal power density QN can be defined for the value um = p=2 QN =

2Pr pRB

ð55Þ

As QN decreases, we would obviously expect the performance of a machine to improve, assuming that dynamics or kinematic limits are not reached, and it may be of interest to relate the power density to the specific energy. The process-specific energy Es for the RC tool is its net power output divided by the volumetric excavation rate Es =

Pr UdB

ð56Þ

Simulation We give the following simulation in order to illustrate application of the concepts and equations. This is based on an actual engineering problem, but has been simplified so as to concentrate on the key points.

ð48Þ

where the volumetric rate of cutting or excavating v_ is a function of traverse velocity U, width of RC tool B and cutting depth d, that is v_ = UBd

PT = Pr + PH + PL

ð45Þ

From equation (22), the net power Pr required for cutting can be expressed as Pr = Ft ut = 2pfT

The total power output PT comprises the net power Pr, the thrust power PH and the power loss PL

ð49Þ

System description The Underwater Construction Robotics R&D Center (UCRC) at KIOST has been developing an UTV with RC tool which the principal parameters for the simulation and the simulation results are listed in Table 1. As

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Table 1. Principal parameters for the simulation and simulation results.. Principal parameters for the simulation Parameters

Notation

Unit

Value

Range

Maximum cutting depth Tip radius of the cutter tool Maximum transverse speed Weight of the system including UTV and a RC tool Maximum drawbar coefficient Coefficient of the shape cutter tool Coefficient of friction for sliding or rolling of the RC tool against the supporting RC tool Weight of the RC tool Weight of the UTV Width of the RC tool RC tool radius Diameter of the RC tool Buoyancy force of the RC tool Buoyancy force of the UTV Seawater density Cutter tools per revolution in each ring Rings of cutter tool across the width of the RC tool Maximum operating speed Tool speed

d r U W CT K m

m mm m/s kN 2 2 2

0.3 1.2 0.083 294.3 0.6 1.8 0.1

0 4 d 4 0.3 0.13 4 r 4 1.5 0.00178 4 U 4 0.331 2 0.2 4 CT 4 0.6 14K42 0.05 4 m 4 0.25

Wr Wv B R Dr Br Bv r n m f u

kN kN m m m kN kN kg/m3 2 2 rpm m/s

6.867 264.9 0.5 0.25 0.5 7.182 59.566 1025 10 10 60 1.571

2 2 2 2 2 2 2 2 1 4 n 4 20 1 4 m 4 15 0 4 f 4 190 04u45

kR ku Ft fu fR T Q v_ H V Pr PH

2 2 kN kN kN kN m kW/m2 m3/s kN kN hp hp

Simulation results The proportionality constants with dimensions of the force (representing tool geometry and rock properties) Maximum net overall cutting force of the RC tool Maximum tangential component of the tool force Maximum radial component of the tool force Maximum torque of the sprocket Maximum power density Maximum volumetric production rate Maximum tractive thrust Maximum down thrust Maximum specific energy Maximum thrust power

1292 781 94.863 4.8658 8.7558 23.72 672.7 0.01245 177.96 29.486 202.7 20.1

UTV: underwater tracked vehicle; RC: rock crushing; rpm: round per minute.

shown in Table 1, where the width of the RC tool is 0.5 m, its drum RC tool diameter is 0.5 m, and its weight is 6867 N in air without water ballast, the RC tool is targeted to cut a 0.5 m width and 0.3 m cutting depth. The working environment is on a ground (or seabed) with the maximum uniaxial compressive strength of 20 MPa, drawbar coefficient in range of 0.3–0.6, and normal friction coefficient of 0.1. Based on a series of reports covering mechanics of various cutting and boring machines in Mellor,13 the ranges of the principal parameters for designing an RC tool can be defined (Table 1). Figure 8 shows the RC tool, where we assume the cutter tools on a wide, rigid drum RC tool are disposed in m = 10 rings across the width of the drum, and that there are n = 10 uniformly spaced cutter tools in each ring. The RC tool is pushed into the material, and as it rotates while the UTV moves forward, the cutter tools (bits) cut through the rock material. The RC tool usually operates at its full capacity, and ideally, the tangential velocity of the cutter tool tip remains constant (1.571 m/s). The travel speed of the system is automatically kept at the

Figure 8. The RC tool. Source: Sandvik.12

maximum level, which depends on the dimensions of the RC tool, and the resistance of the rock mass. The travel speed of the UTV varies in the range from 0 to 300 m/h.

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  1=2  # kR R kR d 2R d um  1 A= 1 + 2 R d R ku d ku

Steps of calculation for the design process The analysis process can be summarized in the following steps: Step 1. Start with a given set of performance requirements: RC tool size (width and diameter of RC tool), rock strength, and the volumetric production rate (Table 1). Step 2. Provide the necessary RC tool dimensions along with the ability to cut the given rock strength (Table 1).  Acquire technical specifications, such as the tooth pattern layout on the RC tool, tool speed, and chipping depth. 1. Estimate of the designed parameters of the RC tool including m, n, kR, and(see sections ‘‘Forces on individual cutter tools,’’ and ‘‘Net overall cutting force and tool force,’’ and equations (9)–(11) and (21)). Step 3. Calculate the cutting forces:  Assume a simplified structure of teeth as rows of blades (number of cutter tracks) dependent on the spacing between the tracking cutters (see section ‘‘Net overall cutting force and tool force,’’ and Table 1).  Determine the tangential force and radial force distribution of the individual cutter tools that depend on the number of teeth engaged in the work and penetration depth of teeth (chipping depth) (see sections ‘‘Forces on individual cutter tools,’’ and equations (9) and (10)) and then determine the tangential force and radial force of the entire RC tool per unit angle (see sections ‘‘Tractive thrust and down thrust forces on the RC tool axis,’’ and equations (29) and (30)).  Determine the forces including the tractive thrust H, the down thrust V, and net overall cutting force Ft (see section ‘‘Tractive thrust and down thrust forces on the RC tool axis,’’ and ‘‘Net overall cutting force and tool force,’’ equations (21), (31)–(34)). (a) Integrate the resolved incremental components (including the tangential force and radial force of the entire RC tool per unit angle), to determine the tractive thrust and down thrust (see equations (31) and (33)). (b) Estimate net overall cutting force Ftmax (see section ‘‘Analysis of the traction of the UTV,’’ and equation (57), where we set up CT = 0.43 because the desired transverse velocity was quite small, that is, the transverse velocity U = 0.083 m/s). FT max =

where

2(W  CT  BT  CT ) A + B  CT

ð57Þ

"

1=2 #     kR d R d 2R 1 B= 2  um + 1  R d R d ku



Add the estimated hydrodynamics forces and resistance force (see section ‘‘Resistance forces,’’ and equations (35)–(37)). Step 4. Determine the moment of the overall system, and the structural as well as deployment actuation forces for the RC tool to UTV.  Determine the position at which the RC tool is located. (a) Design the moment of the RC tool (see section ‘‘Analysis of the moment of the RC tool,’’ and equation (44)). Step 5. Determine the power and energy of the overall system.  Determine the power and energy including the volumetric production rate, RC tool power, machine power, torque of sprocket, specific energy, and power density (see section ‘‘Analysis of the energy and power of the RC tool,’’ and equations (45)– (56)). Step 6. Ensure that the current selection is adequate.  Check the torque and moment requirements of the system.  Check that the spacing/penetration ratio is reasonable.  Ensure the structure, UTV system, and actuators can provide the thrust forces needed.

Results and discussion In this section, a number of numerical simulations are performed using the derived approach to demonstrate the application of the various equations to practical problems of machine design or performance analysis. This is based on an actual engineering problem but has been simplified so as to concentrate on the key points. The information generated from the proposed method in this article is used to estimate important variables such as cutter forces, overall forces (tractive thrust force and down thrust force), the torque, and power requirements of the RC tool, and the angular position of RC tool. Figure 9(a) shows the relationship between the angular position u and cutting depth d as defined from equation (14), where the RC tool radius R is assumed of fixed value. In the slot-milling mode, the RC tool cuts across its complete semi-circumference. This means that

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Figure 9. Angular position of the RC tool, tool forces, and tractive thrust and down thrust of RC tool: (a) angular position u, (b) the tool forces fu and fR, and (c) tractive thrust H and down thrust V (with K = 1.8).

the maximum depth equals the diameter of RC tool (or d = 2R), or the maximum angular position u = 180°. With a typical up-milling RC tool and we assume that the RC is more than axle deep in the work (d/R . 1), as is discussed in section ‘‘Terminology,’’ the RC tool is sunk into the work to a depth less than the diameter (or d \ 2R), we can define the ratio between cutting depth d and radius of RC tool R in the range of 1 \ d/R \ 2 and the angular position is in the range of 90° \ u \ 180°. From this figure, it can be seen that one RC tool revolution is from 0° (at d/R = 0) to 180° (at d/ R = 2). The figure also shows that the angular position u varies from zero at point of entry u = 0 to a maximum value at the point of exit um = cos1 (1  d=R). The maximum value can occur at the maximum angular position umax \ p=2 with d/R \ 1 or umax = p=2 with d/R = 1, or umax \ p=2 with d/R . 1. For the tool forces analysis, we used equations (9) and (10) to determine the radial component force fR and the tangential component force fu as functions of chipping depth. Since the chipping depth ‘ varies systematically through the working sweep on a transverse rotation machine, fu and fR are functions of angular position. Figure 9(b) shows the relationship between

the tool forces of each individual cutter tool fu , fR, and the variation of the cutting depth d at a fixed RC tool radius R, respectively. The result shows that the radial component of cutting force fR always increases more rapidly than the tangential component fu , and the ratio of the maximum radial component fR to the maximum tangential component fR is 1.8 because the quite worn cutter tools are used in this study. Moreover, in an upmilling RC tool if cutting depth is greater than radius of RC tool (d . R), fu will rise from zero at point of entry u = 0 to a maximum at a point roughly halfway u = p=2 through the working stroke (because the chipping depth of each individual tool is proportion to sin u at intervening positions and it reaches the maximum value ‘max at u = p=2), before decreasing back to zero at u = cos1 (1  d=R) and then remaining zero as shown in Figure 9(b). Figure 9(c) shows the relationship of the RC tool force components H and V against the ratio between d and R as defined from equations (32) and (34). We observed that with the RC tool only slightly touching the workspace, that is, with d close to zero, the resistance to forward motion H and the required down thrust V would be zero because of the cutting depth

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Figure 10. The torque of the sprocket.

d = 0 m. Setting the RC tool deeper into the work would produce increases in both H and V. As d is increased, the tractive thrust H would increase very significantly and reach a maximum value of H = 232 kN at d/R = 1.9. If d is further increased, the value of H then falls off again. The normal reaction V would not increase very much, and when the RC tool is set to a depth equal to about 50% of the effective radius, that is, at d/R = 0.5, it would reach a maximum value of V = 26 kN. If d is further increased, the value of V then falls off again. If d is increased even more, V continues to decrease, dropping to zero at d/R = 1.1 and then becoming negative at greater depths. The torque of the sprocket is the one of main important variables for designing RC tool that describes the operating torque input applying to the RC tool to be able to work in a certain target working environment. Figure 10 shows the relationship between torque of the sprocket and cutting depth. We simulated the results from the values in Table 1 and from equation (22). The result above shows that while increasing the cutting depth d, the torque of the sprocket of the RC tool increases linearly because it is a linear function of the cutting depth d as shown in equations (21) and (22), and the maximum value is T = 40 kN m at the ratio between d and R, d/R = 2. It is important to know in advance the capacity of the carrier vehicle so that the trenching operation can be executed properly on certain ground/soil condition. While power density of RC tool is used to denote the RC tool power per unit area of cutting surface, the net power and thruster power are the power consumed by the RC tool for cutting and the power used to overcome cutting resistance in the direction of advance, respectively. In order to determine the net power, thrust power, and power density with variation of the depth cutting, we use equations (46), (47), and (54), respectively. For the energy of RC tool analysis purpose, Figure 11 shows the power and power density

Figure 11. The thrust power, power density, and specific energy.

against the various cutting depth. It was observed that the proposed system has built to trench the depth of cutting from d = 0 m to a maximum depth d = 2R m. The figure also shows that with the RC tool only slightly touching the workspace, that is, with d close to zero, all of the net power, thrust power, and power density would be zero. Driving the RC tool deeper into the rock material, finally it reaches at the maximum cutting depth d = 2R are 230 kW, 525 kW/m2, and 11.5 kW, respectively.

Conclusion In this article, we conducted analyses on the design and mechanics of a developing UTV with a rotating RC tool for rock excavation. We analyzed the parameters that affect the performance including the cutting forces, torque, and power requirements of the UTV with the RC tool in rock conditions for designing. Also, we analyzed the parameters that affect cutting performance of the designing RC tool, so as to provide improved RC tool performance prediction modeling. As a study, we derived the mathematical expression of the mechanics, relating the forces and moments of the RC tool to the UTV. For this, starting from an analysis on the forces developed on individual cutter tools, when they are mounted on a RC tool, and then we analyzed the net overall cutting force on a complete RC tool. Through these analyses, we present the calculation process of the required tractive thrust and down thrust forces of the combined UTV system with a RC tool. We present the design scheme of the rotor carriage considering the moment caused by the cutting system. Similarly, we estimate the energy and the power needed to actuate the RC tool under the water in relation to the tool characteristics. The equations developed in this article might be usefully applied for determining the design appropriate data of the overall system and its performance. Since we analyzed a UTV with underwater RC tool, the

Vu et al. effect of buoyancy force on the analysis of the traction of the UTV as well as hydrodynamic resistance and friction resistance during underwater cutting are presented. Furthermore, we used our methods for the analysis of a developing UTV system with the RC tool. The typical steps of the analysis process to be followed for a combined UTV system with RC tool are presented in this article. These typical steps of design process are useful using for reference when designing a trencher machine. Finally, we conducted a number of simulations using the presented equations for practical problems in the design and the analysis of the RC tool.

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11.

Declaration of Conflicting Interests

12.

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

13.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research is a part of project titled ‘‘R&D center for underwater construction robotics,’’ South Korea (PJT200539), funded by the Ministry of Oceans and Fisheries (MOF), and a part of the Basic Science Research Program through the National Research Foundation of Korea (2016R1 A2B4011875) funded by the Ministry of Science, ICT & Future Planning. References 1. Muro T and O’Brien J. Terramechanics: land locomotion mechanics. Abingdon: Taylor & Francis, 2004. 2. Rubinstein D and Hitron R. A detailed multi-body model for dynamic simulation of off-road tracked vehicles. J Terramechanics 2004; 41: 163–173. 3. Solis J and Longoria R. Modeling track–terrain interaction for transient robotic vehicle maneuvers. J Terramechanics 2008; 45: 65–78. 4. Hong S, Kim H and Choi J. Transient dynamic analysis of tracked vehicles on extremely soft cohesive soil. In: Proceedings of the 5th international society of offshore and polar engineers Pacific/Asia offshore mechanics symposium, Daejeon, Korea, 17–20 November 2002, pp.100– 107. Mountain View, CA: International Society of Offshore and Polar Engineers. 5. Morgan N and Cathie D. Tracked subsea trencher mobility and operation in soft clays. In: Proceedings of the 17th ISOPE conference, Lisbon, 1–6 July 2007, pp.1366– 1373. Mountain View, CA: International Society of Offshore and Polar Engineers. 6. Nishimatsu Y. The mechanics of rock cutting. Int J Rock Mech Min 1972; 9: 261–270. 7. Jonak J and Podgo´rski J. Mathematical model and results of rock cutting modeling. J Min Sci + 2001; 37: 615–618. 8. Inyang HI. Drag bit cutting: a conception of rock deformation processes that correspond to observed forcedistance plots. In: Proceedings of the 2nd international

symposium on mine planning and equipment selection, Calgary, AB, Canada, 7–9 November 1990, pp.373–378. Rotterdam: A.A. Balkema. Inyang HI. A computational scheme for estimating the cutting rate of ladder-type excavators in hard rock. In: Proceedings of the 8th international conference on computer methods and advances in geomechanics, Morgantown, WV, USA, 22–28 May 1994, pp.2583–2585. Bailey JJ and Dean RC. Rock mechanics and the evolution of improved rock cutting methods. In: Proceedings of the 8th joint symposium on rock mechanics, University of Minnesota, Minneapolis, MN, 15–17 September 1996. Alexandria, VA: American Rock Mechanics Association. Rabia H. Specific energy as a criterion for drill performance prediction. Int J Rock Mech Min 1982; 19: 39–42. Sandvik, http://www.miningandconstruction.sandvik. com/ (2013, accessed 20 December 2014). Mellor M. Mechanics of cutting and boring. Hanover: Cold Regions Research and Engineering Laboratory, 1975.

Appendix 1 Notation a b A B Br BT Bv CD CT d Dp Es EST f fR fR0 fu fu0 FF Ft FW g h hct H kR, ku K ‘ m n Na P

dimensionless exponents dimensionless exponents frontal area of the tool and its mount width of RC tool buoyancy force of the RC tool buoyancy force of the combined UTV system with RC tool buoyancy force of the UTV drag coefficient of order unity drawbar coefficient cutting depth drawbar pull specific energy overall specific energy rotational frequency of the RC tool radial component of tool force radial cutting force per unit angle tangential component of tool force tangential cutting force per unit angle force needed to overcome the friction net overall cutting force hydrodynamic resistance acceleration due to the gravity effective tool length height of tool tractive thrust proportionality constants with dimensions of force ratio of tool force components chipping depth of the cutter tool rings of cutter tools across the width of the RC tool number of the tracking cutters in a complete revolution of the RC tool number of cutter tools in the work machine power

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Proc IMechE Part M: J Engineering for the Maritime Environment 00(0)

Wr Wv

power for accelerating cuttings power needed to overcome friction thrust power power loss RC tool power (net power) total power output of the machine power density radius of curvature of the cutter tool tip RC tool radius lateral tooth spacing distance between tracking cutters (circumferential tooth spacing) RC tool cutting torque tool speed tool velocity at the point of exit tangential velocity of the RC tool radial acceleration traverse velocity down thrust volume of the RC tool volumetric rate of cutting or excavating width of tool combined UTV system with RC tool weight weight of the RC tool weight of the UTV

u

angular position of the RC tool

PA PF PH PL Pr PT Q r R s S T u ue ut u2t =R U V Vr v_ wct W

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m r tw v

rock-to-rock friction coefficient fluid density mean shear stress angular velocity of the RC tool

Subscripts A ct d D e F H L p r R s ST t ta T v w W u

accelerating cutter tool drawbar drag exit friction thrust loss pull rock crushing radial component specific overall specific torque tangential total underwater tracked vehicle mean hydrodynamic tangential component