Final Version Report

Mathematical Modelling Report • February 9, 2016• Coursework II

An Unconveyorentional System: Reducing theLoss of Ore in an Industrial Conveyor Belt

Model

Frank Brooks-Tyreman

University of Loughborough,

Department of Mathematical Sciences

February 9, 2016

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Contents

1 Introduction 4

2 Method 52.1 Model 1: The Buckets and the Bouncing Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Model 1.1 A Single Bucket Attached by a Spring to the Conveyor Belt . . . . . . . . . . . . . . . . 52.1.2 Model 1.2: The Three Bucket System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 Model 1.3: Jerk and the Three Bucket System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Model 2: The Swinging Bucket System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Model 2.1: A Single Swinging Bucket with Air Resistance Proportional to Velocity . . . . . . . . 62.2.2 Model 2.2: The Double Mass System Joined by a Spring . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Model 2:3 The 3 Mass System Joined by Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Method Investigating Mass Loss in Model 2 83.1 Model 3: Artificial Mass Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1.1 Model 3.1: The Single Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.2 Model 3.2: The Double Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Model 4: Natural Mass Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.1 Model 4.1: The Double Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.2 Model 4.2: The Triple Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Results 94.1 Model 1: The Buckets and the Bouncing Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.1 Model 1.1 A Single Bucket Attached by a Spring to the Conveyor Belt . . . . . . . . . . . . . . . . 94.1.2 Model 1.2: The Three Bucket System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1.3 Model 1.3: Jerk and the Three Bucket System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Model 2: The Swinging Bucket System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.1 Model 2.1: A Single Swinging Bucket with Air Resistance Proportional to Velocity . . . . . . . . 104.2.2 Model 2.2: The Double Mass System Joined by a Spring . . . . . . . . . . . . . . . . . . . . . . . . 104.2.3 Model 2:3 The 3 Mass System Joined by Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Results for Mass Loss for Model 2 115.1 Artificial Mass Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.1.1 Model 3.1: The Single Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.1.2 Model 3.2: The Double Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5.2 Results for Model 4: Natural Mass Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2.1 Model 4.1: The Double Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2.2 Model 4.2: The Triple Mass System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6 Conclusions 11

7 Further Research 12

8 Acknowledgements 12

9 Diagrams and Plots 139.1 Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139.2 Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Appendices 20

Appendix A The Drag Constant 20

Appendix B Derivation of the Mathematical Pendulum Equations 20

Appendix C Non-Dimensionalising the Single Bucket System 20

Appendix D Creation of Diagrams and Plots 21

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List of Figures

9.1.1 Diagram of the Buckets used in our System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

9.1.2 Diagrams to show the Two Models we Investigated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

9.1.3 Diagrams of the systems used in Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

9.1.4 Diagrams of the first two systems used in Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

9.1.5 Diagram of the 3 Bucket System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

9.2.1 Modelling the Extension of the Single Bouncing Bucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

9.2.2 Graphs of the Displacement in the Spring for the Three Bucket System . . . . . . . . . . . . . . . . . . . . 15

9.2.3 Graphs of the ‘Jerk’ in the Spring for the Three Bucket System . . . . . . . . . . . . . . . . . . . . . . . . . 15

9.2.4 Graphs of the Dimensional and Non-dimensional Models for the Motion of the Bucket. . . . . . . . . . . 16

9.2.5 Graphs for the Damping for the Two and Three Bucket Systems . . . . . . . . . . . . . . . . . . . . . . . . 16

9.2.6 Graphs for the Artificial Mass Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

9.2.7 Graphs to show how Mass Loss is Related to the Angle of Oscillation for the 2 Bucket System. . . . . . 17

9.2.8 Graphs to show how Mass Loss is Related to the Angle of Oscillation for the Three Bucket System. . . . 18

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Abstract

The aim of this report is to give suggestions on how to prevent ore loss from a conveyor belt system consisting of springs and buckets.Our method consists of two models: the first was the initial conveyor-belt system presented to us by ‘BN Industries’. The problem weinvestigate is the amount of ore lost from the buckets due to bouncing caused by adding ore to other buckets in the system. Oursuggestions are fairly straightforward though completely change the model. We found that we should stabilise the system by addingextra supports and replace the springs connecting the buckets to the conveyor belt with inextensible rods. Both changes wouldremove the ‘bouncing’ effect.The second model did not entirely prevent mass loss; when we modelled it we found that the system moved with pendulum likemotion and ore was lost when the bucket went above a certain angle. The major result from the second model is that the initial swingof the bucket system causes the most mass loss. Therefore we should increase damping to reduce the initial swing.

1 Introduction

We were approached by BN Industries regarding their recent conveyor system as shown in figure (9.1.2a) which they useto transfer iron ore from their dock yard to their cargo ships. We were asked to make recommendations on changeswhich would reduce the loss of ore in the system. Figure (9.1.2a) shows the unconventional ‘hanging basket’ system ofa series of buckets attached together by a springs and hanging from a conveyor-belt. Ore is dropped into buckets at oneside and deposits its load in the ship at the other. The initial system was unstable so the buckets ‘bounced’ wheneverore was added to a new bucket. The initial changes we suggested were:

• Introduce supports for the buckets unattached to the conveyor system, and

• Replace spring supports with inextensible metal rods.

These changes improve the overall sturdiness of the model. However, new problems arose in the new conveyor-beltsystem shown in figure (9.1.2b). Instead of ‘Bouncing,’ the buckets now ‘swung’ with pendulum like motion and asthe buckets went above the angle π/6 radians, they lost ore. We consider this new system and concentrate on naturaldamping (air resistance) to investigate how loss of mass and air resistance affect the the buckets angle of oscillation.

The buckets shown in figure (9.1.1) have dimensions of length x meters, width, x/2m and height x/3m. For the bucketswe use, x = 5m. A trapdoor at the base opens to deposit the ore into the ship. We model the buckets experiencingdamping due to air resistance which we assumed was proportional to velocity.In our models we use the values a, b, c and d for varying values of the damping constant D (Appendix (A). For model 1,A is the area of reference of the bucket in the vertical direction so A = 25/3 ≈ 8.33m2; for model 2, A = 25/6 ≈ 4.16m2.Standard results we use are: ρ (density of the fluid that the bucket is moving in to) which for air is 1.2tn/m3 ([1]) andCd = 2.1 ([1]) which is the standard coefficient of drag for a rectangular box.

We also consider the type of ore ‘BN Industries’ transport. “The chemical composition of Precambrian iron-formationsfrom Australia, Brazil, South Africa, and North America shows that . . . iron varies between 20% and 40% and isgenerally in magnetite or hematite.” [2]. We used this to find the mass of the bucket (assuming all of the bucket isfilled with ore). In order to do this we measured the volume of the bucket as 20.83 m3 and also needed the specificgravity ([3]) ρ of the magnetite / hematiteores. For these two ores, the specific gravity (relative density) is usually aroundρ = 5.2 = 5.2tn/m3 [4]; thus the mass of the buckets was 108.33tonnes.The buckets are connected to the conveyor-belt by springs (in model 1) of natural length l = 6m or rods (in model 2) oflength l = 5m.

For the ships used in our model; the most common carrier ship is the ‘handysize’ or ‘handymax’ which “form themajority of ocean going cargo vessels in the world,” ([5]) and “are primarily used for carrying . . . cargo such as ironore.” ([6]) These ships hold between 10, 000 and 50, 000 Deadweight Tonnes (DWT). The conveyor system is based uponcommon dock conveyor systems “that can load at a stunning 16, 000 tons per hour” ([7]) Thus, our model concerned16, 000/108.33 = 147.7 ≈ 148 buckets per hour or approximately 3 buckets per minute. The conveyor belt system wasset to be moving at a safe velocity of around 5m/s.

We explore two areas: the first (model 1), where the buckets are attached by springs to the conveyor-belt and to eachother. We model both a simple single mass model and three bucket system to investigate the ‘bouncing’ effect. We alsoinvestigate the effect of ‘jerk’ upon the displacement of the bucket. The second (model 2) explores ‘swinging’ motion.Like the model 1, model 2 is a progression from single mass to multi-mass systems investigating mass loss. Finally

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we investigate how mass loss affects ‘swing’ in two ways: ‘artificially,’ i.e creating a function to model mass loss; and‘naturally,’ i.e model a function which ‘best fits’ our collected mass loss data.

Our final solutions are:

• Insert extra supports for all buckets,

• Replace springs with rods for the vertical supports, and

• Introduce artificial damping.

All of which we believed would reduce the mass loss of the system so although initially expensive would make a benefitin the long run.

2 Method

2.1 Model 1: The Buckets and the Bouncing Spring

Model 1 investigates the ‘bouncing’ effect caused by adding ore to the buckets. We assume there is no horizontal‘swinging’ in model 1 and mass is lost only through vertical movement. We explore the effect of ‘bounce’ on the boththe supported ‘end’ buckets and the ‘free’ central buckets for the three bucket system. We also investigate the role of‘jerk’ (

...y (t)) in mass loss from the system.

Some important values we need: ρ = 1.2, Cd = 2.1 , A = 8.33m2, and v = 1.5m/s (found by measuring the velocity ofthe bucket when ore was added to it). Thus air resistance b (for all buckets) is 23.53.

2.1.1 Model 1.1 A Single Bucket Attached by a Spring to the Conveyor Belt

We consider the single bucket system in figure (9.1.3a) and model the effect adding ore has upon displacement. Addingore to the bucket causes the ‘bouncing’ effect which we want to prevent. To create our governing equations we useNewton’s 2nd Law ([8]) and Hooke’s Law ([9]). The origin is taken as where the spring joins the conveyor belt ‘O’ so theinitial displacement of the bucket is the natural length of the spring, l = 6m. The bucket has no initial vertical velocitybut when loaded with ore it has a velocity of 1.5m/s so b = 23.53mtn.s2. The forces acting on the bucket are its weight,tension in the spring, and the damping force caused by air resistance. We measure with downwards as the positivedirection as this is the initial direction the bucket is moving in. Thus our governing equations are:

y(t) = g− km(y(t)− l)− b

my(t) (2.1.1.1)

with initial conditions:

y(0) = l y(0) = 0

We then modelled the three bucket system to investigate a bigger system and explore the ‘bouncing’ from the ‘free’bucket.

2.1.2 Model 1.2: The Three Bucket System

The second model is as shown in figure (9.1.3b). The three bucket system has two components: the two outside buckets,i.e those buckets connected to the conveyor belt by springs; and the central bucket, i.e the ‘free’ unconnected bucket. Weassume the outside buckets act like model 1.1 although the ‘bouncing’ would be reduced due to the connection to the‘free’ bucket. We also assume the free bucket, which hangs at an angle of α = π/6 has a much greater ‘bounce’; its lackof support accentuates its ‘bounce’.There are differences to model1.1: the outside bucket also experiences tension (T1 and T4) caused by the spring thatconnects to the ‘free’ bucket. We add this as an extra force acting upon the system so for the outside buckets thegoverning equation is:

y(t) = g +km((y(t)− l − lα)α− (y(t) + l))− b

my(t) (2.1.2.1)

with initial conditions

y(0) = l y(0) = 0

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We assume the initial displacement of the bucket is the natural extension of the spring (l = 6m) and that the bucket hasno velocity in the vertical direction. For the ‘free’ bucket we assume both springs which connect it to the other bucketsact identically. Therefore we assume that the forces for the tensions in the spring T2 and T3 are equivalent. Taking thisinto consideration our governing equations for the free buckets are:

y(t) = g− 2km((y(t)− l − lα)α)− b

my(t) (2.1.2.2)

with initial conditions

y(0) = l(1 + α) y(0) = 0

The displacement of the bucket is the natural length of the spring for the first bucket and the vertical distance of theconnecting spring to the free bucket. This distance is l sin(α) approximated to be lα. Again, the bucket has no initialvertical velocity.

2.1.3 Model 1.3: Jerk and the Three Bucket System

From modelling displacement in model’s 1.1 and 1.2 we noticed the initial ‘bounce’ had the biggest displacement. Werealised that by plotting the ‘jerk’ in the spring we could predict how much ‘bounce’ would happen. We found that ifthe ‘jerk’ was such that −5m/s3 >

...y (t) > 5m/s3 then ore would be lost. The plots in our results section, figures (9.2.3a)

and (9.2.3b) relate to the figures (9.2.2a) and (9.2.2b) directly. There are no explicit governing equations for the plotsof the ‘jerk’ as they use the governing equations for the three bucket system from model 1.2: for the outside buckets(figure (9.2.3a))

...y (t) is derived from equation (2.1.2.1) and for the central bucket (figure (9.2.3b))

...y (t) is derived from

equation (2.1.2.2). It is from these results that we were able to draw up our solutions to the bouncing problem and giveour suggestions:

• Add extra connections to stop the free movement of the central buckets, and

• Replace springs with inextensible metal rods.

We notice a general pattern in models 1 and 2; the initial movement of the system is predominantly the cause of massloss. We discuss this predominantly in model 2 although it has connotations for both.

2.2 Model 2: The Swinging Bucket System

From our changes model 1 we end up with model 2 (figure (9.1.2b)). We removed the ability for the buckets to ‘bounce’,however the new system ‘swung’ with a pendulum like motion. Investigations revealed that ore is still lost from thenew model. Thus from the following models we investigate how the angle of oscillation caused mass loss.Unlike the first model where we consider the bucket moving in the Cartesian coordinates x and y, model 2 uses radialand tangential coordinates by changing Cartesian to Polar coordinates.

2.2.1 Model 2.1: A Single Swinging Bucket with Air Resistance Proportional to Velocity

We look first at a single bucket in motion (figure (9.1.4a)). From Hookes’ Law and Newton’s 2nd Law we found thegoverning equation to model the swing of the bucket (Appendix B). Air resistance is proportional to velocity i.e ar ∝ φ(t)so ar = bφ(t) where b is the drag constant with units meters·tonnes/seconds2 (Appendix A). Here we set ρ = 1.2tn/m3,A = 4.16m2 (the surface area of the side of the bucket 5/2× 5/3), v = 6.5m/s (the velocity of the conveyor belt) andCd = 2.1ms ([1]), so b = 221.46mtn/s2

The governing equation for our single ‘swinging’ bucket model is

mlφ(t) = −(mg sin(φ(t)) + bφ(t)) (2.2.1.1)

We set sin(φ(t)) ≈ φ(t) for small oscillations so our governing equation became:

mlφ(t) = −(mg(φ(t)) + bφ(t)) (2.2.1.2)


φ(0) = 0 φ(0) = V0l

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After some rearranging we get the equation:

φ(t) = − gl

φ− bml

φ(t) (2.2.1.3)

We then set g/l = ω2 and b/ml = λ so our final equation for the pendulum with damping became

φ(t) + λφ(t) + ω2φ = 0 (2.2.1.4)

which has the analytic solution

φ(t) = e−λ2 t(

V0

αlsin(αt)

)(2.2.1.5)

where α =

√(4ω2−λ2)

2

We assume that the bucket is set with no initial angle so φ(0) = 0 and the bucket moves with initial velocity V0 = 6.5m/sas we assume the buckets move with the same velocity as the conveyor-belt.

For efficiency we non-dimensionalise to reduce the number of parameters in our model (Appendix C). Thus our model’sgoverning equation becomes:

φ(t) + λφ(t) + φ(t) = 0 (2.2.1.6)


φ(0) = 0 φ(0) = 1

Non-dimensionalisation means our model depends only on the damping constant λ = b/(m√

gl). Comparing with thedimensional model, the non-dimensional damping constant is dependent on m, l and b and is also dependent on g. λwill therefore be smaller than the dimensional quantity λ.

2.2.2 Model 2.2: The Double Mass System Joined by a Spring

The natural progression of our model was to add a second bucket to the system as shown in figure (9.1.4b) joined by aspring of natural length l = 5m with spring constant k = 30kN, the velocity v = 5m/s.The values are smaller than in model 1 because this time the springs only connect the buckets, they don’t need tosupport them. The velocity is smaller as the safe speed of the conveyor-belt reduces with extra buckets added. With theaddition of the extra bucket we consider additional forces acting on each bucket. The governing equations are:

m1φ1(t) = −m1g

lsin(φ1(t))− k(sin(φ1(t))− sin(φ1(t))− bφ2(t) (2.2.2.1)

m2φ2(t) = −m2g

lsin(φ2(t))− k(sin(φ2(t))− sin(φ2(t))− cφ1(t) (2.2.2.2)


φ1(0) = 0 , φ2(0) = 0 , φ1(0) = 5 , φ2(0) = 5

This double mass system has a number of assumptions:

• All springs are uniform and have the same length and spring constant, and

• We assume the second bucket (m1) has reduced damping due to a slip-streaming effect caused by the first bucket(m2).

We assume reduced air resistance due to ‘slipstreaming’. Whilst unable to find previous research on the effects of‘slip-streaming’ in conveyor-belt models we simplified by assuming the same effect on our system as you would withcyclists. In Blocken et al (2013) [10] we found that, “the few published studies on drafting all confirm the large dragreduction for the trailing riders (is) up to 30-40%.” [10]. We apply this idea to all future models.

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2.2.3 Model 2:3 The 3 Mass System Joined by Springs

Model 2.3 is a three bucket conveyor-belt system shown in figure (9.1.5). We consider what will happen to all buckets insystem but especially if there is any significant effect upon the central bucket. The 2nd bucket is connected by 2 springsto the end buckets and so there is extra force from the tension in the added spring. For the middle bucket we applythe force from the spring to the right −k(sin(φ2(t))− sin(φ1(t)) and also from the left −k(sin(φ2(t))− sin(φ3(t)). Thusour equations are:

m1φ1(t) = −m1g

lsin(φ1(t))− k(sin(φ1(t))− sin(φ2(t))− cφ1(t) (2.2.3.1)

m2φ2(t) = −m2g

lsin(φ2(t))− k(2 sin(φ2(t))− sin(φ1(t)− sin(φ3(t))− bφ2(t) (2.2.3.2)

m3φ3(t) = −m3g

lsin(φ3(t))− k(sin(φ3(t))− sin(φ2(t))− aφ3(t) (2.2.3.3)


φ1(0) = 0 , φ2(0) = 0 , φ3(0) = 0 , φ1(0) = 5 , φ2(0) = 5 , φ3(0) = 5

As usual the buckets hang straight down initially, move at the speed of the conveyor belt 5m/s and we apply slip-streaming.

3 Method Investigating Mass Loss in Model 2

We were asked to relate how much ore is spilled from the buckets and consider how the construction of the conveyorcould be changed to improve performance. We therefore modelled mass as a function of time, allowing us to see therelationship between the swinging motion of the bucket and the loss of mass.

3.1 Model 3: Artificial Mass Loss

We model mass as a function of time by creating an ’artificial’ mass loss function to see how generic mass loss affectedthe swing of the pendulum. We describe it as ’artificial’ because we made sure the mass did not fall below a certainamount. We used a paper by Digilov, Reiner, and Weizman (2005) [11] as the basis for our model. Digilov, Reiner, andZ. Weizman (2005) modelled a system “of damped oscillations of a variable mass on a spring pendulum, with themass decreasing at a constant rate,” [11] where they explored “the effect of the mass loss rate on the magnitude of thedamping parameters.” [11] Their model has similarities to ours so we used it as a guide.

In our model we decide a linear function is inappropriate as not all mass is lost instead we use an exponential function.We felt this would be a good initial model as it allows us to explore how mass loss affects swing whilst having controlover how much mass is lost. Making sure not all the mass was lost is sensible as from previous models, it is usuallyonly the first oscillations where mass is lost, with most mass retained.

3.1.1 Model 3.1: The Single Mass System

From model 2.1 we replace the constant mass in equation (2.2.1.1) with mass as a function of time given by

m(t) = 33.33e−t + 75 (3.1.1.1)


φ(0) = 0 , φ(0) = 5 , m(0) = 108.33

3.1.2 Model 3.2: The Double Mass System

Like with model 3.1 we replace constant mass in equations (2.2.2.1) and (2.2.2.2) with our artificial mass loss equations:

m1(t) = 38.33e−t + 70 (3.1.2.1)

m2(t) = 33.33e−t + 75 (3.1.2.2)


φ1(0) = 0 , φ2(0) = 0 , φ1(0) = 5 , φ2(0) = 5 , m1(0) = 108.33 , m2(0) = 108.33

The equation for m1 loses more mass. This represents the slipstream effect affecting the second bucket by reducing airresistance.

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3.2 Model 4: Natural Mass Loss

Natural mass loss is more difficult to model. We created a function which lost mass when −π/6 > φ2(t) or φ2(t) > π/6but retained mass otherwise. We created a step function with symmetric logarithmic component to model the mass lossin the system. The function itself behaved very closely to the data for mass loss we recorded during our research; wehad to multiply by 10 to get similar values.


We replace constant mass in equations (2.2.2.1) and (2.2.2.2) with our equations modelling mass loss:

m1(t) =

{−10|ln(|φ1(t)|)| if − π/6 > φ1(t) or φ1(t) > π/60 if − π/6 ≤ φ1(t) ≤ π/6

(3.2.1.1)

m2(t) =


(3.2.1.2)


φ1(0) = 0 , φ2(0) = 0 , φ1(0) = 5 , φ2(0) = 5 , m1(0) = 108.33 , m2(0) = 108.33

we then do this for the three bucket system.

3.2.2 Model 4.2: The Triple Mass System

As with model 4.1 w model mass as a function of φ and not as a constant, replacing the mass in equations (??) , (??),and (??) with the equations:

m1(t) =


(3.2.2.1)

m2(t) =


(3.2.2.2)

m3(t) =


(3.2.2.3)


φ1(0) = 0 , φ2(0) = 0 , φ3(0) = 0 , φ1(0) = 5 , φ2(0) = 5 , φ3(0) = 5 , m1(0) = 108.33 , m2(0) = 108.33 , m3(0) = 108.33

4 Results

4.1 Model 1: The Buckets and the Bouncing Spring

4.1.1 Model 1.1 A Single Bucket Attached by a Spring to the Conveyor Belt

From figure (9.2.1) we see the bucket clearly models the ‘bouncing’ effect. Due to damping the bucket eventually stops‘bouncing’, however it’s the initial ‘bounce’ which problematic. Adding mass causes a large bouncing effect resulting inspilled ore. Another problem is that a single mass doesn’t give us an accurate representation. The single bucket can’tshow us the effect other buckets have in terms of accentuating or reducing ‘bounce’. Another problem is that we modelthe system over a period of 20s, however, in reality buckets are stopped above the ship before depositing their loads.The plot allows the system to carry but in reality it stops at intervals like when ore is added to another bucket or thebuckets reach the ship.

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4.1.2 Model 1.2: The Three Bucket System

The ‘outside’ buckets act like the singular bucket in model 1.1. Differences occur due to extra forces acting on the systemi.e the addition of the central bucket allows less freedom to ‘bounce’. This is represented in the graph where dampinghas a much larger effect. The bucket’s displacement eventually evens out to be around 8m which shows that extra weightin its own bucket and the central bucket causes the extension of the spring to be stretched above its natural length l = 6m.

In contrast the displacement for the central bucket is much greater. This is a result of no support spring connecting thebucket to the conveyor-belt. We expect a lot of ore to be spilled from these ‘free’ buckets. Another noticeable elementof the graph is how little effect damping has upon the ‘free’ bucket. ‘Bouncing’ occurs frequently and is still fairlylarge after 10s. This violent ‘bouncing’ caused ‘BN Industries’ to ask for our suggestions on ways to reduce this loss of ore.

On comparison between the two plots we see the supported bucket moves less freely and ore is lost only during theinitial ‘bounce’. This comparison means that extra supports need to be added to connect the central buckets to theconveyor belt system.

4.1.3 Model 1.3: Jerk and the Three Bucket System

Figure (9.2.3a) shows the jerk for the outside buckets and relates directly to figure (9.2.2a). There is only one instancewhen the jerk is violent enough to spill ore and that is during the initial ‘bounce’. We can ascertain that the bucketsconnected to the conveyor belts are much less likely to ‘bounce’.

To reinforce this thinking, figure (9.2.3b) shows how unstable the ‘free’ bucket is. There are three instances where massis lost from the system; the first, second and third bounces. Relating this back to figure (9.2.2b) we see that large ‘jerks’in the spring mean greater displacement.

By comparing ‘bounce’ and ‘jerk’ we see a clear relation between the two: the greater the jerk in the system, the greaterthe displacement. To stop this jerk from happening the simplest solution is to remove the springs from the systementirely and replace them with inextensible rods.

4.2 Model 2: The Swinging Bucket System

4.2.1 Model 2.1: A Single Swinging Bucket with Air Resistance Proportional to Velocity

From figure (9.2.4a) we see a sinusoidal curve showing the oscillations of the bucket. We see that the damping effectreduces the ‘swinging’ motion of the bucket. It is only when the bucket initially ‘swings’ that ore is lost.

λ = b/ml so damping is dependent on a number of factors. By decreasing the length l of the rod we increase thedamping and reduce the size of the oscillations and therefore reduce ore loss. λ is also implicitly dependent on velocity(b is a function of velocity). A small increase in the velocity of the conveyor belt increases the damping in the system;however, we need to be careful suggesting this change. As mentioned in the introduction, the conveyor system movesat velocity 5m/s; we are only able to increase the velocity for the single bucket model as less stress is placed on theconveyor system. More buckets means an increase in velocity will damage the system and is not an economically orsafe option.

Figure (9.2.4b) shows a similar curve to figure (9.2.4a). There is less damping, due to the dependency on gravity of thedimensionless model. Despite this only the initial oscillation has an angle greater than π/6 radians and loses ore. Thedimensionless model is a lot easier to investigate, the only parameter we need to change is the damping parameterwhich is the area of the model we are most interesting in.

4.2.2 Model 2.2: The Double Mass System Joined by a Spring

From figure (9.1.4b) we see the damping effect. The second bucket experiences less air resistance due to the slip-streaming effect and has larger oscillations than the first bucket. Despite this only the initial movement of the bucketwhich causes mass loss. This regular result we found during our modelling task suggests we need to consider increasingdamping in order to prevent mass loss.

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A positive to take from this plot is how quickly damping takes effect. In other models, one issue is that it takes a whilefor damping to reduce the movement. In this model however damping is shown to have an effect around 3s which isfitting with reality; buckets do not move for very long due to the stop start process of the conveyor belt.

4.2.3 Model 2:3 The 3 Mass System Joined by Springs

Similar results for figure (9.2.5b) are found to those which appear in figure (9.2.5a). A similar conclusion can be drawnfrom the results too; damping is the dominant term in the equations. This is shown clearly by the curves of the angle ofoscillation for the buckets. The smaller the damping, the larger the angle of oscillation.A positive from these results is that it is only the initial oscillation in which mass is lost; however, considering this is forall the buckets, extrapolating for more buckets this is still a lot of ore being lost.

5 Results for Mass Loss for Model 2

5.1 Artificial Mass Loss

5.1.1 Model 3.1: The Single Mass System

From figure (9.2.6a) we can’t see how much angle of oscillation is related to mass loss. Once mass reaches an equilibrium,oscillations of the bucket significantly decrease. However, all previous models also show that the initial oscillation islargest. We can argue that the highest oscillations relate to their being a larger amount of mass in the system; howeverwe do not have enough evidence to back this claim. If similar results appear in later models then we have a significantresult and another factor which we could change to decrease mass loss.


Unlike model 3.1 there is more evidence to suggest that mass has an effect upon the angle of oscillation in the model.Comparing figures (9.2.6b) and (9.2.5a) we see that oscillations take longer so force of the swing is dissipated resultingin smaller angles of oscillation. Both models have the same damping effects yet their angle of oscillation curves aredifferent; this highlights an effect of not only damping but also mass loss. An anomaly we can’t explain is why theinitial angle of oscillation for model 3.2’s first ‘swing’ is greater than that of model 2.2. We assume that mass being lostfrom the bucket gives extra force to the bucket and increases the ‘swing’ although this we did not investigate this.

5.2 Results for Model 4: Natural Mass Loss


The results from figure (9.2.7) we got were inconclusive. Whilst the second bucket in the system acted like we expected- the initial angle is large enough to lose mass before the oscillations are reduced due to damping - the first bucketdid not act like this at all. Our results show that in a two bucket system the first bucket, despite experiencing largerdamping would continue to increase its oscillations and therefore lose mass throughout its movement.

We could draw no conclusions from these results, however this did not deter us as we also knew that a two bucketmodel would not be used in real life.

5.2.2 Model 4.2: The Triple Mass System

The results from figure (9.2.8) were much more reliable; they not only acted like how we expected but also gave usresults corresponding to previous models. The most noticeable result from the plots is that movement is mostly dictatedby damping. We can see that the larger the damping in the system, the smaller the angle of oscillation. As with all othermodels we found that the initial swing was the most likely to lose mass in the system; this confirmed all our ideas thatwe need to consider ways of artificially increasing damping in order to reduce the initial movement of the bucket andhence prevent mass loss.

6 Conclusions

Firstly, it is worth noting that the conveyor system we considered is highly unorthodox and is very unlikely to be usedelsewhere by other businesses. However, our aim was to see how we could improve the system in order to reduce the

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amount of ore lost due to the ‘bouncing’ effect and in our second model due to ‘swinging’.

From our initial model we came up with some alterations which would improve the overall stability of the model. Firstly,by replacing the connecting springs with rods we would remove the ‘bouncing’ effect entirely. Secondly, although itwould increase costs for ‘B.N Industries’ adding extra supports so that all buckets are connected to the conveyor systemwould remove the ‘bouncing’ effect and overall instability of the ‘free’ buckets. Lastly, all the systems experiencednatural damping caused by air resistance which we explored and generally the systems reduced their ‘bounce’ oroscillations relatively quickly. However, in our second model we found out that in all our models it was that first initialoscillation which was always violent enough to cause mass loss. Hence we would argue that introducing artificialdamping to the system, i.e by fortifying the rods connection to the conveyor belt would reduce the system so that themodel would no longer lose mass at any time.

We believed that although these changes would be initially costly, in the long run a more stable conveyor belt systemwould be produced and the loss of mass minimised effectively enough to increase profit.

7 Further Research

In real life the conveyor belt system would consist of many buckets, we only modelled a maximum of 3 buckets. Weassumed that we would be able to extrapolate the data for a many bucketed system however, there would be morethings to consider: we would maybe see the system become more stable with more buckets added and also how wouldour initial conditions would be effected by more buckets, would the velocity of the conveyor belt reduce with addedstrain to the system and how would this change our results?

In our second model we managed to model mass loss dependent upon the angle of oscillation whereby if the angle ofoscillation φ(t) was such that −π/6 > φ(t) > π/6 then mass would be lost from the bucket. In our first model it wouldhave been very interesting to model how mass would be lost dependent on the force of the ‘jerk’ in the system. Hencewe would assume that the more violent the jerk in the system, the more mass would be lost and we would model this tosee if our hypothesis would be correct.

We made a huge assumption for the effect of ‘slip-streaming’ on the air resistance of the buckets. By assuming thateach bucket received a 30% reduction in air resistance was an oversimplification of the system. If we had more time wewould explore how much air resistance was actually reduced by drafting behind the bucket in front of it.

Other points of note are that we did not consider any other damping techniques except air resistance. Natural resistancewould be found in the springs which is also something we could have investigated. We could also have used abroadening area of mathematical research which is delay differential equations, we would have been able to model orebeing added to different buckets at different times and see what the effect would be then.

8 Acknowledgements

I would like to thank Tim Blades for all his help with the work we did with the modelling. I would like to thank DrDavid Sibley for all the advice and support he has given myself and Tim during this project. Lastly I would like tothank Kieron Lafferty for his help on relative density and for his engineering know-how.

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9 Diagrams and Plots

9.1 Diagrams

Figure 9.1.1: Diagram of the Buckets used in our System.The buckets have open tops allowing ore to be added quickly and efficiently. It also allows ore to be spilled due to ‘bouncing’ in

model 1 or ‘swinging’ in model 2. The base has an inbuilt ‘trapdoor’ which releases the load into the ships. Standard ore bucketshave dimensions xm× x/2m× x/3m Our model uses buckets with length x = 5m.

(a) Diagram of the Initial Conveyor System.This is the initial conveyor-belt system. Buckets are

suspended from the conveyor-belt and attached to each otherby springs with spring constant k and natural length l. Weassume the springs are uniform and therefore all have the

same value of k. We assume all buckets are completely filledso the total mass of the buckets is 108.332tn. The

unsupported buckets hang lower than the supported ones atan angle of α radians to both buckets.

(b) Diagram of the Secondary Conveyor System.We replace the springs in model1 with inextensible rods of

length l. This stabilises the system and removes the‘bouncing’. We also added extra supports so all buckets areconnected to the conveyor-belt removing excess ‘bouncing’from the ‘free’ buckets. All other assumptions remain i.e

uniform spring constants k and full buckets of 108.332tnmass.

Figure 9.1.2: Diagrams to show the Two Models we Investigated.

(a) Diagram of the Bucket on a Spring Model.Our initial model is a single bucket attached by a spring to the

conveyor-belt. The spring has natural length l and springconstant k, the bucket has mass m. Once mass is added to the

bucket the spring ‘bounces’ vertically and spills ore.

(b) Diagram of the Three Bucket System.The three bucket system has two buckets connected to the

conveyor-belt, the other bucket is ‘free’ (it is connected only tothe other buckets). The ‘free’ bucket is so assume it loses thelargest amounts of mass. All springs have uniform springconstant k = 200kN and natural length l = 6m. The ‘free’bucket hangs at an angle α = π/6. For initial displacementy(t) of the ‘free’ bucket y(0) = l + l sin(α), approximated asy(0) = l + lα. We assume mass is not lost by ‘swinging’. We

also assume that the forces T1 = T4 and T2 = T3

Figure 9.1.3: Diagrams of the systems used in Model 1

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(a) Diagram of the Single Bucket System.The single bucket system isattached to the origin O by a spring of length l. We consider radialand tangential coordinates as the pendulum moves in a circular motionthus we have radial er and tangential eφ unit vectors. Forces actingupon the bucket are: mg, the weight of the bucket, T, the tension of thespring and ar, the air resistance.

(b) Diagram showing the Two Mass System.The two buckets are connected by a spring with k = 30kN. Thetwo buckets are attached to the conveyor-belt by an inextensiblerod. We model what the effect of attaching the buckets togetherhas upon the mass loss. We also consider how the effect of air

resistance is changed with the addition of the extra bucket. Weassume that ‘slipstreaming’ reduces air resistance on the second

bucket (m1).

Figure 9.1.4: Diagrams of the first two systems used in Model 2

Figure 9.1.5: Diagram of the 3 Bucket System.We consider the three bucket system to investigate the effect of the two springs on the ‘middle’ bucket. We explore if it makes it morestable. As usual the system had rods of length l = 5m, springs with k = 30kN and buckets of mass m1 = m2 = m3 = 108.33tn.We apply the ‘slip-stream’ effect to all masses so ar1 = 0.49ar3 and ar2 = 0.7ar3 . Each following bucket receives a 30% reduction in

air resistance (geometric progression.

9.2 Plots

Figure 9.2.1: Modelling the Extension of the Single Bouncing BucketA plot of the displacement; the damping effect reduces the ‘bounce’ of the bucket over time and the springs extension levels out to bearound 7.25m. We assume the extension of the spring to a length of 7.25m is caused by the mass added to the bucket. In this case

our numeric values are g = 9.81m/s, l = 6m, k = 200kN, m = 108.33tn and b = 23.63mtn/s2.

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(a) Graph for the Displacement of the Outside Buckets.The outside buckets have a large initial ‘bounce’ before

stabilising around 8m, an extension caused by the extraweight in the bucket. We assume the ‘free’ bucket adds extra

resistance to the ‘bouncing’ effect preventing excessive‘bouncing’. We assume that A = 8.33m2, ρ = 1.2tn/m3,

Cd = 2.1, v = 1.5m/s so b = 23.63mtn/s2 andk = 200kN with l = 6m and α = π/6 for buckets with

mass m = 108.33tn

(b) Graph for the Displacement of the Central Bucket.Unlike the outside buckets, the central bucket has less

stability. We found that ‘bounce’ is much greater and ittakes longer for damping to have an effect. We assume thatwithout being connected to the conveyor-belt there would be

more ‘bouncing’ which we can see in the plot. As usual,A = 8.33m2, ρ = 1.2tn/m3, Cd = 2.1, v = 1.5m/s so

b = 23.63mtn/s2 and k = 200kN with l = 6m andα = π/6 for buckets with mass m = 108.33tn

Figure 9.2.2: Graphs of the Displacement in the Spring for the Three Bucket System

(a) Graph of the Jerk (...y (t)) of the Outside Buckets in the Three

Bucket SystemFrom our assumptions that mass is lost when

−5m/s3 >...y (t) > 5m/s3 we see that for the outside

buckets mass is lost only during the initial ‘bounce’. Thisresult follows a trend that we found in all our models

namely that it is this initial ‘bounce’ (or in model 2’s case -‘swing’) which cause mass loss. We directly relate this plot

back to figure (9.2.3a) to see that the jerks in the springcorrespond to how large the ‘bounce’ is.

(b) Graph of the Jerk (...y (t)) of the Central Bucket in the Three

Bucket SystemWe see that the ‘jerks’ in the central buckets are large andoften. This reflects in the plot of figure (9.2.3b) where the

bucket is free to bounce. From our research that mass is lostwhen −5m/s3 >

...y (t) > 5m/s3 we see that ore is lost

from the bucket in 3 occasions. This plot is the final evidenceneeded to implement our suggestion that all the buckets

should be attached to the conveyor belt.

Figure 9.2.3: Graphs of the ‘Jerk’ in the Spring for the Three Bucket System

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(a) Plot showing the Motion of the Bucket when the Air Resistanceis Proportional to Velocity.We assume there is immediate damping because the bucket isnon-aerodynamic so would encounter resistance as soon as it

started moving. This model was an investigation intowhether mass loss happened even under damping. We

assume mass loss occurs if the angle of the oscillation wasgreater than π/6 radians. We see that only the first

oscillation has an angle greater than pi/6 and when thebucket would lose mass. The values we needed were

V0 = 6.5m/s , l = 5m , g = 9.81m/s2, b = 221.46mtn/s, ω =

√g/l and λ = b/ml = 0.41.

(b) Plot showing the Motion of the Bucket after Non-dimensionalisation.

The model now only depends on the parameterλ = b/(m

√gl) = 221.46/(7× 108.33) = 0.29. The plot

is similar to the dimensional models plot; only the initialoscillation is large enough for the system to lose ore. We areinterested in seeing the effect of damping upon the system

and the dimensionless model allows us to concentrate solelyon how increasing the damping in the system would result

in less mass loss.

Figure 9.2.4: Graphs of the Dimensional and Non-dimensional Models for the Motion of the Bucket.

(a) Graph to show the Motion of a 2 Bucket System with Damping.The model follows the pattern in previous plots where the firstoscillation is greater than π/6. The damping effect happenspretty quickly though and the bucket stops ‘swinging’ after a

relatively short amount of time, 4seconds. The rods are of lengthl = 5m, the mass of the buckets is m1 = m2 = 108.33tn and

the spring has k = 30kN. Applying the reduction in airresistance from Blocken et al (2013) to the drag constant;

b = 132.10 and c = 188.7 where b = 0.7c (a reduction of 30%).

(b) Graph to show the Motion of the 3 Bucket System with Damping.We see that the damping effect has the strongest impact on the

oscillations of the buckets. The middle bucket has a similarcurve as the other two buckets which we assume is down to the

connecting springs keeping the system ‘rigid.’a = 188.70mtn/s2, b = 132.10mtn/s2 and c = 92.46mtn/s2

which confirm that the higher the damping constant the fasterthe bucket stops oscillating. Despite this, damping is still not

big enough to stop ore being spilled in that initial oscillation.

Figure 9.2.5: Graphs for the Damping for the Two and Three Bucket Systems

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(a) Graph to show the Effect of Mass Loss upon the Oscillations of theBucket.We can’t confirm our assumption that the size of the oscillationof the bucket is related to the weight of the bucket. There is no

certainty that by reducing the mass in the bucket we reduce the‘swinging’ in the system. We use b = 567.84mtn/s2 and therod is of length l = 6m. We see that by artificially removing

mass, the bucket stops oscillating violently and once massreaches an equilibrium level at around t = 2, damping takes

over and the oscillations reduce.We have scaled down the mass values in order to fit both curves

on the same graph.

(b) Graph to show the Effect of Mass Loss upon the Oscillations of aDouble Bucket System.

This plot shows us the relationship between the swing of thependulum and the amount of mass in the bucket. We less massloss in the first bucket m2 as there is greater damping. A two

mass system is less dependent on mass loss than the singlemodel system. Despite losing less mass, the curve of

equation (2.2.2.2) has smaller initial oscillation, thus thedamping is the dominant component rather than mass loss.

However around 3s, the curve of equation (2.2.2.1) reaches anangle of 0 quickest. This implies mass loss has an effect on the‘swing’ of the bucket. Bucket 2 has higher mass and thus takeslonger to stop oscillating than bucket 1. We use values: l = 6m,

g = 9.81m/s2, k = 30kN, b = 132.10mtn/s2 andc = 188.70mtn/s2.

We have scaled down the mass loss curves as to fit them on asingle graph.

Figure 9.2.6: Graphs for the Artificial Mass Model

(a) Graph of the Mass Loss Dependent on the Angle. (b) Graph of the Angle of Oscillation for the Two Buckets.

Figure 9.2.7: Graphs to show how Mass Loss is Related to the Angle of Oscillation for the 2 Bucket System.The two bucket system has a few problems. The first bucket doesn’t follow our hypothesis that less mass means smaller oscillations.The values of φ2(t) increase as mass is lost. The second bucket acted as we would have expected the system to; mass is lost until thesystem reaches an equilibrium where the oscillations are not big enough to lose mass. There are some natural boundary conditions

we don’t consider, in the real system we stop the buckets in order to deposit the mass into the ship. This condition prevents theincrease of angle of oscillations. From our results we suggest a system of only two buckets doesn’t work. It wouldn’t be economically

logical and is too unpredictable; there are unexplained questions relating to why the oscillations increase despite experiencingdamping

The values we used for this model were l = 5m, k = 30kN, b = 91.73mtn/s2 and c = 131.04mtn/s2.

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(a) Graph of the Loss of Mass due to the Angle. (b) Graph of the Angle of Oscillation for the Three Buckets.

Figure 9.2.8: Graphs to show how Mass Loss is Related to the Angle of Oscillation for the Three Bucket System.The 3 mass system works as we expected. There is a clear relationship between the amount of mass lost and how large the angle ofoscillation is. We see that once the system is increased beyond two buckets it becomes regular and the damping effect reduces the

angle of oscillation. We also see that the bigger the oscillation the more mass is lost; another expectation of the system. If the bucketis at a higher angle then more mass is lost, assuming that the bucket is initially full. We gain understanding of how systems

including more buckets will work. However, there is more exploration needed to see how to reduce the initial oscillation. All ourmodels systems initial oscillations remain resolutely large enough for the system to lose mass.

We used the values l = 5m, k = 30kN, b = 64.2096mtn/s2, c = 91.728mtn/s2 and d = 131.04mtn/s2.

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References

[1] The Engineering Toolbox. Drag Coefficient: The drag coefficient expresses the drag of an object in a moving fluid.

[2] Wells MA Ramanaidou ER. Sedimentary hosted iron ores. Holland HD, Turekian KK, editors. Treatise on geochemistry.2nd ed, pages 313–355, 2014.

[3] Encyclopaedia Britannica Online. Specific Gravity, 2015.

[4] Reade Advanced Materials. Specific Gravity Table For Ceramics, Metals & Minerals, 2015.

[5] Maritime Connector. Ship Sizes, 2015.

[6] Maritime Connector. Handymax, 2015.

[7] Maritime Connector. Bulk Carrier, 2015.

[8] Steven Holzner. Physics for dummies, pages 173–174. Wiley, 2006.

[9] Steven Holzner. Physics for dummies, pages 190–191. Wiley, 2006.

[10] Bert Blocken, Thijs Defraeye, Erwin Koninckx, Jan Carmeliet, and Peter Hespel. Numerical study on the aerodynamicdrag of drafting cyclist groups. 6th European and African Conference on Wind Engineering, pages 1–5, 2013.

[11] Rafael M. Digilov, M. Reiner, and Z. Weizman. Damping in a variable mass on a spring pendulum. Am. J. Phys.,73(10):901–905, 2005.

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Appendix A The Drag Constant

The drag coefficient is defined as:

D = ρ · A · v2

2· Cd

where

• ρ is the density of the fluid the bucket is moving through, i.e air where ρ = 1.2tn/m3,

• A is the reference area of the bucket, i.e the area of the face of the bucket moving into the air ,

• v is the velocity at which the tank is moving, and

• Cd is the coefficient of drag i.e a set constant relating to the shape of the object and how streamlined it is.

Appendix B Derivation of the Mathematical Pendulum Equations

We use both Newton’s 2nd Law and Hookes’ Law for the derivation. First we can assume that from figure (9.1.4a) that

T = ky = kr cos(φ)

where k is the ‘spring constant’ and y = r cos(φ) is the Cartesian to Polar change in coordinates. Now, from Newton’s2nd Law F = ma, radially:

mr = (mg cos(φ)− kr cos(φ))er − (mg sin(φ) + ar)eφ

We can define:

Position r = rer

Velocity r = rer + rφeφ

Acceleration r = (r− rφ2)er + (rφ + 2rφ)eφ

For the motion on a circle with radius r = l, then r = r = 0, so the acceleration is

r = −lφ2er + lφeφ

So the radial and tangential components are:

−mlφ2 = mg cos(φ)− kr cos(φ)mlφ = −(mg sin(φ) + ar)

and with a constant radius r = l we get the 2 equations

−mlφ2 = mg cos(φ)− kl cos(φ)mlφ = −(mg sin(φ) + ar)

The second of which we will use to solve the pendulum problem.

Appendix C Non-Dimensionalising the Single Bucket System

We introduce the dimensionless parameters φ(t) and t such that φ(t) = φ0φ(t) and t = t0 t where φ0 and t0 aredimensional constants. We then substitute these into our governing equation:

φ(t) = − gl

φ− bml

φ(t)

and initial conditions

φ(0) = 0 φ(0) = V0/l

to getd2φ(t)

dt2= − g

lt20φ(t)− b

mlt0

dφ(t)dt

We set the dimensional constants to be

t0 =√

l/g φ0 =√

gl/V0 and set λ = bm√

gl

and so once we drop the hats we have our required governing equation and initial conditions.

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Appendix D Creation of Diagrams and Plots

The diagrams in this project were created on the programme ‘Ipe’ which is an extensible drawing editor which canbe used for creating figures in LATEX. All the plots included in this project were created using ‘Maple’ which is amathematical computer algebra system. We input specific numeric values as shown in the text into our governingequations and initial conditions. Maple solved these ODE’s and system of ODE’s using either standard techniquesfor constant coefficient linear ordinary differential equations (when mass is constant), or by converting to a first-ordersystem Y′(x) = f (x, Y(x)) namely with Y′ represented by YP and solving using standard methods.

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