Rollover Prediction for Ro-Ro Cargo Trailers

Anders Huss

anders@mhuss.se

SA104X Degree Project in Engineering Physics, Candidate Thesis

Supervisor: Hanno Essén, Gunnar Maxe

Department of Mechanics

Royal Institute of Technology (KTH)

Stockholm Sweden

May 2013

1

Abstract

‘High and heavy’ cargo units are carried around the world on roll-on-roll-off (ro-ro) ships. These cargo units are loaded and unloaded using rolltrailers and the risk of rollover for the trailer-cargo unit, during port manoeuvre, often needs to be assessed in short time. On request by Wallenius Marine AB in Stockholm, this thesis examines the dynamics of the rolltrailer and different methods to predict the risk of rollover. The objective is to present a theoretical foundation for a toolbox that can support the assessment of trailer-load combinations.

The study focuses on the effect of running over ‘bumps’. An analytic model, based on assumptions of the transfer of energy in the rollover scenario, as well as a simulation, within the SimMechanics environment, was developed and the two were compared. The models produced almost identical output under the circumstances specified. The conclusion was that the assumptions made in the analytic model were reasonable and that the model can be used to characterize the trailer-cargo unit’s stability. The major reservation was that some dynamic properties of the trailer had been neglected and that the model does not account for a sequence of inputs and resonance effects.

The recommendation given is to develop the analytic model to an application for practical use. A developed SimMechanics simulation, on the other hand, can be used to examine additional dynamic aspects, resonance effects and other types of influences. Such results could support the decision making regarding what safety margin should be applied to the analytic result.

2

Table of Contents

1   Introduction .................................................................................................................. 3  

1.1   Background ............................................................................................................ 3  

1.2   Objective ................................................................................................................ 3  

1.3   Scope ...................................................................................................................... 3  

1.4   Concept of Study ................................................................................................... 4  

2   Method Description ...................................................................................................... 5  

2.1   Coordinate Conventions ........................................................................................ 5  

2.2   Trailer Construction .............................................................................................. 5  

2.3   Trailer Dynamics ................................................................................................... 5  

2.4   Rollover Criteria .................................................................................................... 7  

2.5   Comments on the Assumptions Made ................................................................... 7  

2.6   Types of External Influences ................................................................................. 8  

2.7   Two Dimensional Analytic Model ......................................................................... 9  

2.8   Three Dimensional Analytic Model ...................................................................... 13  

2.9   SimMechanics Model ............................................................................................ 19  

2.10   Comparison Technique ....................................................................................... 25  

3   Result .......................................................................................................................... 28  

3.1   Variation of Mass and Geometry ......................................................................... 28  

3.2   Variation of Ground Contact Parameters ............................................................ 31  

4   Conclusion ................................................................................................................... 34  

4.1   Limitations of Validity and Further Research ..................................................... 34  

4.2   Enhanced SimMechanics Model ........................................................................... 35  

4.3   Practical Application ............................................................................................ 37  

5   References .................................................................................................................... 38  

6   Appendix ..................................................................................................................... 39  

6.1   Ground Contact Function: Code .......................................................................... 39  

6.2   Derivation of Normal Forces: ............................................................................... 40  

3

1 Introduction

1.1 Background

‘High and heavy’ cargo units, such as small ships, generators etc., are carried around the world on so called roll-on-roll-off (ro-ro) ships. These cargo units, which can be rather extreme in terms of weight and size, are loaded and unloaded using Rolltrailers, on which they are also kept on-board during the overseas transport.

Wallenius Marine AB in Stockholm is responsible for design and technical management of a large number of PCTC (pure car and truck carriers) that operates worldwide. Wallenius ship design department also supports the ship operator WWL with technical analysis for trailer transports during loading and unloading of ships at ports.

The risk of rollover for a trailer-cargo unit during port manoeuvre often needs to be assessed in short time. Today this is done on request with simplified quasi-static analysis that does not fully describes the characteristics of the unit. In order to increase the quality and speed up the analysis process Wallenius is in need of a simulation model that better accounts for the geometrical and mechanical properties of a specific unit including a dynamic description of the rollover scenario.1

1.2 Objective

The study examines the dynamic behaviour of the trailer-cargo unit (referred to as ‘the unit’), under external influence critical to its stability. The aim is to present a theoretical foundation for an analysis toolbox that can support the assessment of trailer-load combinations.

1.3 Scope

The study focuses on trailers of ’semi-trailer’ type that has no front axel but is hung onto the pulling truck with a ’gooseneck’ configuration. For detailed description see section 2.2 and 2.3.

Detailed analysis is limited to the influence of uneven ground and running over ‘bumps’. Other types of external influence will be briefly discussed.

This study does not include the completion of an analysis toolbox for practical use. However, directions for completing such an application will be proposed, based on the results of the study.

1 Based on ”project sheet” Wallenius Marine AB 2012-11-14

4

1.4 Concept of Study

Step 1: The trailer construction was analysed in order to define its general geometry and dynamics, as well as a criteria for rollover. For this purpose, the technical descriptions and drawings of the trailer type were used.

Step 2: The operational environment was examined and different types of external influence, critical to the roll stability, was specified. The analysis was based on pictures of trailers in operation, as well as oral descriptions of the loading process by Roger Palmqvist at Wallenius Marine AB.

Step 3: A model of the cargo unit under influence of uneven ground was developed - for prediction of the maximum tilt angle reached when driving down a bump with one side of the trailer. An analytic solution as well as a simulation of the scenario was produced and compared. The analytic solution was derived from the trailer’s geometrical properties and observations of the transfer of energy throughout the dynamic process. The simulation was carried out using SimMechanics™, a multibody simulation environment for 3D mechanical systems, which is part of the MATLAB® and Simulink®

software.

Two separate methods were used for several reasons: The analytic solution contributes with theoretical understanding and states the dependency of the variables involved explicitly. The simulation, on the other hand, allows ’tuning’ of parameters that had to be idealized in the analytic solution. The simulation can also, due to its modular design, easily be extended to a more detailed and complex model of the trailer (which was done in step 4). Furthermore, the possibility to crosscheck the results from the different approaches was essential for the reliability of the results, since no experimental verification could be carried out.

Step 4. The Simulation was developed into a more detailed model of the trailer. This was done to examine the influence of the internal dynamics of the running gear. It also allows for a combined and more realistic input of external influence. This work will only be briefly described in this paper.

5

2 Method Description

2.1 Coordinate Conventions

The following coordinate system is used, unless otherwise stated. The x-axis is in the trailers lateral direction, left to right; y-axis in the longitudinal direction, rear to front; z-axis in the vertical direction, from ground upwards.

2.2 Trailer Construction

The trailers studied (shown in Table 1) are of semi-trailer type, which has no front axel but is hung onto the pulling truck with a ’gooseneck’ configuration. It consists of a platform, with a coupling for the gooseneck in the front, resting on a running gear in the rear. The platform is made up of a framework of steel beams with a wooden decking. The running gear consists of two sets of four wheels. Each set consists of one rocker arm, oscillating about the x-axis, with two wheel axels, oscillating about the rocker arms longitudinal axis, connected to 4 solid rubber tires (see Figure 2-3).

Table 1: List of specific trailers studied.

2.3 Trailer Dynamics

The freely oscillating rocker arms and axels in the running gear allows for a completely even distribution of the load between the four wheels within each set. As a result of this, point A and B, marked in Figure 2-1 to Figure 2-3, can be considered as pivot points, about which the trailer is free to rotate (within a certain range, see discussion in section 2.5). For some of the running gear types there is a slight vertical displacement between the oscillation axis for the rocker arm and for the wheel axels. In the analytic model and simplified simulation, described in section 2.7 to 2.9, this is approximated to a single pivot point, which should be at the height of the wheel axels’ oscillation axes since mainly the roll motion of the unit is examined.

Producer Reference in Tech. drawing. Houcon: RR-GC-30ft/140t

RR-GC-40ft/120t CIMIC: Rolltrailer 62’/90T

Rolltrailer 72’/80T ROKO: ROLLTRAILER 40 f SWL 100 t

6

The only ’suspension’ present is in the compression of the rubber tires. There has been no data available on the characteristics of this compression. However, the thickness of the tires is about 7 cm in unloaded condition, and it can be assumed to vary no more than a few cm under a certain load during manoeuvring.

The total load is distributed to the ground via the running gear but also partly to the towing truck via the gooseneck. Point C, according to Figure 2-1, is the third point about which the unit is assumed to pivot.

The platform itself is not completely torsional stiff, but it is assumed to gain a lot of stiffness from the cargo carried, which is strapped onto the platform

Based on the preceding observations the general approach to modelling the unit is to consider the platform with cargo as a rigid body free to pivot about point A, B and C.

Figure 2-1: Side view of trailer and truck.

Figure 2-2: Rear view of Running Gear.

A

C

A B

7

Figure 2-3: Different views of one ’Set’ of the Running Gear

2.4 Rollover Criteria

Based on the modelling approach described in the previous section, rollover occur when the centre of gravity (denoted point G) passes outside the lines A-C or B-C as a result of some external influence.

2.5 Comments on the Assumptions Made

The trailer platform has a certain amount of torsional flexibility but as long as the cargo is placed above, or close to, point A and B they are considered rigidly interconnected. Since the torsional resistance in pivot point C is assumed to be considerably less than in rest of the structure in front of the cargo, rotation is assumed to take place about point C. However, there is certain flexibility in the gooseneck configuration that allows for translational displacement between the front of the platform and point C. This is not taken into account in the analytic model and simplified simulation, described in section 2.7 to 2.9.

The rocker arm and wheel axles are only free to oscillate within a certain range; for the wheel axels approximately ± 11˚ about the rocker arm longitudinal axis (see Figure 2-1), but the range varies for the different trailer types. When this limit is reached, in a rolling motion, the point about which rotation is taking place is moved further out from the centre of gravity, which results in a higher moment that counteracts the rotation and

A, B

8

increases stability. This part of the dynamic process is not taken into account in any of the models; if an external influence causes a rollover for a unit that is free to rotate about point A or B, this unit is considered unstable even if it in reality could withstand such influence.

2.6 Types of External Influences

Rolling motion can be induced by any combination of the following types of external influences:

• Wind and collision with other objects

• Centripetal forces when in a turn.

• Acceleration of the towing truck (from low or zero speed) when not aliened with the trailer, causing a sideways acceleration of point C.

• Uneven surface; bumps and tilting surfaces, including resonance effects.

2.6.1 The ‘Bump Problem’

In the scope of this study, only the influence of running over sudden ground level changes, or ‘bumps’, is studied. The analysis is further restricted to driving down an edge or bump and single level changes – followed by a brief comment on the effects of driving up an edge and sequential level changes. The scenario studied is referred to as the ’Bump Problem’.

In the Bump Problem the maximum tilt angle of the unit is studied when the ground is ‘instantly lowered’ on the right side of the trailer, that is, when point B is allowed to ’fall freely’ a certain distance. The height of point B is proportional to the average height of the four tires (see Figure 2-4). Since the mass of the platform and load is considerable higher than the mass of the running gear, letting any combination of the four tires fall freely some short distances is essentially equivalent to letting point B fall the average distance.

Figure 2-4

dh dh / 2 dh / 4

9

2.7 Two Dimensional Analytic Model

The cargo unit is represented as a point mass rigidly connected by two (massless) supports to point A and B (as defined in section 2.3). The maximum value for ϕB is

sought, as point B is allowed to fall freely a distance d. It is assumed that the following motion can be described as a sequence of two rotations; about A until B ‘hits’ the ground (state 0 - 1), and thereafter about B until it reaches the ‘turning point’ (state 1-2), in other words:

ϕ

B2≡ ϕ

B ,max, see Figure 2-5 below.

Figure 2-5: Illustration of the ‘Bump’ Problem.

2.7.1 First Approach

If the total energy is conserved through out the process the solution is simple; the potential energy of the system when ϕB2

is reached must be equal to the initial potential

energy, since the kinetic energy is zero at both instances.

V2

=V0

rB

sinϕB2−d = r

Asinϕ

A0

ϕB2

= arcsinrA

rB

sinϕA0

+drB

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

(1)

The first approach ignores how support B interacts with the ground; apart from that ’it stays where it hits the ground’. It is hard to construct a completely elastic process where this would be the case.

ϕA0

Δϕ

G0

ϕB0

A0,A1 B1,B2 dB0

A2

G1 G2

10

2.7.2 Second Approach

The model is extended with the assumption that what causes support B to stay at the point of impact is perfect dampening through the support, perfect in the sense that the dampening is achieved without compression of the support, and a high enough lateral static friction. This cancels out the velocity component parallel to support B.

Figure 2-6: Illustration of the ‘Bump Problem’ with damping

vG1 is calculated from the transfer of energy:

V1

+T1

=V0

mgrAsinϕ

A1+

12mv

G12 = mgr

Asinϕ

A0

vG12 = 2gr

Asinϕ

A0− sinϕ

A1( ) (2)

The geometry implies that:

v⊥

= vG1

cos ϕA1−ϕ

B1( ) (3)

The case where cos ϕ

A1−ϕ

B1( ) < 0 must be excluded since this causes a rotation back

towards support A. The remaining kinetic energy is:

T⊥

=12mv

⊥

2 =12mv2 cos2 ϕ

A1−ϕ

B1( ) = mgrA

sinϕA0− sinϕ

A1( )cos2 ϕA1−ϕ

B1( ) (4)

No torque acts on point B from the ground and after impact, gravity is the only force doing work on the system:

ϕA1ϕB1

d

vG1

v⊥G1

11

V2−V

1=T

⊥

mgrB

sinϕB2− sinϕ

B1( ) = mgrA

sinϕA0− sinϕ

A1( )cos2 ϕA1−ϕ

B1( )ϕ

B2= arcsin sinϕ

B1+

rA

rB

sinϕA0− sinϕ

A1( )cos2 ϕA1−ϕ

B1( )⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

(5)

Presumably this better describes what happens when a vehicle with damped but relatively stiff suspension is put in the bump test. Still, this approach does not account for the distribution of the mass.

2.7.3 Third Approach

Figure 2-7: Illustration of the ‘Bump Problem’ with damping and inertia.

With the same premises as in approach 2, a distributed mass is now allowed for, resulting in an additional rotational component of the kinetic energy:

T =T

trans+T

rot=

12mv

G2 +

12I

Gω2 (6)

where IG is the moment of inertia (about the z-axis) relative the centre of gravity. It then follows:

V1

+T1

=V0

mgrAsinϕ

A1+

12mv

G12 +

12I

Gω

12 = mgr

Asinϕ

A0

m rAω

1( )2

+ IGω

12 = 2mgr

Asinϕ

A0− sinϕ

A1( )

(7)

ϕA1ϕB1

d

vG1

v⊥G1IG

12

vG12 = r

Aω

1( )2

=2mgr

A3 sinϕ

A0− sinϕ

A1( )mr

A2 + I

G

Ttrans,⊥

=12mv

⊥

2 =m2gr

A3 sinϕ

A0− sinϕ

A1( )mr

A2 + I

G

cos2 ϕA1−ϕ

B1( ) (8)

As in the previous case, there is no torque acting on point B, however the reaction force (or ‘impulse’) acting on point B from ground has one component parallel to the support B, assumed to cancel out the velocity component parallel to the support, as well as a component perpendicular to the support. The perpendicular component will result in dampening of the rotation since it exerts torque on the centre of gravity, counteracting the units’ rotation. At the same time, this force accelerates the centre of gravity in the direction of motion after B’s impact with ground. Based on this observation, at least some of the rotational energy before impact, Trot, contributes to the total kinetic energy after impact. It is not trivial to derive how much of Trot that is ‘preserved’ and it would require a more detailed model of how point B interacts with ground, therefore a conservative assumption is made; that all of Trot contributes to the total kinetic energy after impact:

V2−V

1=T

trans,⊥+T

rot

mgrB

sinϕB2− sinϕ

B1( ) =12mr

A2ω

12 cos2 ϕ

A1−ϕ

B1( )+12I

Gω

12

2mgrB

sinϕB2− sinϕ

B1( ) = mrA2 cos2 ϕ

A1−ϕ

B1( )+ IG( )

2mgrA

sinϕA0− sinϕ

A1( )mr

A2 + I

G

ϕB2

= arcsin sinϕB1

+rA

rB

sinϕA0− sinϕ

A1( )rA2 cos2 ϕ

A1−ϕ

B1( )+ IG

m( )rA2 + I

Gm

⎡

⎣

⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥

(9)

13

2.8 Three Dimensional Analytic Model

The third approach in the 2D analysis was further extended to 3D and the mass of the sets of wheels, at support A and B was added to the model as point masses (their respective centre of gravity is rigidly connected to point A and B, but they do not necessarily rotate with the rest of the configuration). As in the 2D approach, the transfer and absorption of energy was used to calculate the maximum tilt angle

Figure 2-8: Illustration of the 3D ‘Bump Problem’.

Through out the calculations, rotation-matrix operators were used to handle rotations about defined axes. By ”Rodrigues’ rotation formula”2 one can construct the matrix that rotates a vector an angle θ about an arbitrary axis

u = ( u

xu

yu

z)T :

Ru

=1 0 00 1 00 0 1

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟cosθ+ sin θ

0 −uz

uy

uz

0 −ux

−uy

ux

0

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

+ 1− cosθ( )u

x2 u

xu

yu

xu

z

uxu

yu

y2 u

yu

z

uxu

zu

yu

zu

z2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

(10)

To calculate changes in potential energy one must also be able to determine the angle θ by which a vector is rotated about a defined axis so that it moves a distance Δz , i.e. solve the equation:

Ru

(θ)r ⋅ z = r ⋅ z +Δz (11)

2 Source: http://en.wikipedia.org/wiki/Rodrigues'_rotation_formula#Matrix_notation, 2013-04-16. (The same formula is also found in: Råde, L (2004) p. 112.)

θ0−1

G0

A0,A1B0

B1

C0,C1,C2

IG

mA mB

mG

d

14

This was done using MATLAB’s symbolic variable tool ‘syms’ which gives θ as an explicit function of the other variables3:

Δθ =Δθ(u, r,Δz) (12) With these tools at hand one can take on the specific problem. Rotations about two axes only is studied and to simplify the equations the following notation is introduced:

rXY 1≡ R

AC 0(Δθ

0−1)r

XY 0

rXY 2≡ R

BC1(Δθ

1−2)r

XY 1

[r]z≡ r ⋅ z

(13)

Where rXYS represents a vector from point X to point Y at state S; R

XYS(Δθ

S−(S+1))

represents the matrix the rotates a vector an angle Δθ

S − (S+1)about the axis defined by

the vector rXYS.

2.8.1 Rotation 0-1

Rotation angle about axis rAC 0:

Δθ0−1=Δθ(r

AC 0, r

AB0,−d) (14)

Conservation of energy:

V1−V

0=T

G ,rot,1+T

G ,trans,1+T

B ,trans,1

[rAG0− r

AG1]zm

Gg +dm

Bg =

12wT I

Gw +

12m

Gv

G⋅v

G+

12m

Bv

B⋅v

B

=12ω

12 rT

AC 0I r

AC 0+m

GrAG2 +m

BrAB2( )

(15)

Which gives:

ω12 = 2g

[rAG0− r

AG1]zm

G+dm

B( )rT

AC 0IG

rAC 0

+mGrAG2 +m

BrAB2( )

≡ 2gA

(16)

3 An alternative would be to find R numerically and get θ using 1+ 2cosθ = trace(R) , Råde, L (2004) p. 112

15

The available part of the kinetic energy (not absorbed at impact) is given by the expression:

Eavailabe,1

=TG ,rot,1

rAC1⋅ r

BC1( )2

+TG ,trans,1

nACG1⋅ n

BCG1( )2

=12ω

12 rT

AC 0I r

AC 0rAC1⋅ r

BC1( )2

+mGrAG2 r

AC1× r

AG1( ) ⋅ rBC1× r

BG1( )( )2⎛

⎝⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

≡ gAB

(17)

It should be noted that all of the rotational energy cannot be conserved in the 3D case. Since there are three pivot points involved there is an angular impulse transferred to ground at impact and any rotation non-parallel to the second rotation axis, rBC, is assumed to be absorbed.4

2.8.2 Rotation 1-2

The available kinetic energy is converted to potential energy only, when the maximum tilt angle is reached.

V2−V

1= E

availabe,1

mGg[r

BG2− r

BG1]z

+mAg[r

BA2− r

BA1]z

= gAB⇒

RBC1

(Δθ1−2

) mGrBG1

+mArBA1( ) ⋅ z = m

GrBG1

+mArBA1( ) ⋅ z + AB

(18)

With the definition rB(AG)1

≡ mGrBG1

+mArBA1( ) the following equation is obtained:

R

BC1(Δθ

1−2)r

B(AG)1⋅ z = r

B(AG)1⋅ z + AB (19)

Which is of the same form as (11) with the solution according (12):

Δθ

1−2=Δθ(r

BC1, r

B(AC )1, AB) (20)

Now rB(AG)1

, pointing in the direction of the combined centre of gravity for mG and mA,

can be calculated:

rB(AG)2

= RBC1

(Δθ1−2

)rB(AG)1

(21)

4 An intuitive illustration of this is that if there is a right angle between rAC and rBC the rotational energy about rAC will not contribute to the rotation about rBC. In this aspect the 3D model converge to the 2D model as C is moved along the y-axis to ’infinity’.

16

Figure 2-9: The centre of gravity for A and G combined must stay inside line B-C

2.8.3 Maximum Tilt Angle

In the two dimensional case ϕB is the angle between support B and the x-axis. It serves

as a measurement of the stability since as long as ϕB does not exceed 90° the

configuration will not roll over. In the three dimensional case there are several possible ways of defining an angle with this quality, of which the following was chosen:

ϕ = arccos

l1

l2

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟ (22)

Where l1 is the shortest distance between a vertical line from the centre of gravity to the rotation axis (B-C) and l2 is the distance between the point L, where the line l1 connects to the rotation axis, and the centre of gravity (see Figure 2-10).

To find a general expression for l1 and l2 the vectors a, b, and c are introduced according to Figure 2-11 and Figure 2-12. The following applies:

a =

mGrBG

+mArBA

mG

+mA

=rB(AG)

mG

+mA

(23)

b = a− a ⋅ e

z( ) ez (24)

ϕmax

B2

C2

mA

mG

A2

G2

17

c = k ⋅ rBC

c ⋅ rBCprojXY

= a ⋅ rBCprojXY

where : rBCprojXY

≡rBC−[r

BC]ze

z

rBC−[r

BC]ze

z

k rBC⋅ r

BCprojXY( ) = a ⋅ rBCprojXY

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎫

⎬

⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

=a ⋅ r

BCprojXY

rBC⋅ r

BCprojXY

rBC

(25)

l1

= a−c

l2

= b− b ⋅ rBCprojXY( ) rBCprojXY

(26)

Altogether, ϕ2 is given by (22), with

l1

= l1(r

B(AG)2,r

BC1) ,

l2

= l2(r

B(AG)2,r

BC1) according to

(23) to (26).

Figure 2-10: Illustration of ϕ in 3D.

mtotg

B

C

ϕ

l2

l1 L

18

Figure 2-11: Components (a, b and c) for determination of ϕ , view 1.

Figure 2-12: Components (a, b and c) for determination of ϕ , view 2.

B

C

a

c

b

z

⊥ z ⊥ rBC( )

z

rBCprojXY( )

a

c

bB

C

19

2.9 SimMechanics Model

2.9.1 SimMechanics Environment

SimMechanics™ is a part of the Simulink® product family, which is tightly integrated with the MATLAB programming environment. Simulink is a graphical programming tool where block diagrams are used to model dynamical systems. SimMechanics is an ‘add-on library’ for physical mechanical modelling. Below is the developer MathWorks’ description of its functionality and use:

SimMechanics™ provides a multibody simulation environment for 3D mechanical systems (...). You model the multibody system using blocks representing bodies, joints, constraints, and force elements, and then SimMechanics formulates and solves the equations of motion for the complete mechanical system.5

This means that as long as the user sets up the ’physical network’ and defines the initial states, the equations of motion are derived and solved automatically.

There is a predecessor to this environment, SimMechanics 1st Generation; therefore the environment used is sometimes referred to as ’2nd Generation’, but often just SimMechanics.

2.9.2 Model Description

The unit consists of three bodies; a ’box’ (grey), representing the mass and inertia of the trailer platform and the cargo combined, and two small spheres at point A and B (blue), representing the aft pivot points and the mass of the respective running gear set. Point C is not a body (no mass or inertia) but purely a graphical representation of the third pivot point (red). These are all rigidly interconnected. A fourth body, the ground reference (brown), is added. It is rigidly connected to the world frame. In point C the unit is connected to the world frame by a spherical joint allowing any rotational motion but restricting translational motion at this point. Point A and B are ‘connected’ to the ground reference by a custom made Ground Contact Block described in the following section.

5 http://www.mathworks.se/products/simmechanics/, 2013-04-12

20

Figure 2-13: SimMechanics 3D view of the model.

Figure 2-14: SimMechanics Block diagram of the model.

21

Figure 2-15: SimMechanics block diagram of the subsystem ‘Bodies’.

2.9.2.1 Ground Contact Block

This block represents the wheels’ contact with ground and exerts normal and friction forces on body A and B as a function of their position and velocity relative ground. It connects the respective point to ground with a ’6-DOF Joint’ that allow any motion but can also exert forces between its ‘Base’ and ’Follower’ port according to received input signals. The ‘contactForce’ MATLAB function produces these signals, based on its input from a ‘Transform Sensor’. The Transform Sensor delivers parameters describing the position, speed and rotation of the wheels’ reference frame relative the world reference frame.

22

Figure 2-16: SimMechanics block diagram of subsystem ‘Ground Contact’.

The normal force was modelled as a linear spring and dampener when the wheel reference is at or ‘below’ ground (tire is compressed). Otherwise it is zero. It should also be noticed that the wheel cannot be ’pulled down’ to the ground by the dampening force:

Fz

=

00

zWheel

> zGround

k zGround

− zWheel( )≤ c z

k zGround

− zWheel( )−c z z

Wheel≤ z

Ground k z

Ground− z

Wheel( ) > c z

⎫

⎬

⎪⎪⎪⎪⎪⎪

⎭

⎪⎪⎪⎪⎪⎪

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

[N ] (27)

Friction was modelled reusing an approach from the ‘Translational Friction Block’ in SimMechanics 1st Generation (in this environment each degree of freedom is handled separately and in the 2nd Generation there are no predefined blocks for modelling friction).6

A common model of translational friction is shown in Figure 2-17. This is however ’too idealistic’ and the discontinuity at v = 0 causes computational problems. There are several ways of avoiding this discontinuity and the Translational Friction Block implements one of the simplest ways, defined in equation (28) and (29) and illustrated in Figure 2-18.

6 MATLAB Help documentation: Simscape/Blocks/Foundation/Mechanical/Translational Elements/Translational Friction.

23

F =

FC

+ Fbrk−F

C( )e −cv v( )⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟sign(v)+ fv v ≥ v

th

vv

th

FC

+ Fbrk−F

C( )e −cvvth( ) + fvth

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ v < v

th

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

(28)

Fbrk

= Breakaway friction [N ]F

C=Columb friction [N ]

f = viscouse friction coefficient [N /(m / s)]v

th= velocity thershold [m / s]

cv

= transition approximation coefficient [s /m]

(29)

Figure 2-17: Ideal discontinuous model of

friction.7

Figure 2-18: Adjusted continuous model of friction.8

This basic approach was used but with some enhancements to fit the 3D model. The speed v relevant for the friction in (28) is the magnitude of: the actual 3D velocity (vactual) of the wheel reference point, with the component in the wheel rolling direction removed (vRD) and then projected to the ground plane (vF).

7 MATLAB Help documentation: Simscape/Blocks/Foundation/Mechanical/Translational Elements/Translational Friction. 8 Ibid.

24

Figure 2-19: Illustration of transformation of the reference velocity for friction.

Furthermore, the total friction was set proportional to the normal force Fz. The constants were set to pre-set values for the Translational Friction Block, with FC and f normalised to the break-off friction (Fbrk = 1), and the expression was multiplied with Fz and a break off friction coefficient cbrk that relates the normal force to the break off force. All in all, the friction force was defined as follows:

vred

= vact−v

act⋅ r

RD

vF

= vred−v

red⋅ z

F =

cbrk

Fz⋅ F

C+ 1−F

C( )e −cv vF( )⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟⎟+ f v

Fv

F≥ v

th

cbrk

Fz⋅v

F

vth

FC

+ 1−FC( )e −cvvth( ) + fv

th

⎛⎝⎜⎜⎜

⎞⎠⎟⎟⎟ v

F< v

th

⎧

⎨

⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

Ffriction

=−Fv

F

vF

(30)

FC

= 0,8f = 4

vth

= 10−4

cv

= 10c

brk= 1

[1][1 /(m / s)][m / s][s /m][1]

(31)

That cbrk = 1 means that the breakaway friction is equal to the normal force. For further details on the Ground Contact Block see appendix 6.1.

rRD vactual

v redvF

y

x

z

r

v red

rRD

25

2.10 Comparison Technique

The analytic solution was compared with the SimMechanics model as described below:

2.10.1 Variation of Mass and Geometry

The parameters in both models were related to a set of ‘tuneable’ variables of interest9:

rA

= 0 0 0( )rB

= wheelbase _w 0 0( )rC

= wheelbase _w / 2 rC _y rC _ z( )rG

= wheelbase _w / 2 +Gdisp _ x rG _y rG _ z( )m

A= m

B= sup_m

mG

= load _m

IG

=load _m

12

(2m)2 + (6m)2 0 00 load _w 2 + (2m)2 00 0 load _w 2 + (6m)2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

(32)

The introduced variables was set to the following values, typical for the type of trailer studied10:

wheelbase _w = 1,1 [m]rC _y = 10 [m]rC _ z = 0 [m]Gdisp _ x = 0 [m]rG _y = 1 [m]rG _ z = 1,5 [m]sup_m = 1 [kg ]load _m = 100 [kg ]load _w = 2 [m]dz = 0,05 [m]

(33)

9 IG is the inertia matrix for a ‘box’ with width (x) load_w m, depth (y) 6 m and height (z) 2 m.

10 The absolute mass is not typical but only the relation between masses and spring/dampening coefficients is relevant: in the analytic model it does not affect the result if all mass is multiplied by a constant: in the simulation the spring/dampening coefficients were set in relation to the mass, as described in the following page.

26

In the SimMechanics model three additional parameters had to be set for the Ground Contact Block:

cbrk

= 1 [1]k = 3 ⋅105 [N /m]c = 5477 [N /(m / s)]

(34)

The value for k was chosen to get as ’stiff’ contact as possible without making the simulations too time consuming. With the chosen value, a displacement of about 3 mm at one contact point results in a normal force corresponding to the total weight. The dampening coefficient was thereafter set based on the assumption that the dampening was ‘nearly critical’.

ζ =

c2 km

(35)

If the whole load_m = 100kg is oscillating about point B the values chosen gives a damping ratio (35) ζ = 0,5 . A better way, however, to define ζ in this case is to relate k and c to the mass supported by A and B respectively when stationary on level ground:

ζeff ,B

=c

2 kmeff ,B

=c

2 k NB

/ g( )

(36)

With NB according to (40) in appendix 0. For the standard set up described

m

eff ,A= m

eff ,B= 46 kg , which gives

ζ

eff= 0,74 .

Two constants, dzA and dzB according to (42) in appendix 0, were added to the ‘ground level’ at support A and B in the SimMechanics model, so that the configuration rest exactly at ”zero level” when in steady state before the ”bump”.

From the standard setup one of the tuneable variables were varied at a time. The resulting ϕ2

from the analytic solution and the simulation were compared.

Figure 2-20: Illustration of oscillation patterns for different damping ratios.11

11 Source: http://upload.wikimedia.org/wikipedia/commons/9/94/2nd_Order_Damping_Ratios.svg

27

2.10.2 Variation of Ground Contact Parameters

With the standard setup from the previous section ϕ2 was compared for a variation of

dz ∈ [0.07m, 0.12m] , for a set of ground contact parameters according to Table 2 to 4. As a reference, the deflection (‘compression of tires’) is presented for steady state on level ground, dz0 (load 46 kg), and with the whole mass (load 102 kg) in equilibrium

over point B, dzmtot.

Table 2: Variation of k, ζ

eff= 0,5

Table 3: Variation of ζ

eff, k = 300000.

k [N/m] c [Ns/m] ζ

eff

ζmtot dz0

[cm] dzmtot [cm]

300000 4458 0,60 0,40 0,15 0,33 300000 2972 0,40 0,27 0,15 0,33 300000 1486 0,20 0,13 0,15 0,33 300000 743 0,10 0,07 0,15 0,33

Table 4: Variation of ζ

eff, k = 15000.

k [N/m] c [Ns/m] ζ

eff

ζmtot dz0

[cm] dzmtot [cm]

15000 996,8 0,60 0,40 3,01 6,67 15000 664,5 0,40 0,27 3,01 6,67 15000 332,3 0,20 0,13 3,01 6,67 15000 166,1 0,10 0,07 3,01 6,67

k [N/m] c [Ns/m] ζ

eff

ζmtot dz0

[cm] dzmtot [cm]

300000 3714 0,50 0,34 0,15 0,33 30000 1174 0,50 0,34 1,50 3,34 15000 830 0,50 0,34 3,01 6,67 7000 567 0,50 0,34 6,45 14,29

28

3 Result 3.1 Variation of Mass and Geometry

Figure 3-1

Figure 3-2

Figure 3-3

Figure 3-4

Figure 3-5

Figure 3-6

0 0.02 0.04 0.06 0.08 0.1 0.1270

72

74

76

78

80

82

84

86

88

90k = 300 000, c = 5477, ceff = 0.74

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.108 0.1085 0.109 0.1095 0.11 0.1105 0.111

87

87.5

88

88.5

89

89.5

90k = 300 000, c = 5477, ceff = 0.74

dz [m]an

gle

[deg

]

q1q2 analytic

q2 simulation

0.7 0.8 0.9 1 1.1 1.2 1.3 1.472

74

76

78

80

82

84

86

88

90k = 300 000, c = 5477, ceff = 0.74

wheelbase_w [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.76 0.765 0.77 0.775 0.78

86.5

87

87.5

88

88.5

89

89.5

90k = 300 000, c = 5477, ceff = 0.74

wheelbase_w [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

1.5 2 2.5 3 3.572

74

76

78

80

82

84

86

88

90k = 300 000, c = 5477, ceff = 0.74

rG_z [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0 1 2 3 4 5 672

74

76

78

80

82

84

86

88

90k = 300 000, c = 5477, (ceff varies)

rG_y [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

29

Figure 3-7

Figure 3-8

Figure 3-9

Figure 3-10

Figure 3-11

Figure 3-12

0 0.05 0.1 0.15 0.272

74

76

78

80

82

84

86

88

90k = 300 000, c = 5477, (ceff varies)

Gdisp_x [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0 1 2 3 4 5 6 7 873.5

74

74.5

75

75.5

76

76.5

77

77.5k = 300 000, c = 5477, ceff = 0.74

load_w [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0 5 10 1572

74

76

78

80

82

84

86k = 300 000, c = 5477, (ceff varies)

rC_y [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

−2 −1 0 1 2 369

70

71

72

73

74

75

76

77

78

79k = 300 000, c = 5477, ceff = 0.74

rC_z [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0 50 100 150 20070

71

72

73

74

75

76

77k = 300 000, c = 5477, (ceff varies)

load_m [kg]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0 5 10 15 20 25 3062

64

66

68

70

72

74

76

78k = 300 000, c = 5477, (ceff varies)

sup_m [kg]

angl

e [d

eg]

q1q2 analytic

q2 simulation

30

The maximum tilt angle ϕ2 increases with increasing dz, rG_z, rG_y G_disp, load_w,

and decreases with increasing wheelbase_w, as expected (see Figure 3-1 to Figure 3-8). Increasing load_w has little effect for the setup and range studied. This is because most of the available energy (equation (17)) is due to translational energy for the relatively ‘high’ load. For wide and low cargo the contribution from the rotational energy is considerably higher, see Figure 3-13 below. For the wider load (8 m) ϕ2

−ϕ1≈ 4 ,

compared to 1˚for the narrower load (0,5 m). There is no risk for this unit to roll over, but ϕ2

can still be of interest for determining if the wide load will touch the ground at

its outer edges.

Figure 3-13: Variation of load_w for a lower centre of gravity and higher bump

relative the standard setup

Figure 3-9 shows that rC_y has little effect on ϕ2 within the range for the trailers

studied (9-15 m). However, the distance between G and C is relevant, since shorter distance increases ϕ2

. Figure 3-10 shows that both the absolute value of ϕ2 and the

difference ϕ2−ϕ

1 increase as rC_z decreases indicating less stability when going down a

ramp compared to going up a ramp, for example.

The dependency of load_m and sup_m is a result of the displacement of the total centre of gravity. For the simulation, an increasing total mass affects the interaction with ground (‘higher compression of tires’) but the effect of this is small and not observable within the range studied – since the analytic and simulation results follow each other tightly (Figure 3-11 to Figure 3-12).

Overall, the dependency of the tuneable variables, as well as the absolute value of ϕ2, is

very similar for the analytic solution and the simulation.

0 1 2 3 4 5 6 7 854

54.5

55

55.5

56

56.5

57

57.5

58

58.5

k = 300 000, c = 5477, ceff = 0.74Changes from Standard Setup: dz = 0.08, rGz = 0.6

load_w [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

3.2 Variation of Ground Contact Parameters

Figure 3-14

Figure 3-15

Figure 3-16

Figure 3-17

Figure 3-14 to Figure 3-16 indicates that ϕ2 for the simulation generally decreases as k

decreases with constant ζ

eff= 0,5 . Figure 3-17, however, shows that this dependency is

not consistent since ϕ2 has increased for the lower values of dz. A closer look at the

lower dz values (Figure 3-18) shows that it still stays close to the analytic solution.

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 300 000, c = 3715, ceff = 0,5

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 30 000, c = 1175, ceff = 0,5

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 15 000, c = 830.7, ceff = 0.5

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 7000, c = 567.5, ceff = 0.5

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

32

Figure 3-18

Figure 3-19

Figure 3-20

Figure 3-21

Figure 3-22

Figure 3-19 to Figure 3-22 show that for the simulation increases as decreases for

constant .

0.02 0.04 0.06 0.08 0.1 0.1272

74

76

78

80

82

84

86

88

90k = 7000, c = 567.5, ceff = 0.5

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 300 000, c = 4458, ceff = 0.6

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 300 000, c = 2972, ceff = 0.4

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 300 000, c = 1486, ceff = 0.2

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 300 000, c = 743, ceff = 0.1

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

ϕ2 ζ

eff

k = 3 ⋅105

33

Figure 3-23

Figure 3-24

Figure 3-25

Figure 3-26

Figure 3-23 to Figure 3-26 show that ϕ2 for the simulation increases as

ζ

effdecreases also

for the lower k = 3 ⋅105 .

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 15 000, c = 996.8, ceff = 0.6

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 15 000, c = 664.5, ceff = 0.4

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 15 000, c = 332.3, ceff = 0.2

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

0.08 0.09 0.1 0.11 0.1275

80

85

90k = 15 000, c = 166.1, ceff = 0.1

dz [m]

angl

e [d

eg]

q1q2 analytic

q2 simulation

34

4 Conclusion

With high stiffness and nearly critical dampening in the simulation (setup according to section 2.10.1) the analytic and simulation results are almost identical. Since the simulation was modelled independently of the assumptions on energy transfer, the result indicates that these were reasonable. Also when the ground contact parameters are varied, the results stay relatively close together. The case presented where the simulation overshoot the analytic solution the most is with

ζ

eff= 0,1 and k = 15 000 (Figure 3-26).

The simulation then predicts rollover for a dz of about 1 cm less than the analytic solution, or put in another way, it predicts up to about 5˚greater tilt angle. The value for k in this case is judged to be rather representative (based on the resulting compression of tires, see Table 4) whereas the damping ratio is assumed to be too low. In reality, the normal force is necessarily a highly nonlinear function of the compression of the tire, but for stiff tires as in this case, it is assessed to be a sufficient representation.

The overall conclusion is that the analytic model offers a relevant tool for assessing the stability of the trailer-cargo unit under the type of influence studied. There are many uncertainties in what combinations of external influences actually will occur and in the description of the load etc. In relation to this the precision in the analytic model is high. A certain safety margin will of course have to be adapted and a possible application is to use this tool for standardized categorization of the units, where units within a ‘critical range’ are only accepted when certain external conditions can be assured.

4.1 Limitations of Validity and Further Research

Only the effect of letting point B ‘fall freely’ has been studied, this is unlikely to actually happen but it’s a conservative approximation of driving down an edge. To calculate the effects of driving up an edge one has to use a different approach to determine the angular velocity and available energy at state 1 (where the wheel is all the way up on the edge), that is: another method up to equation (17) in section 0. This can be estimated by a geometrical observation of the tire and profile of the bump, along with a model for the compression of the tires (vertical velocity of point B in state 1 will depend on the forward speed of the trailer). It should be stated that driving up an edge on one side of the trailer could cause greater rotational acceleration than driving down the same bump.

When estimating the effect of driving over an edge at one side of the trailer it is when the second wheel pair passes that is most critical. As described in section 2.6.1 the falling distance of point B is the same when the front and rear wheel pair passes but the second time it is from an already tilting (and non stationary) state. The fact that a sequence of ‘two level changes’ for point B is expected is important since they can ‘amplify’ each

35

other; there will be a critical speed at which the level changes coincide with the unit’s angular ‘resonance frequency’.

For a more thorough analysis of the effect of resonance, other and combined external influence as well as the flexibility in the front of the trailer and gooseneck, one can use a developed SimMechanics model.

4.2 Enhanced SimMechanics Model

In a further development of the SimMechanics model all moving parts in the running gear should be defined separately and interconnected with appropriate joints. The ground contact function should be ‘moved’ to each individual tire’s contact point with ground and the normal force could easily be adjusted to a more realistic nonlinear function of the compression of the tires. Flexibility in the front of the trailer can also be modelled by a subdivision of the platform and gooseneck into elements that are interconnected with torsional ‘spring-dampener loaded joints’.

A ‘draft’ of such model is illustrated in Figure 4-1 to Figure 4-3.

Figure 4-1

36

Figure 4-2

Figure 4-3

37

4.3 Practical Application

Based on the presented result it is proposed that a practical application is based on the analytic model. The calculation is fast and it can relatively easily be programmed and developed into a standalone application. It is possible to develop the analytic model to also take into account some combinations of external influence, e.g. hitting a bump when in a turn.

The enhanced SimMechanics model is appropriate for further detailed analysis of the type of influence not included in the analytic solution. To make use of the flexibility of the SimMechanics model, one is restricted to using the MATLAB environment; it is not easily deployed into a standalone application. Furthermore, the processing of a simulation of a few seconds take several minutes up to an hour, which makes the tool less practical. It can however be used to set the result of the analytic model in relation to that of other influences studied by simulation, and thereby support the decision making regarding what safety margin should be applied to the analytic result.

An application based on the analytic model could also be extended with extra functionality such as determine whether there is a risk for low and wide cargo to ‘hit the ground’.

38

5 References

Apazidis, N. (2008), Rigid Body and Analytical Mechanics. Stockholm, Sweden: Institution of Mechanics KTH.

Råde, L. Westergren, B. (2004), Mathematics Handbook for Science and Engineering, Lund, Sweden: Studentlitteratur.

Wood, Giles D. Kennedy, Dallas C. (2003), Simulating Mechanical Systems in Simulink with SimMechanics, Natick, MA, USA: The MathWorks. Paper by the developer available at: http://www.mathworks.com/tagteam/12634_SimMechanics.pdf (2013-04-16)

http://en.wikipedia.org/wiki/Rodrigues'_rotation_formula#Matrix_notation (2013-04-16)

Math Works help documentation: Translational Friction. Available at: http://www.mathworks.se/help/physmod/simscape/ref/translationalfriction.html?searchHighlight=Translational+Friction+block, (2013-04-16)

For learning how to perform mechanical modelling within the Math Works software an extensive use of the product information and help documentation has been necessary and very helpful. Roots for the pages used below.

http://www.mathworks.se/products/simmechanics/

http://www.mathworks.se/products/simulink/

http://www.mathworks.se/products/matlab/

http://www.mathworks.se/help/index.html

39

6 Appendix

6.1 Ground Contact Function: Code

40

6.2 Derivation of Normal Forces:

Moment equation about y-axis in point C:

x

C− x

A( )N 'A+ x

C− x

B( )N 'B

+ xG− x

C( )mGg = 0 (37)

Moment equation about x-axis in point C:

y

C−y

A( )N 'A+ y

C−y

B( )N 'B

+ yG−y

C( )mGg = 0 (38)

Which gives the result:

N 'A

= mGg

xB

yC−y

G( )+ xC

yG−y

B( )+ xG

yB−y

C( )x

Ay

B−y

C( )+ xB

yC−y

A( )+ xC

yA−y

B( ) (39)

N 'B

= mGg

xA

yC−y

G( )+ xC

yG−y

A( )+ xG

yA−y

C( )x

Ay

C−y

B( )+ xB

yA−y

C( )+ xC

yB−y

A( ) (40)

NB

= N 'B

+mBg

NA

= N 'A+m

Ag

(41)

dzA

= NA

kdz

B= N

Bk

(42)

Figure 6-1: Normal Forces, view of y-z-plane

Figure 6-2: Normal Forces, view of x-z-plane

N 'AN 'B

mGg

NC

C

z

y

NC N 'BN 'A

C

mGg

z

x