Rigid-Body Modelling of Forklift Masts and Mast Sway Simulations

(1)

Master of Science Thesis in Electrical Engineering

Department of Electrical Engineering, Linköping University, 2016

Rigid-Body Modelling of

Forklift Masts and Mast

Sway Simulations

(2)

Master of Science Thesis in Electrical Engineering

Rigid-Body Modelling of Forklift Masts and Mast Sway Simulations

Minh Le Tran LiTH-ISY-EX--16/4960--SE Supervisor: PhD Sertac Erdemir

isy, Linköpings universitet

Jim Fredin

A forklift manufacturer in Mjölby

Examiner: Prof. Svante Gunnarsson

isy_{, Linköpings universitet}

Division of Automatic Control Department of Electrical Engineering

(3)

Abstract

Reach truck masts are subjected to oscillations, which have significant impacts on the dynamics of the entire vehicle. Mast oscillations can cause undesirable outcomes in extreme situations and therefore it is desirable to be able to predict these outcomes before they occur. A forklift manufacturer in Mjölby initiated a thesis with the intention to obtain a model that can simulate mast sway for situations where oscillations occur. The objective of the thesis was to create a model of Triplex masts and find dependencies between model parameters and variables such as fork height and load.

The thesis was conducted modelling the mast with a rigid multibody approach where torsion springs and dampers were used between mast parts to simulate mast elasticity. Clearance at the connections were considered and included in the model. The obtained model constitutes of 8 parameters that could be tuned to at-tain different oscillation characteristics. Parametric optimisation was carried out to find optimal sets of parameters for compliance with sway measurement tests with different load and fork height cases.

The thesis has resulted in a model that is able to simulate mast sway with different oscillation characteristics depending on model parameters. Performed parametric optimisation resulted in parameters that reveal useful information about how model parameters depend on load and fork height. The method used for obtaining optimal parameters can likewise be applied to other mast models in order to gain insight into model parameters as functions of load and fork height.

(4)

(5)

Acknowledgments

I would like to dedicate the greatest and most grateful acknowledgements to-wards my supervisor Jim Fredin who has offered me useful advice and indispens-able assistance during my thesis work. Richard Fyhr also deserves my dearest gratitude for allowing me the opportunity to work in the facilities of the forklift manufacturer in Mjölby. The employees who have shared the same working at-mosphere with me have received me with much affection and warmth, and for that I owe them my sincere gratitude.

I must not forget to tender my thanks to Fredrik Sjögren from MSC Software, who offered me his eminent expertise. His advice have been of great assistance. Christian Walette and Johan Häretun also deserve my acknowledgements for as-sisting me with perilous sway tests. Their kind and helpful conduct during the measurement sessions occasioned me much pleasure.

I owe Svante Gunnarson my profuse gratitude for undertaking the task to serve as my examiner. His kind and considerate conduct during our encounters have encouraged me immensely. I would also like to thank Sertac Edemir who served as my supervisor at the university.

Finally I would like to devote my endless gratitude towards my family who has supported and encouraged me during harsh time in life. Without them, I would not be able to reap the fruits of today’s success.

Linköping, June 2016 Minh Le Tran

(6)

(7)

viii Contents 5.3 Test Procedure . . . 32 5.4 Results . . . 33 6 Parameter Analysis 35 6.1 Procedure . . . 35 6.2 Tests . . . 35 6.3 Results . . . 36 7 Parametric Optimisation 37 7.1 Procedure . . . 37 7.2 Parameter Spaces . . . 38 7.3 Results . . . 38 8 Discussion 43 8.1 Modelling Approach . . . 43 8.2 Sway Measurements . . . 44 8.3 Parameter Analysis . . . 44 8.4 Parametric Optimisation . . . 45 8.5 Conclusions . . . 46 8.6 Future Work . . . 46 A Functions 51

B Parameter Analysis Results 59

C Parameter Spaces 67

(9)

Notation

Variables Variable Description τ Exerted torque θ Angular displacement ˙ θ Angular velocity κ Torsion coefficient c Rotational coefficient θc Critical angle α Clearance d Roller overlap h Fork height

hf Free lift height

F Force

E Young’s modulus

A0 Initial cross-sectional area

L0 Initial length

∆L Difference between initial length and actual length xf Free lift piston stroke

xm Main piston stroke

m Mass ρ Density J∗ Criterion function ζ Damping ratio f Frequency A Initial deflection ix

(10)

x Notation

Abbreviation

Abbreviation Description

qtm _{Qualysis Track Manager} mbs _{Multi-Body Systems} fem _{Finite Element Method} dof _{Degree of Freedom}

r_- _Revolute

(11)

1

Introduction

Reach trucks are one type of forklifts used for moving and lifting objects. Equipped with a mast to which a set of forks are attached, they are able to place and retrieve objects at different heights. The mast is however subjected to oscillations, which have significant impacts on the dynamics of the entire vehicle and could cause undesirable outcomes in extreme cases. The characteristics of the oscillations are dependent on a number of factors, such as fork heights, load weights, etc. There are a number of scenarios during which mast sway is generated, e.g. during load lifting, reach motions, deceleration, etc.

It is necessary to predict mast sway and its impact on forklifts in order to prevent undesirable outcomes. Simulations of mast sway is helpful in this matter. This requires a sufficiently detailed model of the mast. A thesis work is conducted in collaboration with a forklift manufacturer in Mjölby in an attempt to develop a model for simulations of mast sway.

1.1 Problem Description

It is known that masts undergo oscillations during numerous situations, such as during change of fork height, reach motions, deceleration, etc. Since mast oscil-lations affect the dynamics of the entire reach truck, it is necessary to simulate mast sway in these situations. Presently, simulations can be done with help of existing software tools to predict the behaviour of the forklift.

The forklift manufacturer is currently using a model of one specific mast which a number of reach trucks are equipped with. The model has been vali-dated successfully with loads at the highest fork height. It is however not certain whether the model is sufficiently accurate for different loads on different fork heights. The forklift manufacturer wishes to obtain a model which can represent all possible mast configurations.

(12)

2 1 Introduction

Table 1.1:A table showing available lift heights for different mast classes. Mast Class 1.6t 2.0t 2.0t HD 2.5t Mast height [mm] 4400 x 4600 x 4800 x x x 5400 x x x 5700 x x 6300 x x 6750 x 7000 x x 7150 x 7500 x 8000 x x 8500 x x 9000 x x 9500 x x 10000 x x x 10500 x x x 10800 x 11000 x 11500 x 12000 x 12500 x

The forklift manufacturer provides a wide range of reach trucks in order to comply with the demands from their customers. There are an extensive amount of factors that can vary between trucks such as weight, chassis, mast dimensions, batteries, and many more. Masts are provided with many variations in load ca-pacity, mast dimension, and mast type.

There are three types of masts, namely Duplex Tele, Triplex Tele and Triplex HiLo. Duplex masts are composed of two guides while the Triplex masts are composed of three guides. Tele and Hilo types differ in the manner which the forks are raised and lowered. Each mast type also comes with a load capacity. For instance, Triplex masts come with a load capacity range of 1.6, 2.0 and 2.5 tons. In Table 1.1, mast heights for different Triplex masts are shown. Variations in mast class and mast height make up numerous combinations. It is therefore desirable to obtain one model that can represent all mast configurations.

1.2 Objective and Scope

The objective of this thesis is to develop a model in SimXpert Motion for simula-tions of mast sway. In addition, an attempt to determine dependencies between

(13)

1.3 Resources 3

(a) A reach truck seen from above.

(b) A reach truck seen from the right.

Figure 1.1:Reach truck seen from two different angles.

model parameters and variables such as fork height, load mass and others will be done, assuming that dependencies exist. A method to determine model param-eters that yield compliance with sway measurement data should be used. The mast will be modelled as a rigid multibody system (MBS) with flexibility in the connections between different parts of the mast. Remaining parts of the reach truck will not be considered in the model except the reach carriage which will be set to ground. Clearance in the connections between mast parts should be consid-ered in the modelling procedure. Oscillations in other directions than the global x-direction, according to Figure 1.1, are neglected.

Validation will be done with sway measurements solely with 1.6t masts since resources and time for measurements with all mast classes are limited. Measure-ments will be conducted in a manner that isolates the mast sway from the rest of the forklift to the highest possible extent. Since the situations in which the mast undergoes oscillations generally have a duration of only a few seconds, it is mostly relevant for simulations to comply with the initial seconds of the oscilla-tions.

1.3 Resources

A brief description of the main tools used in this thesis are presented below.

1.3.1 SimXpert

SimXpert is an integrated multidisciplinary simulation solution provided by MSC Software [1]. It enables users to work in a unified environment where analysts

(14)

4 1 Introduction

and designers can share critical information without duplicating work. Engineers can choose from various workspaces like Structures, Explicit, Motion, and Sys-tems and Controls, which provide the tools required to complete all stages of the simulation process. Features such as import of CAD models and possibility to view, manipulate and organize models graphically are also provided among oth-ers. SimXpert Motion is used for modelling and simulation of mast sway in this thesis work.

1.3.2 Qualisys Track Manager

Qualisys Track Manager (QTM) [2] is a motion capture software for tracking movements. The system allows users to perform 6DOF capture of data in real-time using cameras and markers. A few of the main applications of QTM are: Sport science, industrial, marine and underwater, among others. QTM is used for data acquisition in mast sway tests in this thesis work.

(a) Retracted state with forks at their lowest height.

(b) Retracted state with forks raised from lowest height.

(c) Retracted state with forks at free lift height.

(d) Extended state.

Figure 1.2:Illustration of a mast in four different states which is equivalent to forks at four different heights.

(15)

1.4 Brief Description of the Mast 5

1.4 Brief Description of the Mast

The masts modelled in this thesis are 11-metres Triplex masts belonging to the mast class 1.6t, which are masts with a load capacity of 1600 kg. The masts mod-elled in this thesis are not yet available on the market. There is an extensive vari-ation of masts. The different available masts are presented in Table 1.1. They are composed of three main parts: a mast frame, a middle guide and an inner guide. All main parts are connected to their neighbouring part via rollers. Rollers con-stitute connections between the mast frame and the middle guide, and between the middle guide and the inner guide. Connected to the inner guide is the fork carriage, which the forks are attached to. The mast frame is attached to the reach carriage, which is connected to the vehicle. The reach carriage can be pushed away from the vehicle causing the mast and forks to move forward. This motion is called a reach motion and it is a common feature for all reach trucks.

Figure 1.2a depicts a mast where the forks are at their lowest height. The fork carriage is able to translate vertically along the inner guide. The fork height is controlled by a hydraulic piston that pushes a pulley to which a chain is attached when the mast is in its retracted state. One end of the chain is attached to the fork carriage, and the other end is attached to the cylinder of the piston. Figure 1.3a illustrates the mechanism that raises and lowers the forks along the inner guide. Figure 1.2 illustrates the different states of the mast during raising of the forks.

(a) Illustration of the mechanism that raises and lowers the forks along the inner guide.

(b) Illustration of the mecha-nism that raises the inner guide with twice the speed of that of the middle guide during mast extension.

Figure 1.3:Mechanisms that raise and lower the forks.

To raise the forks above the free lift height, the mast extension phase is en-tered. Mast extension cannot be initiated unless the forks are at their free lift height. Moreover, the forks can neither be lowered relative to the inner guide when the mast is in its extended state. This is a feature of Triplex HiLo masts. The middle guide is pushed upwards during mast extension by two hydraulic pistons. The mast extension mechanism causes the inner guide to translate along the middle guide with twice the speed of the middle guide. Figure 1.3b illustrates this mechanism.

(16)

6 1 Introduction

1.5 Brief Description of the Current Model

The model which is presently employed by the forklift manufacturer is developed in SimXpert by Fredin [3]. This section serves to give a brief overview over the modelling approach used in developing this model.

The mast is modelled as a rigid MBS with flexibility at the connections be-tween the bodies. The flexibility is attained by the use of bushings. According to Fredin [3], the mast model can be divided into three main areas:

1. The connection between the mast frame and the reach carriage 2. The connection between the three main mast parts (rollers)

3. The connection between the fork carriage and the inner guide (rollers) The location of each area is shown in Figure 1.4a. There are four bushings at location number one, eight bushings at location number two, and four bushings at location number three. Each bushing constitutes a set of 12 parameters which can be seen in Table 1.2. This model constitutes of 192 parameters in total.

(a)Bushing locations (b)Schematic figure that illustrates how the connection between roller and mast part is modelled.

Figure 1.4: The left figure points out the locations where the bushings are placed. The right figure illustrates how the connections between mast parts are modelled.

The connections are modelled with bushings and point/curve contacts. Each roller is connected with a mast part via bushing and a point/curve-contact, ac-cording to Figure 1.4b. The figure shows the mast from the side where only two rollers are seen. In reality there are two additional rollers on the opposite side

(17)

1.6 Outline 7 Table 1.2:A table showing bushing properties.

Translational Rotational # of parameters

Stiffness x y z x y z 6

Damping x y z x y z 6

=12

of each location which are modelled in the same manner. Opposite rollers have identical bushing properties. Furthermore, several of the bushing properties are trivial and set to a fixed value. In practice, the model constitutes of less than a half of the number of parameters previously mentioned. Regardless, tuning of model parameters is cumbersome.

1.6 Outline

This section discloses the outline of this report. The contents of each chapter is summarised below:

• Chapter 1 - Introduction

A description about the objective and scope of the thesis is given together with a brief presentation about the mast and the model that is presently employed by the forklift manufacturer.

• Chapter 2 - Literature Review

This chapter reveals the findings of the literature study conducted prior to the thesis work.

• Chapter 3 - Modelling Approach

An account of the procedure for obtaining the model is given in this chapter. • Chapter 4 - Modelling and Simulation with SimXpert

A description of the obtained model is presented. • Chapter 5 - Sway Measurements

A description of the test procedure is presented together with pieces of test results.

• Chapter 6 - Parameter Analysis

This chapter gives an account of the parameter analysis conducted and also presents the results.

• Chapter 7 - Parametric Optimisation

The procedure and results of the parametric optimisation conducted to find the optimal parameters are presented.

• Chapter 8 - Discussion

This chapter gives a discussion regarding the content of thesis. Results and conclusions are also presented.

(18)

(19)

2

Literature Review

Prior to the thesis, a pre-study was made to ensure that the author of this thesis possesses sufficient information regarding the work in order to perform it in an appropriate manner. This chapter presents previous work on simulation of mast sway. Furthermore, a review on topics that are related to this thesis is presented.

2.1 Previous Work

A thesis work which included modelling of forklift masts was conducted in the past by Norrby [4]. With the intention of finding a way to reduce oscillations in the mast, a model of the forklift was created in order to simulate the dynamics. The model consisted of a carriage, a mast, a load and a control system. Norrby used a rigid-body modelling approach where each deformable part of the mast was divided into several rigid-body parts, each with a revolute joint. This ap-proach of modelling appeared to be successful after validation with experimental results.

Inspired by Norrby [4], a model of the mast intended to simulate mast sway was developed by Fredin [3]. As opposed to Norrby’s approach, the deformable beams were not divided into smaller rigid parts. These bodies were modelled as rigid-bodies with bushings at the connections instead. The bushings were placed between the three mast parts, between the mast frame and the reach carriage, and between the inner guide and the fork carriage, which constitute 16 connections. Each bushing had six stiffness parameters and six damping parameters, which results in a total of 192 parameters. This model has been successfully validated with a number of tests despite the high number of parameters.

Norrby claimed in his thesis [4] that oscillations in the mast are generated by accelerations from the forklift. Another work on simulation of forklifts [5], con-ducted by Doci and Imeri, indicated that the lifting mast undergoes the highest

(20)

10 2 Literature Review

oscillations and amplitudes compared to any other part of the vehicle. Doci and Imeri concluded this after investigating stresses on different parts of a forklift.

2.2 Related Research

Research has been done on modelling of deformable bodies and has resulted in a number of methods. One legitimate method of modelling the elasticity of a flexible beam is to use a classical Finite Element Methods (FEM) approach [6]. This approach is however unavoidable costly in terms of computational time [7] and adaptations of the original FEM formulations have been encouraged in order to attain better performances. One alternative to FEM is to adopt the assumed modes method, which can be done using so called shape functions [7]. Shabana used shape functions and shape vectors to describe elastic deformation of non-linear inertia variant MBS in [8]. Hale and Meirovitch used similar methods for dynamic analysis of complex linear structures in [9]. Valembois, Fisette and Samin investigated several discretization techniques of flexible beams, in a pure multibody context and showed that shape functions is a good alternative to finite element analysis in [10].

Alternative methods for dealing with deformable beams are methods based on rigid body systems. The finite segment approach deals with a deformable beam by dividing it into several finite segments connected to one another via springs and dampers [11] [12]. The incorporated stiffness and damping between the segments accommodate the movement of the beam. Norrby used a similar method for modelling elastic bodies in his thesis [4], dividing bodies into several equally long bodies each with a revolute joint.

In ideal multibody dynamics systems, joints are modelled without friction and clearance. In reality, however, this is not the case. Seneviratne, Earles, and Fenner claim in [13] that joints can suffer from contact loss which can lead to noise generation and joint deterioration. In order to predict the behaviour of clearance joints, researchers believe that detailed modelling is required. Since joint backlash implies loss of contact, it is not unreasonable to believe that this phenomenon also can have an overall impact on the system. Researchers have found a number of methods on modelling joint clearance. Makkonen’s literature review in his doctoral thesis [14] shows that there are at least four approaches on how to model joint clearance :

1. Zero clearance approach 2. Massless link approach 3. Spring damper approach 4. Impact model approach

The zero clearance approach considers the connecting points of two bodies linked by a joint are coincident. In the massless link approach, the presence of clear-ance is modelled as a massless link with constant length connecting the bodies

(21)

2.3 Other Sources 11

together. This adds an additional degree of freedom to the system. In the spring damper approach, the clearance is modelled with an element with stiffness and damping properties connecting the bodies. This model simulates the surface’s elasticity. These methods are mentioned and briefly described in [13], [15] and [16]. Makkonen presents in [15] a method for MBS simulation of planar joint clearance using the impact model approach for radially compliant joints . This model is established by using a vector force element for the contact’s internal support force. The joint contact points are uncoupled and the impact force is ac-tivated when contact is established between the two elastic bodies where the force is a function of the penetration depth. Makkonen claims that the impact model approach is the only method of the four mentioned above, which can simulate loss of contact with zero force.

2.3 Other Sources

Other sources have been used in conducting this thesis. Basic information re-garding rigid MBS is found in [17], [6], and [18]. Relevant theory concerning oscillations characteristics are found in [19] and [20]. Theory regarding elastic material is obtained from [21].

(22)

(23)

3

Modelling Approach

This chapter provides a description of the approach used for obtaining a model of the mast. A method for dealing with mast elasticity is first elaborated upon followed by an account of the method for representing the clearance between roller and mast. Lastly, an account of the approach for modelling mast extension is given.

3.1 Mast Elasticity

Forklift masts deform during mast sway and there are several methods to deal with elasticity of beams. A few methods are mentioned in chapter 2. The method for dealing with body deformation adopted in this thesis is to consider the entire system as a rigid MBS with the flexibility placed at the connection points between bodies using springs and dampers.

Triplex masts are comprised of three main parts, namely a mast frame, a mid-dle guide and an inner guide. In an MBS, these parts represent separate bodies. The connection between each body can be represented by joints. Using revolute (R-)joints at the connections would allow the possibility to generate mast sway by application of torsion springs and dampers about the joints. The torsion springs and dampers exert a torque which is dependent on the angle and angular velocity between one body relative to its neighbouring body which it is connected to via the joint. The torque is exerted according to:

τ(θ, ˙θ) = −κθ − c ˙θ (3.1)

where τ represents exerted torque, θ angular displacement, and ˙θ angular

veloc-ity. The coefficients κ and c represent the torsion coefficient and the rotational friction, respectively. These coefficients must be tuned so that the model com-plies with sway measurements. Figure 3.1 illustrates the modelling approach for

(24)

14 3 Modelling Approach

(a) Connection between two mast parts.

(b) Connection between the reach carriage and the mast frame.

Figure 3.1:Schematic figures illustrating how the connections between bod-ies are modelled. The curved arrows represent torques exerted by torsion springs and dampers.

the connection points between bodies. Each roller depicted in Figure 3.1a is con-sidered as one pair of rollers. The joint shown in the same figure is modelled as a revolute-prismatic (RP-)joint and is placed in the centre between the two roller pairs. The distance between the rollers depicted in Figure 3.1a vary depending on the fork height caused by mast extension. The RP-joint possesses two degrees of freedom and apart from serving as a pivot point, it is permitted to translate along mast part 1 so that it remains at the centre of the rollers regardless of the fork height. The joint in Figure 3.1b on the other hand serves only as an R-joint with one degree of freedom since the connection point is fixed. In reality, the mast frame is attached to the carriage at four points. The R-joint is therefore placed in the centre between these points.

3.2 Clearance Between Roller and Mast Part

The thesis objective compels the inclusion of clearance in the model. The clear-ance that is present between rollers and mast occasions mast sway to appear dif-ferently compared to the ideal case, i.e. the case of no clearance. A depiction of the concept of clearance applied to the situation between roller and mast is shown in Figure 3.2. Each roller is constrained within a track along the mast. The model includes the clearance as an angle that is dependent on the clearance and the roller overlap. The angle is calculated according to:

θc= arctan(

α

(25)

3.2 Clearance Between Roller and Mast Part 15

Figure 3.2:Illustration of clearance between roller and mast part. α repre-sents the clearance and d the roller overlap.

where θcrepresents the clearance expressed as an angle, α the actual clearance, and d the roller overlap. The angle, θc, makes up a boundary between two re-gions depicted in Figure 3.3. The mast part rotates freely as long as its angle of rotation is less than the critical angle, θc. A torque is applied about the joint when the angle of rotation exceeds the critical angle, θc. The torque is applied according to: τ(θ, ˙θ) =        0, |_{θ| ≤ θ}_c −_{κθ − c ˙}_θ, |_{θ| > θ}_c (3.3) The roller overlap is dependent on the fork height according to:

Figure 3.3: The sector depicts the region in which the mast part is moving freely.

(26)

16 3 Modelling Approach d =        dmax, h ≤ hf dmin+ hmax −_h 2 , h > hf (3.4) where d is the roller overlap, h the fork height, and hf the maximal fork height in the mast’s retracted state.

3.3 Free Lift and Mast Extension

The model should be able to simulate sway at different fork height. This can be attained by allowing the model the possibility of raising and lowering the forks. A connection between the inner guide and the fork carriage that allows the forks to translate along the inner guide is defined. The chain forces keep the forks connected to the inner guide. The chain forces are defined to act on the forks and are calculated according to Hooke’s law for elastic material:

F = (EA0 L0

)∆L (3.5)

where F is the force exerted by the chain, E is the Young’s modulus, A0 is the

chain’s cross-sectional area, L0 is the initial length of the chain, and ∆L is the

difference between the actual length and the initial length. The position of the forks along the inner guide depends on the stroke of the cylinder illustrated in Figure 1.3a. This stroke will be referred to as the free lift stroke. The fork height is a function of the free lift stroke according to

h = 2xf + hmin (3.6)

in the mast’s retracted state, where h is the fork height, and xf is the free lift stroke. Maximal free lift stroke is attained when the free lift fork height is reached. Equation 3.6 is only valid in the mast’s retracted state. The forks height in the extended state is a function of the stroke of the pistons illustrated in Fig-ure 1.3b. This stroke will be referred to as the the main stroke. The fork height is related to the main stroke according to:

h = 2xm+ hf (3.7)

in the mast’s extended state. In equation 3.7, xm denotes the main stroke. This equation is only valid if xf equals xf ,max. This is essential in order for the model to comply with the description of Triplex HiLo masts described in chapter 1.

The model must permit translation between mast parts in order for mast ex-tension to be possible. Section 3.1 reveals that mast parts are connected with RP-joints which allow rotation about the connection point and translation along a surface. SimXpert does not provide the user with an RP-joint directly. It can nevertheless be accomplished using two dummy parts as shown in Figure 3.4. In order to prevent the dummy parts from falling without restrictions due to gravity during simulations, the translational joint between the mast frame and dummy

(27)

3.3 Free Lift and Mast Extension 17

Figure 3.4:This is a schematic figure showing the manner of modelling the connection point between two mast parts using a revolute-prismatic joint.

part is defined as a motion driver. In addition, the other translational joints are coupled to the motion driver.

The connection between the fork carriage and inner guide is actually mod-elled as an RP-joint. However, it is not accomplished in the same manner as previously mentioned. In this case the joint is modelled as shown in Figure 3.5. A torque is applied to the R-joint as described in section 3.1.

(28)

(29)

4

Modelling and Simulation with

SimXpert

This chapter presents a detailed description of how the obtained model is repre-sented in SimXpert.

4.1 Geometry and Components

The model is comprised of a number of geometric models provided by the forklift manufacturer. They are imported to SimXpert and placed so that the initial state of the model appears as in Figure 4.1. Each such entity represents a geometry of a specific part. The main parts are disclosed in Table 4.1 together with their properties.

Each component has a source of inertia which is user specified. All compo-nents possess moments of inertia computed by SimXpert based on their geome-tries. The mass, density and centre of mass of each main part are specified accord-ing to Table 4.1. Masses are computed by SimXpert based on geometries given the density of each part. The following code snippet sets the mass of the mast frame given its density:

# Get part partMastFrame = MotionUtl.Modelmanager.findPartByName(’Mast Frame’) # Update part partMassFrame.setMassPropertiesFromGeometry(1) partMassFrame.setMaterialDensity(density) partMassFrame.setMassPropertiesFromGeometry(0)

Remaining components are presented in Table 4.2. Their masses are set to a small value in order to avoid division by zero. The centres of mass presented in Table 4.1 and Table 4.2 are initial locations given in global coordinates. Each part

(30)

20 4 Modelling and Simulation with SimXpert

(31)

4.1 Geometry and Components 21 Table 4.1: Properties of main components. CM denotes the centre of mass and its values are presented in millimetres.

Mass[kg] ρ [kg/mm3] CMx CMy CMz Reach Carriage 335.0 1.343 · 10−5 708 9 333 Mast Frame 551.8 8.273 · 10−6 873 0 2384 Middle Guide 413.9 8.676 · 10−₆ 893 1 2186 Inner Guide 354.4 8.156 · 10−6 891 1 2052 Forks 232.3 8.202 · 10−6 1172 2 250

Table 4.2: Properties of other components. CM denotes the centre of mass and its values are presented in millimetres.

CMx CMy CMz

Load 1729 0 670

Initial Lift Piston 825 113 3227 L Roller Mast Frame 891 282 4407 R Roller Mast Frame 891 −₂₈₂ ₄₄₀₇ Lower L Roller Mid. G 917 284 116 Lower R Roller Mid. G 917 −₂₈₄ ₁₁₆ Upper L Roller Mid. G 917 231 4407 Upper R Roller Mid. G 917 −₂₃₁ ₄₄₀₇

L Roller Inn. G 891 232 116

R Roller Inn. G 891 −₂₃₂ ₁₁₆

Dummy Mast Frame 896 0 2262

Dummy Mid. G 1 896 0 2262

Dummy Mid. G 2 888 −₃ ₂₂₆₁

Dummy Inn. G 888 −₃ ₂₂₆₁

Dummy Fork Carriage 0 0 0

has a local coordinate system that possesses the same orientation as the global coordinate system initially. Dummy parts have been introduced to this model to represent RP-joints. More about RP-joints are presented in sections below.

(32)

4.2 Connectors

There is a number of connectors defined preventing the model from falling apart during simulation due to gravity. The model includes a gravity of 9800 mm/s2in the negative global z-direction. In order to keep the model stationary in space, the reach carriage is set to ground. Other components are connected to the car-riage directly or indirectly in a number of ways.

The mast frame is the only part that is directly connected to the reach carriage. Section 3.1 describes the principle for modelling the connection between these parts. The two separate bodies are connected with an R-joint that is positioned in between the real connection points. Figure 4.2 shows where the connection points are situated in reality. There are two more connection points on the oppo-site side of the ones shown in the figure. The R-joint allows rotation around its y-axis which coincides with the direction of the global y-axis.

Figure 4.2: The arrows disclose the positions of the connection points be-tween the carriage and mast frame in reality. There are two more points on the opposite side that are not visible in the figure.

The mast frame is also connected to the middle guide with an RP-joint. The joint is initially placed in the centre between the rollers on the mast frame and the two bottom rollers of the middle guide. Chapter 3 describes the principle of modelling this joint. It is accomplished in SimXpert by introducing two dummy parts, that are placed at the location of the RP-joint. This location constitute an R-joint that connects the two dummy parts. Each dummy part is connected to the mast frame and middle guide, respectively, allowing translation between dummy part and mast part. The middle guide is connected to the inner guide in the same manner. The connectors defined in the model are presented in Table 4.3.

(33)

4.2 Connectors 23

Table 4.3: List of defined connectors. Connector Type: Fixed Joint

Part 1 Part 2 Location[mm]

Mast Frame L Roller Mast Frame [891.1, 279.9, 4407.0] Mast Frame R Roller Mast Frame [891.1, -279.9, 4407.0] Middle Guide Lower L Roller [917.2, 285.9, 116.0] Middle Guide Lower R Roller [917.2, -286.4, 116.0] Middle Guide Upper L Roller [917.2, 228.6, 4407.0] Middle Guide Upper R Roller [917.2, -229.1, 4407.0] Inner Guide L Roller Inn. G [891.0, 234.1, 116.0] Inner Guide R Roller Inn. G [891.0, -234.1, 116.0] Connector Type: Translational Joint

Dummy Mast Frame Mast Frame [896.2, -0.5, 2261.5] Middle Guide Dummy Mid. G 1 [896.2, -0.5, 2261.5] Dummy Mid. G 2 Middle Guide [887.9, -3.2, 2261.2]

Inner Guide Dummy Inn G [887.9, -3.2, 2261.3]

Dummy Fork Carriage Inner Guide [916.5, 0.2, 351.5] Init. Lift Piston Inner Guide [823.9, 113.5, 2362.0] Connector Type: Revolute Joint

Mast Frame Reach Carriage [926.0, -0.0, 440.0] Dummy Mid. G 1 Dummy Mast Frame [896.2, -0.5, 2261.5] Dummy Inn. G Dummy Mid. G 2 [887.9, -3.2, 2261.3] Forks Dummy Fork Carriage [916.5, 0.2, 351.5]

(34)

4.3 Forces

The chains are modelled as forces as described in section 3.3. Table 4.4 shows the defined force entities. The force entities shown in the table represent the forces exerted by the mechanisms described in section 1.4 for raising and lowering forks as well as extending and retracting masts. The locations indicate the forces acting points. A general force is also used for pulling the load in order to simulate tests described in chapter 5.

Table 4.4:List of forces Applied Forces

Free Lift

Part 1 Location 1 Part 2 Location 2

Inner Guide [784.5, 152.3, 2377.0] Init Lift Piston [791.5, 145.9, 2282.0] Init Lift Piston [856.3, 81.1, 2282.0] Inner Guide [866.8, 70.0, 4352.0] Inner Guide [931.6, 5.1, 4352.0] Forks [936.5, 0.2, 395.5] Mast Extension

Mast Frame [767.6, 294.2, 4211.0] Middle Guide [777.4, 278.5, 4326.0] Middle Guide [823.9, 194.5, 4326.0] Inner Guide [828.9, 186.5, 369.0] Mast Frame [767.6, -294.2, 4211.0] Middle Guide [777.4, -278.6, 4326.0] Middle Guide [823.9, -194.6, 4326.0] Inner Guide [828.9, -186.5, 369.0] Other Forces

Load [1728.8, 0.2, 670.3] Ground

-The forces exerted by the free lift chain is computed according to Equation 3.5. This is expressed in SimXpert as

2e6*10*MAX(0,1-6122.0269/MAX(1,VARVAL(.RR900B. Total_ChainLength_Initial)))

and describes all Free Lift forces in Table 4.4. The variableTotal_ChainLength_Initial

sums the measured distances between the locations that defines these force enti-ties. Figures 4.3 - 4.5 display how the chains are represented in SimXpert.

Figure 4.6 shows the chain forces involved in the mast extension mechanism. There is one chain on each side of the mast as can be seen in the figure. The forces exerted by these chains are also computed according to Equation 3.5. They are expressed in SimXpert as follows:

2e6*10*MAX(0,1-4073.4932/MAX(1,VARVAL(.RR900B. Total_ChainLength_Main)))

The variableTotal_ChainLength_Main sums the measured distances between the

(35)

4.3 Forces 25

Figure 4.3:A display of the exerted chain forces acting on the piston.

Figure 4.4:A display of the exerted chain forces acting on the pulley which is attached to the mast frame.

(36)

Figure 4.5: A display of the exerted chain forces from the free lift mecha-nism.

Figure 4.6: A display of the exerted chain forces from the mast extension mechanism.

(37)

4.4 Torques 27

4.4 Torques

Four torque elements are defined in the model. Each R-joint is subjected to a torque. Chapter 3 accounts for the torques applied about the joints. Each torque element is expressed with the following function:

BI ST OP (x, ˙x, x1, x2, k, e, cmax, χ)

TheBISTOP-function is a built-in function in SimXpert and is used for modelling

gap elements. Table 4.5 presents the arguments. This function can be mathemat-Table 4.5:Arguments of the BISTOP-funtion

Argument Description

x Displacement variable. ˙x Time derivation of x.

x1 The lower bound of x.

x2 The upper bound of x.

k A non-negative value that specifies the stiffness of the boundary

surface interaction.

e A positive value that specifies the exponent of the force deforma-tion characteristic. For a stiffening spring characteristic, e >1.0. For a softening spring characteristic, 0 <e <1.0.

cmax A non-negative variable that specifies the maximum damping co-efficient.

χ A positive real variable that specifies the penetration at which the full damping coefficient is applied.

ically expressed as follows:

BI ST OP =            MAX(k · (x1−x)e−ST EP (x, x1−χ, cmax, x1, 0)) · ˙x, x < x1 0, x1≤x ≤ x2 MI N (−k · (x − x2)e−ST EP (x, x2, 0, x2+ χ, cmax) · ˙x), x > x2 (4.1) TheSTEP-function is also a built-in function in SimXpert and is used for

accom-plishing a Heaviside-step. See appendix A for details about theSTEP-function

provided in SimXpert. In the case of computing a torque according to the con-tents of chapter 3, theBISTOP-function appears as follows:

−_{BI ST OP (θ, ˙}_{θ, −θ}_c_{, θ}_c_{, κ, 1, c, 1d)}

A penetration depth of one degree is assumed in order to not cause discontinuity. The exponent of the force deformation characteristics is assumed to be one. The torsion coefficient, κ, and rotational friction, c, serve as model parameters. Table 4.3 shows that there are four R-joints, which indicates four torque entities and eight model parameters.

(38)

4.5 Applied Motions and Couplers

Motion drivers are accomplished by introducing applied motions. The user must be able to raise and lower the forks as desired during simulations. In order to allow the user to translate the forks relative the inner guide, a translational joint motion is applied on the translational joint between initial lift piston and the inner guide. Using Equation 3.6, desired fork height relative the inner guide is obtained.

Mast extension is accomplished by an applied motion on the translational joint between the mast frame and its connecting dummy part. In order to re-strict the R-joints between mast parts to remain in intended positions described in chapter 3, couplers are used. The couplers defined in this model are presented in Table 4.6.

Four rotational motions are defined. They allow the user to constrain the R-joints. It is convenient to be able to keep the mast still in some simulation cases. The purpose of these motions are discussed further below.

Table 4.6:List of couplers

Coupler Part 1 Part 2

1 Joint 1 Dummy Mast Frame Mast Frame

Joint 2 Middle Guide Dummy Mid. G 1

Joint 2 Dummy Mid. G 2 Middle Guide

Joint 2 Inner Guide Dummy Inn. G

4.6 Simulation

Simulations can be run directly in SimXpert Motion. This section describes the steps to simulate the tests described in chapter 5.

A scripted simulation is created to simulate mast sway according to chapter 5. The following script is used:

SIMULATE/STATIC

SIMULATE/DYNAMIC, DURATION=7, DTOUT=0.016666667 DEACTIVATE/MOTION, ID=898

DEACTIVATE/MOTION, ID=916 DEACTIVATE/MOTION, ID=924 DEACTIVATE/MOTION, ID=938 SIMULATE/STATIC

SIMULATE/DYNAMIC, DURATION=4, DTOUT=0.016666667 SIMULATE/STATIC

(39)

4.6 Simulation 29

DEACTIVATE/GFORCE, ID=7

SIMULATE/DYNAMIC, DURATION=10, DTOUT=0.016666667

The script initially brings the system to equilibrium before dynamic simulations starts. Dynamic simulations occur with a frequency of 60 H z in order to match the sample frequency from conducted tests. See section 5.2 for test specifications. After seven seconds of simulation, four motion constraints are deactivated in or-der to generate initial deflection. The first seven seconds simulate the raising of the forks to desired height. A static simulation is performed directly after deac-tivation of motion constraints in order to attain equilibrium. Once equilibrium is attained, the simulation proceeds where the load is pulled with a force. Equi-librium is attained once more before the mast is released. This is achieved by deactivating the force. Subsequently, the mast is allowed to sway for ten seconds before the simulation ends.

The simulation script above devotes seven seconds for attaining desired lift height. However, this process could have a different duration depending on how the motion is defined. Two motions are involved in this process, namely the mo-tion that raises the forks relative the inner guide, and the momo-tion that occasions mast extension. To comply with the behaviour of a Triplex HiLo mast, the forks are raised before mast extension initiates. The motion that raises the forks is expressed in SimXpert as follows:

STEP5(time,0.5,0,3,VARVAL(.RR900B.IntitialCylStroke)) This function occasions the model to perform a full piston stroke to attain free lift height. The stroke initiates after 0.5 seconds of simulation and completes after 3 seconds of simulation. See appendix A for description of theSTEP5-function.

The motion that occasions mast extension is expressed in SimXpert as follows: STEP5(time,3,0,7,VARVAL(.RR900B.MainCylStroke)/2)

The motion performs half the main stroke, since it is applied to the translational joint between the mast frame and its connected dummy part. The dummy part travels with half the speed of the middle guide during mast extension in order to achieve the behaviour described in chapter 3. The couplers in Table 4.6 ensure that mast extension occurs as intended.

The motion constraints that are deactivated during simulation are rotational motion constraints applied on the revolute joints. They are defined to constrain the rotations caused by external forces. Deactivation of these constraints must happen in order to simulate mast sway as described in section 3.1. The force that pulls the load are expressed in SimXpert as follows:

STEP5(time,7.5,0,10,VARVAL(.RR900B.pullingForce))

The sway starts once the force is deactivated. The script shown above simulates ten seconds of mast sway. The duration of mast sway simulation can be adjusted if so is desired.

(40)

(41)

5

Sway Measurements

This chapter provides an account for the measurement procedure for acquiring data. Tests and data collection are performed in the test facilities of the forklift manufacturer.

5.1 Set-Up

Sway tests are conducted in the laboratory provided by the forklift manufacturer. QTM is used for acquiring measurement data and requires a number of IR cam-eras installed so that the test object can be seen during the entire measurement process. At the time this thesis is written, there are eleven IR cameras installed. The cameras are all connected to a computer where QTM is installed. Calibration must be done and a global coordinate system must be determined prior to first time motion capturing.

Prior to measurements, the test object is placed on the platform where sway measurements are performed. The test object constitutes a reach truck in this the-sis. The coordinate system must be calibrated so that the x-direction corresponds to the coordinate system in Figure 1.1a. The global coordinate can be adjusted by setting its pitch and yaw with QTM.

To ensure that the test object is stationary during the measurement process, wedges are placed between the chassis and the platform on which the test object is standing to lift the wheels off the ground. Furthermore, the reach carriage is wedged to the chassis in an attempt to isolate the mast from the rest of the vehicle.

Markers are placed at the locations on the test object which are wished to be tracked. In the case of this thesis, one marker is attached to the top of the mast frame, the middle guide and the inner guide. It must be ensured that each marker is visible for at least two cameras during the entire measurement process in order

(42)

32 5 Sway Measurements

for the QTM system to be able to determine the position of the marker at each sample.

5.2 Test Specifications

Tests are conducted on test objects with the specifications shown in Table 5.1. Table 5.1:Test specifications.

Sway Test 1

Test Object: RRE

Measurements Sample Frequency: 60 H z Pull Force: 1000 N Height [m] Load [kg] 7 9.5 11 0 x x x 500 x x x 1000 x x x

5.3 Test Procedure

Tests are executed according to the following steps: 1. Load the forks with a pallet/pallets of desired mass. 2. Fasten the pallet/pallets to the forks.

3. Raise the forks to desired height and wait for equilibrium.

4. Adjust the coordinate system of the markers so that the position at equilib-rium becomes the reference point.

5. Start recording with QTM.

6. Pull the pallet/pallets with desired force and wait for equilibrium. 7. Release the force.

8. Stop recording when oscillations stop.

If tests are conducted with zero load mass, the forks are not loaded. The fork carriage is pulled in step 6.

(43)

5.4 Results 33

5.4 Results

Below are the measurement results for a few measurements. Data have been processed so that oscillations occur about the x-axis. The x-axis represents the time and the y-axis represents the deflection. Figure 5.1 presents measured os-cillations of a mast at 11 metres fork height with different loads. The diagrams presented below do not reveal the axes for reasons of secrecy. Figure 5.2 shows

Figure 5.1: Measurement data from measurements with 11 metres fork height. Upper diagram shows measurements with 1000kg. Bottom diagram shows measurements with 500kg.

the deflection of the inner guide at 11 metres fork height with different loads in the same diagram. The oscillations appear differently for different load cases.

(44)

34 5 Sway Measurements

Figure 5.2:Comparison of sway with different loads at 11 metres fork height.

Figure 5.3 shows the deflection of the inner guide at different fork heights with a load of 1000 kg. Deflection and frequency vary with different heights.

Figure 5.3: Comparison of sway with a load of 1000kg at different fork height.

(45)

6

Parameter Analysis

An analysis is conducted to find how each parameter affects the outcome of mast sway. This chapter describes the procedure and presents the results of the analy-sis. The results are used for deciding sets of parameters for parametric optimisa-tion.

6.1 Procedure

The procedure of the analysis is presented below:

1. Find a combination of parameters that generates mast sway. 2. Decide which parameter to test.

3. Run several simulations with the parameters found in step 1 and vary the parameter chosen in step 2 for each simulation.

4. Plot the simulation results.

5. Go to step 2 if there are more parameters to test.

6.2 Tests

Parameter analysis was conducted with simulations of mast sway with a fork height of 11 metres and a load of 1000 kg. The parameter configuration found in step one is shown in Table 6.1. Table 6.3 presents test simulations for each param-eter except κ4 and c4 since they were not tested. The index number represents

the joint number seen in Table 6.2.

(46)

36 6 Parameter Analysis

Table 6.1: Parameter configuration obtained in step one.

κ1 c1 κ2 c2 κ3 c3 κ4 c4

1 · 109 _{1 · 10}8 _{1 · 10}9 _{1 · 10}8 _{1 · 10}9 _{1 · 10}8 _{5 · 10}8 ₁

Table 6.2: List of defined connectors. Connector Type: Revolute Joint

Joint Nr. Part 1 Part 2 Location[mm]

1 Mast Frame Reach Carriage [926.0, -0.0, 440.0] 2 Dummy Mid. G 1 Dummy Mast Frame [896.2, -0.5, 2261.5] 3 Dummy Inn. G Dummy Mid. G 2 [887.9, -3.2, 2261.3] 4 Forks Dummy Fork Carriage [916.5, 0.2, 351.5]

Table 6.3:List of values tested.

Test Variable Values # of Sim.

κ1 [1 · 108] , [5 · 108, 1 · 109, 1.5 · 109, ..., 5 · 109] 11 c1 [1, 1 · 101, 1 · 102, ..., 1 · 1010] 11 κ2 [1 · 108] , [2 · 108, 4 · 108, 6 · 108, ..., 20 · 108] 11 c2 [1, 1 · 101, 1 · 102, ..., 1 · 1010] 11 κ3 [1 · 108] , [2 · 108, 4 · 108, 6 · 108, ..., 20 · 108] 11 c3 [1, 1 · 101, 1 · 102, ..., 1 · 1010] 11

6.3 Results

The plots obtained in this analysis are presented in appendix B. The x-axis and y-axis are not revealed in the diagrams for reasons of secrecy.

The results serve as a foundation for understanding how each parameter affect the outcome of simulated oscillations. The figures reveal that the stiffness param-eters affect the initial deflection of the inner guide and also oscillation frequency. The damping parameters affect the damping characteristic. A more thorough discussion is given in chapter 8.

(47)

7

Parametric Optimisation

Parametric optimisation was conducted in order to find the optimal parameters. This chapter describes the procedure. The results of the parametric optimisation are also presented below.

7.1 Procedure

It is wise to only dedicate the parametric optimisation on the scope of parameter values where the optimal parameters are likely to be found. The algorithm to find the most proper parameter space is as follows:

1. Tune the parameters manually and simulate until results fairly comply vi-sually with the initial deflection.

2. The parameters are thereafter tuned so that the resulting oscillation fre-quency fairly complies with the frefre-quency of measurement data.

3. Tune the damping parameters so that the damping characteristic is fairly compliant.

4. Define the range and dedicate the values within that range for the optimi-sation process.

These steps are followed for each mast configuration before the actual optimi-sation process begins. The chosen combinations of parameter values are used for simulations. The optimal set of parameters are determined according to the following criterion:

(48)

38 7 Parametric Optimisation J∗= argmin (κ,c)T_∈Θ N X i=0 ( ˆyi−yi)2 (7.1) κ =         κ1 κ2 κ3         , c =         c1 c2 c3         (7.2)

where, J∗is the criterion function, y is the measurement value, ˆy is the simulated

value, N is the total number of samples, and Θ is the parameter space.

7.2 Parameter Spaces

This section presents the parameter space for each mast configuration tested. See configurations in test specifications in Table 5.1 in section 5.2. The parameter spaces involved in parameter optimisation for each configuration are presented in Table 7.1 and in appendix C. The parameter space, Θ, that the parameters are optimised over can be expressed as follows:

Θ= Θ1∪ Θ2

Table 7.1:Parameter space in each run of parametric optimisation.

11m 1000kg c1= 1, c2= 1 Θ1 Θ2 κ1 [4 · 109: 5 · 108: 6.5 · 109] [6.6 · 109: 2 · 108: 7.5 · 109] κ2 [4 · 108: 2 · 108: 2 · 109] [1.6 · 109: 2 · 108: 2.4 · 109] κ3 [5 · 108: 5 · 108: 4 · 109] [1 · 108: 1 · 108: 1 · 109] c3 [2 · 108: 2 · 108: 2 · 109] [2 · 106: 2 · 106: 2 · 107] 11m 500kg c1= 1, c2= 1 Θ1 Θ2 κ1 [4.9 · 109: 1 · 108: 4.4 · 109] [4.7 · 109: 1 · 108: 5.2 · 109] κ2 [1 · 109: 1 · 108: 1.9 · 109] [1.6 · 109: 1 · 108: 2 · 109] κ3 [4 · 108: 2 · 108: 1 · 109] [6 · 108: 1 · 108: 1 · 109] c3 [1 · 107: 1 · 107: 1 · 108] [8 · 107: 1 · 107: 1.2 · 108]

7.3 Results

Below are the results from the parametric optimisation where optimal parame-ters are determined for compliance with oscillations of the mast frame. Optimal

(49)

7.3 Results 39 Table 7.2: Results from parameter optimisation for compliance with the mast frame. h= 7 9.5 Load i= 1 2 3 1 2 3 500 κi 5 · 10 9 _{5.3 · 10}9 _{8.7 · 10}9 _{4.5 · 10}9 _{2.6 · 10}9 _{3.8 · 10}9 ci 1 1 2.06 · 1011 1 1 1.8 · 1010 1000 κi 4.9 · 10 9 _{6.5 · 10}9 _{7.4 · 10}9 _{4.4 · 10}9 _{3.6 · 10}9 _{2.4 · 10}9 ci 1 1 8 · 1010 1 1 8 · 108 h= 11 Load i= 1 2 3 500 κi 5.1 · 109 1.7 · 109 8 · 108 ci 1 1 1.8 · 1010 1000 κi 4 · 10 9 _{1.8 · 10}9 _{2 · 10}9 ci 1 1 1 · 109

parameters are presented in Table 7.2. A 3D-plot based on the content of Table 7.2 is presented in Figure 7.1. A comparison between measured data and simu-lated data are shown in Figure 7.2.

(50)

40 7 Parametric Optimisation

Figure 7.1: 3D plot of each parameter as function of mass and fork height. Plotted with parameters from Table 7.2

(51)

7.3 Results 41

Figure 7.2: Comparison between measured data and simulated data. Fork height = 11 metres, Load = 1000 kg.

(52)

(53)

8

Discussion

This chapter provides a discussion about the methods adopted in this thesis. Dis-cussions about the results are also presented followed by a presentation of the conclusions. The chapter ends with a discussion about future work.

8.1 Modelling Approach

The modelling approach presented in chapter 3 was adopted with the intention to obtain a model with decreased number of parameters compared to the current model. R-joints were therefore used as connectors between mast parts together with applied torques exerted according to Equation 3.1, in an attempt to obtain a simpler model. The joints were placed in the centre between connection points in order to avoid asymmetries in the model as much as possible. This entails the need for the connectors to move relative the mast parts during mast extension. The currently employed model does not have this problem. RP-joints were used permitting the connection points of the model to change location during simula-tions.

The inclusion of clearance was enforced into the model by creating a region where the mast parts are not subjected to applied torques. The clearance is 0.2 millimetres for 1.6t masts.

The method of modelling the chains described in section 3.3 was used in the model developed by Fredin [3]. It appeared to yield desired behaviour and was therefore incorporated into this model.

(54)

44 8 Discussion

8.2 Sway Measurements

The procedure described in section 5.3 is an established method of conducting mast sway measurements by the forklift manufacturer. All sway tests are carried out in this manner. Initially, it was intended that measurements would be con-ducted with other mast classes as well. The plans were however cancelled due to limited time and resources.

Regarding the results, Figure 5.2 shows that the damping characteristic is different between measurements with 1000 kg and 500 kg. Moreover, the fre-quency appears to decrease with increasing load. The following dependencies are concluded:

ζ = ζ(m) (8.1)

f = f (m) (8.2)

where ζ is the damping ratio, m the mass of the load, and f the oscillation fre-quency.

Figure 5.3 shows deflection of the inner guide at different fork heights carry-ing a load of 1000 kg. It can be seen that the deflection increases with increascarry-ing fork height, which is plausible since the guides become less stiff with decreasing roller overlap. Equation 3.4 shows how the overlap is dependent on fork height. The following dependencies are concluded from this figure:

f = f (h) (8.3)

A = A(h) (8.4)

where A is the initial deflection, and h is the fork height.

Due to reasons of secrecy, all acquired data cannot be presented in this report.

8.3 Parameter Analysis

The torsion and rotational coefficients of the torsion spring between the fork car-riage and the inner guide were not tested. It was concluded that these parameters had an extensively small effect on the oscillation outcome would the torsion coef-ficient be sufcoef-ficiently large. It would be better had this joint been modelled as a fixed joint initially.

The results presented in appendix B reveal a number of things about the simu-lated oscillations. Sway was generated setting the load to 1000 kg and fork height to 11 metres. It can be seen that the torsion coefficients, κi, influence the initial deflection of the mast. Furthermore, the frequency increases with increasing κi. Figure B.1 shows an increase of frequency as κ1 increases. Increased κ2and κ3

also yield an increase in frequency, according to Figure B.3 and Figure B.5. The damping characteristic is dependent on the rotational coefficient, ci. However, it is found that c1has no effect on the oscillations, which can be seen in Figure

B.2. It was also found that c2 and c3 influence the damping characteristic only

in a certain range. As can be seen in Figure B.4 and Figure B.6, small rotational coefficients have no effect on the outcome. However, larger values of rotational

(55)

8.4 Parametric Optimisation 45 coefficients do affect the damping characteristics. It can also be seen in the fig-ures that the rotational coefficients, c2and c3, change the frequency when c2>κ2

or c3>κ3.

To summarise, there are three different ranges in which each of the rotational coefficients, c2and c3, affects the oscillations.

Case 1(ci << κi, where i = 2, 3):

c2and c3have not effect on the oscillation characteristic.

Case 2(cc< ci < κi, where i = 2, 3 and ccis a boundary value):

ci affects the damping characteristic. Case 3(ci > κi, where i = 2, 3):

ci affects both damping characteristic and frequency.

8.4 Parametric Optimisation

Parameter spaces were determined from the steps in section 5.3. The second rota-tional coefficient, c2, was set to 1, in order to decrease the number of simulation.

It seemed to be a plausible approximation since the parameter analysis showed that c3was capable to influence the damping characteristic on its own.

Parametric optimisation resulted in the optimal parameters presented in Ta-ble 7.2. Since time was limited, optimal parameters were not determined for the case where the load was 0 kg. Furthermore, parametric optimisation was carried out over a time span of 6 seconds from the time when oscillations start. This time span was arbitrarily chosen given that a too long time span would be costly in terms of computational time but more accurate, and that a too short time span would not yield sufficiently satisfactory compliance but shorter computational time.

Figure 7.1 shows parameters as function of load and fork height. It can be seen that κ1has relatively flat surface, which means that it does not change drastically

with respect to fork height and load. The other two torsion coefficients, κ2 and

κ3, appear to be more load and fork height dependent. The position of the joints

which κ2 and κ3 belong to changes with respect to fork height. It seems that

smaller rotational coefficients are required for higher fork heights in order to permit larger deflections.

The rotational coefficient, c3, appeared to decrease as fork height increases.

Since the measurements revealed that the oscillation was less damped at a fork height of 11 metres, it seems plausible that the damping coefficient decreases with respect to increasing fork height. However, the rotational coefficient is ex-tremely large at 7 metres, even exceeding κ3. The slope from 7 metres to 9.5

metres is much steeper compared to the slope from 9.5 metres to 11 metres. One speculation is that the function is definitely not linear. However, it is known that would c2have values as in case 2, which is described in the previous section 8.3,

(56)

46 8 Discussion

of this thesis has manually tuned parameters setting c2 = c3, and obtained

sim-ulated oscillations which are visually somewhat compliant with measured data. These simulations are however not presented in this thesis since parametric op-timisation has not been carried out to show a more optimal set of parameters than those presented. Assuming optimality exists for c2possessing values of case

2, the optimal c3 would probably be smaller since c2 would also be involved in

causing the oscillations to appear more damped.

Figure 7.2 presents a comparison between measured sway and simulated sway obtained by optimal parameters. The x-axis constitutes the time axis, and y-axis reveals mast deflection. Simulated mast frame oscillations comply visually well with measured oscillations for the initial seconds as opposed to simulated inner guide oscillations.

8.5 Conclusions

This thesis was initiated with the purpose to obtain a model that could be used for simulation of mast sway. In this perspective, the thesis has resulted in success. The obtained model allows the user to adjust parameters in order to obtain oscil-lations with different characteristics. However, the obtained optimal parameters have yielded simulations that are not visually compliant with oscillations of all mast parts simultaneously. See Figure 7.2. It can be suspected that the model is not sufficiently detailed and that additional factors should be considered. Model changes can nevertheless be introduced would it be necessary.

Concerning dependencies, Figure 7.1 reveals useful information about the pa-rameters with respect to load and fork height. The optimal papa-rameters presented were determined based on the parameter spaces. It is reasonable to believe that other optimal parameters could be found would the parameter spaces be deter-mined with more precision. The same conclusions would probably have been made concerning parameter dependencies regardless. Certainly it would be pos-sible to determine parameters as functions with respect to load and fork height by interpolation on the acquired data. However, more measurement points would have been required to determine functions with higher accuracy.

The method used for determining optimal parameters is intended for the ob-tained model. It can nevertheless be applied to other mast models as well. This thesis has provided a method that can be used for evaluation of models to decide which model is the most appropriate to use in different cases. Whether the ob-tained model is sufficiently detailed depends on different situations and are to be decided by the forklift manufacturer.

8.6 Future Work

More investigation remains to be done in order to obtain a model that can repre-sent masts of all possible types and classes. The obtained model can be further improved in a number of ways. Additional measurements with different load and forklift cases would make it possible to determine model parameters as function

(57)

8.6 Future Work 47

of load and fork height. Furthermore, the model can be extended so that it repre-sents masts of other types and classes as well. This requires validation with sway measurements with masts of other types and classes.

Fredin’s model Fredin [3] can be validated with sway measurements acquired in this thesis. Performing the same procedure to find optimal parameters for Fredin’s model will offer the forklift manufacturer the possibility to evaluate which model of the two is the most suitable for different cases.

The results of this thesis can serve as foundation for future work on models of other types of masts not intended for reach trucks. Other types of forklifts which are equipped with masts are, e.g. power stacker trucks, order picking trucks, aisle trucks, and more. These masts differ in size and dimension and might oscillate differently compared to reach truck masts. The modelling method used in this thesis can probably likewise be applied for obtaining models of mentioned masts.

(58)

(59)

(60)

(61)

A

Functions

The functions used for running the scripted simulations are described more thor-oughly below. The material comes from SimXpert’s quick reference guide.