for Heavy Duty Machinery
Master Thesis Ted Samuelsson
Department of Physics Ume˚ a Universitet
June 12, 2015
Supervisor Examiner
Bobbie Frank Claude Dion
bobbie.frank@volvo.com claude.dion@umu.se
The construction equipment industry is developing at a fast pace, increasing the expectation on the next-generation machines. Wheel loaders and backhoe loaders are part of this evolution and all subsystems in those machines need to be developed to meet the high demands in energy efficiency and productivity.
One of the most important parts of the wheel loader is the loading unit. This is traditionally designed using highly experienced engineers and CAD software. To simplify the early stages of this process was an optimization tool developed to gen- erate a design outlay. The optimization will minimize the mass of the linkage since unnecessary weight will lower the efficiency. The minimum can be found by moving the joints and adjusting the shape of the device. The optimization will also include constraints to assure the correct performance of the final linkage.
Since there are a high number of design variables, a gradient-based optimization method was used. A finite element solver was also implemented to calculate the forces and stresses in the linkage.
The linkages studied in this report are one from a typical wheel loader and one from a backhoe loader. Since these machines are extremely versatile, and used for many different tasks, two sets of constraints are compiled. One of the constraint sets yields a linkage suitable for machines only equipped with bucket, while the other results in an all-round linkage suitable for most tools and applications.
The optimized linkages are compared to existing devices. The results show that there are some improvements possible and that the software could be used to help designers. However, the optimization problem is hard to solve due to non-smooth constraints functions and numerical instabilities. This issue could be overcome by different means, like using automatic differentiation, a non-gradient based optimiza- tion method, decreasing the number of constraints or decreasing the number of design variables.
Sammanfattning
Utvecklingen av anl¨aggningsmaskiner sker i snabb takt och detta ¨okar f¨orv¨antningarna p˚a framtidens maskiner. En stor andel av alla anl¨aggningsmaskiner ¨ar hjullastare och traktorgr¨avare och alla delsystem p˚a dessa maskiner m˚aste f¨olja med i utvecklingen.
En av de viktigaste delarna p˚a en hjullastare ¨ar lastaggregatet. Det designas traditionellt med hj¨alp av CAD mjukvara och mycket erfarna konstrukt¨orer. F¨or att underl¨atta denna process har en optimeringsrutin utvecklats, som generarar ett design f¨orslag. Optimeringen minskar l¨ankagets massa genom att flytta lagringspositioner och ¨andra delarnas dimensioner. Detta ¨okar effektiviteten hos maskinen eftersom den slipper k¨ora runt p˚a on¨odig vikt. Optimeringen inneh˚aller ¨aven villkor f¨or att s¨akerst¨alla god prestanda hos det optimerade aggregatet.
Eftersom det ing˚ar v¨aldigt m˚anga designvariabler i optimeringen anv¨ands en gra- dientbaserad metod. En finita element approximation anv¨ands f¨or att ber¨akna krafter och sp¨anningar i l¨ankaget.
De l¨ankage som unders¨oks i detta projekt ¨ar ett typsikt hjullastaraggregat och ett typiskt traktorgr¨avaraggregat. Eftersom dessa maskiner ¨ar v¨aldigt m˚angsidiga sammanst¨alldes tv˚a olika upps¨attningar av villkor. Den ena upps¨attningen anv¨ands f¨or att optimera ett aggregat som endast ska anv¨andas med skopa, medan den andra upps¨attningen anv¨ands f¨or att ta fram ett mer m˚angsidigt aggregat avs¨att f¨or att kunna klara av de flesta situationer och verktyg.
De optimerade lastaggregaten ¨ar j¨amf¨orda med produktionsaggregat och det visar sig att vissa f¨orb¨attringar ¨ar m¨ojliga. Slutsattsen ¨ar att optimeringsrutinen kan bli ett bra hj¨alpmedel f¨or konstrukt¨orer men att den beh¨over lite mer verifiering. Villko- rsfunktionen som optimeringen m˚aste l¨osa ¨ar inte helt sl¨at vilket ¨ar ett problem f¨or en gradientbaserade metod och dessutom finns vissa numeriska instabiliteter. Dessa sv˚arigheter kan undkommas p˚a olika s¨att, t.ex. genom att anv¨anda automatisk de- rivering, byta optimeringsalgoritm, minska antalet villkor eller minska antalet vari- abler.
1 Introduction 1
1.1 Problem Definition . . . 1
1.2 Related Work . . . 2
1.3 Machine Design . . . 3
1.4 Linkage Designs . . . 4
2 Optimization problem 7 2.1 Mathematical formulation . . . 8
2.2 Design Variables . . . 9
2.3 Constraints . . . 9
2.3.1 Geometrical . . . 10
2.3.2 Force . . . 12
2.3.3 Stress . . . 14
2.4 Objective Function . . . 14
2.4.1 Preliminary Functions . . . 15
2.4.2 Final Function . . . 16
3 Method Overview 19 3.1 Optimization Algorithm . . . 19
3.2 Stress Calculations . . . 21
4 Result and Analysis 25 4.1 Z-bar linkage . . . 25
4.2 TPC linkage . . . 30
4.3 Comparison . . . 35
4.4 Robustness . . . 36
4.4.1 Objective and Constraint Function . . . 36
4.4.2 Optimization Routine . . . 37
4.4.3 FEM Mesh Resolution . . . 39
5 Discussion 41 5.1 Result Reliability . . . 41
5.2 Future work . . . 42
A Design Variables 45
B Finite Differences 49
C Additional Results 51
D Function Analysis 53
1-1 Schematic figure of a wheel loader. . . 3
1-2 The common Z-bar linkage set-up. . . 4
1-3 Volvo invented TP linkage. . . 5
1-4 Common backhoe loader linkage. . . 6
2-1 Definitions of attachment angles (AA) and hinge pin height (HPH). . 10
2-2 Possible collision for a both a Z-bar and TPC linkage. . . 12
2-3 Geometric performance constraints. . . 13
2-4 Break-out torque definition. . . 13
2-5 Optimization of Z-bar (above) and TPC (below) linkage with mass as objective function. . . 15
2-6 Optimization of Z-bar with respect to tipping force. . . 16
2-7 Forces acting on a stationary machine. . . 17
3-1 Simplified schematic of the optimization routine. . . 20
3-2 Showing elements of initial and optimized Z-bar linkage. . . 21
3-3 The three load cases used in the optimization. . . 23
4-1 Initial state of Z-bar linkage. . . 27
4-2 Optimized Z-bar linkage with all-round conditions. . . 28
4-3 Optimized Z-bar linkage with bucket constraints. . . 29
4-4 Initial state of TPC linkage. . . 32
4-5 Optimized TPC linkage with all-round conditions. . . 33
4-6 Optimized TPC linkage with bucket constraints. . . 34
4-7 Analysis of the objective function for a Z-bar linkage. . . 37
4-8 Analysis of the constraint function for Z-bar linkage. . . 38
4-9 Influence of FEM mesh resolution on execution time and calculated mass. . . 39
B-1 Optimized TPC linkage with all-round conditions. . . 49
C-1 All-round optimized Z-bar linkage. . . 51
C-2 Bucket optimized Z-bar linkage. . . 52
C-3 All-round optimized TPC linkage. . . 52
C-4 Bucket optimized TPC linkage. . . 52
D-1 Z-bar constraint functions for problematic design variables. . . 53
D-2 Z-bar constraint functions for problematic design variables. . . 54
D-3 Evaluation of TPC objective function. . . 55
D-4 Z-bar constraint functions for problematic design variables. . . 55
2.1 Minimal dump and rollback angles at different hinge pin heights. . . . 10
2.2 Critical geometrical collisions which need to be considered during the optimization of a Z-bar linkage. . . 11
2.3 Critical geometrical collisions which needs to be considered during the optimization of a TPC linkage. . . 11
2.4 Performance metrics used to validate the correct function of the linkage. 12 2.5 Breakout torque constraints for an all-round and a bucket machine at different linkage positions. . . 14
2.6 Lift force constraint for both bucket and all-round machine. . . 14
4.1 Resulting masses for optimized Z-bar. . . 26
4.2 Force, torque and attachment angle constraint fulfilment for Z-bar link- age. . . 26
4.3 Performance constraint fulfilment for Z-bar linkage. . . 26
4.4 Resulting masses for optimized TPC. . . 30
4.5 Force, torque and attachment angle constraint fulfilment for TPC link- age. . . 31
4.6 Performance constraint fulfilment for TPC linkage. . . 31
4.7 Linkage masses inclusive cylinders. . . 35
4.8 Adjusted linkage masses. . . 35
4.9 Sensitivity analysis for Z-bar linkage. . . 38
4.10 Sensitivity analysis for TPC linkage. . . 39
A.1 Design variables for Z-bar linkage. . . 46
A.2 Design variables for TPC linkage. . . 47
Chapter 1 Introduction
This project is carried out at Volvo Construction Equipment who are developing and manufacturing wheel loaders, excavators, articulated haulers and other types of heavy duty machinery. The Eskilstuna site, were this project is based, is primary focusing on wheel loaders, as is this report.
High energy efficiency and productivity is one of the most important attributes of a modern wheel loader. To achieve this in next-generation machines must all subsystems be optimized to fulfil these expectations. However, the loading unit (i.e.
the device holding the bucket, also called linkage or working device) of the current machines are traditionally made using CAD tools and experienced designers [1, 2].
Shorter project times, higher material costs and higher demands on performance call for a more effective design process. Therefore aims this report at improving the early stages of this process by implementing an design optimization tool. The tool will optimize the bearing positions and linkage dimensions with respect to a specified objective. This will streamline the process and at the same time strive for a better result.
1.1 Problem Definition
The optimization problem consists of adjusting all bearing positions of the loading unit to minimize the linkage mass. Furthermore, the width and height of all arms
1.2. RELATED WORK
should be adjusted to hold any likely external load. An attempt of this has been done by Volvo CE in 2009 [1] and is used as a starting point for this work.
The linkages considered in this report will be one for a typical wheel loader and one for a backhoe loader. These two linkages have different design, which will be further described in section 1.4.
1.2 Related Work
There is a great interest in linkage optimization for heavy duty machines and some of the research will be presented here. An optimization of lift cylinders placement and size has been done by Volvo CE in 2003 [3]. A complex optimization method and an exhaustive search method were used. Both worked well for the problem which only consists of four design variables (two positions in a two dimensional domain).
Another attempt using an exhaustive search was done in [4], optimizing the power consumption. Others have also tried to optimize the design of the linkage using a genetic algorithm on a backhoe loader [5] and a wheel loader [6]. A sequential quadratic programming method was used in [7].
However, the above-mentioned authors do not consider the internal stresses in the linkage and treat the problem as two dimensional. This approach is not good enough to create a trustworthy and usable result but could be used for evaluating new concepts. A 3D parametric finite element approach has been performed, but only by minimizing the material on an existing excavator working device (no coupling between joints) [8].
The method described by Ekevid in [1] is three-dimensional and has both joint positions and arm dimensions as parameters resulting in more than hundred design variables, more than 5 times the amount of the other papers. The stresses in all linkage parts are calculated using a finite element method (FEM) and the result looks very promising. However, the project did not really get all the way to achieve a usable software. One of the largest issues was the objective function, which did not include all required physics. This work adresses this issue and the changes to the
function are presented in section 2.4.
1.3 Machine Design
Wheel loaders are multi-purpose vehicles which are used for many different tasks, e.g.
transportation of gravel, pallet handling or logging applications. This is especially true for small and mid-sized machines. They are often equipped with an attachment bracket for easy exchange of buckets, forks, handling arms, logging grapples, etc.
The largest vehicles are, to the contrary, often used for a specific task, like bucket handling.
The backhoe loader is also a versatile vehicle, but in general somewhat smaller.
The backhoe loader has, in addition to the front working device, also a rear-mounted excavator linkage. To simplify the comparison between the linkages, only one kind of machine will be used in the optimizations. This is possible since the machines are very similar, not least since this report only considers the front devices. Therefore, the 20 ton Volvo L120H wheel loader is chosen as the reference machine for the optimization.
Front Frame
Counter Weight Rear Frame Operator Cabin
Loader Linkage
Hinge
Figure 1-1: Schematic figure of a wheel loader.
The complete machine can be divided into a small number of basic parts, see Figure 1-1. There is a front and a rear frame joined by a hinge used for the articulated steering. On the rear frame a cabin and a counter weight are attached. The counter weight prevents the machine from tipping forward when a large force is exerted on
1.4. LINKAGE DESIGNS
the bucket, e.g. when a heavy load is lifted. The rear frame includes most of the parts related to the engine and hydraulic system. On each frame there is also one pair of wheels and their related axles. The loader linkage and bucket are positioned on the front frame.
1.4 Linkage Designs
There are many different linkage designs available for these kind of machines. They differ in complexity, weight and performance. They can, with the same hydraulic system, have different lift force, breakout torque and parallel alignment. Good parallel alignment means that the tool keeps its angle to the ground while the linkage is raised or lowered, important when using forks and logging grapples.
The most common linkage on a wheel loader is called the Z-bar, see Figure 1-2.
It is used by most manufacturers and is standard on Volvo’s largest models.
E
P
C
F D O
G A
J
Figure 1-2: The common Z-bar linkage set-up.
The design owns its popularity to a simple and light construction with few joints, but also because of its strong lift force. The downsides are bad parallel alignment and low breakout torque at some positions. This type of linkage works great for bucket applications, but less for fork handling.
A linkage used on most of Volvo’s loaders is called TP (Torque Parallel), see Figure 1-3. It is a heavier and more complicated setup but offers excellent parallel
alignment and at the same time high breakout torque throughout the lifting range.
This linkage is suitable when good parallel alignment is of importance, like fork and logging grapple applications. However, the higher weight decreases the maximum possible bucket load. This linkage will not be considered in this report more than as a reference.
E
P
C
F O
G
A J H
I
B D
Figure 1-3: Volvo invented TP linkage.
The third linkage, common on backhoe loaders, is called TPC and utilizes a simple uncoupled design. This linkage has no mechanical coupling adjusting the parallel alignment of the bucket when the unit is raised or lowered. Instead it often uses hydraulic coupling to address the problem.
This linkage can, contrary to the others, roll-back the bucket enough to dump material over the cabin. This is a safety issue that needs to be dealt with.
The version considered in this report will be a single boom unit, i.e. it only has one lift arm instead of the more common set-up with two arms. The boom will also feature a hollow-cross section and one centred lift cylinder. This yields a compact and light-weight linkage without decreasing the performance.
1.4. LINKAGE DESIGNS
E
P
C
F O
G
A J
Figure 1-4: Common backhoe loader linkage.
Chapter 2
Optimization problem
The optimization problem consists in minimizing the total weight of the machine and at the same time improve the operating performance, like lift force and maximum dump angle. To achieve this, the placement of the joints in the loading unit (defined in Figures 1-2 and 1-4) are moved towards their optimum, with respect to a specified goal. All joints are bounded to boxes (often, but not restricted to, 2 dimensional).
At the same time the dimensions of the different parts in the linkage are adjusted to confine the von Mises stress [9]
σevM = r1
2(σx+ σy)2+ (σy + σz)2 + (σz+ σx)2 . (2.1) The linkage weight is then calculated and the counter weight is sized to satisfy the tipping force.
2.1. MATHEMATICAL FORMULATION
2.1 Mathematical formulation
Let f (x) be the objective function that should be minimized, where x exists in the domain Ω. Then the optimization problem can be stated as
minimize f (x) subject to
c(x) ≤ 0 ceq(x) = 0 xlb ≤ x ≤ xub.
(2.2)
where c(x) ≤ 0 and ceq(x) = 0 are the inequality and equality constraints to the prob- lem, respectively. The upper and lower bound are defined by xlb and xub. Inequality constraints can be rewritten as equality constraints on the form c(x) + s = 0, where s ≥ 0 are called slack variables.
Most gradient based optimization algorithms for constrained non-linear problems use the Lagrangian
L(x, λ, λlb, λub, λeq, s, slb, sub) = f (x) + λT(c(x) + s) + λTeq(ceq(x) + s) + λTlb(xlbx + slb) + λTub(xubx + sub)
(2.3)
where λ ≥ 0, λlb ≥ 0,λub ≥ 0 and λeq ≥ 0 are called Lagrange multipliers. The s, slb and sub are the slack variables.
Minimizing the Lagrangian is considerably easier since the constraints are in- corporated in the Lagrangian function. At an optimum, the Karush-Kuhn-Tucker conditions [10]
∂L
∂x = 0, ∂L
∂λ = 0, ∂L
∂s = 0 (2.4)
λ ≥ 0, s ≥ 0
c(x) ≤ 0, ceq(x) = 0 xlb− x ≤ 0, x − xlb ≤ 0
(2.5)
λT(c(x) + s) = 0 λTlb(xlb− x + slb) = 0 λTub(x − xub+ sub) = 0
(2.6)
must be satisfied [11].
2.2 Design Variables
The model parameters, which should be optimized, are called design variables. These should describe the linkage without any additional information. They will, under the optimization, be adjusted to satisfy equation 2.2 and will then be returned as the solution.
For this linkage optimization there are four different groups of variables: Joint positions, cylinder dimensions, element sizes and counter weight size. The joint po- sitions describe the simple shape and lengths of all linkage parts. They are defined in each linkage parts local coordinate system. Parameters for the cylinders are for instance their extended and contracted lengths as well as diameters. The element sizes, which are the largest part of the design variables, describe the width and height of all linkage parts. They will also describe the thickness were it applies, e.g. the hollow boom of the TPC. The number of variables in this group is dependent on the FEM mesh resolution. The last group only consists of the counter weight mass.
The variables for the joints, cylinder lengths and counter weight are stated in Appendix A.
2.3 Constraints
The linkage needs to fulfil a certain number of constraints related to geometrical conditions, lift forces and stresses. To deal with the multi-purpose behavior of the machine, two sets of constraints have been compiled. One for an all-round machine, which need high overall lift performance and breakout torque, and one for a bucket handling machine, which only needs high performance at ground level when operating
2.3. CONSTRAINTS
in a pile of material. The two different sets of constraints will henceforth be referred to as all-round and bucket.
2.3.1 Geometrical
The geometrical constraints consist of three subgroups: Attachment angles, collisions and linkage performances. None of these will differ between the two versions of the constraints. However, there will be some difference between the Z-bar and the TPC linkage due to the variation in design.
Attachment Angles
The machine needs to be able to tilt the bucket sufficiently so that the operator can easily empty the material. It must also be able to tilt the bucket backwards to keep the material in the bucket while travelling on rough roads. To assure these properties the linkage must manage the dump and roll-back angles (see Figure 2-1) stated in Table 2.1.
Table 2.1: Minimal dump and rollback angles at different hinge pin heights.
HPH [m] Min rollback angle [°] Min dump angle [°]
min -55 35
0.58 -55 49
2.00 -55 60
max -45 58
Roll-back Angle Dump Angle
Hinge Pin Height
Figure 2-1: Definitions of attachment angles (AA) and hinge pin height (HPH).
Collisions
To assure that the optimized design works satisfactorily, no linkage parts are allowed to collide with each other when the machine is operated. Therefore, it is important to localize the critical linkage positions were this could happen. These different cases are visualized in Figure 2-2 and Tables 2.2 - 2.3.
Table 2.2: Critical geometrical collisions which need to be considered during the optimization of a Z-bar linkage.
No Description Condition [m]
A1 Bucket A ↔ Bucket ridge > 0.10 A2 Bucket J ↔ Bucket ridge > 0.10 A3 GDF (D) ↔ Bucket ridge > 0.20 A4 GDF (DF) ↔ Bucket top > 0.20
A5 GDF (G) ↔ Front axle > 0.34
A6 Lift cylinder ↔ Front axle > 0.25
Table 2.3: Critical geometrical collisions which needs to be considered during the optimization of a TPC linkage.
No Description Condition [m]
B1 Bucket A ↔ Bucket ridge > 0.05
B2 Bucket J ↔ Bucket ridge > 0.05
B3 Tilt cylinder (EF) ↔ DEG (G) > 0.20 B4 Tilt cylinder (E) ↔ Boom (OC) > 0.20 B5 Lift cylinder (C) ↔ DEG (D) > 0.34 B6 Lift cylinder ↔ Front axle > 0.25
In addition to the constraints stated above, the cylinders inner diameter must not become larger than the outer diameter since this would yield a negative mass.
Performance
The linkage needs to handle some basic tasks. For instance, it needs to be able to lift the bucket enough to dump in a hauler and also lower the bucket enough for some digging capabilities. These situations are defined in Table 2.4 and Figure 2-3.
2.3. CONSTRAINTS
A4 A3
A6 A5
A2
A1
B3
B4 B5
B6
B2
B1
Figure 2-2: Possible collision for a both a Z-bar and TPC linkage.
Table 2.4: Performance metrics used to validate the correct function of the linkage.
Measure Value [m]
Digging depth < -0.10
Reach > 1.20
Dump Clearance > 2.88
2.3.2 Force
The force constraints treat the different forces and torques the linkage need to be able to handle. Those are divided into two sub categories: Lift forces and breakout torques.
Breakout Torque
The breakout torque is the maximum torque available at the bucket tip and is defined as
M = |r1× F | = |r2× R| (2.7)
where F is the reaction force when a tilt cylinder force is present, see Figure 2-4. R is the corresponding force in the GJ link.
The breakout torque is mostly used at ground level when filling the bucket. How- ever, when the machine is used with other equipment, like logging grapples, other positions are also of interest. The constrains are collected in Table 2.5.
Digging Depth Reach
Dump Clearance
Figure 2-3: Geometric performance constraints.
r
1r
2F
R
Figure 2-4: Break-out torque definition.
Lift Force
Lift force is defined as the force exerted by the linkage on the load, measured at the loads center of gravity. To achieve the wanted lift behavior three constraints for the minimum lift force are sufficient. These are collected in Table 2.6.
The tipping force is also of interest, in addition to of the constraint above. A large tipping force makes it possible to lift a heavy load without risking that the rear wheels lift of the ground. The tipping force is calculated with the linkage positioned so that the bucket load is as far forward as possible. For this size of machine, the tipping force should be at least 13380 N including the bucket.
2.4. OBJECTIVE FUNCTION
Table 2.5: Breakout torque constraints for an all-round and a bucket machine at different linkage positions.
HPH [m] Angle [°] Torque All-round Bucket
0 min 1.00 0.45
0 0 1.00 1.12
0 max 1.00 0.49
max min 1.00 0.70
max 0 1.00 0.50
The values have been normalized for each position to protect proprietary information.
Table 2.6: Lift force constraint for both bucket and all-round machine.
HPH [m] Lift force All-round Bucket
min 1.00 0.88
2 1.00 0.97
max 1.00 0.88
The values have been normalized for each position to protect proprietary information.
2.3.3 Stress
To assure that no parts of the linkage break must the von Mises stress, due to twisting and bending, be under a predefined value. This value is chosen according to the materials properties. During the optimization the stresses in the linkage are calculated by a finite element method described in section 3.2.
2.4 Objective Function
The function defining the goal of the optimization is called objective function. Given a set of design variables the function should return a scalar value, representing the fitness of the input. Often an optimization has more than one goal, e.g. both min- imizing the mass and maximizing the rigidity of a mechanical structure. This can be handled by combining the goals into one objective function. The different goals
can then be weighted individually dependent on their importance. For n goals the function becomes
f (x) =
n
X
i=1
wigi(x) (2.8)
where g(x) is the individual objective functions and wi are each goals weight.
The weighting process can however be an issue since the exact relative importance between the goals often are hard to estimate or even unknown. To avoid this issue, some goals can be converted to constraints. Then the issue instead becomes choosing the constraint limitation value, which sometimes is easier [12].
2.4.1 Preliminary Functions
The first model minimized the total mass of the linkage, including cylinders. The result from that kind of formulation is not satisfactory even though it becomes light weight. The problem is that there are no tendencies to move the center of gravity towards the front axle and thus increasing the tipping force. Instead the linkage becomes rather long and slim, see Figure 2-5. This is not the appearance, nor the performance, a linkage should have and therefore another objective function was developed.
−3 −2 −1 0 1 2 3 4 5
0 1 2 3
x
z
−3 −2 −1 0 1 2 3 4 5
0 1 2 3
x
z
Figure 2-5: Optimization of Z-bar (above) and TPC (below) linkage with mass as objective function.
2.4. OBJECTIVE FUNCTION
The second function maximized the vehicles tipping force, i.e maximized the pos- sible load in the bucket before the vehicles rear wheels ease from the ground. This is done by calculating each linkage parts contribution to a moment around the front wheel. This causes the optimization to push the linkage towards the rear. The place- ments of the joints looks better with this model, however all elements behind the tipping point will be maximized in size resulting in a very odd looking and heavy linkage, see Figure 2-6.
−4 −3 −2 −1 0 1 2 3 4 5
0 1 2 3
Figure 2-6: Optimization of Z-bar with respect to tipping force.
2.4.2 Final Function
The final function uses the above concepts and adds a new part to the optimization, a counter weight. This yields the correct behavior; with a beefy, short and back-pushed linkage. Since the size of the counter weight decides the tipping force, it must be considered during the optimization process. In a normal design process the counter weight is actually designed in parallel with the linkage to yield the expected tipping force.
The tipping load is now set as a constraint. Consequently, the linkage must be able to handle a specific load and the center of gravity will be pushed backwards. The optimization function then consists of minimizing the total weight of the linkage and the counter weight. Since the counter weight is farther from the tipping point (triangle in Figure 2-7) it will be cheaper to add weight there than on the linkage. Since the
Fcw
Frear
Fload Flink
Figure 2-7: Forces acting on a stationary machine.
function is a summation of two masses these can be individually weighted, which will be considered in the result section. However, to minimize the actual physical weight of the machine the ratio should be set to one. To minimize the product cost another ratio could be used to move weight to the cheaper manufactured counter weight. The final function is
f (x) = w1X
i
mi+ w2mcw (2.9)
where mi is the mass of linkage element i, mcw is the counter weight mass and w = (w1, w2) are the goal weights.
2.4. OBJECTIVE FUNCTION
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Chapter 3
Method Overview
The optimization routine will under the optimization evaluate the objective function and check the constraints. The first task is accomplished by passing the design variables to the objective function which then evaluates and returns the fitness value.
The constraints are checked by passing the design variables to a constraint evaluation function. This function checks all constraints by evaluating the different constraint types individually. The stress constraint evaluation part invokes a FEM routine for the stress calculations. The FEM routine also generates important information used by the force constraint evaluation function. The constraints are then combined together and returned to the main optimization routine, which uses the information to decide if the optimization should terminate. Termination is done if the current design variables satisfy the constraints and represent a minimum for the objective function, otherwise the routine is iterated with modified design variables. An overview of the system is represented in Figure 3-1.
3.1 Optimization Algorithm
How to move the design variables inside the calculation domain, at each iteration, de- pends on the chosen optimization algorithm. There exist several algorithms to choose from for constrained non-linear optimization, with different advantages. For the task in this report a gradient based algorithm has been chosen, called interior-point. It
3.1. OPTIMIZATION ALGORITHM
Design Variables
Objective Function Constraint Evaluation
Geometrical Constraints Evaluation
Stress Constraints
Evaluation
Force Constraints
Evaluation
FEM model
Optimization Routine OptimizedDesign
Variables
DV DV DV
DV
DV DV
CG CS CF
C
f d
R Material
Properties Load Cases
Terminate?
No Yes
Figure 3-1: Simplified schematic of the optimization routine.
moves the design variables in the negative objective gradient of the current point.
If an expression for the gradient is unknown can it often be approximated by some numerical method. In this report, the gradient is approximated with central finite differences. Forward finite differences were investigated since they are less compu- tationally heavy but they introduced numerical disturbances in the calculation, see Appendix B. The gradient-based method is a good choice due to its robustness and possibility to handle the large number of variables. However, this method only works on problems with an objective function and constraint functions that have continuous derivatives, see Section 4.4.1 for an analysis of the functions used. Other alternatives considered were the Complex method, genetic algorithms and exhaustive search. Both the exhaustive search and the complex method are not well suited for the size of this system, and were therefore discarded. The genetic algorithm is a suitable choice but
was unfortunately not tested.
Feeding the algorithm with analytic Jacobian and Hessian will decrease the compu- tational time considerably. However, the expressions are often complex and sometimes they become too large to be used. A work-around is to use automatic differentiation software, e.g. the ADMAT and ADiMat toolboxes for MATLAB [13]. This technique automatically differentiates a function by using the chain rule on its elementary arith- metic operations. The calculations have an accuracy of working precision and uses only a constant factor operations more than the original function, often resulting in shorter computational time than using analytic expressions [14]. Such software would also add robustness and accuracy to the calculations of the derivatives compared with usual finite differences, but were not used due to the limited project time.
3.2 Stress Calculations
The stresses in the linkage are calculated using a finite element method. The linkage parts are divided into small beam elements, as seen in Figure 3-2. The calculation is provided with information about the linkage: details about all part positions and material properties. It also includes all external forces present and boundary condi- tions. The displacement of each element is returned, which can be recalculated to stresses and forces in the linkage.
Figure 3-2: Showing elements of initial and optimized Z-bar linkage.
3.2. STRESS CALCULATIONS
The elements are approximated by Euler-Bernoulli beam elements [15]. It is a classical beam theory for elastic beams subject to forces yielding small deflections.
The Euler-Bernoulli equation is defined as d2
dx2(EId2w
dx2) = q (3.1)
where w is the deflection, q is the force per unit length, I is the second moment of inertia and E is Young’s modulus [9]. For a complete calculation the displacement equation
d
dx(EAdu
dx) = U (3.2)
and the equation for Saint-Venant’s torsion d2
dx2(GKv
dτ
dx) = T (3.3)
are also needed. A is the cross-section area, U is the displacement along the beam, Kv is the Saint-Venants torsion constant, G is shear modulus and T is the torque along the beam. For a deeper insight on how to formulate the finite element approximation, see [16].
With the use of the bending moment
M = −EId2w
dx2 (3.4)
the normal stress can be calculated as
σx = N A σy = M b
2Iz σz = M h
2Iy
(3.5)
where h is the element height, b is the element width and N is the normal force.
The shear stress is calculated with
τ = 1 Wx
dM
dx (3.6)
where Wx is the section modulus [9].
Since the stress in the linkage is dependent on the cross-section area will the optimization add material where the stress is high and cut off where the stress is low.
This will make the optimization strive towards minimizing the stress and by that satisfy the goal of decreasing the mass.
To calculate the maximum stress that the linkage may be subject to under normal operation are a number of so called load cases used. Load cases are extreme situations that could happen when a wheel loader is operated. They state external forces, hydraulic cylinder pressures and linkage positions [17]. In addition to the load cases, the stress is also measured during the evaluation of the force and torque constraints.
The load cases with the most significant influence, i.e. yielding the largest stresses, on the linkage geometry were used during the optimization and are defined in Figure 3-3.
Fh Fd
Fd
Fh
(a) The machine is crowded with the bucket in dump position. External force on bucket tip center. Not against mechanical stop.
Fh Fh
Fv Fd
(b) The machine is crowded with level bucket at ground.
Bucket is lifted with max- imal pressure. External force on bucket tip.
Fh Fh Fv Fd
Fs
(c) The machine is crowded with level bucket at ground.
Bucket is lifted with max- imal pressure. Assymet- ric external force on bucket tip.
Figure 3-3: The three load cases used in the optimization.
3.2. STRESS CALCULATIONS
THIS PAGE INTENTIONALLY LEFT BLANK
Chapter 4
Result and Analysis
In this chapter, the results from the optimizations are presented. The mass of each linkage part are stated as well as a constraint fulfilment review.
There will also be a comparison between the different types of linkages and con- straints. However, the quality of the result will be further discussed in chapter 5. An analysis of the robustness and performance are included in section 4.4.
4.1 Z-bar linkage
The final Z-bar linkage masses are stated in Table 4.1. The differences between the constraints can easily be observed. In Table 4.2 and 4.3 can the constraint fulfilment be studied. Additional result, containing data about variable positions and constraint convergence, is presented in appendix C.
The initial shape of the linkage is showed in Figure 4-1 and the final design of the linkages is visualized in Figure 4-2 and 4-3.
There is some unexpected behavior in the results. Even though the all-round linkage has a higher breakout torque for all attachment angles and a higher lift force for all hinge pin heights it also has a lower weight. This indicates that the optimization is sensitive to the specified constraints. However, the hydraulics are much lighter for the bucket version and the maximum dump and roll-back angles are a bit better.
4.1. Z-BAR LINKAGE
Table 4.1: Resulting masses for optimized Z-bar.
Part Original Mass All-round Bucket
Optimized Mass Optimized Mass
Boom 100% 78% 80%
GDF 100% 58% 65%
GJ 100% 127% 156%
Lift Cylinder 100% 91% 80%
Tilt Cylinder 100% 170% 72%
Total, w/o cylinders 100% 77% 81%
Total, w/ cylinders 100% 83% 80%
The masses are presented in percent of original mass to protect proprietary information.
Table 4.2: Force, torque and attachment angle constraint fulfilment for Z-bar linkage.
Measure AA HPH Value
All-round Bucket Unit
Tilt Torque 0 0 1.88 1.00 kNm
max 0 1.09 1.40 kNm
min 0 1.00 1.00 kNm
0 max 1.39 1.28 kNm
Lift Force max min 1.00 1.11 kN
max 2 1.01 1.00 kN
max min 1.00 1.02 kN
Tipping Force - - 1.00 1.00 kN
Attachment Angle roll-back min 35.30 35.30 deg
dump min 56.60 56.60 deg
roll-back 0.7 53.30 54.60 deg
dump 0.7 55.00 55.10 deg
roll-back 2 60.00 60.00 deg
dump 2 55.00 55.10 deg
roll-back max 58.00 58.00 deg
dump max 40.90 42.00 deg
Table 4.3: Performance constraint fulfilment for Z-bar linkage.
Measure Value [m]
All-round Bucket Digging depth -0.10 -0.10
Reach 1.48 1.48
Dump Clearance 2.88 2.88
−3 −2 −1 0 1 2 3 4 5 0
1 2 3
−3 −2
−1 0 1
2 3
4 5
−1 0
1 0
1 2 3
Figure 4-1: Initial state of Z-bar linkage.
4.1. Z-BAR LINKAGE
−3 −2 −1 0 1 2 3 4 5
0 1 2 3
−3 −2
−1 0
1 2
3 4
5
−1 0
1 0
1 2 3
Figure 4-2: Optimized Z-bar linkage with all-round conditions.
−3 −2 −1 0 1 2 3 4 5 0
1 2 3
−3 −2
−1 0 1
2 3
4 5
−1 0
1 0
1 2 3
Figure 4-3: Optimized Z-bar linkage with bucket constraints.
4.2. TPC LINKAGE
4.2 TPC linkage
The TPC linkage results are stated in Table 4.1. The constraint fulfilment can be observed in Tables 4.5 and 4.6. Constraint convergence and design variable domain position are availble in appendix C.
Table 4.4: Resulting masses for optimized TPC.
Part Original Mass All-round Bucket
Optimized Mass Optimized Mass
Boom 100% 53% 55%
GDF 100% 56% 67%
GJ 100% 20% 21%
Lift Cylinder 100% 109% 115%
Tilt Cylinder 100% 72% 97%
Total, w/o cylinders 100% 51% 53%
Total, w/ cylinders 100% 65% 72%
The masses are presented in percent of original mass to protect proprietary information.
The final design of the linkages is visualized in Figure 4-5 and 4-6. Also on this linkage there are some disturbance in the stress calculations.
Also for this linkage are the bucket version heavier, but it also performs better for force and torque constraints at some positions than the all-round version. Another interesting result is that the boom masses for the optimized linkages are about half, compared with the original boom. The optimization algorithm does not take into account for needed padding around the bearings and link ends. That would approxi- mately add about 10% to the mass. Another reason for the original linkage boom to be so much heavier is that it was designed to be equipped with a hydraulic parallel alignment system. Last the optimizations are considering a molded boom in contrast to the original welded boom.
Table 4.5: Force, torque and attachment angle constraint fulfilment for TPC linkage.
Measure AA HPH Value
All-round Bucket Unit
Tilt Torque 0 0 1.20 1.00 kNm
max 0 1.31 2.19 kNm
min 0 1.00 2.45 kNm
0 max 1.87 2.75 kNm
Lift Force max min 1.15 1.32 kN
max 2 1.11 1.09 kN
max min 1.00 1.00 kN
Tipping Force - - 1.15 1.01 kN
Attachment Angle roll-back min 35.30 35.30 deg
dump min 56.50 56.50 deg
roll-back 0.7 50.00 50.00 deg
dump 0.7 56.50 56.50 deg
roll-back 2 63.30 63.30 deg
dump 2 56.50 56.50 deg
roll-back max 58.30 58.30 deg
dump max 40.50 40.50 deg
Table 4.6: Performance constraint fulfilment for TPC linkage.
Measure Value [m]
All-round Bucket Digging depth -0.21 -0.21
Reach 1.59 1.48
Dump Clearance 2.89 2.88
4.2. TPC LINKAGE
−3 −2 −1 0 1 2 3 4 5
0 1 2 3
−3 −2
−1 0 1
2 3
4 5
−1 0
1 0
1 2 3
Figure 4-4: Initial state of TPC linkage.
−3 −2 −1 0 1 2 3 4 5 0
1 2 3
−3 −2
−1 0 1
2 3
4 5
−1 0
1 0
1 2 3
Figure 4-5: Optimized TPC linkage with all-round conditions.
4.2. TPC LINKAGE
−3 −2 −1 0 1 2 3 4 5
0 1 2 3
−3 −2
−1 0 1
2 3
4 5
−1 0
1 0
1 2 3
Figure 4-6: Optimized TPC linkage with bucket constraints.
4.3 Comparison
The assembled results are compared and analyzed here. From Table 4.7 it can be seen that the TPC linkage is lighter for all categories. But bear in mind that the TPC linkage needs some kind of help for the parallel adjustment. A hydraulic help system would add about 15% in weight. An electronic help system would not add as much weight, but has poor energy efficiency.
Table 4.7: Linkage masses inclusive cylinders.
Type TP Z-bar TPC
Original 100% 93% 90%
Optimized All-round 77% 58%
Optimized Bucket 74% 65%
For a more accurate comparison the bearing padding, mentioned in section 4.2, must also be considered in the result. The maximum added weight due to this is calculated to 10%. The final adjusted masses for all linkages, also including the hydraulic help system for the TPC, are stated in Table 4.8.
Table 4.8: Adjusted linkage masses.
Type TP Z-bar TPC
Original 100% 93% 104%
Optimized All-round 85% 74%
Optimized Bucket 81% 82%
Added bearing padding(+10%) and parallel alignment hydraulics (+15%, TPC only).
It is easy to see that a mass reduction is possible, for both Z-bar and TPC, without decreasing the performance of the linkages. The best choice for bucket applications is the Z-bar were the mass reduction is 10%. For a all-round machine the TPC seem to be lightest (reduction of about 20%) and therefore the best choice. However, the linkages are modeled as fully molded, meaning that no welded seams are present. The yielding strength for a welded seam is about half the strength of a molded. Since real linkages needs to be welded in some sections must material be added to strengthen
4.4. ROBUSTNESS
these parts. For the TPC linkage considered in this report the boom is welded at several large seams.
But trying to get an exact mass from this optimization is not the point of the software. The software should not substitute a design engineer, instead it should generate an optimal design layout for the engineer to use in the linkage development.
This is an important distinction that needs to be understood.
4.4 Robustness
To verify the reliability and correctness of the used approach the robustness of the optimization was analyzed. The objective and constraint functions behaviour were inspected since these need to be smooth and have continuous derivatives. Later the complete optimization routine is analyzed. Different starting points, domain sizes and goal weights are tested to investigate the impact on the result. In section 4.4.3 different mesh resolution are compared to see when the FEM calculations converges.
4.4.1 Objective and Constraint Function
A parametric analysis of the constraint function was performed by studying one design variable at a time. The function was evaluated with the variable at different points between the lower and upper boundaries. This was done for all design variables with 20 evenly spaced points on the interval [xlb, xub]. The other variables were fixed at their center position. The top row in Figure 4-7 shows the result for the variables handling the joint positions, cylinders and counter weight. For some variable sets the linkage can not be geometrically assembled by the software, this can be viewed in Figure 4-7 as missing data. In graph II the last variable, the counter weight, is removed and the z-axis is zoomed to make it easier to see the curvature of the other variables (since they have much smaller amplitude). In III, the 15th variable (stabilizers z position) is also removed. The second row shows the variables connected to the beam elements. In graph V the 60th variable is removed.
All the curves analyzed seem to be continuous and smooth with the step size
0 10 20 30 40 50 0
5 10 15 20 0.4 0.6 0.8 1 1.2 1.4 1.6
x 104
DV − position DV #
f(DV)
0 10
20 30
40
0 5 10 15 20 1.06 1.07 1.08 1.09 1.1 1.11 1.12
x 104
DV − position DV #
f(DV)
0 10
20 30
40
0 5 10 15 20 1.065 1.07 1.075 1.08 1.085 1.09
x 104
DV − position DV #
f(DV)
40 60
80 100
120
0 5 10 15 20 1.05 1.1 1.15 1.2 1.25 1.3 1.35
x 104
DV − position DV #
f(DV)
40
60 80
100 120
0 5 10 15 20 1.07 1.08 1.09 1.1 1.11 1.12 1.13
x 104
DV − position DV #
f(DV)
I II III
IV V
Figure 4-7: Analysis of the objective function for a Z-bar linkage.
used. That means that the derivatives should be easy to compute and generate a fairly accurate result. However, since the analysis was only considering one variable at a time this result should be considered with caution.
The constraint function was analyzed in a similar way, but there were some non- smooth behaviour. A few of these problematic variables are plotted in Figure 4-8, the rest are collected in appendix D. Some of the variables seem to have a singular point in the interval [xlb, xub], they include the booms Ax and Ay variables (see left graph in Figure 4-8). The singularities do presumably depend on physical conditions of the linkage and may be avoided by adjusting the boundaries (if they cause problems to the solver). Other variables, e.g. bucket x-coordinates, were subject to numerical perturbations (see center and right graph in Figure 4-8). The FEM calculations are probably the reason for these instabilities, however, it is hard to establish the exact cause without a more thorough investigation (which is beyond the scope of this report).
4.4.2 Optimization Routine
The sensitivity of the optimization has been analyzed by changing the domain size, goal weights and initial starting variables. As reference are the all-round optimized
4.4. ROBUSTNESS
−6
−5
−4
−3
−2
−1 0 1 2 3
−10 0 10 20 30 40 50 60 70 80 90
Constraint Value
−7
−6
−5
−4
−3
−2
−1 0 1 2 3
Constraint Value Constraint Value
xlb xub xlb xub xlb xub
Figure 4-8: Analysis of the constraint function for Z-bar linkage.
versions for each linkage used.
Z
The analysis for the Z-bar linkage is collected in Table 4.9. With expanded domain size the optimization does converge but to another minimum. The run with altered goal weight to 1:1.2 is the one nearest to the normal optimized linkage, with a maximum joint position offset of 12%. The optimization started from the upper boundary did not, unfortunately, converge during the first 5000 function evaluations but the constraint violation become very low.
Table 4.9: Sensitivity analysis for Z-bar linkage.
Domain Size Goal
Initial Variables Constraint Max variable No of variables
Weight Violation Deviation > 1% dev.*
Large (+100%) 1:1 Original Z-bar 0% 66.0% 26
Normal 1:1.2 Original Z-bar 0% 12.3% 9
Normal 1:5 Original Z-bar 0% 25.0% 16
Normal 1:1 Lower Bound 0% 58.0% 14
Normal 1:1 Upper Bound 1% 44.0% 19
*Number of variables with constraint deviation larger than 1%
TPC
The analysis for the TPC linkage is presented in Table 4.10. Increasing the domain size causes the optimization to fail in founding a feasible solution (with a constraint violation of 6%). Changing the goal weight does not influence the result much, just
moving a few joints some millimeters. The same happens when alternating the start- ing point to any of the boundaries. This is an important feature for a stable solution and implies that the minimum may be global on the domain.
Table 4.10: Sensitivity analysis for TPC linkage.
Domain Size Goal
Initial Variables Constraint Max variable No of variables
Weight Violation Deviation > 1% dev.*
Large (+100%) 1:1 Original TPC 7% 25.0% 11
Normal 1:1.2 Original TPC 0% 7.7% 4
Normal 1:5 Original TPC 0% 3.1% 3
Normal 1:1 Lower Bound 0% 8.3% 4
Normal 1:1 Upper Bound 0% 7.0% 4
*Number of variables with constraint deviation larger than 1%
4.4.3 FEM Mesh Resolution
The FEM mesh resolution for the stress calculations needs to be fine enough to give a reliable result. Therefore are 5 different resolutions tested. The finest resolution is 8 times denser than the coarsest. The result from the measurements are plotted in Figure 4-9.
Coarse Normal Fine Extra Fine Extremly Fine
0 0.5 1 1.5 2 2.5 3 3.5 4
Mesh Resolution Mesh Resolution Analysis
Execution time Optimized Mass
Figure 4-9: Influence of FEM mesh resolution on execution time and calculated mass.
4.4. ROBUSTNESS
The best mesh to use, with respect to both execution time and result reliability, is the ”Fine” mesh which is 4 times denser than the coarsest one. It is the mesh with the lowest execution time where the computational results had converged satisfactory.
That mesh was used for all the results in this chapter.
Chapter 5 Discussion
This chapter will cover a discussion about the overall performance of the optimization software developed and the result it produces. Last are some suggestions in improving the software.
5.1 Result Reliability
The sensitivity to the initial conditions (see Section 4.4.2) is quite low and suggests that the optimization works as intended to. That is an important feature to mini- mize the time spent on tuning the optimization. The domain size seem to influence the result most. This is expected since some variables are on the boundary for the optimized linkages, see Appendix C.
That the all-round linkage is lighter than the bucket version is surprising and reduces the trust in the software, which seems to be sensitive to changes in the constraint function. However, to analyze the result more thoroughly a CAD model of the result must be done, were all conditions can be verified and the mass calculations confirmed.
5.2. FUTURE WORK
5.2 Future work
For a more complete optimization of the TPC linkage must a help system for the par- allel alignment should be implemented, preferably a system with hydraulic coupling between lift and tilt. This can be done by adding extra cylinders, mechanically at- tached between the boom and front frame and coupled to the tilt cylinders hydraulic system. Since the final loading unit must include such a system, it would be of interest to include it in the optimization directly instead of correcting for it afterwards.
The parallel alignment is of high importance for some applications, e.g. fork handling, and it may be interesting to include this metric in the optimization. This could easily be implemented by just evaluating and comparing the attachment angle at different hinge pin heights. For this to make any sense on the TPC linkage must the above mentioned tilt alignment system be implemented in advance.
It could also be interesting to include the standard Volvo TP linkage in the op- timization. That way an even more thorough comparison could be done. There are probably other linkages that also could be of interest, for the same reason.
Another interesting implementation is to expand the software to consider the full machine. Then the optimization could use real drive cycles to optimize the linkage for use in a certain application (e.g. short load cycles), material (e.g. large rocks or fine sand) or with a certain tool (e.g. fork). It makes it also possible to change the objective function to minimize the fuel consumption, productivity or some other machine related physical property. More information on the subject can be found in [18].
It could also be interesting to implement automatic differentiation (as suggested in section 3.1) to decrease the execution time and increase the reliability of the result.
Changing to a non gradient based optimization method, i.e. genetic method, could also increase the performance of the software, if the rough constraint functions are causing problems.
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[2] J. Unneb¨ack. How to make a loader linkage. Technical report, Volvo Construction Equipment, 2003. Internal Volvo document.
[3] R. Filla. Optimisation of lift cylinder size and placement in the conceptual design of wheel loaders. Technical report, Volvo Construction Equipment, 2003. Internal Volvo document.
[4] H. Shin Y. Yoo J. Kim K. Shin, S. Lee. Coupled linkage system optimization for minimum power consumption. Journal of Mechanical Science and Technology, 26(2), 2011.
[5] L. Ipek. Optimization of backhoe-loader mechanisms. Master thesis, Middle East Technical University, 2006.
[6] Q. Bi J. Shen, G. Wang and J. Qu. A comprehensive genetic algorithm for design optimization of z-bar loader. Journal of Mechanical Science and Technology, 27(11), 2013.
[7] L. Shen Y. Yu and M. Li. Optimum design of working device of wheel loader.
Mechanic Automation and Control Engineering (MACE), 2010.
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[11] S. Wright J. Nocedal. Numerical Optimization. Springer, second edition, 2006.
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BIBLIOGRAPHY
[13] A. Vehreschild C. Bischof, B. Lang. Automatic differentiation for matlab pro- grams. Proceedings in Applied Mathematics and Mechanics (PAMM), 2:50–53, 2003.
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Appendix A
Design Variables
Table A.1: Design variables for Z-bar linkage.
No Description Direction Coordinate System State
1 Lower waist x Front frame Fix
2 Lower waist z Front frame Fix
3 Upper waist x Front frame Fix
4 Upper waist z Front frame Fix
5 Front axle x Front frame Fix
6 Front axle z Front frame Fix
7 P x Front frame Variable
8 P z Front frame Variable
9 E x Front frame Variable
10 E z Front frame Variable
11 O x Front frame Variable
12 O z Front frame Variable
13 Width y All Fix
14 Stabilizer x Boom Variable
15 Stabilizer z Boom Variable
16 A x Boom Variable
17 A z Boom Variable
18 C x Boom Variable
19 C z Boom Variable
20 D x Boom Variable
21 D z Boom Variable
22 GDF F x GDF Variable
23 GDF G x GDF Variable
24 GDF G z GDF Variable
25 GJ J x GJ Variable
26 Bucket A x Bucket Variable
27 Bucket A z Bucket Variable
28 Bucket J x Bucket Variable
29 Bucket J z Bucket Variable
30 Design pt 1 x Boom Variable
31 Design pt 1 z Boom Variable
32 Design pt 2 x Boom Variable
33 Design pt 2 z Boom Variable
34 Rear axle x Rear frame Fix
35 Rear axle z Rear frame Fix
36 Width D y Boom Fix
37 Lift cylinder min Variable
38 Lift cylinder max Variable
39 Tilt cylinder min Variable
40 Tilt cylinder max Variable
41 Counter Weight Variable
