System modelling of a tightening tool

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System Modelling of a Tightening Tool

CAROLINE ERIKSSON

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

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R

OYAL

I

NSTITUTE OF

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ECHNOLOGY

MASTER’S THESIS

System modelling of a tightening tool

Author:

Caroline Eriksson

Supervisors:

Kate Eriksson Prof. Jonas Faleskog

A thesis submitted in fulfillment of the requirements for the degree of Master in engineering

in the

Department of Solid Mechanics

June 15, 2020

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Abstract

With the help of modern computational power it has become much more common to employ simulation driven development. When performing simulations before creating prototypes and actual test, companies can reduce both the time spent on the development and the cost.

To get an as accurate model as possible one would ideally include all of the physics involved - in order to allow simulation of all potential scenarios and hence create a digital twin. When having a sufficiently complex model it is easy to perform simulations and thereafter give feedback to the designer, without having to build prototypes.

The aim of this project has been to create a model which captures the multiphysics of a handheld tightening tool provided by Atlas Copco. The tool contains an electric AC permanent magnet motor which through electromagnetism creates an oscillating torque acting on the rotor. The mechanical response of the driveline in the tool due to the variations in the torque was to be analysed.

The modelling of the system was split into two parts; one electromagnetic analysis concerning just the motor and one dynamic analysis concerning the driveline of the tool in order to find the arising vibrations. To detect which frequencies the system should be sensitive to, a modal analysis was performed - yielding among others a torsional mode at 3276 Hz. For investigating for which frequencies the model - with applied forces and boundary conditions - is sensitive to, a velocity ramp up test was performed. A prominent peak was detected at 3200 Hz which corresponds to the previously observed torsional mode at the same frequency.

It could thus be concluded that the system is sensitive to rotor velocities of 32 000 rpm since this causes axial vibrations in the bevel gear of the tool. Some improve- ments to the model could be made, but in order to make an even more accurate model of the tool a time span exceeding the one intended for the project would be required.

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Sammanfattning

Tack vare dagens moderna datorhjälpmedel har det blivit mycket mer vanligt att an- vända sig av simuleringsdriven utveckling. När simulering utförs innan skapandet av prototyper och riktiga tester kan företag reducera både tiden som spenderas på utvecklingen samt den totala kostnaden.

För att skapa en så precis modell som möjligt är det idealt att inkludera all fysik som är inblandad - för att möjliggöra simulering av alla potentiella scenarion. När en tillräckligt komplex modell är skapad är det enkelt att utföra simuleringar och därefter ge återkoppling till konstruktören, utan att behöva bygga prototyper.

Syftet med det här projektet var att skapa en modell som fångar multifysiken hos ett handhållet åtdragningsverktyg, försett av Atlas Copco. Verktyget innehåller en elektrisk, AC, permanent magnet motor som genom elektromagnetism skapar ett os- cillerande moment verkande på rotorn. Den mekaniska responsen som uppkommer i verktygets drivlina på grund av variationerna i momentet skulle analyseras.

Modelleringen av systemet delades in i två delar; en elektromagnetisk modell som endast innefattade motorn samt en dynamisk modell som innefattade drivlinan i verktyget för att kunna observera de uppkommande vibrationerna. För att detek- tera vilka frekvenser systemet bör vara känsligt mot genomfördes en modalanalys - vilken uppvisade bland annat en torsionsmod vid 3276 Hz. För att sedan kunna undersöka för vilka frekvenser modellen - med applicerade krafter och randvillkor - är känsligt mot, genomfördes ett test där vinkelhastigheten på rotorn rampades upp. En framträdande resonanstopp upptäcktes vid 3200 Hz, vilket sammanfaller med den tidigare observerade torsionsmoden vid samma frekvens.

Det kunde därför som slutsats konstateras att systemet är känsligt mot rotorhastigheter på 32 000 rpm då detta orsakar axiella vibrationer i vinkelväxeln i verktyget. Några förbättringar på modellen kan genomföras för att skapa en ännu mer precis modell av verktyget, men detta skulle kräva ett längre tidsspann för projektet än vad intentionen var.

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ficient ideas and support during this project has made the thesis genuinely pleasant to perform. I have always felt welcome and it has been really rewarding to work with the two of you during these months.

A great thank you to Niklas Sehlstedt, Martin Karlsson and Patrik Brauer from Atlas Copco for your expertise and for providing such a fascinating project. The many interesting discussions over the course of this project have certainly gained my knowledge in both dynamics and electromagnetism.

I am also very grateful to the guidance from Erik Magnemark, Altair. Without all of your assistance regarding the multibody model, the finalization of this project would have been considerably more difficult.

Last but not least I would like to thank professor Jonas Faleskog from KTH for your interesting thoughts and useful proposals concerning the project. Your advice concerning the structuring of the project has been really useful as well.

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Introduction

With the advanced technology of today it has become much more common with

“Simulation driven development”. This is very handy since it allows fast iterations during the product development, which is both time saving and cost saving.

As shown in Figure 1.1, simulations are preferably performed before at an early stage in the product development since the ability to impact cost is still high.

FIGURE1.1: Graph created by Altair [16] showing the benefits of simulation driven development.

The goal when performing simulation driven development is to create a model which captures – ideally - all the physics involved. This allows for simulating a product in its operating conditions and thereafter give valuable feedback to the designer. When carrying out simulation driven design engineers are able to propose the most advantageous design solutions that display the best arrangement of multiple engineering functions and constraints [9].

With the challenges of the future regarding the demand on electrification, a new requirement of highly efficient electrical machines is taking place. Electric motors are used in a great number of applications, such as electric or hybrid vehicles, fans, appliances and power tools. It is a powerful device which converts electrical energy into mechanical energy through electromagnetism [11]. Figure 1.2 displays the fundamental workings of a DC motor.

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Chapter 1. Introduction 2

FIGURE 1.2: Image from Magnet Academy [2] showing the basic functionality of a DC motor.

The black arrows in the image shows the direction of the current and the two green arrows acting on the coil presents the direction of motion of the corresponding part of the coil.

Nowadays, when constructing electric motors, more and more constraints are to be fulfilled at the same time. The motor has to be highly efficient, not too noisy, avoid vibration and at the same time - neither too hot nor too expensive. In order to be able to achieve as much of these goals as possible in the development process, multiphysic simulations has to be performed. Coupled simulations regarding the electromagetics of the motor and the mechanics of the surroundings are required since many motors have torque ripple which may cause torsional vibrations. This could generate discomfort or even harm to the user during long time use.

The purpose of this project is thus to create a model in which it is possible to investigate how the torque variations in the motor leads to vibrations which can spread and magnify through the surroundings.

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Modelling with Altair’s softwares

In this chapter the possibilities with Altair’s products are described. When performing multiphysic simulations there are many different aspects to be considered, which is why it is convenient to have different softwares attending to the different areas.

During this project mainly five different softwares created by Altair will be used for building the model. A flow chart for the simulation process is presented in Figure 2.1

FIGURE2.1: Flow chart of the simulation process in this project.

As can be seen in the flow chart, Altair Flux [3] will be used first in order to perform an electromagnetic analysis. Flux is a simulation software which is able to capture the complexity of electromechanical equipment in order to improve their efficiency, cost, dimensions or performance. The conditions viable to simulate are magneto static, steady state as well as transient, in conjunction with thermal and electrical properties [8]. For this project no thermal properties will be included.

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Chapter 2. Modelling with Altair’s softwares 4

Following the electromagnetic analysis, any modifications performed on the geometry will be done in Altair Inspire [5]. This software is very powerful when it comes to quickly create, modify or optimize a design or a topology. When the geometry is ready for meshing, the finite element pre-processor Altair HyperMesh [4] is used.

The solvers which thereafter may be used are Altair OptiStruct [7] and Altair Motion- Solve [6]. The structural solver OptiStruct has accurate, extensive and scalable solutions for linear and nonlinear analyses while MotionSolve executes 3D multibody system simulations to anticipate the dynamic response and improve the performance of mechanisms.

For analyzing the results from the simulations performed in OptiStruct as well as MotionSolve, the post-processing and visualization environment Altair HyperView as well as the plotting and data analysis tool Altair HyperGraph are suitable.

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Application to a tightening tool

The tool that will be used in this project is a handheld tightening tool from Atlas Copco, displayed in Figure 3.1 below. The tool is often used in the automotive in- dustry for tightening wheel bolts, among other things. It is 0.5 m long and weighs 2.6 kg. The driveline of the tightening tool studied here consists of an electrical motor driving a multistage planetary gear connected to a bevel gear.

FIGURE3.1: The tightening tool used in the project, provided from Atlas Copco [10].

Electric motors often have torque ripple and this may cause vibrations in the rest of the tool – which can lead to discomfort for the user or a reduced life span of the tool.

The aim of this project is to couple the electromagnetics of the tool to the mechanical response. The goal is thus to create a sufficiently complex model which captures the coupling between torque variations arising from the motor and the vibrations arising in the driveline of the tool.

Some limitations which have been made in this project are listed below:

• The planetary gear is not modelled in detail since it does not contribute signif- icantly to the vibrations.

• The bevel gear is not modelled in detail since no such geometry was available.

• The housing of the tool is not included.

• The stiffness and damping for the different bearings and joints are estimations.

• No thermal properties will be accounted for.

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Chapter 3. Application to a tightening tool 6

3.1 Electromagnetic analysis

In this chapter different electromagnetic analyses will be performed by the use of Altair Flux and some theory behind the modelling will be explained. The motor used in the tightening tool is a brushless, slotless, AC, permanent magnet motor.

A slotless type of motor has little to no cogging torque - as regular slotted motors usually have - and is able to achieve a higher torque due to the increased radius obtained when the teeth of the stator are removed. The motor is comprised of three main components: a fixed part including the stator and the windings, an air gap and lastly a moveable rotor with surface mounted magnets.

All of the data used for the electric motor in this analysis is provided by Atlas Copco, where the performance output data has been obtained in a motordesign program.

3.1.1 Geometry and mesh

Flux provides a library of predefined motor templates which simplifies the stage of the geometry construction and the mesh building. In the motor template editor a motor and winding type are chosen from the library and the parameters for the specific motor are entered [13]. The characteristics entered are:

• Rotor - Shape, dimensions and number of poles.

• Stator - Shape, dimensions and number of slots.

• Winding - Distribution of the phases in the slots.

The achieved geometry is presented in Figure 3.2.

FIGURE3.2: Half of the cross-section of the motor.

The gray parts in Figure 3.2 consists of air, the dark blue part is the stator, the red part is one of the two magnets and the yellow part is the shaft. The green, turquoise and pink parts corresponds to the three-phase windings, where the turquoise section is the negative phase.

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FIGURE3.3: The mesh used in all of the electromagnetic analyses.

3.1.2 Motor data

In Table 3.3 and 3.2 below the material data used in the coming analyses are presented.

TABLE3.1: Material data for the steel.

Relative permeability Saturation Magnetization Isotropic resistivity Density

1.50 [-] 2.07 [T] 5.0·10⁻⁷[Ω m] 7650 [kg/m³]

TABLE3.2: Material data for the magnets.

Relative permeability Remanent flux density Isotropic resistivity Density

1.49 [-] 1.35 [T] 1.6·10⁻⁶[Ω m] 7700 [kg/m³]

The magnet on the displayed half of the motor is oriented in a positive radial direction, meaning that due to the odd periodicity about the z-axis the other pole is oriented in a negative radial direction.

3.1.3 Back EMF

As the armature rotates inside the magnetic field in electric motor operation, a current is induced in the coils and thus a voltage is produced. Since the produced voltage acts against the voltage driving the motor, it is commonly referred to as back EMF (electromotive force). By computing the back EMF in the coils the initial position of the rotor can be optimized in order to align the magnetic field in the magnets

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with the magnetic field in the coils.

The back EMF can be described by the combination of Faraday’s law and Lenz’s law as

E_B = −NdΦ

dt . (3.1)

Faraday’s law relates the rate of change of magnetic flux, ^dΦ_dt, through a loop to the magnitude of the EMF, E_B, induced in the loop. Since multiple coils of wire are used the number of coil turns, N, is also a contributing factor.

Lenz’s law states that the direction of the back EMF is always such that it will oppose the change in flux which produced it, meaning it tells us the direction that current will flow. This means that any magnetic field produced by an induced current will be in the opposite direction to the change in the original field [1].

The study to be performed is a transient magnetic 2D analysis where the depth of the domain is set to the active stator length; 64.5 mm. The back EMF is computed with the speed of 6000 rpm and external circuit connections over 1 electric cycle. It corresponds to the motor being in generator mode at no load.

The circuit created for this case is presented in Figure 3.4. The coils and inductances in the circuit are corresponding to the stator coils while the resistances are represen- tative of the load of the generator. In order to compute the back EMF, the values of resistances are designated large values [13].

FIGURE3.4: The electrical circuit used for the back EMF analysis.

TABLE3.3: Characteristics of the circuit components.

Component Resistance per phase Inductance Resistance

Coilconductors 0.9387[_Ω] − −

Inductors − 0.268e−3[H] −

Resistors − − 1e4[_Ω]

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the electrical angle and 360^◦rotation of the rotor is required in the simulation.

After having performed the transient electromagnetic simulation, the optimal initial position for the rotor can be determined from the graph of the induced voltages in the coils versus the angle of the rotor which is presented in Figure 3.5. From the graph in this figure it is possible to determine the angle of the different phase currents through the zero-cross over of the induced back EMF waveform for each phase. The zero-cross over of the first phase corresponds to the optimal initial angle of the rotor. Hence, the rotor should be turned 30^◦ when starting the following simulations.

FIGURE 3.5: The voltages produced in the coils when rotating the rotor.

TABLE3.4: Mechanical- and electrical angles for the three phases.

Phase Mechanical angle Electrical angle

1 30 30

2 150 150

3 270 270

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3.1.4 Constant speed

After identifying the optimal initial position for the rotor, the next step is to determine the current control angle. In this analysis the motor is driven with a 3-phase cosine current running at constant speed. The simulated motor performances are used to compute the shaft torque and thus the torque ripples. The current functions used in the simulation are given in Eqs. 3.3 - 3.5, where the angular velocity ω equals 36 krpm.

I₁ = Imaxcos(ωt+γπ/180) (3.3) I2 = Imaxcos(ωt+γπ/180−2π/3) (3.4) I3 = Imaxcos(ωt+γπ/180−4π/3) (3.5) The shift angle (current control angle) is expressed as γ and it is the angle between the phase current and corresponding induced voltage in the phase. The peak value of the current, I_max, is in this study set equal to 2.635 A. By performing a static analysis with the rotor fixed at the initial position determined in Section 3.1.3, the shift angle giving the maximum torque on the rotor can be determined.

The circuit from the previous study needs to be altered such that it includes current sources for every phase. The eddy current patterns in each magnet are included in this model and are enforced with the zero total current constraint. This is done by representing the magnet as a solid conductor connected in series with a large resistance of the same value as the previous ones. Figure 3.6 displays the renewed circuit.

FIGURE3.6: The altered electrical circuit.

The static analysis with the rotor fixed at 30^◦generates the graph of the torque versus the control angle displayed in Figure 3.7. As the figure implies, the maximum torque is obtained when the shift angle is 270^◦and hence this is the angle γ is set as.

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FIGURE3.7: The torque on the rotor versus the shift angle.

After entering the determined value for the shift angle a second simulation is performed - this time a transient. The initial rotor position is still set to 30^◦ so that the phase current is aligned with the phase back emf when γ is zero. Figure 3.8 shows the achieved torque on the magnets with the given data after carrying out the transient analysis.

FIGURE3.8: The torque on the magnets versus the rotation angle.

The mean value of the torque is equal to 0.12046 Nm, which corresponds quite well to the given data from previous simulations in a different program performed by Atlas Copco. In the given performance output data a mean torque of 0.15 Nm had been obtained when the windings where fed with a current having a peak value of 2.635 A. The difference of about 0.03 Nm might be because of differences in the mesh used, details in the windings or slight differences in the material used. Since the focus in this project is not on the performance of the motor, this small distinction is acceptable. In the above graph one can also see that the amplitude of the torque

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is 0.0018 Nm and the frequency of the torque is six times the frequency of the rotor, since the torque curve performs three full cosines on half a revolution. The following relation is obtained

ftorque=6·frotor. (3.6)

The obtained flux in the motor is displayed in Figure 3.9.

FIGURE3.9: The magnetic flux density and the isolines of the magnetic vector potential normal component.

The figure also shows the isolines of the magnetic vector potential normal component, A_n. In 2D, the magnetic vector potential is reduced to its normal component perpendicular to the represented cross section. In the area where the lines are crowded, it corresponds to high values of flux density (highly saturated). In the area where the lines are sparse, it corresponds to lower values of flux density [13].

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very complex and is not a crucial factor for the simulations to be performed in this project. As can be seen in the image as well, the structural model is limited to the driveline itself and will not include any surrounding structure.

FIGURE3.10: A cross section image of the geometry provided by At- las Copco with the performed modifications.

3.2.1 Finite element model

Finite element models of structures can involve a substantial number of degrees of freedom (DOF) which entails extensive computational effort. Substructuring and component mode synthesis techniques involve separation of the whole structure into several substructures or components. The main idea is to derive the behavior of the entire assembly from its constituents. The individual substructure problems are first solved and then the coupling of interfaces is posed. The benefit of partitioning a single large problem into several reduced order problems is reduced simulation time and size of the model [15].

The geometry is divided into five different substructures - depending on their angular velocity. The substructures are then independently meshed in HyperMesh with mainly hexahedral elements of an approximate size of 0.96 mm, displayed in Figure 3.11.

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FIGURE3.11: The simplified and meshed geometry used in the analyses.

At every location in the model where a load or a constraint is to be applied, multi point constraints in the form of RBE3 "spiders" are used.

RBE’s are rigid geometrically one-dimensional elements, where there is a link from one node to one or multiple other nodes and the motion of the node(s) is governed by the degrees of freedom one choose to connect. Both RBE2 and RBE3 are used to transfer loads from one part to another, but unlike RBE2 elements which do not allow any relative motion between the nodes they are attached to (creating rigid body displacement), RBE3 spiders does not add stiffness to the structure.

The RBE3 “rigid” element is thus not actually rigid, however, it also is used to connect the DOF’s at one or (typically) multiple independent nodes to a single dependent node. In practice, the RBE3 is often used to distribute the effect of a force, mass or constraint acting at the dependent node over the independent nodes.

The motion of the single dependent node in an RBE3 element is a weighted average of the motions at the independent nodes – based on the weights one choose for the independent nodes corresponding to the proportion of the load they take as well as the DOF’s one choose to connect [12]. In Figure 3.12 below some of the RBE3 elements used are displayed.

FIGURE3.12: The RBE3 elements displayed as colorful "spiders".

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vious section a simple model containing just the motor and the first planetary gear step is created to begin with.

FIGURE3.13: The simplified model used for validity check.

The left end of the planetary gear stage is kept fixed. After performing a modal analysis on this substructure a first eigenvalue of approximately 1700 Hz was found.

To check the validity of this value, an analytical torsion analysis of the same substructure was performed. The analytical model is displayed in figure 3.14 and the equations of motion used are presented in Eqs. 3.7 and 3.8.

FIGURE3.14: Analytical torsion model representing the rotor and the first planetary gear stage.

J₁¨θ₁+ (k_θ,1+k_θ,2)θ₁−k_θ,2θ₂=0 (3.7) J₂¨θ₂−k_θ,2θ₁+k_θ,2θ₂ =0 (3.8)

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Mass number one corresponds to the first planetary gear stage and mass number two corresponds to the motor and the shaft. The values for the torsional stiffness, k_θ, and the inertias, J, are determined from the CAD software Altair Inspire which was used for modifying the geometry of the tool. When solving for the eigenvalues of the problem a first eigenfrequency of approximately 1200 Hz was obtained. This is not completely the same as obtained in the numerical analysis presented above, but the difference is reasonable since the analytical model is simple and likely to be less stiff than the actual model due to the placement of mass number two. It is thus safe to say that the model is valid.

After concluding that the flexible bodies are accurate, the rest of the tool is built and is displayed in figure 3.15.

FIGURE 3.15: The entire multibody modell with applied boundary conditions.

At each location where a bushing or a constraint is to be applied, revolute joints are used. The revolute joints are compliant and assumed to have a stiffness of 10⁵N/mm in the translational radial directions, 10⁶N/mm in the translational axial directions and 10⁹N/mm in rotation about the radial directions. They also have a damping of 100 Ns/mm in the translational directions and 1000 Ns/mm in the rotations about the radial directions.

In the planetary gear as well as the bevel gear, exchange of torque is taking place.

The torque is increased for each stage and the angular velocity is hence reduced for each stage. This exchange is in the software modelled with a "coupler".The coupler is a constraint between two joints in two different bodies and can be expressed as

θ₁·x+θ₂=0, (3.9)

where θ₁and θ₂are the rotations of the two bodies at the specified joints and x is the gearing ratio.

3.2.3 Bevel gear

A detailed CAD geometry of the bevel gear was not available. Therefore, the goal is to create a simplified but still accurate model of the torque transfer was created as well as the corresponding unbalance forces occurring at the wheel engagement. In a real bevel gear the wheel engagement point moves during the time of contact of the two teeth, but in this simplification the point is assumed to be fixed.

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The constraint implies that the rotation about the global y-axis of the bevel, θ_B,Y, is 1.83 times less than the rotation about the global z-axis of the pinion, θP,Z.

As mentioned, when the gears rotate, unbalance forces occur in the axial and radial directions on both gears, see Figure 3.16. These unbalance forces create lateral vibrations in the tool.

FIGURE3.16: Image from KHK Gears [14] showing the forces acting on the bevel gear.

With data obtained from Atlas Copco it was found that for a driving torque on the pinion equal to 1 Nm, the axial force acting on the pinion is 49.5 N and the radial force is 38.6 N. As these unbalance forces are related to the torque on the pinion, the forces will oscillate with the same frequency as the torque ripple.

To be able to apply the axial and radial forces on just the part of the gears where the teeth are in contact - despite the fact that the gears rotate - deformable curves are created. These curves are attached to ASET nodes created on the wheel engagement planes, thus creating one ring on the bevel and one ring on the pinion. Since the curves are attached to the ASET nodes, they will rotate with the bodies.

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FIGURE3.17: Close up of the bevel gear showing the rings of ASET nodes and the "inplane" joint used for the dummy bodies.

Two dummy point mass bodies with negligible mass are then applied at the location for the wheel engagement point on the curves. To attach the dummy bodies to the curves so called "point-to-deformable curve"-joints are used. These joints allows the dummy bodies to slide along the deformable curves. The dummy bodies are also constrained with an ”inplane joint” fixing the translation in the tangential direction, to make sure that they do not move in the tangential direction. Meaning – in this case – that the dummy bodies will stay in place as the curves rotate with the gear bodies.

3.2.4 Modal analysis

In order to perform a modal analysis of the system, a combination of MotionSolve and OptiStruct had to be used. This is due to the fact that the multibody model has constraints defined for the intersection point in the bevel gear in MotionSolve, but this software does not allow for performing modal analyses of an entire system - which instead can be done in OptiStruct. To know how to connect the intersection points of the bevel and the pinion in order to get an as accurate modal analysis as possible, the stiffness of the two points with the constraints defined in MotionSolve is measured. This is done by applying a torque of 1 Nm on the pinon and then keeping the bevel fixed close to deformable curves, making only the top part of the bevel able to rotate. The obtained stiffness is then used as the tangential stiffness for a spring element applied between the intersection points in OptiStruct. The obtained eigenfrequencies after performing the modal analysis are presented in Table 3.5.

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Mode 1 and mode 3 are torsional modes while mode 2 and mode 4 are bending modes.

3.3 Results

After modelling the driveline in the tool with the multibody simulation software, Altair MotionSolve, the next step was to perform an analysis where an angular velocity and a resistive torque were applied to investigate lateral vibrations. From the electromagnetic constant speed test in section 3.1.4, a relation between the frequency of the torque ripple and the frequency of the rotor was obtained, as well as the mean value and the amplitude of the torque. This can thus be expressed as:

T =0.12+0.0018·sin(2π ftorque·time) (3.11) Where T is the resitive torque to be applied to the output shaft and f_torqueis presented in equation 3.6.

A velocity ramp up test is performed where the angular velocity applied at the rotor is increased from 0 rad/s to 5000 rad/s in 1.5 s with a time step of 10⁻⁵ s. During this analysis the mean torque and the amplitude of the torque is kept constant. When observing the resulting axial displacement of a point on the deformable curve on the pinion and performing a fourier transform in order to get the result in frequency domain, the graph in Figure 3.18. The same fourier transform is performed on the axial displacement of the bevel, resulting in the peak shown in Figure 3.19.

FIGURE3.18: The displacement in the axial direction of an intersection point on the pinion.

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FIGURE3.19: The displacement in the axial direction of an intersection point on the bevel.

The prominent peak at approximately 3200 Hz, seen in Figure 3.18 corresponds to a rotor velocity of 32 000 rpm. The same critical frequency can be found in Figure 3.19 for the bevel, although the peak is not as prominent.

When comparing these results to the results obtained from the modal analysis, it can be seen that it corresponds to the torsional mode at 3276 Hz. Since the unbalance forces on the bevel gear are dependent on the torque ripple (as mentioned in section 3.2.3), it is reasonable that these vibrations in the axial directions can be caused by a torsional mode. It can thus be concluded that the system is sensitive to rotational velocities of 32 000 rpm since this velocity causes magnified axial vibrations for the bevel gear.

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Discussion

4.1 Application discussion

When looking at the electromagnetic analysis, there are several aspects which most likely has an impact on the result. If the motor is driven with a perfect cosine shaped current - as it has been in this project - it is not likely that large vibrations will arise in the motor. The air gap between the phases could also have an impact on the result as this was not specified.

The gears in this model have flexible bodies but do not exhibit the non-linear behaviour that occurs in real gears. When gears are rotating the point of contact moves along the teeth during the wheel engagement. If such behaviour would have been implemented, the results would have been more accurate. In reality small moments where no teeth are in contact occur which affects the stiffness of the gears as well.

The transfer of torque is in the model divided equally over the surface of the gears, but this is not an ideal way of modelling since the torque is actually only transferred through the tangential force acting at the wheel engagement point. A better rep- resentation of the torque transfer could be made but would be difficult since the tangential force needs to be applied at the correct location and at the correct time.

The simplification is thus appropriate for this project.

The values of the stiffness and damping of certain components and in particular the bearings have been assumed to some values, as accurate estimation of these are out of scope for the current project. To know if the results from the simulations are valid real life tests should be performed to have something to compare the simulations to. The results from the velocity ramp up test in section 3.3 is justifiable, but if the damping or stiffness would be changed additional results might be obtained. In this test a time step of 10⁻⁵seconds was used which is quite small - however it could be worth reducing the time step even more.

Future work on this model could be to include a more detailed geometry of the bevel gear and the planetary gear. By having detailed geometry with teeth, the problem with the moving contact point for the bevel gear could be solved. It would also be of interest to consider thermal effects in order to make the model more accurate, as well as a control system. Some vibrations in the real tool might be due to imperfec- tions in the control system, hence it would improve the model if a control system was included. Future work could also be to create a "dashboard" where the different physics are coupled and it is easy to enter different values for the relevant parameters, e.g. motor type, winding specifics, angular velocity, required torque or bearing stiffness.

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Chapter 4. Discussion 22

4.2 General discussion

To perform multiphysic simulations as done in this project is very complex. The aim of creating a digital twin is hard to achieve since creating a completely accurate model involves so many aspects.

When producing electric motors in real life the tolerances allows for slight variations. These variations could as an example be the placement of the rotor - it is not always perfectly concentric. The eccentricity might create torque ripple which is not possible to see in the perfectly concentric simulations. Lateral vibrations spreading to the rest of the tool are likely to occur as well. The same goes for the windings in the motor; during application they may be pressed together or be skewed, which affects the directions of the electromagnetic fields induced in the coils, hence also the torque on the rotor.

Generally there is always some uncertainty about the tolerance in the production of products which makes it difficult to perfectly predict the behaviour of the product with simulations. However, substantial information is usually gained from creating sufficiently complex models and performing as many analyses as the time limit allows. Simulation driven development is thus something which should be practiced.

A model of a product can easily be expanded in order to make it more complex and capture more real life aspects.

When performing simulations it is always advantageous to have real test results to compare the simulation results to - which was not available in this project. Actual measurements of stiffness and damping properties on a system level will help create a more accurate model and validate the observed outcome.

It is hence possible to create very complex models, but be that as it may, with increased model complexity the simulation time increases in like manner. In order to capture the linear and non-linear behaviours as well as all of the physics involved, one have to brace the simulation time required.

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Conclusions

The work is summarized and concluded as follows:

• The model of the tightening tool is sufficiently complex to capture the effect of the electric motor on the driveline.

• The system has torsional modes at 664 Hz and 3276 Hz. Bending modes oc- curred at 1940 Hz as well as 5763 Hz.

• The system is sensitive to rotor velocities of 32 000 rpm.

• Magnified vibrations occur in the axial directions of the bevel and the pinion at the above mentioned critical velocity, which can be related to the torsional mode at 3276 Hz.

• The model can easily be expanded, which would benefit the accuracy of the results.

• Having a really complex model is unquestionably ideal when conducting design driven development, but the modelling and simulation time are important aspects that needs consideration.

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System modelling of a tightening tool

System Modelling of a Tightening Tool

CAROLINE ERIKSSON

R

I

T

System modelling of a tightening tool

Abstract

Sammanfattning

Contents

Introduction

Modelling with Altair’s softwares

Application to a tightening tool

Discussion

Conclusions

Bibliography