Dynamic behaviour of high-speed rotor

Alícia Santarén Rodríguez Dynamic behaviour of high-speed rotor Thesis submitted as partial fulfilment of the requirements for the degree of Masters in Science (Technology) Espoo 27.03.2017 Supervisor: Professor Kari Tammi AALTO UNIVERSITY School of Engineering Department of Engineering Design and Production 1 Author Alícia Santarén Rodríguez Title of thesis Dynamic behaviour of high-speed rotor Degree programme Master’s Programme in Mechanical Engineering Major Mechanical Engineering Code IA3027 Thesis supervisor Professor Kari Tammi Thesis advisor Professor Kari Tammi Date 27.03.2017 Number of pages 57 Language English Abstract High-speed electrical motors are increasing the application fields where they are used, and vehicle applications are one of them. These motors offer advantages when compared to conventional ones, mainly because of the fact that they offer a high power in a relative small size. For that reason, their application in mobility systems is increasingly under study, since they arise as a powerful alternative for sustainable transportation systems. This thesis is part of a project in which a high-speed electrical motor for vehicle applications will be designed. In the development of the motor there are several design fields that need to converge, mainly the mechanical, electrical, structural and thermal aspects, always considering the cost-efficiency of the project. In this thesis, the analysis of the dynamic behaviour of one part of the motor, actually the rotor, will be conducted. In order to do that, the rotordynamic aspects which have great influence in the behaviour of the motor will be described. The most important dynamic features, such as the critical speed, the natural frequencies and the vibration modes, as well as the stress due to centrifugal forces that take place in the operational speed range will be determined. Keywords High-speed rotor, dynamic behaviour, critical speed, natural frequency, centrifugal force, Von Misses Stress, FEM 2 Acknowledgement This Master’s Thesis was written in the School of Engineering at Aalto University as part of a mobility program. I would like to thank my thesis supervisor, Professor Kari Tammi, for giving me the opportunity to be part of his project, and for all the guidance and support since my arrival to Finland. I also would like to thank master students Jesse Mäntylä and Kevin Campbell for all the knowledge and help they offered during this thesis. Finally, special thanks to my lovely family, who have unconditionally supported me, to my friends, for their help and positivity, and to all the amazing people I have met during my stay in Helsinki. Alícia Santarén Rodríguez Espoo, 27.03.2017 3 Table of contents Abstract Acknowledgement Table of contents .................................................................................................................... 3 Abbreviations and Symbols ................................................................................................... 4 1 Introduction .................................................................................................................... 5 1.1 Background ............................................................................................................. 5 1.2 Research problem .................................................................................................... 5 1.3 Objective ................................................................................................................. 6 1.4 Scope of the study ................................................................................................... 6 2 State of the art ................................................................................................................ 8 2.1 The high-speed electrical motor .............................................................................. 8 2.1.1 Introduction ...................................................................................................... 8 2.1.2 The designed motor ......................................................................................... 9 2.2 Theory of the dynamic behaviour ......................................................................... 14 2.2.1 General aspects of vibrations ......................................................................... 14 2.2.2 Introduction to rotordynamics ....................................................................... 18 2.2.3 Single degree of freedom rotor model ........................................................... 18 2.2.4 Rankine rotor model ...................................................................................... 20 2.2.5 Jeffcott rotor model ........................................................................................ 20 2.2.6 Multi-DOF system ......................................................................................... 23 2.2.7 The Finite Element Method ........................................................................... 24 2.2.8 Dynamic factors limiting the rotor design ..................................................... 24 3 The computational analysis ......................................................................................... 26 3.1 Computational simulation ..................................................................................... 26 3.1.1 Computational analysis process ..................................................................... 26 3.1.2 Methods of discretization ............................................................................... 27 3.1.3 Tools and software ......................................................................................... 28 3.2 The rotor model ..................................................................................................... 29 3.3 The mesh ............................................................................................................... 31 4 Dynamic analysis and results ....................................................................................... 32 4.1 Natural frequencies and modes ............................................................................. 32 4.1.1 Free rotor model ............................................................................................. 32 4.1.2 Mesh generation ............................................................................................. 33 4.1.3 Analysis and results ....................................................................................... 40 4.2 Critical speed ......................................................................................................... 46 4.2.1 Rotor model with bearings and mesh ............................................................ 46 4.2.2 Analysis and results ....................................................................................... 47 4.3 Effect of centrifugal force ..................................................................................... 49 4.3.1 Analysis and results ....................................................................................... 49 5 Discussion and further studies ..................................................................................... 54 6 Conclusion ................................................................................................................... 56 References ............................................................................................................................ 57 4 Abbreviations and Symbols 2D Two dimensions 3D Three dimensions CAD Computational Aided Design DOF Degree Of Freedom FA amplification factor FEM Finite Element Method RPM Revolutions Per Minute STEP Standard for the Exchange of Product c damping C symmetric damping matrix f(t) time-dependant vector with the forcing functions Fp force amplitude f0 natural frequency f oscillation frequency E Young’s modulus G skew-symmetric gyroscope matrix H skew-symmetric circulatory matrix I moment of inertia k stiffness K symmetric stiffness matrix m effective mass M symmetric mass matrix N shape functions n1 first critical speed q(t) vector with the generalized coordinates – inertial frame r radius of the rotor u vector of displacement xp amplitude of movement xstatic static movement x(t) movement function 𝜑 initial phase angle ζ damping ratio σ stress ρ frequency ratio ρ density of the material ν Poisson’s coefficient ω angular velocity of the rotor 5 1 Introduction 1.1 Background The creation of alternative ways of transportation is a matter of study in our society. Nowadays, we are seeking for lowering the ecological footprint that we humans produce on the Earth, as we are now more aware of the environmental impact that we can cause to our planet. The main goal is to reduce the pollutant emissions that we produce with our activity and one of the key factors to achieve that is by making some changes in the transport system. Traditionally, motor vehicles have worked with combustion engines which energy source is fossil fuel and which have high emission levels. The trend for improving the efficiency of these systems is the introduction of efficient electrical machines. Electrical machines can replace traditional combustion engines and they can also become integrated and directly coupled with mechanical systems, which can improve greatly the efficiency of the system [1]. A big step in the improvement of electrical motors is the development of the high-speed. High-speed machines have been under study and development for a long time for the great advantages they offer compared to the machines with lower rotating speed. A high-speed machine used for power generation can convert more power because it is rotating faster, although it is using the same torque [2]. For that reason, when comparing a high-speed machine with a conventional machine, one of the main advantages of a high-speed machine is that for delivering the same amount of power, the system weight can be considerably lower [3]. Moreover, high-speed machines are direct drive applications, which eliminates the need of intermediate gearing and then they are considered to be more reliable machines [3]. The use of high-speed in electrical motors has several advantages. The rotating speed increases, and so does the power and torque density, which means weight savings and a reduction in the motor size [1]. These facts create a light and powerful motor, which can be ideal for its use in vehicle systems. 1.2 Research problem High-speed electrical machines working principle is based on the high energy densities which are achieved thanks to the high speed of the rotor. This high speed can lead to some problems regarding the high inertial loads as well as problems with the shaft whirl, vibrations and rotordynamic instability [4]. There can also appear problems regarding the cooling and lubrication of the whole motor, which can produce high losses in the motor. In order to achieve a proper design for the high-speed electrical motor under development in this project, it is imperative to integrate all the aspects that may affect the motor into the study. The design of the electromagnetic area, the mechanical area, and the thermal area need to be carefully developed, as well as the analysis of the rotordynamic and structural aspects. All these design areas cannot be developed separately, since there is a high de- 6 pendence of all the fields. For that reason, the design of the motor is considered to be a strongly iterative process. One of the biggest challenges in the design of a high-speed electrical motor that differ from the design of a conventional electrical motor is the importance of the thermal and mechanical properties. These aspects need to have more importance in the design, since due to the high rotating speed the centrifugal forces in the rotor may be unusually high and that elevated rotating speed can also lead to a sudden overheating of the motor, with can cause several and irreversible damages in the motor. For the reasons exposed above, the design of a high-speed electrical motor is different from the design of a conventional electrical motor, and new procedures and analysis in the design need to be performed. 1.3 Objective The current Master Thesis will give continuity to the study and development of a highspeed electrical motor for electrical vehicle applications, and will follow the work done before by master student Mäntylä on electro-magnetic design, where the design of the rotor was performed. From all the fields that need to be covered to develop the design of a highspeed electrical motor, this thesis will focus on the analysis of the dynamics of the designed rotor. One of the objectives of this project is to develop an analysis of the dynamic behaviour of the high-speed rotor and to validate and change the pertinent design choices of the preliminary design based on the results obtained. The most important and basic aspects for assuring a good behaviour of the rotor should be determined, and the main aspects that need to be covered are the determination of the critical speed of the rotor, the natural frequencies, the vibration modes and the stress caused by the centrifugal forces that occur when the rotor is working in its speed range. In order to conduct a proper analysis and to give answer to the aspects listed above, a literature review concerning the most important rotordynamic aspects will be performed and some different ways for carrying out the analysis of the rotor behaviour will be described. The other objective of this thesis is the use of Finite Element Method (FEM) tools to develop the analysis of the dynamic behaviour of the rotor and discuss the suitability of the use of this method in the determination of the dynamical behaviour of the rotor. 1.4 Scope of the study As described in the previous section, this study will focus on some of the dynamic aspects of the rotor. The rotordynamic field has infinity of features that can be determined and the analysis to determine them can be highly complex. The motor under development is still in a previous design phase, and the design is expected to experience changes in future studies. 7 For that reason, there is no need to develop a complex study in this design step, and the determination of some of the basic and most important features in the dynamic behaviour will be enough for the purpose of this thesis. The mechanical behaviour of the whole motor will not be covered in this thesis, as it will only focus on the rotor part. In further studies, when the design of the whole motor is completed, a wide rotordynamic study should be developed in order to assure a proper behaviour of the motor as a whole. Moreover, the analysis of other design fields which need to be covered in the design of the motor, such as the gearing system, the cooling system and the lubrication system is out of scope of this thesis, although they should be studied in further studies. 8 2 State of the art The aim of this chapter is to provide insight into the high-speed electrical motor, by analyzing its most important aspects as well as its functions. Moreover, the theoretical basis for a suitable dynamical analysis will be established. 2.1 The high-speed electrical motor 2.1.1 Introduction A motor is considered to be a high-speed one when its surface speed is over 150 m/s, when its rotational speed is over 10.000 rpm and, usually, it works with high control frequencies [5]. The main advantage that the high-speed offers is a high power density in the motor, since compared with a standard motor of the same size, the output power is higher. Nevertheless, the global efficiency in a high-speed motor can be lower than in a standard motor. For that reason, when designing a high-speed motor, one of the main objectives is to keep a power/mass ratio as high as possible, to obtain the maximum power at the highest speed with the lowest motor size and weight. But the maximum power and the speed of the device are limited by mechanical, thermal and material constraints [6]. Due to its compact size, the high-speed electric motor is of special interest for direct drive applications. When connecting the electric motor directly to the device, there is a huge reduction of unnecessary gearing and that fact allows avoiding part of the friction losses. The design of a high-speed motor can be separated into 5 categories: the electromagnetic design, which focuses mainly in the design of the stator and the rotor; the mechanical design, which includes the gearing and bearing systems, sealing and casing; the rotordynamic design, which approaches the analysis of the natural frequencies and critical speeds; the structural analysis, which analyzes the centrifugal forces and the stress of the motor; and the thermal design, which includes the design of the cooling system. For a proper behaviour of the motor, the thermal and mechanical properties should be carefully analysed. Due to the reduced dimensions of the motor and the high rotating speed, the heat losses can be important, and the thermal properties of the motor should be accurately considered. Moreover, it is important to develop a proper refrigeration system, since an unexpected overheating can cause an irreversible damage to the motor. Furthermore, the mechanical properties have to be specially considered because factors such as stress and vibrations can cause the failure of the machine [1], and the high-speed of the motor can produce high centrifugal forces. Another aspect to be considered is the natural frequencies, which should not be near the operational range of the motor. 15 When the system is in motion, the oscillation takes place at the natural frequency f0, which is a property of the system, 𝑓0=1 2𝜋 𝑘 𝑚 (2) The movement of this kind of vibrations, which are the simplest ones, is a harmonic periodic movement, which can be described from the sinusoidal function, 𝑥(𝑡) = 𝑥𝑝cos⁡(2π𝑓0𝑡+𝜑) (3) But in real systems there are dissipative forces which introduce energy losses to the system, and this fact decreases exponentially the amplitude of free vibrations over time. The dissipative forces are modelled with a damper c. Figure 6. Mass-spring-damper model, free vibration In Figure 6, the damping constant is added to the model in equation 1, and the equation of motion is, 𝑚𝑥 +𝑐𝑥 +𝑘𝑥= 0 (4) The damping ratio ζ is, ζ=𝑐 2 𝑘𝑚 (5) The oscillation frequency f is, 𝑓=𝑓0 1−ζ2 (6) And the movement function of the free vibration is, 𝑥(𝑡) = 𝑥𝑝𝑒−ζ2π𝑓0𝑡cos⁡(2π𝑓𝑡+𝜑) (7) 16 As it can be seen in equation 7, the movement of the system depends on the oscillation frequency, on the rigidness constant, the damping constant and the inertia. The bigger the damping of the system is, the faster the amplitude decreases, and the oscillations are slower. 2.2.1.2 Forced vibrations Forced vibrations occur when a motor or an actuator cause an oscillation to the system. The oscillating force can appear due to the fast motion of the masses in an alternative movement, or due to the static and dynamic unbalance of a rotor. When a system is forced to vibrate at the same frequency as that of the excitation, appears the resonance. Resonance has to be avoided for the safety of the structure. Damping has a great influence on the amplitude and phase angle in the frequency near resonance. To avoid resonance, damping forces can be added to the system. Figure 7. Mass-spring-damper system, forced vibration The equation of motion of the system in Figure 7 is, 𝑚𝑥 +𝑐𝑥 +𝑘𝑥=𝐹(𝑡) (8) The equation of motion can be solved as the superposition of two parts. In the first part, the homogeneous differential equation is solved for F(t)=0, which corresponds to the freedamped system of equation 4. The equation is solved for some given initial conditions, which describes the initial transitory state as a free vibration which vanishes. In the second part, the particular solution of the complete function is solved for F(t)≠0, which describes the permanent state meanwhile there is the oscillating force. The excitation force can be sinusoidal with force amplitude of Fp and frequency f. 𝐹(𝑡) = 𝐹𝑝cos⁡(2π𝑓𝑡) (9) The movement function is then, 𝑥(𝑡) = 𝑥𝑝cos⁡(2π𝑓𝑡+𝜑) (10) 17 The movement amplitude xp and the initial phase angle 𝜑 depend on the system parameters and the frequency ratio ρ, which is the relation between the frequency of the excitation force f and the natural frequency f0, 𝜌=𝑓 𝑓0 (11) 𝑥𝑝=𝐹𝑝 𝑘1 (1 −𝜌2)2+ (2𝜁𝜌)2 (12) 𝜑=−arctan⁡ 2𝜁𝜌 1−𝜌2 (13) The amplification factor FA is the relation between the movement amplitude xp and the static movement xstatic, which corresponds to the movement of the system if the excitation is constant. 𝑥𝑠𝑡𝑎𝑡𝑖𝑐 =𝐹𝑝 𝑘 (14) 𝐹𝐴=𝑥𝑃 𝑥𝑠𝑡𝑎𝑡𝑖𝑐 =1 (1 −𝜌2)2+ (2𝜁𝜌)2 (15) The oscillating forces, generally, cannot be avoided, but some design choices can be taken in order to minimize their effects on vibrations and to prevent undesirable situations. In electrical motors, the oscillating forces can be produced mainly due to rotor unbalance. This unbalance is inevitably present in motors, and it can come from the winding, from the non-uniformity of the stator and rotor magnetic fields, from mass unbalance in the manufacture of the rotor, due to the non-uniformity of the materials, etc. 2.2.1.3 Linear and non-linear vibrations Oscillatory systems can be classified as linear or non-linear. For linear systems, the mathematics methods for their solution are well defined, and the superposition principle can be applied. On the other hand, for non-linear systems, the methods for their solutions are not as well known and they are more difficult to apply. 18 2.2.2 Introduction to rotordynamics Rotordynamics can be defined as the field in system dynamics which deals with mechanical devices that have, at least, one part that rotates with significant angular moment, the part known as rotor [2]. Its study is of special importance at the beginning of the design phase of the high-speed machines, since the rotation in such elevated speeds has a great influence on the dynamical behaviour of the rotor. The dynamic behaviour is affected with high inertial loads, and there could appear problems associated with shaft whirl, vibrations, and rotordynamic instability, the main subjects under study in the rotordynamics field [4]. When the proper studies are carried out at the beginning of the design phase, these harmful effects can be limited or even avoided. But the rotordynamic analysis is not only limited to the designing step, it has to be taken into account during the whole useful life of the machines. According to Vance [4], with the use of rotordynamics, critical speeds of the rotor, those speeds at which vibration reaches its maximum because of the rotor imbalance, can be predicted from the design data and therefore, they can be avoided in the operating speed range of the rotating machine. When the rotor needs to be operating in the range of the critical speed, rotordynamics is helpful to determine design modifications in order to establish a new critical speed. It is also useful to predict natural frequencies of torsional vibration when an entire train system is involved, as well as to prognosticate the amplitudes of synchronous vibration and the effects of rotor imbalance and system damping at specific locations. Moreover, for dynamic instability, the threshold speeds and vibration frequencies can be predicted, and design modifications can be done to suppress the instability. Rotor unbalance is an important source of vibrations. There are two types of rotor unbalance [9]: a) Static unbalance: The static unbalance is common in thin rotor disks where the unbalanced masses are all in a single plane and the resultant is a single radial force. This kind of unbalance can be detected without spinning the wheel, because the heavy point will roll to a position below the axle. b) Dynamic unbalance: In the dynamic unbalance is common in long rotors where the unbalanced masses are in more than one plane, and therefore, the resultant is a force and a moment. For detecting this kind of unbalance it is necessary to spin the rotor. 2.2.3 Single degree of freedom rotor model The simplest model to represent the vibration of a rotor-bearing system with just one degree of freedom is the system formed by a rigid mass, the effective mass m, mounted on a linear spring, effective stiffness k, as shown in Figure 8. The system can be represented in two different ways, depending on the stiffness of the rotor: it can be either considered as a rigid rotor, Figure 9 a), or a flexible rotor, Figure 9 b). When the rotor is relatively rigid compared to the bearing supports, the effective mass is the total mass of the rotor and the effective stiffness is the stiffness of all the bearing supports taken in parallel, equation (16). 19 When the rotor is relatively flexible, the stiffness is determined by the bending stiffness of the shaft, equation (17). [4] Figure 8. Spring-mass system 𝑘= 2𝐾 (16) 𝑘=48𝐸𝐼 𝑙3 (17) a) b) Figure 9. Spring-mass system of a) rigid rotor and b) flexible rotor In this single degree of freedom system of the rotor-bearing model, the first critical speed can be approximated by the natural frequency: 𝑛1=60 2𝜋 𝑘 𝑚[rpm] (18) This single DOF model has important limitations, because it can not reflect the real motion of the rotor-bearing system. In this simplified model, the only movement of the system is a translational motion in one direction. The whirl orbits of the rotor-bearing system are not represented in the model, so it is not suitable for rotordynamic analysis. 20 2.2.4 Rankine rotor model The Rankine rotor model represents an improvement to overcome the limitations of the single degree of freedom system. The Rankine model is a spring-mass system with two degrees of freedom that allows the vibration in two directions simultaneously, which can reproduce different patterns of motion of the mass such as circular orbits, elliptical orbits or translational motion. [4] Figure 10. Rankine rotor model. Adapted from [4] The model is formed by a rigid mass whirling in a circular orbit with an elastic spring acting in the radial direction, as shown in Figure 10. Rankine model proved that the frequency at which the shaft vibrated transversely was the same as the whirling speed and defined this speed as the limit speed for the stability of the rotor [11]. He concluded that the first critical speed could never be exceeded in a rotating machine, statement that was erroneous. 2.2.5 Jeffcott rotor model The Jeffcott rotor is the most used model to describe rotordynamics and is a tool to analyse the response of high speed rotating machines to rotor imbalance. It is a simple model represented by a point mass attached to a massless shaft that is used to study the flexural behaviour of rotors. Although the model is a simplification of real rotors, it offers a good understanding of some important characteristics of rotordynamics. The model consists of an unbalanced disk, with mass m, mounted in the middle of a flexible shaft of negligible mass which is supported by stiff bearings, as represented in Figure 11.The model is axially symmetrical and there are no damping effects associated to the bearings or the shaft. The only damping present in the model is viscous damping c, introduced by the action of air drag on the whirling disk and shaft. The stiffness k in the model is provided by the stiffness of the shaft or the supporting structure. [2][4] 21 Figure 11. Jeffcott rotor model. Adapted from[11] Figure 12. X-Y view of the Jeffcott rotor. Adapted from [4] In the Jeffcott rotor model, the spin angular velocity ω is assumed to be constant, which reduces the system from a 3 DOF system to a 2 DOF system. As it is shown in Figure 12, an X-Y view from the model, the rotor is represented as an unbalanced disk which has the centre of the mass in the point M. The eccentricity of the centre of the mass M in relation to the geometric centre of the disk C, corresponds to the static imbalance, represented by the vector u. The dynamic loads cause the shaft bending deflection from the shaft centre O to C, represented by vector r. Gravity loads are usually omitted in this kind of analysis because, compared with the dynamic loads, they are insignificant. [4] 22 The differential equations of motion, equations (19 and (20, the solution for the synchronous whirling, equations 21 to 23, and the shaft deflection, equation 24, for the Jeffcott rotor of Figure 12 are presented below, from the analysis carried out by Vance [4]: 𝑚𝑋 +𝑐𝑋 +𝑘𝑋=𝑚𝜔2𝑢 cos(𝜔𝑡) (19) 𝑚𝑌 +𝑐𝑌 +𝑘𝑌=𝑚𝜔2𝑢 sin(𝜔𝑡) (20) 𝑋=𝜔2𝑢 (𝑘 𝑚−𝜔2) 2+ (𝑐𝜔 𝑚) 2 cos(𝜔𝑡−𝛽𝑠) (21) 𝑌=𝜔2𝑢 (𝑘 𝑚−𝜔2) 2+ (𝑐𝜔 𝑚) 2 sin(𝜔𝑡−𝛽𝑠) (22) 𝛽𝑠= tan−1 𝑐𝜔 𝑚(𝑘 𝑚−𝜔2) (23) 𝑟= 𝑋2+𝑌2 (24) A graphic representation of the response of the Jeffcott rotor enables to understand the behaviour of the model. Figure 13. Imbalance response of Jeffcott rotor. Adapted from[4] As it can be seen in Figure 13, the critical speed is the speed at which the synchronous response to imbalance is maximum. It can be observed that the closer the speed is to critical speed, the higher the amplitude of the synchronous whirl is, and this amplitude can only be reduced by adding more damping to the model. Once the critical speed is reached, the rotor is operating at a supercritical speed and the amplitude decreases until it reaches a value 23 close to the static imbalance. As it can be seen, the smaller the static imbalance is, the smaller the synchronous whirl amplitude will be at supercritical speeds. In order to reduce the whirl amplitude, a proper balancing of the rotor can be done. Also, the operational speed of the machine can be changed in order to set this speed away from the critical speed. The operational speed can be changed itself, or the critical speed can be changed by modifying the stiffness of the supports of the rotor. Another option to reduce the whirl amplitude is to add damping to the system by increasing the damping of the bearings supports. It can be concluded that a rotor can operate above its critical speed because of the phenomenon of critical speed inversion, where the rotor whirl amplitude diminishes as the critical speed is exceeded due to imbalance. [4] 2.2.6 Multi-DOF system Multi-DOF systems are usually referred as N-DOF systems, where N corresponds to the number of coordinates required to describe the system. The simplest multi-DOF system is a 2-DOF system. An N-DOF system has N equations of motion, with N natural frequencies, and each of them corresponds to a normal mode. The normal mode is the natural state of vibration with a displacement configuration, and it is related to eigenvalues and eigenvectors. The normal modes of vibration are free undamped vibrations, which depend only on the mass and stiffness of the system and the initial conditions. When the excitation frequency of the system is the same as the natural frequency, resonance takes place. [9] The Euler equations for the rotational degrees of freedom describe the motion of a rigid body in a nonlinear way. However, it is possible to assume two simplifications that allow to obtain a linearized model. The first assumption is that the rotor is perfectly balanced, or that the deviation is small. The second assumption is that the linear and angular displacements and velocities, excepting the rotation angle and angular velocity, are small too. With these two assumptions, the linearized equation of motion is the following [2]: 𝑀𝑞 𝑡 + 𝐶+𝐺 𝑞 𝑡 + 𝐾+𝐻 𝑞 𝑡 =𝑓(𝑡) (25) In equation 25, M is the symmetric mass matrix, C is the symmetric damping matrix and K is the symmetric stiffness matrix. The damping and stiffness matrices may depend on the spin speed. G is the skew-symmetric gyroscopic matrix and H is the skew-symmetric circulatory matrix. The gyroscopic matrix contains inertial terms, such as Coriolis acceleration, related to gyroscopic moments present in rotating machines. The circulatory matrix contains nonconservative terms, such as the internal damping of rotating elements. Both gyroscopic and circulatory matrices are proportional to the spin speed, and when this speed tends to zero, both terms are omitted. For a system having N-DOF, the size of the matrices are NxN. The vector q(t) is the inertial frame, and contains the generalised coordinates, and the vector f(t) contains forcing functions. [2] The definition of the matrices is not a trivial issue, and several literature can be found in order to define them, for example in the work of Vance [2], Genta [4] or Thomson [10]. 24 2.2.7 The Finite Element Method Continuous models have an infinite number of degrees of freedom. These models are discretised and thus converted into a finite number DOF system. There are some discretization techniques that assure the accuracy of the results. The most common method used in rotordynamics for the discretization of a model is the Finite Element Method (FEM). In FEM, the body under study is divided into a number of regions, the finite elements. Each element has a finite number of degrees of freedom, and the differential equations are solved in the selected points of each element, the nodes. In each node, the displacement ū of the coordinates x, y, z, is approximated by the linear combination of the shape functions n, as can be seen in equation 26, with as many rows and columns as DOF in the system. [2] 𝐮 𝑥,𝑦,𝑧,𝑡 =𝐍 𝑥,𝑦,𝑧 𝒒(𝑡) (26) It is usual to simplify the model to a three DOF system, equation 27, with the displacements in the directions x, y and z. In this case, the displacements in each direction are functions of the nodal displacement in the same direction only [2]. 𝒖𝒙(𝑥,𝑦,𝑧,𝑡) 𝒖𝒚(𝑥,𝑦,𝑧,𝑡) 𝒖𝒛(𝑥,𝑦,𝑧,𝑡) = 𝐍(𝑥,𝑦,𝑧)𝟎 𝟎 𝟎 𝐍(𝑥,𝑦,𝑧)𝟎 𝟎 𝟎 𝐍(𝑥,𝑦,𝑧) 𝒒𝒙(𝑡) 𝒒𝒚(𝑡) 𝒒𝒛(𝑡) (27) The use of FEM allows the simulation of the behaviour of rotors with lower numerical costs and in less time. There exist different formulations to simulate the rotor system, but the most common elements used in rotor dynamics are the beam, the spring, and concentrated mass elements [8]. 2.2.8 Dynamic factors limiting the rotor design The main dimensions of the rotor, i.e. the rotor diameter and the rotor length, are parameters which are influenced by dynamic aspects [12]. Their design may have serious effects on stress in the rotor and on natural frequencies. The length of the rotor is influenced by natural frequencies. When the rotor operating speed coincides with critical speed, the rotor may enter in a resonance state. In order to avoid the resonance, the rotor should operate away from the critical speed, either below the first critical speed, or between the first and second critical speed if the construction and bearings of the rotor allow to pass the first critical speed quickly. During operation stage of rotor, critical speed can be moved away by adding auxiliary supports which provide an increment in the effective stiffness of the rotor and increases the critical speed. During design stage, critical speed can be changed by changing the rotor mass, which is related to changes in the dimensions of the rotor. The first critical speed is influenced by rotor length, so in order to modify the first critical speed, the length of the rotor can be modified. For a given critical speed, the maximum length is the following: 31 The bearings for the system need to be chosen according to dynamical requirements. They should be able to stand high-speed. Deep groove ball bearings are a good option for the model, because they are suitable for high and very high speeds [14]. A model such as the model number 61806 of deep groove ball bearing of SFK can be selected [15]. The limiting speed for this bearing is 20.000 rpm, so this model may be a good option for the rotor. It would be interesting to determine the stiffness of this bearing, but stiffness cannot be determined without mounting the bearing. Usual values for stiffness of this kind of bearings go from the 104 N/mm to 106 N/mm [14]. 3.3 The mesh The mesh is one of the most important parts, if not the most, in a computational simulation because it ensures the accuracy of the results obtained. For creating a correct mesh for the model, the mesh elements and the number of nodes need to be chosen according to the quality of the results, but also considering the computational time. A high number of nodes usually lead to an accurate result, but the computational time can be very high, which is translated into a high computational cost. A balance needs to be found between the quality of the mesh, the accuracy of the results and the computational cost. The first step in every computational analysis of a model should be the determination of the proper mesh. For every geometry model and depending on the kind of the analysis that is intended to be developed, an analysis of the mesh should be carried out. Here below, the principal aspects to set the proper mesh in ANSYS Workbench are described, and in Chapter 4, the mesh settings for the rotor model will be analysed. The settings configuration in ANSYS Workbench can be divided into two main aspects. The first aspect to choose when designing a mesh is the shape of the mesh elements, which is defined by the method of meshing, and the second aspect is the mesh density, which is related to the size of the elements and the number of elements in the mesh. ANSYS Workbench provides of five meshing methods that the user can choose and set in order to control the mesh: a) Tetrahedral Method b) Hexahedral Method c) Automatic Method d) Sweep Method e) MultiZone Method There are several parameters than can be modified in the size control of the mesh in order to adapt the mesh to the user’s requirements. Parameters such as element size, edge length and curvature angle can be modified. All the meshing parameters available are widely described in the Meshing User’s Guide [16]. 32 4 Dynamic analysis and results In this chapter the analyses to determine the natural frequencies and mode shapes, the critical speed and the stress produced by the centrifugal force at the maximum speed will be developed. All the setting for preparing the simulations will be described, and the results obtained will be analysed. 4.1 Natural frequencies and modes The first study carried out was the determination of the natural frequencies of the modelled rotor introduced in Chapter 3.2. Several simulations were developed in order to determine the validity of the mesh used and to validate the results obtained. The aim of this study was to determine the natural frequencies of the rotor and their mode shapes. In order to develop this study, the rotor system is simulated as a free-free rotor, i.e. without bearings. The analysis carried out is the Modal Analysis from ANSYS Workbench. The modal analysis determines the vibration characteristics of structural components. The vibrations characteristics are the natural frequencies and mode shapes. A continuous structure has an infinite number of degrees of freedom, but with the use of FEM, the real structure can be approximated with a finite number of DOF. 4.1.1 Free rotor model The first step in the analysis of the natural frequencies was to import the geometry of Chapter 3.2 in a STEP format into the Design Modeler of ANSYS Workbench. In this platform, the imported geometry was divided into 7 bodies. It is recommended to make such a division in the transitions of changes in the geometry, in order to make the meshing step easier. If the whole geometry is considered as a unique part, there may appear difficulties, or even errors, in an abrupt change in the geometry. For this reason, the model was divided in these parts where there is a change in the diameter. Moreover, a separation in the place where the bearings will be mounted was also added, although bearings have no effect in the determination of the natural frequencies of the free rotor. Divisions in the geometry can be seen in Figure 17. 33 Figure 17. Free rotor model divided into 7 bodies 4.1.2 Mesh generation A study of the mesh was carried out in order to generate the proper mesh for the analysis and to give validity to the obtained results, and the meshing method as well as the mesh size were chosen. As it has been mentioned before, a balance between the quality of the results and the computational time needs to be found in order to keep the computational cost low. Different meshing methods were tested for the model in Figure 17. The meshing methods that ANSYS Workbench incorporates are Tetrahedral Method, Hexahedral Dominant Method, Automatic Method, Sweep Method, and MultiZone Method. In order to compare the effect that the method has in the mesh, just the parameters related to the meshing methods were modified and all the parameters related to mesh size were the same for all methods, in order to compare the methods on the same basis. The main parameters of the methods were chosen or left in Program Controlled option. A visual inspection of the different meshing methods on the geometry can be seen in Figure 18. Tetrahedral Method mesh to Figure 22, and the different methods are compared in Table 2. 34 Table 2. Properties of different mesh methods Method Number of nodes Number of elements Tetrahedral 12.049 6.582 Hexahedral 21.943 6.290 Automatic 21.565 4.340 Sweep 24.902 5.054 MultiZone 21.759 4.373 Figure 18. Tetrahedral Method mesh 35 Figure 19. Hexahedral Method mesh Figure 20. Automatic Method mesh 36 Figure 21. Sweep Method mesh Figure 22. MultiZone Method mesh In all the figures above, it can be observed the effect that the separation of the geometry into different bodies presented in previous chapter has in the meshing. For every different 37 body, an independent mesh is created, and it allows a better adaptation of the mesh in the different parts of the geometry, i.e. to have a more accurate mesh in the parts of the geometry that are smaller, and thus more critical, and a coarser mesh in those parts that are bigger and less critical. It can be observed that MultiZone Method and Automatic Method meshes are quite similar, in the shape of the mesh and in the number of nodes and elements. Sweep Method mesh is also very similar to that ones, although the number of nodes and elements are a bit higher. The reason for the similarity in these three meshing methods is that most of the parameters of the MultiZone Method and the Sweep Method can be left to Program Controlled option. This fact makes the algorithm method work under similar parameters as the Automatic Method, and the resulting mesh is practically the same. For this reason, MultiZone Method and Sweep Method will be omitted in the choice of a meshing method for this thesis, since they can both be approximated to Automatic Method. On the other hand, huge differences can be observed between Tetrahedral Method and Hexahedral Method. Under the same size parameters controls, tetrahedral mesh contains almost the half of the number of nodes than hexahedral mesh, but more mesh elements. That implies that the solution of the equations of the analysis when using the tetrahedral mesh would be carried out in a fewer number of points, and the results could be considerably less accurate than for the hexahedral mesh. Since it is preferable to have a higher number of nodes in the meshing, the Tetrahedral Method mesh is dismissed. Therefore, the two better methods for meshing the model are Hexahedral Method and Automatic Method. It was observed that the computational time for generating the mesh with the Hexahedral Method was considerably higher than the time for generating the mesh with the Automatic Method, which could lead to high computational time when developing the analyses. For this reason, and in order to leverage the potential that ANSYS Workbench program offers, it is considered a good option to choose the Automatic Method for meshing. This method provides the best combination of different element types and ensures the best adaptation of the mesh elements in the geometry, and has a reasonable number of nodes with a lower number of elements. Once the meshing method was chosen, the size controls of the mesh were analysed. The main parameters of size control were modified to obtain different meshes with different mesh densities. The mesh density is especially related to number of nodes and number of elements, and it determines the validity of the results. The results obtained using a mesh with a low number of nodes could be erroneous, but if a mesh with a high number of nodes is used, then the computational time for developing the analysis could be excessively high. For this reason, a study of convergence of the results should be carried out, in order to ensure that the proper mesh density is chosen and to validate the results. To develop the study of convergence of the results, the analysis of the first natural frequency of the free rotor was carried out with different mesh densities. The analysis of the natural frequencies will be widely explained in Chapter 4.1.3. To generate meshes with different densities, sizing parameters such as Mesh Relevance, Relevance Center, or Element Size, were modified, and different values were given to these parameters in order to obtain meshes with different number of nodes. The Modal Analysis was then solved for each mesh, and the first natural frequency of the free rotor was determined. In Table 3 the results of the first natural frequency of the different analysis carried 38 out are presented, as well as the number of nodes and computational time associated to each analysis. Table 3. First natural frequency for different mesh densities Analysis Number of nodes First natural frequency (Hz) Computational Time (min) 1 4.582 6.265 0:06 2 5.934 6.195 0:11 3 7.209 6.102 0:20 4 8.672 6.066 0:28 5 9.971 5.998 0:35 6 13.793 5.995 0:50 7 18.000 5.993 1:40 8 22.767 5.992 2:50 9 25.358 5.992 3:11 10 27.179 5.997 6:34 The different analyses were performed increasing the number of nodes, i.e. making the density of nodes in the mesh higher. The results of Table 3 are plotted in Figure 23 and Figure 24 in order to analyse the convergence of the results and the influence of the number of nodes in the computational time. Figure 23. Convergence of results 5950 6000 6050 6100 6150 6200 6250 6300 0 5000 10000 15000 20000 25000 30000 First Natural Frequency (Hz) Number of nodes 39 Figure 24. Influence of number of nodes in computational time It can be seen that when the number of nodes is increased, the variation in the results obtained decreases. In Figure 23 it can be observed that when the number of nodes is lower than 10.000 nodes, the result of the analysis carried is quite different from the previous and the next result. On the contrary, when the number of nodes is higher than 10.000, a convergence in the results can be seen, and, although there is a dispersion in the results obtained, the estimation of the first natural frequency can be determined to be quite accurate, because the results are of the same order. It can be determined, that for the rotor model under study, a mesh with a number of nodes higher than approximately 10.000 nodes is a valid mesh for developing the analyses and that the results will be valid. The computational time associated with each analysis was also studied, because it helps to decide the density of the mesh. As it can be seen in Figure 24, the higher the number of nodes is, the higher the computational time. A tendency line is added to the plot in order to determine the tendency of the computational time. It can be clearly seen that the computational time increases exponentially with the number of nodes, so that time needs to be kept as low as possible. Analysing the factors described above, it seems a reasonable option to choose a mesh density which contains a number of around 15.000 nodes. With 15.000 nodes, there is a convergence of the results, which assures the validity of the results obtained. Moreover, the computational time at 15.000 nodes is still quite low, so the computational costs will not be that high. The main parameters selected to generate the chosen mesh are listed in Table 4, and the mesh can be seen in Figure 25. 0:00 1:12 2:24 3:36 4:48 6:00 7:12 0 5000 10000 15000 20000 25000 30000 Time (min) Number of nodes 40 Table 4. Main mesh parameters Parameters Value Method Automatic Physics Preference Mechanical Relevance 30 Element Midside Nodes Program Controlled Size Function Adaptive Relevance Centre Coarse Smoothing Medium Minimum Edge Length 5mm Figure 25. Selected mesh 4.1.3 Analysis and results Once the proper mesh was selected, the conditions for developing the analysis were chosen. The aim of this study is to find the natural frequencies and the mode shapes of the rotor under the assumption of free rotor, therefore, for this analysis the bearings will not be considered. 47 4.2.2 Analysis and results The critical speed analysis was developed with ANSYS Workbench software. The program used to carry out the analysis was the Modal Analysis program, as for the analysis of natural frequencies, but in this case the conditions of the analysis were different. The damping is important in the critical speed analysis, and it was introduced to the model. A constant damping was added to model, exactly a 2% of material damping. Stiffness is the other factor important for the determination of the critical speed. The bearings add stiffness to the model, but this value cannot be exactly determined without the physical model, because it depends on factors related to the mounting of the bearings. As it was explained in Chapter 3.2, usual values of stiffness for deep groove ball bearings are from 104 to 106 N/mm. A stiffness value of 50 kN/mm in the bearings was selected for the analysis. Results showed the critical speed at 23.035 rpm, as represented in the Campbell Diagram of Figure 35. Figure 35. Campbell Diagram for k=50kN/mm The maximum operational speed of rotor is 18.000 rpm, and the critical speed found for the given parameters is around 23.000 rpm. It can be concluded that the critical speed is higher than the maximum operational speed, but it could be adequate to move away the critical speed. In order to do that, more stiffness can be added to the bearings. When a stiffness value of 100 kN/mm is applied to the bearings, the critical speed is 31.873 rpm, as can be seen in the Campbell Diagram of Figure 36. 48 Figure 36. Campbell Diagram for k=100kN/mm It can be observed that increasing the stiffness of the bearings has a direct effect on the critical speed. Values of stiffness around 50 and 100 kN/mm should be achieved to ensure that the rotor is operating far from the critical speed. It is also interesting to determine which is the minimum stiffness that the bearings should have. The minimum stiffness should be the stiffness at which the critical speed is equal to the critical speed. Different values of bearing stiffness were added to the program, and it was determined that for a stiffness of 32.500 N/mm, the critical speed becomes 18.000 rpm. Therefore, the minimum bearing stiffness for the system has to be 32,5 kN/mm. 49 4.3 Effect of centrifugal force Although the rotor is quite small, the high operational speeds at which the rotor will whirl create the need to evaluate centrifugal forces in the rotor. At the maximum rotational speed of 18.000 rpm (300 s-1), the centrifugal force, is approximately of 75 kN. A centrifugal force of such dimensions cannot be neglected. This rotational behaviour may lead to high stresses in the rotor that should be considered, especially in its inner part. 4.3.1 Analysis and results For evaluating the effect of centrifugal force in the rotor, the solid rotor model and the model with the rotor and the shaft are used. Similar mesh as studied in previous chapters is used. In order to determine the stress that the rotor may suffer at the maximum operational speed, a Structural Static analysis was developed in ANSYS Workbench software. The analysis was developed at the maximum operational speed of the rotor, 18.000 rpm. First, the effect of centrifugal force in the solid rotor was determined. In order to do that, the maximum rotational velocity was applied to the rotor model, and a condition of cylindrical support was added to the solid rotor, as shown in Figure 37. Figure 37. Solid rotor boundary conditions Results of the performed analysis can be seen in Figure 38, were the equivalent stress (Von Misses Stress) is represented. The maximum stress that the rotor may suffer is determined, with a value of 180,65 MPa. It can be observed that in the inner part of the rotor, stress 50 can reach values from 100 MPa. As expected, the higher stresses of the rotor are in the inner part. Figure 38. Stress in solid rotor The effect of centrifugal force in the model with the rotor and the shaft used in previous chapters was also determined. Results obtained of the developed analysis are shown in next figures. To analyse the stress that the rotor suffer, the equivalent stress (Von-Misses Stress) is displayed in the pictures. The values of stress are high, although lower than when the solid rotor in analysed alone. The equivalent stress achieved a value of 90 MPa in the most critical part, which is in the exterior part of the inner diameter of the rotor. In the exterior part of the rotor, the stress is around 20 MPa, and in the inner part of the rotor, the stress range is about 50 and 70 MPa. It can be visually observed how the centrifugal force affects the rotor and the shaft. In Figure 39 to Figure 43 the Von-Misses Stress is represented in all the bodies under analysis. It can be seen that the lateral faces of the rotor are deformed due to the high stresses, as well as the shaft itself. Such a high stress can be dangerous for the operation of the rotor and the shaft. 51 Figure 39. Equivalent stress of the model Figure 40. Equivalent Stress of the rotor 52 Figure 41. Equivalent Stress of the shaft Figure 42. Left view of the equivalent stress of the rotor and shaft 53 Figure 43. Right view of the Equivalent Stress of the rotor and shaft 54 5 Discussion and further studies The dynamic behaviour of the rotor has been analysed and the natural frequencies, critical speed and the effect of centrifugal force have been determined. The natural frequencies were obtained in Chapter 4.1. When the simulation of the free-free rotor was carried out, the values for the natural frequencies and mode shapes were determined. The value of the first bending mode obtained was around 6.000 Hz, and the values for the second and third bending mode and for the first torsional mode were even higher. The values of the natural frequencies are quite high, but it is usual in rotors of such dimensions. It was observed that the rotational speed has no influence in the determination of the natural frequency, because the results obtained without rotation and with a rotating speed of 18.000 rpm are almost the same. The critical speed was determined in Chapter 4.2. The critical speed was determined to be above from the maximum operational speed of the rotor. Some stiffness values for the deep groove ball bearings were tested to prove the effect of this value on critical speed. It was observed the influence of the stiffness in the critical speed. With a stiffness value of 50 kN/mm the critical speed was obtained to be around 23.000 rpm, and with a stiffness of 100 kN/mm the critical speed was around 31.000 rpm. The minimum stiffness that the bearings should have was also analysed, and a value of 32,5 kN/mm was determined as the minimum value. Therefore, values of stiffness above 50 kN/mm are recommended to ensure that the rotor is working far enough from the critical speed. The effect of the centrifugal force in the model and in the solid rotor was analysed in Chapter 0. It was calculated that the centrifugal force that the rotor and shaft experience at the maximum speed of 18.000 rpm was around 75 kN. This elevated centrifugal force creates high stresses in the rotor. The equivalent stress, i.e. the Von-Misses Stress, was determined with the analyses. It was observed that in the inner part of the solid rotor, the equivalent stress had values between 80 and 180 MPa, and in the inner part of the rotor in the model with the rotor and the shaft, the rotor part in contact with the shaft, the equivalent stress had values between 50 and 70 MPa. The maximum value of stress in the solid rotor model was of 180 MPa, and in the rotor and shaft model was of 90 MPa, and in both cases it took place in a lateral part of the rotor, close to the inner diameter. From the results obtained of the analyses carried out, the geometry of the rotor is validated to work under the required operational characteristics. In further studies a more accurate analysis of the rotor with the exact geometry should be developed. Due to computational limitations, it was not possible to develop an analysis with the detailed geometry in this thesis, because for a complex geometry the computational costs can increase exponentially, and powerful tools should be used. The advantages of computational simulation were experienced in this project. It allowed to explore a range of alternatives because changes in the geometry or in the simulation conditions could be easily done. Also, for the same geometry, the effect of different conditions could be analysed. But the computational simulation may have errors in the modelization of the physical problem and it produces uncertainty. And although the computational 55 analysis provides accuracy in the results of the simulations, these results need to be compared with experimental test of a prototype. For this reason, an experimental analysis with the rotor prototype should be developed. Although experimental analyses may be long and they may imply an elevated cost, it is mandatory to develop this kind of tests because they allow to incorporate to the analyses effects of the real model that are not considered in the idealised model, such as geometrical imperfections, mass unbalance, etc. These features may have effects on the model that can only be detected in a test with the real prototype. Once the whole structure of the motor is designed, an integrate study containing all the bodies of the motor should be developed. The study should involve electromagnetic, rotordynamic, structural and thermal analyses to give validity to the whole motor. 56 6 Conclusion The objective of this Master Thesis was to develop a proper dynamic analysis to determine the behaviour of the high-speed rotor under design. The rotor under study was first described, and it was modelled to adequate the rotor geometry to the analysis that was going to be developed. The theoretical aspects that are involved with the rotordynamic analysis were reviewed. General aspects of vibrations were summarised, and the models that are used in rotor dynamics were described, from the simplest model that could be used, the single-degree of freedom model; the most popular model that is used to describe the main aspects of rotordynamics, the Jeffcott rotor; to the most useful method that is used nowadays to solve the multi degree of freedom system, the Finite Element Method. Finite Element Method was used to develop the analysis of the dynamic behaviour of the rotor. This method was found suitable for determining the dynamic behaviour of the rotor. ANSYS Workbench software was satisfactorily used in the analysis. It can be concluded that this software was an intuitive and easy tool to develop the first dynamic analysis of the rotor model.