The Pennsylvania State University. The Graduate School. The Department of Mechanical Engineering

Transcription

1 The Pennsylvania State University The Graduate School The Department of Mechanical Engineering HIGH FLEXIBILITY ROTORCRAFT DRIVESHAFTS USING FLEXIBLE MATRIX COMPOSITES AND ACTIVE BEARING CONTROL A Thesis in Mechanical Engineering by Murat Ocalan 2002 Murat Ocalan Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2002

2 iii ABSTRACT Current state-of-the-art rotorcraft drivelines are made up of relatively rigid segmented aluminum shafting, linked by flexible couplings and hanger bearings. This design transmits the power along the driveline while accommodating the misalignment caused by the rotorcrafts structural deflections. While this type of design can transmit power and torque effectively, the use of mechanical components, such as couplings and bearings, significantly increases the weight of the driveline, reduces the system reliability, and increases the manufacturing and maintenance costs. In this thesis, new drive shaft concepts using flexible composite material technology have been explored to solve the problems of the current driveline. The idea is to tailor the composite shaft design such that it can withstand the torque load while having flexible bending properties; thus, making the requirement for the added bending compliance (bearings and couplings) obsolete. Active bearings are being proposed to control the stability and vibration of the highly flexible driveline. The objective of this thesis is to provide a thorough preliminary study of the newly developed driveline concept, studying the effects of design parameters. The results of this study not only provide a feasibility study, but also identify areas that need to be addressed for replacement of the current driveline with the flexible matrix composite shafting. In this investigation, a finite element model of a tailboom and the tailrotor

3 iv driveline system is developed. This model is used to design the flexible matrix composite shaft and the active control system. This control system uses the proportionalderivative control scheme. A thermal finite difference model of the driveline is developed. This model is used with the mechanical finite element model. The heat generation and temperature distribution in the system is analyzed using these coupled models. The whirl stability of the current and the newly designed driveline is also analyzed with linearized eigenvalue analysis using a time-domain damping model. The results show that the flexible matrix composite shaft is comparable with the existing state-of-the-art driveline in mechanical, thermal and stability aspects. Safety factor for the design was calculated to be as high as 3.5. (55 /-55 ) s and (55 /-55 /0 /0 ) s ply orientations were found to give the highest performance to the application. Furthermore including 0 in the composite laminate was shown to improve stability of the driveline. Temperature of the designed driveline was calculated to be only 8 C higher than the ambient. Considering the improvements in the reduction in weight, cost and maintenance, the new design shows promise to be the future of rotorcraft drivelines.

4 v TABLE OF CONTENTS LIST OF FIGURES...vii LIST OF TABLES...ix NOMENCLATURE...x ACKNOWLEDGEMENTS...xvii Chapter 1 Introduction Background and Literature Review Problem Statement and Research Goal Summary of the Thesis...11 Chapter 2 Model of the System Composite Shaft Equivalent Engineering Properties Stiffness Properties Damping Properties Discretized Finite Element Equations of Motion Shaft Element Beam Element for Tailboom Bearing Element End Inertia Assembly of the Driveshaft-Tailboom System Model External Loads Equivalent Aerodynamic and Maneuver Loading Torque Loading Shaft Imbalance Loading Superposition of Response Composite Strength Calculations Torsional Instability Analysis of the Driveline Thermal Analysis Thermal Model of Tailboom Thermal Finite Difference Model of Shaft...38 Chapter 3 Analysis and Synthesis of the High Flexibility Driveline...42

5 vi 3.1 Finite Element Model Steady State Operation Ply Orientation of the Composite Shaft Wall Thickness of the Shaft Control Gains of the Active Actuators Torsional Stability of the Driveline Spin-Up and Spin-Down Operations Thermal Design...66 Chapter 4 Whirl Stability Analysis of Driveline Time Domain Shaft Finite Element Model Model of the System Stability of Driveline...72 Chapter 5 Conclusions and Recommendations Conclusions Recommendations for Future Work...78 BIBLIOGRAPHY...80

6 vii LIST OF FIGURES Figure 1.1 Current tailrotor driveline system...1 Figure 1.2 Misalignment of current tailrotor driveline system due to external loading of tailboom structure...2 Figure 1.3 Young's modulus of a symmetrical two-ply composite laminate...5 Figure 1.4 Shear modulus of a symmetrical two-ply composite laminate...6 Figure 2.1 In-plane force resultants of the composite cross-section...14 Figure 2.2 Cross-section of the composite laminate...14 Figure degree-of-freedom, 2 node shaft element...17 Figure degree-of-freedom beam element modeling the tailboom...20 Figure 2.5 Bearing element...22 Figure 2.6 Model of end inertia...24 Figure 2.7 Finite element model...25 Figure 2.8 Hollow shaft under torsion...33 Figure 2.9 Energy exchange in the system...34 Figure 2.10 Heat generation mechanisms...38 Figure 2.11 Finite difference element of the shaft...40 Figure 3.1 Dimensions of the tailrotor driveline...43 Figure 3.2 Safety factor vs. ply orientation, one ply orientation variable without imbalance loading...47 Figure 3.3 Safety factor vs. ply orientation, two ply orientation variables without imbalance loading...48 Figure 3.4 Safety factor vs. wall thickness...50 Figure 3.5 Maximum shaft response with 1mm and 4mm wall...51

7 Figure 3.6 Effect of proportional gain on the safety factor of the composite shaft...54 Figure 3.7 Tailboom and shaft horizontal deflection curves with and without midspan bearings...55 Figure 3.8 Actuator gap of a magnetic bearing...56 Figure 3.9 Maximum frequency response function of the shaft, critical/ noncritical operating condition...57 Figure 3.10 Torsional buckling safety factor...58 Figure 3.11 Effect of derivative gain with high and low proportional gains...60 Figure 3.12 Maximum spin up shaft deflection vs. control gains...61 Figure 3.13 Frequency response of driveline with K p = 2x10 6 N/m...62 Figure 3.14 Frequency response of driveline with K p = 3x10 5 N/m...63 Figure 3.15 Maximum shaft displacement at the actuator location vs. control gains..64 Figure 3.16 Maximum actuator force vs. control gains...65 Figure 3.17 Tuned control gains...66 Figure 3.18 Effect of wall thickness on temperature profile of driveline...67 Figure 3.19 Effect of convective heat transfer on temperature profile of driveline...68 Figure 4.1 Schematic of the driveline model for stability analysis...71 Figure 4.2 Real parts of eigenvalues of aluminum shaft...73 Figure 4.3 Real parts of eigenvalues of (45 /-45 ) s Toray 700/Adiprene L100- Caytur21 shaft...74 Figure 4.4 Real parts of eigenvalues of (45 /-45 /0 /0 ) s Toray 700/Adiprene L100-Caytur21 shaft...76 viii

8 ix LIST OF TABLES Table 2.1 Parameters used for heat transfer analysis of tailboom...36 Table 3.1 Parameters and dimensions used in the model...44 Table 3.2 Properties of Toray 700/Adiprene L100-Caytur Table 3.3 Most critical loading on composite shaft...48

9 x NOMENCLATURE φ ζ θ b ν A A c a rel Phase of response Area variable Rotational displacement of tailboom end point Dynamic viscosity of air Laminate stiffness matrix Cross-sectional area Relative displacement between the shaft and the tailboom in a-direction (a = u,v,w,β,γ) A sys B sys C f C sys D d act D B D i D o e E State space system matrix State space input matrix Forced convection equation constant Global damping matrix Diameter of the tailboom Vector of shaft displacement at the actuator locations Dissipation function Inner diameter Outer diameter Effective offset Young s Modulus

10 xi e c E f E L E m E T E x F ext Effective eccentricity Young s Modulus of fiber material Young s Modulus of composite laminate in fiber direction Young s Modulus of matrix material Young s Modulus of composite laminate perpendicular to fiber direction Equivalent Young s Modulus External load F ij Strength coefficient (i = 1,2,6; j = 0,1,2,6) F imb G G g G sys G LT G xy H h H(ω) I I d I dy I dz I end Imbalance force Shear modulus Elemental gyroscopic matrix Gravitational acceleration Global gyroscopic matrix Shear modulus of composite laminate Equivalent shear modulus Distance between the neutral axes of shaft and tailboom Average heat transfer coefficient Transfer function matrix Moment of area of finite element cross-section Diametric mass moment of inertia Mass moments of inertia per unit length from the y-axis Mass moments of inertia per unit length from the z-axis Inertia matrix of end mass

11 xii I p I y I z J K K ab k air k ba k c Polar mass moment of inertia Moment of area from the y-axis Moment of area from the z-axis Polar moment of area of finite element cross-section Elemental stiffness matrix Bearing stiffness matrix coupling terms from a element to b element Conductive heat transfer coefficient of air Linear stiffness coefficient in a-direction (a = x,y,z) Conduction heat transfer coefficient of material K Conduction heat transfer coefficient cond K Convective heat transfer coefficient conv K d K p K sys k ta K THR l s M m f m s M sys N S Derivative control gain matrix Proportional control gain matrix Global stiffness matrix Torsional stiffness coefficient in a-direction (a = y,z) Heat transfer matrix Length of element Elemental mass matrix Forced convection equation constant Total mass of shaft Global mass matrix Shape function of twelve degree-of-freedom shaft element

12 xiii N S a,b Shape function associated with a and b variables of twelve degree-offreedom shaft element Nu N x N xy N y Pr P s Q q b q e Q k q k q n Average Nusselt number Force per unit length in x-direction Shear force per unit length in xy-plane Force per unit length in y-direction Prandtl number Transmitted power Forcing matrix Displacement of beam degrees of freedom Elemental displacement vector Stiffness matrix of k th lamina k th degree-of-freedom of finite element Heat generation of n th element Q Radiation heat transfer from atmosphere to tailboom Rad,Atm Q Radiation heat transfer tailboom from to atmosphere Rad,Tail Q Convective heat transfer from tailboom to ambient air Conv,Tail q s q t r R Ra D Displacement of shaft degrees of freedom Total response of system Radius of shaft Safety factor Rayleigh s number

13 r b Re r m Translational displacement of tailboom end point Reynolds number Absolute displacement of effective mass xiv R r s S T t T T s T T n u U U ext u inp U s U y U z v V V B Main rotor radius Length variable of finite element Shear strength Angular transformation matrix Wall thickness Kinetic energy Torque load Ambient temperature Temperature of n th element Displacement in x direction Strain energy of shaft Loading vector associated with external load State space input vector Loading vector associated with torque load Loading vector associated with imbalance load in y direction Loading vector associated with imbalance load in z direction Displacement in y direction Potential energy Bending energy

14 xv V bear w W rc x X Potential energy stored in bearing element Displacement in z direction Weight of rotorcraft Vector of state variables Tensile strength in fiber direction of composite ply X Compression strength in fiber direction of composite ply X e Y Deflection profile of finite element Tensile strength (direction perpendicular to fiber of composite ply) Y Compression strength (direction perpendicular to fiber of composite ply) z z k Β Γ γ γ xy δw ε ε t ε x ε y η η V θ Thickness variable of composite laminate Distance between the median of wall and bottom of k th lamina Rotation around y direction Rotation around z direction Shear strain vector Shear strain in xy-plane Virtual work Linear strain vector Emissivity of tailboom Strain in x-axis Strain in x-axis Damping ratio matrix Viscous damping ratio Rotation around x direction

15 xvi μ ν ρ ρ c σ σ 1 σ 2 σ x σ y σ xy τ 12 Ω Mass per unit length Poisson s Ratio Density of air Density of composite laminate Stefan-Boltzmann Constant Normal stress in 1-direction Normal stress in 2-direction Normal stress in x-direction Normal stress in y-direction Shear stress in the xy-plane Shear stress in 12-plane Shaft rotational speed

16 xvii ACKNOWLEDGEMENTS I would like to thank my advisors, Professor Kon-Well Wang and Professor Edward C. Smith, for their support and guidance towards completion of my thesis. Their generosity has been invaluable. I would also like to thank Professor Olivier Bauchau of Georgia Institute of Technology for his contributions in torsional buckling analysis of composite shafts, Professor Charles E. Bakis, Hans DeSmidt and Ying Shan for their assistance in my work. This work was conducted with support from the U.S. Army Research Office and National Rotorcraft Technology Center.

17 To Sunaina

18 1 Chapter 1 Introduction 1.1 Background and Literature Review The required power to drive a helicopter tailrotor, supplied from the engine, is transferred through a driveline along the tailboom structure. The current driveline system, as illustrated in Figure 1.1, normally consists of segmented metal shafts connected through flexible couplings and supported by hanger bearings. Tail Rotor Hanger Bearing Damper Flexible Coupling Imbalance Loading Tailboom Structure Aerodynamic and Maneuver Loading Figure 1.1 Current tailrotor driveline system The relatively flexible tailboom is subjected to static and low frequency, aerodynamic and maneuver loads, whereas the driveline is loaded with rotating imbalance and torque loading. Maneuver loads are defined as the inertial loads acting on the tailboom structure

19 2 and its components. Aerodynamic loads, distributed throughout the tailboom, are loads caused by the interaction between the tailboom and the different fluid flow across the tailboom (e.g. loading caused by main rotor downwash.) Deflection in the tailboom structure causes an overall misalignment on the driveline as illustrated in Figure 1.2. For this reason, these relatively rigid shafts are connected with flexible couplings accompanied by hanger bearings to accommodate for the misalignments. The hanger bearings and couplings require frequent inspection, maintenance and even replacement in order to ensure reliable operation. Figure 1.2 Misalignment of current tailrotor driveline system due to external loading of tailboom structure The driveline can be designed to be subcritical or supercritical by changing the number of shaft segments; larger number of segments in general means higher driveline natural frequencies. With sufficient number of shaft segments, the lowest natural frequency can be designed higher than the operating speed. This subcritical design avoids spinning-up through resonant frequencies and has lower dynamic loads on the mechanical

20 3 components. Although a subcritical shaft design has its advantages, it employs larger number of mechanical components than a supercritical driveline. Due to these reasons the supercritical shafting can increase reliability and reduce weight significantly. During the starting and stopping of the rotorcraft, supercritical drivelines go through resonant frequencies. This causes the loads on the mechanical components to be very large. To reduce the effects of this problem the current supercritical designs include vibration dampers. The disadvantage of the dampers is that their performance is limited to the neighborhood of resonant frequencies. Since the operating speed is designed to be as far from resonances as possible, the excitations caused from imbalance in the operating speed is damped poorly with the passive system. In the supercritical design, the resonant deflections and loads are very sensitive to the imbalance of the driveline. The manufacturing cost of these shafts is increased significantly by the tolerance requirements, which keeps the imbalance in control. Often the shafts are sent for rebalancing during the operation of the rotorcraft. The dominant steady-state operating loads on the driveline are the torque load, and the bending loads, which are caused by the transmission of forces from the deflecting tailboom structure. Since the tailboom is much stiffer than the shafts, bending can be considered as specified displacement whereas the torque load can be considered as specified forcing. This means that the driveline should be designed to be stiff and strong in torsion but compliant in bending. The flexible couplings give the driveline this

21 property. The couplers can only be removed if the shaft material is strong enough to carry the torque load while having sufficiently flexible bending properties. 4 Isotropic shafts, designed for carrying the torque load have less bending compliance than needed in the tailboom. This requires the use of flexible couplings in the state-of-the-art system. The use of composite materials is very advantageous in the driveline due to the fact that ply orientations in the composite can be tailored to make the shaft stiffer in torsion than in bending. In composite shafting the role of the matrix material is important on the shaft s mechanical properties as illustrated in Figure 1.3 and Figure 1.4. The plots are calculated with using the same fiber material properties (E f = 229GPa) but variable matrix properties. The matrix properties are varied from those of epoxy to those of urethane. The most desirable ply orientation for the torsional rigidity is ±45, where the fibers are in the loading direction for the torsion loads. For the same orientation the elastic modulus of the composite greatly depends on the matrix properties. The flexural compliance of a composite beam increases significantly as the stiffness of the matrix decreases. Thus, as the flexibility of the matrix material increases the level of anisotropy of the composite ply also increases.

22 Figure 1.3 Young's modulus of a symmetrical two-ply composite laminate 5

23 6 Figure 1.4 Shear modulus of a symmetrical two-ply composite laminate Flexible matrix composite materials, manufactured with high strength, high stiffness fibers bonded with flexible matrices, have a very high degree of anisotropy (e.g. E L /G LT =10 3, Hannibal et al. [1983]) when compared to other engineering materials. This property of the flexible matrix composite materials makes them a premium selection for the applications requiring both strength and flexibility from a mechanical component, such as the tailrotor driveline. Hannibal et al. [1985] applied flexible matrix composites to the bearingless rotor systems, making use of the material s highly anisotropic properties. The study was limited to

24 7 mechanical characterization of the material. The results showed that the material was feasible for use in the application by virtue of its mechanical properties. Crane et al. [1994] studied the structural and damping properties of flexible matrix composites. The damping was shown to be significantly larger than that of either steel or rigid matrix composites. The study shows the possible design challenges that can be caused by high damping if the material is used in a dynamic system. Compared to conventional composite materials, the flexible matrix composite materials often have significantly higher damping properties. Damping properties of materials have been studied extensively. Zorzi and Nelson [1976] used a combination of viscous and hysteretic damping in a finite element formulation. Many researchers used various methods to model viscoelasticity. Dewulf and Roeck [2000] demonstrated the effectiveness of the modal strain energy (MSE) method. Another method, Golla- Hughes- McTavish (GHM,) was developed by Golla, Hughes [1985] and McTavish, Hughes [1992]. This method employed additional degrees of freedom with mass, linear damping and stiffness to approximate viscoelastic damping. Lesieutre [1992] suggested augmenting thermodynamic fields (ATF) method. Although these viscoelastic methods mentioned above have the potential of giving more accurate results than other models, there is a need for experimental results for all. Furthermore, when used for composite materials, these experiments need to be conducted for each layer combination, resulting in a very extensive and costly experimental study.

25 8 Rikards [1993] suggested using a hysteretic damping model for composite materials with complex stiffness coefficients. Although the method is limited to the frequency domain, it is very suitable for analytical study of the material characteristics. This method requires only one experiment for each fiber-matrix combination of composite materials. Damping for the laminate is calculated analytically. Although the complex stiffness technique is very effective in modeling hysteretic damping, the limitation of the method to frequency domain narrows the scope of the analysis to steady-state response. In shaft whirl stability analysis; time domain modeling is required for analyzing transient response. Studies on the tailrotor driveline system have been conducted for solving the existing problems and improving the system design. Darlow and Creonte [1995] have studied the design and the testing of the system by replacing the aluminum shafts with rigid matrix composite shafts. The results showed the advantages of the supercritical design in regards to the weight and also showed improvement in the weight reduction further by demonstrating the use of composite shafting. It was noted that employing composite shafting can reduce the number of segments, but the extents of this reduction with the rigid matrix composite technology were not shown. Furthermore the design relied on the current system, without eliminating the manufacturing, maintenance and reliability problems.

26 9 Recently, DeSmidt, Wang and Smith [1998] have explored the use of active magnetic bearings for vibration control of helicopter driveshafts. Results have been promising in terms of vibratory load reduction in mechanical components. Based on the observations, active magnetic bearings have the potential to be used as an actuator and enhance stability, reduce vibration, and thus increase manufacturing tolerances. Furthermore, it is also possible to replace the contact hanger bearings and dampers with non-contact active magnetic bearings to reduce maintenance costs through decreasing wear. Although promising results were shown, the study did not have any suggestions for reducing the number of segments and subsequently the number of flexible couplings. The reduced flexural stiffness decreases the natural frequencies, making vibration control more critical due to the involvement of larger number of modes in operation. Use of flexible matrix composite shafting increases internal damping of the driveline, thus requiring a higher external damping to avoid instability. Magnetic bearings levitate the driveline against the gravitational and bending loads as well as provide the active control action to provide damping and/or reduce system vibrations. These actuators have many properties that are desirable for such applications. They do not need any contact to actuate the control force. Since wear is virtually zero, there is no need for lubrication and periodic replacement of parts, and hence the maintenance cost is reduced effectively. The actuators have very good dynamic characteristics for this application. The time constant of the magnetic forces is sufficiently small so that the actuation of dynamic loads for helicopter driveshaft applications is quasi-static for the actuators. That is,

27 frequency range of interest for the tailboom- driveline system (0~100 Hz.) is in the linear range of the magnetic bearings Problem Statement and Research Goal The tailrotor driveline has been explored and developed by previous researchers. The studies suggested solutions to the existing problems and new improvements for advancing the technology. Although the results showed improvements in the operation of the driveline, none of them simplified the system enough to achieve the bigger goals of obtaining the most reliable, cost effective and lightweight driveline. The newly proposed driveline uses the flexible matrix composite technology to replace the segmented shaft and flexible couplings. The drive shaft is tailored to give the required torsional strength to carry the power across the tail, while being compliant enough in bending, accommodating the misalignment. Active control by magnetic bearings is proposed to assure low vibration levels in both the shaft and the tailboom structure and increase the stability margin of the driveline. The payoffs of the new driveline are numerous. The reduction of the complexity reduces the cost of both, manufacturing as well as operation. It increases reliability and reduces driveline weight. Since the driveline can be made more flexible with fewer components, the tailboom structural design constraint caused by the driveline misalignment is relaxed. The overall weight of the tailboom can be reduced even more with a more flexible and

28 11 lighter design for the tailboom. Furthermore this new driveline will operate at lower vibration levels for both the shaft and the tailboom structure, thus increasing component life and maintenance periods. The objective of this thesis is to provide a thorough preliminary study of the newly developed driveline. The results of this study not only provide a feasibility study, but also identify the areas that need to be addressed for replacement of the current driveline with the flexible matrix composite shafting. This study will show the effects of design parameters of the flexible composite shaft and the control system such as the ply orientations, wall thickness and control gains on the shaft design. These effects will be observed for both a steady-state operating speed and spin-up/spin-down operation. To meet the described objective, a mathematical model of the system is developed. This model includes the driveline as well as the flexible tailboom structure that supports the driveline. A parameter study is carried out on this model for the design of the composite shaft and the control system. To ensure safe operation in this highly damped supercritical driveline, the system is analyzed for whirl instability. 1.3 Summary of the Thesis In the following, the finite element model used to analyze the helicopter tailboom and the tailrotor driveline is described in Chapter 2. Design and parameter study for the driveline

29 are outlined in Chapter 3. In Chapter 4, the whirl stability of the driveline is analyzed. Finally, in Chapter 5, conclusions and recommendations for future work are presented. 12

30 13 Chapter 2 Model of the System In this chapter, the mathematical models for the tailboom structure and the tailrotor driveline are presented. This model includes a finite element model for the dynamics of the systems and a finite difference model for the thermal analysis. 2.1 Composite Shaft Equivalent Engineering Properties Stiffness Properties When both the tailboom and the shaft aspect ratios are considered, one can see that the lengths of both structures are significantly larger than the width and the height of the cross-sections. This observation leads to a conclusion that the behavior of both structures can be approximated by the beam model. For the use of beam model, having the isotropic material assumption, there is a need for calculating the equivalent stiffness properties of the composite laminate.

31 14 z 1 y N y x α N x N xy 2 Figure 2.1 In-plane force resultants of the composite cross-section z 0 z 1 z 1 z Outer Wall Center Line z n z n - 1 n - 1 n Inner Wall Figure 2.2 Cross-section of the composite laminate Classical lamination theory (CLT) is used for calculation of the laminate stiffness matrices from that of orthotropic lamina, as given by Jones [1975]. The in-plane force resultants on the composite laminate can be related to the mid-plane strains with the [A]

32 15 matrix as shown in (2.1). The coupling between the bending and twist terms in the constitutive relations is omitted due to symmetric laminate assumption. A composite laminate is symmetric if its stacking sequence is symmetric from the mid-plane of the laminate. N x ε x Ny = [ A] ε y γ N xy xy (2.1) where, N N N x y xy = zn z0 σ σ σ x y xy dz (2.2) n { k k k-1 } (2.3) -T -1 [ A= ] [ T] [ Q] [ T] ( z-z ) k=1 [A] is a function of all the lamina stiffness matrices, [Q] k, and their respective transformations, [T(α)], caused by the winding angle, α. Thickness variable, z, is illustrated in Figure 2.2. Definition of transformation matrix, [T], is given in (2.3.2). [] T 2 cos α 2 = sin α cosα.sinα sin cos 2 2 α α cosα.sinα 2cosα.sinα 2cosα.sinα 2 2 cos α - sin α (2.3.2) The lamina stiffness matrix, that can be found in literature or calculated from experiments performed on the composite lamina, is used with the described CLT to obtain the laminate stiffness matrix. With (2.4) the equivalent stiffness properties of the composite is calculated.

33 16 1 E ν E 0 * y yx ν E 1 E 0 y y yx * G = t. [ A] 1 (2.4) Thickness of the composite laminate and thus the wall thickness of the shaft is denoted by t. The E * and G * terms are the equivalent Young s modulus and shear modulus respectively, of the shaft. The lamina stiffness matrix, that can be found in literature or calculated from experiments performed on the composite lamina, is used with the described CLT to obtain the laminate stiffness matrix. With (2.4) (Jones [1975]) the equivalent stiffness properties of the composite (E *, G * ) is calculated. The geometric properties of the hollow shaft, given in (2.5), (2.6) and (2.7), are used with the equivalent material stiffness properties to calculate the axial, bending and stiffness properties of the shaft. A c 2 2 ( D D ) π = o i (2.5) ( D o D ) π I = i (2.6) ( D o D ) π J = i (2.7) 32

34 Damping Properties The hysteretic damping of the composite shaft is modeled with complex stiffness coefficients. The equivalent damping of the composite shaft can be calculated using the same equations demonstrated in the previous section. Since the complex stiffness method is used, the lamina stiffness matrices consist of complex coefficients resulting in complex equivalent stiffness moduli. 2.2 Discretized Finite Element Equations of Motion The equations of motion of the system are discretized using the finite element method. In the following sections, elements used to model each component are described Shaft Element q 11 q q 2 4 Γ(s) q q 10 8 q q 7 5 w(s) В(s) q 1 q 3 v(s) u(s) θ(s) q 9 q 12 Ω q 6 s l Figure degree-of-freedom, 2 node shaft element

35 18 The composite shaft is modeled by a twelve degree-of-freedom shaft element, illustrated in Figure 2.3. Angular displacements are related to the translational displacements given by (2.8) and (2.9). w Β= (2.8) s v Γ= s (2.9) Hamilton s Principle, given by (2.10), relates the variation of kinetic, δt, and potential, δ V, energies to the virtual work, δw, done on the element. t2 t1 ( ) δt δ V+δ W dt = 0 (2.10) The virtual work and the kinetic energy terms are formulated by Nelson [1980] as: l T * v δv * * δw = E I + G Jθ ( δθ ) + E A ( Cu δu ) ds (2.11) w δw 0 T l u u T 1 Β Β 2 T= µ v v + Id + Ip( θ) 2IpΩΓΒ ds 2 0 Γ Γ w w (2.12) E * and G * are the equivalent complex elastic and shear moduli respectively, of the composite shaft calculated in the previous section. I and J are the moment and polar moment of area respectively of the shaft cross-section; A c is the cross-sectional area. I d and I p are the diametric and polar mass moments of inertia per unit length respectively; μ is the mass per unit length; Ω is the shaft rotational speed.

36 19 The last term in (2.12) is the Coriolis term that is caused by the rotation of the shaft. Including this energy term allows the model to take the gyroscopic effects of the shaft and the attached rotating elements into account. The variables in (2.11) and (2.12) can be related to the nodal displacements by shape functions given in (2.14). T e { } [ S]{ } v(s,t) w(s,t) u(s,t) Β(s,t) Γ(s,t) θ(s,t) = N q (2.13) 5 6 N S = 0 -Ψ 1 0 Ψ Ψ 3 0 Ψ [ ] Ψ Ψ2 0 Ψ Ψ4 0 0 Ψ1 0 -Ψ Ψ3 0 -Ψ Ψ Ψ Ψ Ψ 2 0 Ψ Ψ Ψ Ψ 6 ( s ) ( s ) Ψ 1= l 2 3 l ( s ) ( s ) 2 Ψ 2 =s l ( s ) ( s ) Ψ 3=3-2 l l 2 3 l ( s ) ( s ) Ψ 4 =l - + l Ψ s 5=l- l 2 3 l (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) Ψ = s (2.20) l 6 Substitution of the shape functions into energy equations and Hamilton s Principle yields the equations of motion as a set of second-degree differential equations given in (2.21).

37 l [ M]{} q Ω [ G]{} q + [ K]{} q = { Q} 20 (2.21) v,w T v,w θ T θ u T u { EI[ Ns ] [ Ns ] + GJ[ Ns ] [ Ns ] + EA[ Ns ] [ Ns ]}ds [K] = (2.22) 0 l u,v,w T u,v,w Β,Γ T Β,Γ θ T θ { μ[ Ns ] [ N ] + Id[ N ] [ N ] Ip[ Ns ] [ Ns ]}ds s s s [M] = + (2.23) 0 [ ] l Γ T Β p S S 0 N= I N N ds (2.24) [ G] = [ N] [ N] T (2.25) Beam Element for Tailboom q 9 q q 2 3 q 1 Γ(s) q q 8 7 q q 6 4 В(s) w(s) v(s) θ(s) q 10 q 5 s l Figure degree-of-freedom beam element modeling the tailboom Due to the elliptical cross-section of the tailboom, the element described in the previous section cannot be used for modeling this structure. The beam element used to model the

38 tailboom has different geometric and inertial properties of cross-section in two transverse directions, which solves the described problem. 21 The axial degree-of-freedom of the tailboom structure does not have a significant effect on the response of the system. Thus, in this element the axial degree-of-freedom is omitted. The virtual work and kinetic energy equations for this element are given in (2.26) and (2.27) respectively. { EI v ( δv ) + EI w ( δw ) + GJθ ( δθ )} l δw = z y ds (2.26) 0 1 T = 2 l 0 v μ w T v + I w dz 2 2 ( Β ) + I ( Γ ) + I () θ dy p 2 ds (2.27) The procedure used in the previous section is applied to this element to formulate the elemental equations of motion Bearing Element The driveline is located on the top of the tailboom cross-section. This causes an offset between the neutral axes of the two structures. This offset couples the bending motion of the tailboom to the axial motion of the shaft and the twisting motion of the tailboom to the bending motion of the shaft. These effects are taken into consideration by the bearing model, which is illustrated in Figure 2.5.

39 22 Γ s s s В s w s Shaft u s v s θ s Ω 6 DOF Spring for Bearing h В b v b Tailboom θ b Γ b s b w b Figure 2.5 Bearing element The potential energy of the bearing is given in (2.28) V bear = kbxu rel +kby v rel +kbzw rel +ktyβ rel +ktzγrel 2 (2.28) Kinematic relations give the relative displacements: u rel = u s + h.γ zb (2.29) v rel = v s v b (2.30) w rel = w s w b + h.θ b (2.31) Β rel = Β s Β b (2.32) Γ rel = Γ s Γ b (2.33)

40 23 Some amount of accuracy will be lost if the bearing locations are assumed to be on nodal points of the finite element mesh. Furthermore such an assumption will lead to unacceptable models, as the number of elements decrease in either of the structures, increasing the distance between the actual location of the bearing to the closest nodal point. To avoid loss of accuracy and to add flexibility to the model, the bearing formulation is done for an arbitrary point in both structures. The displacements for this point are related to the nodal displacements of the associated element by the shape functions. This reduces (2.28) to four matrices, which are illustrated in (2.34). These matrices will be added to the stiffness constants of the associated four nodal displacements { } T [ ]{ } { } T [ ]{ } { } T [ ]{ } { } T [ ]{ } 1 V = q K q + q K q + q K q + q K q 2 bear s ss s s sb b b bs s b bb b (2.34) Stiffness matrices in (2.34) denote the added stiffnesses to two components, the shaft and the tailboom. [K ss ] and [K bb ] are the added stiffness matrices to the shaft and the tailboom, respectively, whereas [K sb ] and [K bs ] are the coupling stiffness terms End Inertia The horizontal and vertical stabilizers, tailrotor and the end vertical portion of the driveline are modeled by an effective mass and offset, as illustrated in Figure 2.6.

41 24 y m Y r m z br e x Nodal DOF Z Inertial Frame X Figure 2.6 Model of end inertia Kinetic energy of the added mass is: 1 T= m( r m.r m) (2.35) 2 The absolute displacement ( r m ) of the mass can be formulated in terms of the nodal displacements of the last node of the tailboom. (2.36) r b and θ b r=r+θ xe (2.36) m b b are the translational and rotational degrees of freedom respectively of the last node; e is the effective offset. (2.36) is inserted into (2.35) and put into matrix form to obtain the inertia matrix which shares the degrees of freedom of the last node:

42 ex ez 0 1 ex 0 ey I 0 e e e e e e e end = m. + e 0 e e e e e e e e ee ee e + e 2 2 x x z y z x y 2 2 x y z x + y x z 2 2 z y x y x z y z (2.37) The off-diagonal terms of (2.37) have a significant role in tailboom dynamics. The coupling of the torsion-bending dynamics of the structure is captured employing this effective offset, thus giving rise to the off-diagonal terms. 2.3 Assembly of the Driveshaft-Tailboom System Model m Bearing Composite Drive-Shaft Active Bearing e Tailrotor Gear Box Tailboom Figure 2.7 Finite element model Following the procedures mentioned in the previous sections, the FEM of the tailboomdriveline structure is generated. The composite shaft is modeled by the twelve degree-of-freedom shaft elements whereas the tailboom is modeled with ten degree-of-freedom beam elements. At the last node of

43 26 the tailboom, the effective mass is added with an offset. Two structures are assembled together with 2 contact bearing elements at each end of the shaft. There are also a variable number of active bearings assembled in the tailboom, providing the required control action. The equations of motion of the elements are assembled together in (2.38). The left hand side of the equation is the sum of the external and control forces [ M ]{ q } + [ G ]{ q } + [ K ]{ q} = { Q} + [ K ]{} q [ K ]{ q} (2.38) sys sys sys d + [K p ] and [K d ] are the proportional and derivative gain matrices respectively. p In the frequency domain the equation of motion (2.39) can be represented with a transfer function, as given in (2.40). {} [ H( ω) ]{} Q 2 [ H( ω) ] ω [ M ] + iω Ω[ G ] [ K ] q = (2.39) { { } + {[ K ] [ K ] } 1 = (2.40) sys Ω is the rotational speed of the driveline in rad/s, whereas ω is the frequency variable of the transfer function. [K p ] and [K d ] are proportional and derivative gain matrices respectively. sys d sys p

44 External Loads Equivalent Aerodynamic and Maneuver Loading There are various aerodynamic loads on the tailboom structure during hover and forward flight. Also, during maneuvering of the aircraft there are the inertial loads causing deflection on both the tailboom and the driveline. Although some of these described forces have dynamic components, an equivalent static load on the tailboom models the forces with sufficient accuracy. The studies of DeSmidt, Wang and Smith [1998] on the forces acting on the tailboom are used as a reference in this thesis Torque Loading The total power transmitted and the shaft rotational speed of the Apache AH-64 is reported by Darlow and Creonte [1995]. The change of rotational speed of the driveline would mean a change of design of other driveline components (e.g. gearboxes, engine), which are not addressed in this thesis. To avoid problems that may be caused by such a change, the rotational speed of the high flexibility driveline is designed to be equal to the current system.

45 Shaft Imbalance Loading Due to the manufacturing and assembly errors, the center of mass of a driveshaft crosssection has an offset from the rotation axis. The force caused during rotation by this eccentricity is formulated in (2.41) 2 F imb =ecmω s (2.41) where, e c is the effective eccentricity. m s is the total mass of the shaft. This total force is equally distributed to the shaft segments between each bearing Although the total force can be calculated using (2.41), the distribution is more uncertain. The designer should carefully try different distributions to see the effect of the placement of the forces. It is seen that using point forces is more critical than distribution over the length. In supercritical designs, attention should be given to the natural frequencies and the mode shapes. If the mode is excited during operation, the forces should not be located at nodal points so that the response to that mode would not be missed. 2.5 Superposition of Response The loads listed in the previous sections cause different deflections on the driveline and the tailboom. Loads other than the imbalance loading are considered to be static and the linear property of the mathematical models allows the response to be added analytically. The dynamic imbalance load is given special attention for the response. The total response to the loads are calculated with (2.42),

46 { } iφ { qt} = Fimb[ H( Ω) ]{ U y} + i.fimb[ H( Ω) ]{ Uz}.e + F [ H() 0 ]{ U } + T [ H( 0) ]{ U } ext ext s s 29 (2.42) where, q T is the total response of the system; F imb, F ext and T s are the imbalance, effective external and the torque loads respectively; { } y U, { U }, { U } and { } z ext U are the loading vectors associated with imbalance load in y direction, imbalance load in z direction, external load and torque load respectively. s The constant i (imaginary parameter) in front of the response of the imbalance load in z- direction is caused by the 90 phase difference from the y-direction. If all the loads on the driveline were static, the displacement field calculated could have been directly checked against the strength of the shaft in order to calculate a safety factor. Since the imbalance loading is dynamic, it should be noted that the superposed response of the driveline is also dynamic. That is, the external and torque loads cause a static displacement field on the shaft and in turn the shaft reciprocates around that field with the imbalance response. The beginning point of the imbalance response cycle, which is the zero phase, superposed to the static response need not be the displacement field that would cause the highest strain. The highest strain could be found in another point in the cycle. Thus, the response of the system due to the imbalance loading is superposed to the response from other excitations, with an artificially added phase of φ. This artificial phase is varied from 0 to 2π with sufficiently small increments to check for the whole cycle of the loading. In this way the most critical strain field of the shaft during the cycle of the imbalance response is calculated.

47 Composite Strength Calculations After the solution of the force response problem, the displacements are used to check the strength of the composite shaft. From the element level nodal displacements, a series of calculations are performed to obtain ply level stresses that can be used in the Tsai-Wu criterion for composite laminates given by Jones [1975]. The axial strain on the wall of the shaft is caused by the axial and two transverse bending nodal displacements given in (2.43). The strain displacement relationship is in terms of the beam shape functions and the polar position (r,β.) The polar position is defined from the center of the shaft cross-section. ε x 2 2 u v ([ N ]) r.sinβ ([ N ]) r.cosβ ([ N ]) { q } e = w s 2 s 2 s (2.43) s s s Due to the thin-wall assumption the maximum distance from the shaft axis, the outer diameter, can be used to calculate strains for the whole thickness. Due to the rotation of the shaft, every point on the shaft rotates through the highest strains. Thus, the maximum value of the strain field will be used for the rest of the calculation (2.44). ε x u v w = ([ N ]) ([ ]) ([ ]) { e s + D o. N s N s q } 2 2 s s + s (2.44) Shear stress is calculated using the related shape function as given in (2.45). γ xy ([ N ]){ q e } = D θ o s (2.45) s The only unknown strain, ε y, can be calculated from the zero transverse stress resultant assumption (2.46).

48 31 ε N x x -1 ε y =A [ ] 0 γ N xy xy (2.46) Unknowns ε y, N x, N xy are solved from this set of equations. For each ply, stresses in the ply axes are calculated from the maximum strain (2.47). σ ε 1 x -T σ 2 =Q [ ][ T] εy τ γ 12 xy (2.47) The Tsai-Wu criterion, Jones [1975] is: ( ) ( ) F σ + F σσ + F σ + F τ R + Fσ + Fσ + Fτ R = 1 (2.48) Where, 1 1 F = 1 X X (2.49) F 11 1 = XX (2.50) 1 1 F = 2 Y Y (2.51) F 22 1 = YY (2.52) F 1 = (2.53) S 66 2 X, X, Y, Y, S are the longitudinal tension, longitudinal compression, transverse tension, transverse compression and shear strength of the composite ply respectively, R is the safety factor of the ply. Failure load of the shaft would result in a safety factor of one

49 (R=1.) Safety factors greater than one (R>1) would be the safe region of operation for the composite shaft. 32 The quadratic equation in (2.48) is solved for each ply of each element to obtain a minimum safety factor for the shaft. 2.7 Torsional Instability Analysis of the Driveline In hollow shafting the mode of failure in torsion is either material failure or torsional buckling. Torsional instability is especially important in applications having thin wall thickness or low transverse shear modulus. The latter is the main cause of this problem in composite shafts since the low modulus matrix material dominates the transverse properties of the composite laminate. For modeling the shaft, illustrated in Figure 2.8, Bauchau et al. [1988] suggested using general shell theory with elastic coupling effects and shearing deformations. Since the ply orientations, that are going to be analyzed in this study are all balanced and symmetric, the elastic coupling effects will not be significant in the analysis.

50 33 Torque Loading R L h Figure 2.8 Hollow shaft under torsion The strain energy of the shaft is given in (2.54). T T ( ε [ Q] ε + γ [ Qs ] γ)dzdζ 1 U = (2.54) 2 ζ z where, z is the thickness variable, ζ is the area variable and [Q s ] is the transverse stiffness matrix of the composite laminate. The calculations are carried out with the Galerkin s method. The torsional stability analysis program, provided by Prof. Bauchau [1988] through personal interaction, was used in this study to check against the torsional buckling criterion. This program, with an input of the composite ply lay-up, outputs the critical buckling load of the hollow shaft. Validation of this program was performed using the examples given in Bauchau et al. [1988] and results were compared.

51 Thermal Analysis The heat generation caused by damping in the flexible matrix composite material is analyzed in this section. The major concern in composite shafting is to keep the shaft temperature below the critical temperature of the matrix material. This critical temperature can be defined differently for each material but is always existent. Most commonly this critical temperature is considered as the glass transition temperature of the matrix material. The energy sources are schematically illustrated in Figure 2.9. Sun + Atmosphere Main Rotor Downwash QGEN Shaft Tailboom Figure 2.9 Energy exchange in the system In operation the tailboom is heated with the radiation heat transfer from the sun and the atmosphere while the main rotor downwash cools it. Inside the tailboom the only significant energy source is the heat generation of the shaft. The dynamic strains of the shaft cause this heat generation due to damping.

52 Thermal Model of Tailboom For steady state temperature calculations of the tailboom the heat transfer in and out from the tailboom is equated by: Q Q = Q (2.55) Rad, Atm Rad,Tail Conv,Tail Q Rad,Atm and Rad, Tail Q are the radiation heat transfer from atmosphere to the tailboom and from the tailboom to atmosphere respectively; Q Conv, Tail is the convective heat transfer from the tailboom to ambient air. Velocity of the air flowing around the tailboom (in hover) is given by momentum theory (Johnson [1980]) to be approximately: W v = (2.56) rc 2 2ρπRr where W rc is the weight of the rotorcraft; ρ is the density of air and R r is the main rotor radius. The elliptical tailboom is approximated with a circular cross-section. With the known velocity, Incropera and DeWitt [1990] suggested using (2.57) for forced convection heat transfer. Nu hd ν mf 1/3 = = Cf Re Pr (2.57) where Nu and h are the average Nusselt number and the average heat transfer coefficient respectively of the forced convection; D is the diameter of the tailboom; ν is

53 the dynamic viscosity of air; C and m are constants; Re and Pr are the dimensionless Reynolds and Prandtl numbers respectively. 36 The heating due to radiation of the sun is given by Incropera and DeWitt as [1990]: G = (2.58) S, o Sc.f.cosθs The most severe conditions are considered for calculation of steady-state temperature of the tailboom. The parameters used are given in Table 2.1. More discussions for the described calculation can be found in Incropera and DeWitt [1990,] section Table 2.1 Parameters used for heat transfer analysis of tailboom m rc 6838 kg ρ kg/m 3 R r m C f m f Pr 0.7 ν x 10-6 m 2 /s ε t 0.97 σ 5.67 x 10-8 W/m 2.K 4 S c 1353 W/m 2 f 1

54 37 θ s 0 T sky 285 K In operating condition, with the forced convection, the steady state temperature of the tailboom is calculated as 44 C. While parking, although the radiation heat transfer takes place, the cooling is free convection instead of forced. For free convection of a horizontal cylinder Incropera and DeWitt [1990] suggested (2.59) for the Nusselt number Ra D Nu = (2.59) ( ) Pr Ra D is the Rayleigh s number for the free convection. 2 In parking condition, the steady state temperature is calculated to be 89 C for the tailboom structure. For the driveline, the operating ambient temperature can be set to that of the tailboom, 44 C, but the parking temperature of 89 C should be taken into consideration while the materials are selected. It is especially important to consider the parking temperature. With the low ambient temperature, the driveline can be designed to operate in much lower

55 temperatures than the parking temperature, but the material should still endure the parking temperature regardless of the operation Thermal Finite Difference Model of Shaft The damping in the shaft causes the mechanical energy to be converted to heat, which results in elevated operating temperatures of the shaft. There are two mechanisms, which result in heat generation in the shaft, illustrated in Figure Mechanism 2 (Static Strains in Rotation) Mechanism 1 (Dynamic Strains) Figure 2.10 Heat generation mechanisms The first mechanism is caused by the time varying response of the shaft. The vibrating shaft dissipates energy due to damping. This energy dissipated in each cycle proportional to the internal damping. The heat generation rate is given by (2.60).

56 39 H ω q e = X e [ I( K) ] X e. (2.60) 2π where, X e is the deflection vector of the finite element and I(K) is the imaginary part of the elemental stiffness matrix. The second mechanism is caused by the static deflection. Although the shaft deflection curve is not varying with time, since the shaft is rotating, the material goes through a tension-compression cycle in each rotation. Although physically the two mechanisms are completely different, the strain fields induced on the material are the same. Thus, the heat generation of a rotating shaft with a static deflection is equal to that of a non-rotating shaft with a vibratory deflection. The only condition is that the vibrating frequency should be equal to the rotating speed. Since the heat generation in each element is known, the finite difference method is used for the thermal model of the shaft. The energy balance for the n th element, illustrated in Figure 2.11, is given in (2.61). K cond ( Tn Tn 1) + K cond. ( Tn Tn 1) + K conv. ( Tn T ) = q n. + (2.61) K cond and K conv are coefficients for conduction and convective heat transfers respectively. T n and q n are the temperature and heat generation of the n th element respectively.

57 40 T T n-1 T n T n+1 Figure 2.11 Finite difference element of the shaft For a hollow shaft, the conduction coefficient is given in (2.62). K cond k π 2 2 ( D D ) c = o i (2.62) ls 4 where k is the conduction heat transfer coefficient of the material; D o and D i are the outer and inner diameters of the shaft respectively; l s is the length of an element. The coefficient of convective heat transfer for a rotating shaft is suggested by Kendoush [1996] as (2.63) k air h = Re.Pr (2.63) D o where, k air is the conductive heat transfer coefficient of air and D o is the outerdiameter of the shaft. Thus the convection coefficient becomes; K = πhd l (2.64) conv o s In matrix form, (2.61) can be represented as: [ K ]{} T = {} q + K { T } THR conv (2.65) The temperatures calculated by (2.65) are used for the thermal analysis of the driveline.

58 41 The coupled thermal and mechanical model of the system can be used to study design parameters of the system and observe the effects of each parameter on temperature distribution of the driveline.

59 42 Chapter 3 Analysis and Synthesis of the High Flexibility Driveline In this chapter, the new driveline utilizing the flexible matrix composite shaft and active bearing technology is synthesized and analyzed. The system design variables are identified and the effects and the role of each are illustrated. The analysis and synthesis processes will be conducted for two operating conditions, namely, steady state and spin-up/spin-down. The two operating conditions will introduce different design criteria. Equal weight will be given to both to ensure the reliable operation of the driveline. Design variables that are used for the synthesis of the tailrotor driveline are: - Number of composite plies - Ply orientations - Wall thickness of hollow shaft - Proportional control gain - Derivative control gain The constraints in this design can be summarized as: - Mechanical failure envelope - Shaft displacement at actuator location - Maximum shaft temperature while operation

60 43 - External damping requirements In the following sections the constraints and effects of the design variables are described in detail. Although not unique, the presented order of the synthesis of the driveline is a logical alternative, which can be applied step-by-step by the designer to achieve a successful design of the tailrotor driveline. 3.1 Finite Element Model The system in the following studies will be the single piece composite shaft fixed to the tailboom from the two ends by hanger bearings as illustrated in Figure 3.1. L t L h L a1 L a2 Hanger Bearing Ω Tailboom Active Bearing M end Composite Shaft Figure 3.1 Dimensions of the tailrotor driveline The dimensions and parameters used in this chapter given in Table 3.1, unless otherwise stated.

61 44 Table 3.1 Parameters and dimensions used in the model L h L a1 L a2 L t D i D o K p 0.79 m 3.21 m 5.63 m 8.05 m m m 1 x 10 7 N/m K d 0 Ω P s 4800 rpm 246 kw Parameters for modeling the tailboom with beam elements are based on the detailed studies of DeSmidt, Wang and Smith [1998]. Most of the following studies are performed by using the Toray 700/Adiprene L100- Caytur 21 system. Properties of this orthotropic lamina at 10Hz frequency are given in Table 3.2. These values are measured at The Penn State Composites Manufacturing Technology Center by Professor Charles Bakis and Ying Shan. Table 3.2 Properties of Toray 700/Adiprene L100-Caytur 21 E L * E T * x x 10 7 i N/m x x 10 7 i N/m 2

62 45 G LT * x x 10 7 i N/m 2 ν LT ρ c 1650 kg/m 3 X 2185 x 10 6 N/m 2 X 2000 x 10 6 N/m 2 Y 5.1 x 10 6 N/m 2 Y 5.5 x 10 6 N/m 2 S 20.0 x 10 6 N/m Steady State Operation A rotorcraft operating in hovering or forward flight usually has the tailrotor rotating with constant speed to provide the required anti-torque force to the vehicle. In this section, the emphasis will be on the effects of design parameters on the performance in this operating condition. The dynamic operating condition (i.e. the unbalance load effect) introduces a challenge for the designer. For a super-critical shaft, keeping the operating speed as far as possible from the neighboring natural frequencies is an objective of the design. The bending stiffness, the mass of the shaft and the control gains have significant effects on the natural frequencies. All of the design parameters have an effect on either or both

63 46 of these variables. Thus, with the large number of design parameters having an effect on the natural frequencies, one can find many areas distributed in the design space, where the operating condition coincides with a natural frequency. This property of the system makes decoupling and studying the design parameters one by one, impossible. In the initial part of the study, for analyzing ply orientation and wall thickness variables, some of the operations are considered without the effect of the dynamic loads (i.e. no unbalance load). This is to simplify the analysis so that the effect of the respective design variable can be more clearly demonstrated Ply Orientation of the Composite Shaft The biggest advantage of composites in comparison to traditional isotropic materials is their designable material properties. The orientation of the constituents of a composite material, the fiber and the matrix, can be engineered to give best performance in a specific application, which is also referred to as elastic tailoring (Smith et. al. [1991], Rehfield [1985].) In this study, the orientation angle of the fibers to the axis of the shaft will be a design parameter. The fiber lay-up determines whether the fiber material or the matrix material dominates the stiffness and strength characteristics of the shaft.

64 47 In Figure 3.2 the safety factor of the design, calculated by (2.48), of the design is plotted against the ply orientation. The composite laminate is considered to be symmetric and made up of four plies oriented in the positive and the negative direction of the respective angles, which can be represented as (α/-α/-α/α). Figure 3.2 Safety factor vs. ply orientation, one ply orientation variable without imbalance loading The results, as expected, show the most favorable ply orientation to be in the region where the torsional characteristics are dependent on the fibers and the bending characteristics on the matrix. Thus the shaft is strong in torsion and flexible in bending. The maximum point for safety factor is observed to be ±55.

65 48 The most critical loadings on the composite shaft are tabulated for few ply orientations in Table 3.3. The values are given in the composite lamina coordinate system, illustrated in Figure 2.1. Table 3.3 Most critical loading on composite shaft Ply Orientation ( ) σ 1 (Pa) σ 2 (Pa) τ 12 (Pa) x x x x x x x x10 6 Figure 3.3 Safety factor vs. ply orientation, two ply orientation variables without imbalance loading

66 In Figure 3.3 the safety factor is plotted versus ply orientations, allowing two different 49 angles for the plies. The maximum safety factor is observed to be in the (55 /-55 /0 /0 ) s ply orientation. This ply orientation will be used for the rest of the studies in this chapter Wall Thickness of the Shaft To have a just comparison between the current and the new driveline, the outer diameter of the shaft is kept constant. Variations on the wall thickness of the shaft are accomplished by changing the internal diameter. The effect of wall thickness on the shaft operation is demonstrated by plotting the safety factor in Figure 3.4. The operation is studied both with and without the imbalance loading.

67 50 Figure 3.4 Safety factor vs. wall thickness The large effect of the imbalance loading on the safety factor happens when the operating speed of the shaft is close to one of the natural frequencies. The effect of the wall thickness is demonstrated with the frequency response function in Figure 3.5

68 51 Figure 3.5 Maximum shaft response with 1mm and 4mm wall The operating speed of 506 rad/s coincides with the natural frequency at a wall thickness of approximately 1.7 mm, which reduces the safety factor of the design. When the wall thickness is increased, the natural frequency steers away from the operating speed resulting in a safer design. The decrease of the safety factor of the design with wall thicknesses larger than 3 mm can also be explained with this figure. The next natural frequency also reduces with the increasing wall thickness and becomes closer to the operating speed, reducing the safety factor.

69 52 In the system, it can be seen that, as the wall thickness increases, most of the natural frequencies of the shaft decrease. Although the stiffness of the shaft increases, the mass per unit length of the shaft also increases. This phenomenon can be explained by analyzing a simply supported beam as an example. One can represent natural frequencies of the beam in flexure by (3.1) where EI ω n = An (3.1) 4 ml ωn is the n th natural frequency, A n is the associated constant and m is the mass per unit length. The area moment of inertia and the mass per unit length of the hollow shaft can be written in terms of the geometric variables. ω n E D D = A n. (3.2) 4 ρl D D 4 o 2 o 4 i 2 i where D o and D i are the outer and inner diameters respectively of the hollow shaft. Factorizing the numerator of the right root and cancellation results in (3.3) E 2 2 ω n = A n. D 4 o + Di (3.3) ρl It can be seen from (3.3) that if the wall thickness is increased by keeping the outer diameter constant, the natural frequencies will decrease. This formulation agrees with the numerical results obtained.

70 53 It is noteworthy that if the inner diameter were to be kept constant while increasing the wall thickness there would be an increase in the natural frequencies. In this case the addition of stiffness is more dominant than the addition of mass. It should also be noted that, the example taken is a simply supported beam. Although the same effect is observed in most of the natural frequencies of the rotorcraft driveline system, the complexity of the driveline is much greater. The effects introduced by the compliant supports (active bearings,) and the compliant base (tailboom) have effects on how the natural frequencies behave. Thus, the results seen in the example cannot be generalized for the driveline system. Wall thickness of 3mm is used for the rest of the studies in this chapter Control Gains of the Active Actuators The control action on the shaft is determined by two variables, namely, proportional (K p ) and derivative (K d ) gains. These variables have great importance in the design of the shaft as well as the dynamic properties of the system. The role of the derivative gain, being more important in spin-up/spin-down conditions, will be discussed in detail in the next section. The effect of proportional gain is illustrated in Figure 3.6 by the plot of the safety factor and the maximum shaft displacement at the actuator locations, {d act }.

71 54 Figure 3.6 Effect of proportional gain on the safety factor of the composite shaft The tailboom has a tapered cross-section whereas the shaft has a uniform cross-section. This difference causes the two structures to have a different deflection curve when they are fixed from both ends and forced from one end, as illustrated in Figure 3.7. As the proportional gain is increased from zero the shaft is forced closer to the tailboom at the active bearing locations, loading the shaft further. This is the reason for the decrease in safety factor in the left end of the curve.

72 55 Figure 3.7 Tailboom and shaft horizontal deflection curves with and without mid-span bearings Similar to the actuator displacement the non-monotonic plot of safety factor is caused by the system resonance when the natural frequencies of the structure coincide with the operating speed. The maximum shaft displacement at the actuator locations is an important property of the control system. Usually the actuators operate with a specified maximum radial displacement as illustrated in Figure 3.8. In the case of magnetic actuators, the shaft comes into contact with an auxiliary bearing after a specified displacement. To avoid such an event, the designer should use high enough proportional gain.

73 56 Actuator Gap y z Magnetic Bearing Figure 3.8 Actuator gap of a magnetic bearing The non-monotonic relation between the actuator displacement and the proportional gain is caused by the system resonance when the natural frequencies of the structure coincide with the operating speed. This trend is presented in Figure 3.9. The maximum frequency response of the shaft is plotted for proportional gains 6.5x10 5 N/m and 1x10 7 N/m.

74 57 Figure 3.9 Maximum frequency response function of the shaft, critical/ non-critical operating condition Torsional Stability of the Driveline In the torsional buckling analysis the program described in Section 2.7 is used. Simply supported edges boundary condition is used in the analysis. The torsional buckling safety factor of the shaft is plotted against the wall thickness in Figure The safety factor is defined as the ratio of the actual loading to the critical loading.

75 58 Figure 3.10 Torsional buckling safety factor As shown in the figure, the system is safe against buckling when the wall thickness is larger than 1mm. This indicates that when close to the design point of 3mm, the system mode of failure will most likely not be due to buckling. Torsional buckling safety factor curve will change with different ply orientations. Although torsional buckling is a mode of failure of the system, the mechanical loading of the shaft requires thicker shafts that are more resistant to this instability. Thus, in most of the cases, torsional instability is not the dominant mode of failure.

76 Spin-Up and Spin-Down Operations In the previous section, the main emphasis was to synthesize the design such that the operating speed is far from the natural frequencies. Although this is very important, every time the helicopter operates, the driveline will still through each of the resonant frequencies that are under the operating speed, during starting and stopping. The severity of the resonance can be greatly reduced by introducing damping to the system. Derivative gain is used in the control system to provide the damping effect. In Figure 3.11, the maximum frequency response function is plotted with proportional gains of 1x10 6 N/m and 5x10 7 N/m. It can be seen that if the proportional gain is selected too high, there is very little damping in the system. The increased actuator stiffness, when increasing the proportional gain, allows very small motion at the actuator location. This reduces the energy dissipation in each cycle, which results in low damping.

77 60 Figure 3.11 Effect of derivative gain with high and low proportional gains On the other hand, if the proportional gain is set too low, high shaft displacement is observed at the actuator location. From this reasoning, it is very evident that the two gains should be designed together, introducing as much damping as possible while staying in the limits of allowable shaft displacement at the actuator location. Change in the maximum shaft deflection (operating in steady state over all the speeds below the nominal operating speed) with various control gains is presented in Figure 3.12.

78 61 Figure 3.12 Maximum spin up shaft deflection vs. control gains It can be seen that the best damping characteristics are achieved as the proportional gain decreases. On the other hand, there is an optimum derivative gain for each proportional gain. To gain valuable insight to the effects of control gains the frequency response of the shaft is illustrated in Figure 3.13 and Figure 3.14 with proportional gains of 2x10 6 N/m and 3x10 5 N/m respectively. A derivative gain of 1170 N.s/m is used in both the plots. The shaft position is defined as the distance of the point from the fuselage side end of the driveline.

79 62 In Figure 3.13 it can be seen that one of the natural frequencies that is passed during spin up is significantly excited. When the proportional gain is reduced, as illustrated in Figure 3.14, the same natural frequency is effectively damped and the deformations are observed to be significantly lower. Figure 3.13 Frequency response of driveline with K p = 2x10 6 N/m

80 63 Figure 3.14 Frequency response of driveline with K p = 3x10 5 N/m The variation in maximum shaft displacement at the actuator location with various control gains is presented in Figure These results are compared with the maximum allowable displacement of the actuator to identify the allowable control gains. Using this criteria and the surface presented in Figure 3.12, an optimum point can be found.

81 64 Figure 3.15 Maximum shaft displacement at the actuator location vs. control gains The variation in actuator force with various control gains is presented in Figure It is seen that reduction in the control gains results in lower actuator requirements and higher energy savings.

82 65 Figure 3.16 Maximum actuator force vs. control gains The tuned gains are demonstrated in Figure The presence of damping is observed by the decreased peek deflections in resonant operation. The lower proportional gain allows the active bearings to damp energy more than the higher proportional gain system. The maximum shaft deflection is decreased from 38 mm to 4 mm with tuning the control gains.