Samane Zeyghami

O Tatin

Effect of non-uniform flexibility on hydrodynamic performance of pitching propulsors

Keywords:

Many aquatic animals propel themselves efficiently through water by oscillating flexible fins. ese fins are, however, not homogeneously flexible, but instead their flexural stiffness varies along their chord and span. Here, we developed a low order model of these functionally-graded materials where the chordwise flexibility of the foil is modeled by one or two torsional springs along the chordline. e torsional spring structural model is then strongly coupled to a boundary element fluid model to simulate the fluid-structure interactions. We show that the effective flexibility of the combined fluid-structure system scales with the ratio of the added mass forces acting on the passive portion of the foil and the elastic forces defined by the torsional spring hinge. We further detail the dependency of the propulsive performance on the flexibility and location of the single torsional spring for a foil that is actively pitching about its leading edge. Our results show that increasing the flexion ratio by moving the spring away from the leading edge leads to enhanced propulsive efficiency, but compromises the thrust production. Proper combination of two flexible hinges, however, can result in a gain in both the thrust production and propulsive efficiency.

INTRODUCTION

Flying and swimming animals propel themselves rapidly and efficiently through a fluid using flexible propulsors. A substantial line of work has already confirmed that flexible propulsors are advantageous to rigid ones in aquatic locomotion, specifically with regard to propulsive efficiency [ , , , ]. Some have argued that the interactions between the fluid and the structure deforms the foil in the direction of the fluid. ese deformations lead to curvature-induced thrust increases [ ] as well as a favorable phase lag between the pitching and heaving motions of the foil which in return enhances propulsive efficiency [ , ]. In addition, the occurrence of resonance is argued to play an important role in enhancing propulsive performance of flexible foils. Previous studies have shown that the efficiency is maximized at or near the resonance frequency of the combined fluid-structure system [ , ]. e resonance frequency of the combined system is a function of the inertial properties of the structure as well as the added mass arising from inertia of the fluid. However, when the flexibility is variable along the chord, the scaling of the resonance frequency of the fluid-structure system is non-trivial, a topic that we will a end to in the present study. e propulsive appendages of swimming and flying animals are made of functionally-graded materials where the flexibility varies both along the chord and span. In fact, Combes and Daniel [ ] measured the flexural stiffness of several insect wings and found that it declines sharply from the wing base to wing tip, in the spanwise direction, and from the leading edge to the trailing edge, in the chordwise direction. Similarly, the flexibility of the propulsive surfaces of swimming animals (such as fluke, fin and tail) appear to be non-uniform and decline from the leading to trailing edge and from the center to the edges [ , , ]. Inspired by these observations, a number of recent studies have suggested that the distribution of the flexibility along the foil, in addition to its overall flexibility, may play an important role in enhancing the propulsive performance. Comparing different distributions of flexibility along a two dimensional thin foil undergoing small amplitude heaving motions, Moore [ ] has suggested that the concentration of the flexibility at the leading edge enhances thrust production. In another study, Riggs et al. [ ] have tested the thrust production of a flexible fin with a standard NACA cross-sectional shape alongside fins with a stiffness profiles mimicking that of a Pumpkinseed Sunfish. ey showed that bio-mimetic fins generate more thrust regardless of the overall stiffness of the fin, showing that the performance improvement is due to the stiffness profile itself and not the flexibility alone. Similar conclusions were found by [ ] in an experimental study on the propulsive performance of robotic fins with variable chordwise flexibility. ey found that fins with variable flexibility outperformed the fins with uniform flexibility with regard to both thrust production and propulsive efficiency. Here we aim to probe the effect of the distribution of flexibility on the propulsive performance of a pitching foil by separating the effect of the overall flexibility of a pitching foil from that of its bending pa ern. We model the chordwise flexibility of the foil with a series of torsional springs with varying flexibility. e effect of the bending pa erns is modeled via changing the location of the spring along the chord. First, we investigate the propulsive performance of a pitching foil with a single flexible joint. We detail the effect of both the overall flexibility and the bending pa ern of the foil. Next, we repeat our numerical experiment on a pitching foil with two flexible joints.

is time the location of the flexible joints are fixed but the distribution of the flexibility is varied. is study is a primary step toward understanding the role the functionally graded materials on the propulsive performance. Results of this study can also inspire design of innovative and non-traditional propulsors.

. PROBLEM DEFINITION

All the numerical experiments are performed on a two dimensional foil where the leading edge of the foil is actively pitching with a peak-to-peak amplitude of 2θ 0 = 10 o . ere are either one or two flexible joints along the chord modeled by torsional springs (figure ). e distance of the flexible joint from the leading edge, normalized by the chord length, is quantified by the flexion ratio, λ. e flapping frequency and swimming velocity are kept constant across these simulations at 2.87 Hz and 0.1 m/s resulting in a reduced frequency of k = 2.87 (defined as k = fc U ) and a Strouhal number (de-fined as St = fA U ) of 0.5. is St is defined for a rigid foil without a flexible joint. However, the real St of the flow is an output of the system and varies with the trailing edge amplitude, which itself is a function of the flexibility and flexion ratio.

For materials with similar densities as that of the surrounding fluid (in the present study ρ s = ρ ), the flexibility of the combined fluid-structure system is a function of the added mass forces of the fluid and the elastic forces of the structure. We define Π k as the ratio of these forces which characterizes the effective flexibility of the combined fluidstructure system.

Π k = (1 -λ 2 ) ρbc 4 f 2 k ( )
where ρ, f , c, and k respectively are the fluid density, pitching frequency, chord length, and the spring stiffness. e numerator is the added mass force represented as a cylinder of fluid with a diameter equal to length of the passive portion of the foil multiplied by a characteristic acceleration. When only added mass forces are modeled, Π k is directly proportional to the ratio of the driving and resonance frequencies.

Both flexibility and flexion ratio are changed and their effect on the propulsive performance is detailed.

e performance is analyzed using thrust and power coefficients as well the propulsive efficiency which are defined below.

C t = T 0.5ρU 2 bc , C p = P 0.5ρU 3 bc ( )
where T, and P are the thrust and input power. b is the span length that is set to . Alternatively, we normalized thrust and power with trailing edge velocity as defined in eqn. . Note that the trailing edge amplitude is an output of a flexible foil system.

C t ′ = T 0.5ρA 2 f 2 bc , C p ′ = P 0.5ρUA 2 f 2 bc ( )
where A is the trailing edge amplitude.

. NUMERICAL METHODS e flow over the foil is modeled using a two-dimensional potential flow method in which the flow is assumed to be irrotational, incompressible and inviscid. We follow [ ] and [ ], in that the general solution to the potential flow problem is reduced to finding a distribution of doublets and sources on the foil surface and in the wake that satisfies no flux boundary condition on the body at each time step. Constant strength source and doublet line elements are distributed over the body and the wake. Each body boundary element is assigned a collocation point which is shi ed a small distance under the body surface (here % of the local thickness of the body). e constant potential Dirichlet condition is enforced at the collocation points to ensure a no flux boundary condition on the body surface. Additionally, at each time step a wake boundary element is shed with a strength that is set by applying an explicit Ku a condition, where the vorticity at the trailing edge is set to zero [ , , ]. A wake rollup algorithm is employed to ensure that the wake does not support any force.

e wake elements advect by the local velocity at the wake panel edge points. During the wake rollup the point vortices, representing the ends of the wake doublet elements, must be de-singularized for numerical stability of the solution [ ].

To do so, at a small cutoff radius of ǫ = 0.05c, the irrotational induced velocities from the point vortices are replaced by a rotational Rankine core model. e tangential perturbation velocity component is calculated by local differentiation of the perturbation potential. Finally, the pressure acting on the body is found via applying the unsteady Bernoulli equation. More details can be found in [ , , ].

e structural flexibility is modeled via torsional springs which connect the structural mesh elements together. e kinematics of the leading structural element is always prescribed. Equation governs the dynamics of the passive structural elements.

I Θ + C Θ + KΘ = N f + N i + N h ( )
where N f is the hydrodynamic moment exerted about the joint location. N i is the inertial moment due to the translational velocity of the center of mass of the corresponding element. N h is the moment exerted by the forces at the joint which keep the elements together. I is the matrix of moments of inertia about the joint points. K and C are the matrices of the structural stiffness and damping, respectively. Θ is a vector containing the orientation of the passive elements.

For a foil with two flexible joints we have:

I ≡ I 1 0 0 I 2 K ≡ k 1 + k 2 -k 2 -k 1 k 2 C ≡ c 1 + c 2 -c 2 -c 1 c 2 ( )
where I i is the moment of inertia of the i th element. c i and k i are the structural damping and the stiffness of the spring a ached to the leading edge of the i th element.

To solve the fluid-structure interaction problem, equation is discritized in time, using the trapezoidal rule (equations and ), and solved within each small time step via a strong coupling between the fluid and structural solvers which is accelerated by the Aitken method. To improve the convergence properties of the solver, while keeping its efficiency, we use two different time step sizes ∆t and ∆t s for the fluid and structure solvers, respectively, where ∆t s = ∆t N s . N s is set to 100 in the present simulations.

Θm+1 = Θm + 1 2 ( Θm + Θm+1 )∆t s ( ) Θm+1 = Θm + 1 2 ( Θm + Θm+1 )∆t s ( )
where superscripts m and m + 1 represent the values at times t m s and t m+1 S , respectively. x represents any variable x within the structure solver. Substituting equation into equation and solving for Θ m+1 we get: Similarly, equation can be rearranged to get an expression for Θm+1 as a function of Θ:

Θm+1 = ( 2 ∆t s ) 2 ( Θm+1 -Θm )-( 4 ∆t s ) Θm - Θm ( ) (a)
Θm+1 = 2 ∆t s ( Θm+1 -Θm )-Θm ( )
Where the right hand sides of both equations and are known from the previous iteration. Substituting equations and into , we can rewrite the governing equations as a linear, but coupled, system of equations as follows: A Θm+1 = bm

A = K + ( 2 ∆t s ) 2 I + ( 2 ∆t s )C bm = ( 2 ∆t s ) 2 I Θm + ( 4 ∆t s )I Θm + I Θm + ( 2 ∆t s )C Θm +C Θm + Nm i + Nm f + Nm h ( )
Equation together with and form a complete set of equations for the structure. e set of structure equations are first initialized by the known solution from the previous time step (of the fluid's solver) and then iterated N s times to advance the solution as much as ∆t.

To improve convergence, equation is uncoupled by employing a Gauss-Seidel formulation where the newly obtained solution for the orientation of the first element is used to obtain the solution for the second element at each time step t s .

e Aitken acceleration method is commonly used in the numerical simulation of fluid-structure interactions and is proven to be sufficiently simple and efficient [ , , ]. is method uses the values from the two previous iterations to correct the new solution. We employ Aitken's method to advance the solution in the fluid's solver based on the residual calculated in the previous two iterations.

e residual is calculated as the difference in the solution obtained in the structure and the fluid solvers, ì r i = Θi -Θ i , where Θ i is a vector representing the orientation of the neutral axis of the foil in the fluid solver.

e solution to the coupled fluid-structure system at each time step t n = n∆t is obtained by following the algorithm below:

. (f) Solve the solid deformations; Θi , Θi , and Θi using equations , , and

i = 0, r 0 = 1, Θ0 = Θ n-1 , Θ0 = Θ n-1 , Θ0 = Θ n-1 , and ω 0 = 1e -2 . While r i >δ (a) i = i + 1 (b) if i > 1 modify the solution; Θi = Θi-1 + ω i-1 r i-1 (
(g) Calculate the residual,

r i = Θi -Θi (h) Calculate Aitken Acceleration factor; if i < 3, ω i = ω 0 else, ω i = ω i-1 ì r i-1 (ì r i-1 -ì r i ) r i-1 -r i 2
. Update the solution for time t n ; Θ n = Θi , Θ n = Θi , and

Θ n = Θi
where δ is set to 10 -8 . When the solution converges within the n th time step, we set n = n + 1 and repeat the steps above to solve for the next time step.

. Discretization independence

. Validation

We tested the accuracy of our numerical model by comparing our results against the analytical results presented in [ ] for a two dimensional thin foil with a torsional spring at the leading edge. A small amplitude (harmonic) heaving motion is enforced at the leading edge.

e foil passively pitches about the leading edge due to the action of fluid, inertial, and elastic forces. We compared both the cycle-averaged thrust and the cycle-averaged power with the analytical solution. Results are shown in Figure . e coincidence of the resonance frequency for all λ's shows that the proposed scaling for the effective flexibility appropriately accounts for the effect of the flexion ratio. However, inspecting the trend of variations in A * with Π k reveals that the effective damping of the combined fluid-structure system increases for larger λ values. It is also worth noting that before and a er resonance, higher flexion ratio foils generally experience larger trailing edge amplitude. Around the resonance the relationship is more nonlinear. For flexion ratios smaller than 0.5 the trailing edge amplitude increases with λ. is relationship is reversed for λ>0.5.

Figure b shows variations in thrust and power coefficients as a function of Π k . Unlike the trailing edge amplitude, the thrust coefficient is generally larger for smaller λ's, except for very rigid foils where increasing λ up to . results in gain in thrust. Inspecting figure c shows that this gain in thrust comes with no additional cost with regard to the power consumption. C t rises up to its peak value at the resonance, for small λ values, and then drops quickly with further increase in flexibility. is is unlike the behavior of C t for large λ values where the thrust plateaus before resonance and then drops with a mild slope when flexibility increases. It is worth noting that for all flexion ratios, C t increases faster than C p until slightly a er resonance. is is reflected in Fig- ure d where we show changes in the propulsive efficiency, defined as η = C t C p , as a function of Π k . ick drop in thrust production a er the resonance for small λ values results in a decline in the propulsive efficiency. In contrast, for large flexion ratios, η keeps increasing with flexibility for a wide range of flexibilities.

In figure a-b, we plo ed C ′ t and C ′ p as the function of Π k . When we normalized the thrust by the trailing edge velocity, the peak in the force production and power consumption disappeared implying that the peak is an artifact of the amplified trailing edge amplitude. Variations in the trailing edge amplitude are responsible for the major changes in the thrust production for λ = 0. is, however, is not the case for other λ values. e fact that the curves for different λ values do not collapse on top of one another implies that the bending e behavior of the C ′ p curves is somewhat different from that of C ′ t . Before resonance, the power coefficient appear to drop quickly by increasing flexibility. A er the resonance, for small λ values, the power coefficient starts to rise again. is results in a drop in efficiency which is due to a simultaneous rise in power consumption and drop in thrust production. For larger λ values, however, as flexibility increases, C ′ p keeps declining where the rate of this decline decreases for large flexibilities. us, the rate of increase in efficiency with Π k (figure d) decreases.

To summarize our findings with regard to the propulsive performance of these flexible foils, we plo ed contours of C ′ t and C ′ p as a function of λ and Π k in figure c-d. Contours of propulsive efficiency are overlayed on both figures with do ed lines. It is worth noting that the contours of efficiency line up with those of power consumption indicating that the propulsive efficiency is mostly governed by the (input) power requirement and not the thrust production. We identified three regions on these contour plots. In region , which contains low Π k and low λ value foils, the thrust coefficient remains relatively constant. e propulsive efficiency can be enhanced by increasing flexibility (contours of propulsive efficiency are almost parallel to the λ-axis). In region , increasing either flexibility or flexion ratio benefits efficiency but compromises thrust production. In region , contours of power consumption, and thus propulsive efficiency, are more or less aligned with the Π k axis, meaning that the efficiency is more sensitive to changing the flexion ratio. us, implying that for largely flexible foils increasing the flexion ratio can result in improved propulsive performance.

. Two flexible joints

In the previous section we showed that changing the bending pa erns of a pitching foil via increasing its flexion ratio is the key to improving the propulsive efficiency. However, there is a trade off to this gain since smaller flexion ratios are required for larger force production. We hypothesize that combining multiple flexible joints may be the key to gaining both in efficiency and thrust magnitude. To test our hypothesis, we repeated our numerical experiment on a pitching foil with two torsional springs located half a chord away from each other at λ 1 = 0.2 and λ 2 = 0.7. e kinematics of the leading edge was kept identical to the cases studied in the previous section.

When multiple flexible joints are allowed not only the flexiblity of the individual joint, but also the profile of flexibility distribution along the chord will affect the foil's deformations and, thus, its propulsive performance. Here, we chose a profile which maintains the effective flexibility of the foil unchanged along the chord. e spring stiffness, thus, declines quadratically with the normalized distance from the leading edge, k ∝(λ -1) 4 . To do so, the stiffness of each one of the springs is determined solely based on its distance from the leading edge using equation . is could be due to increased circulatory effects or existence of nonlinear added mass effects when combining the two joints. In comparison to one flexible joint configurations, the trailing edge amplitude of the present configuration reaches a higher maximum at the resonance, and drops slower a erward. e deformation of the middle element appear to be maximum at the resonance (figure ). However, the amplitude of the motion of the last element continue to increase a er the resonance a enuating the drop in trailing edge amplitude.

In figure b, in black is shown the cycle-averaged thrust coefficients, C t . Similar to the results in the previous section, the peak thrust generation occurs at the resonance but the magnitude of the peak is substantially higher. is is partially due to the increased trailing edge amplitude which leads to enhanced added mass forces and a favorable reorientation of the hydrodynamic force relative to the propulsive direction. For Π k ≤ 0.3, the magnitude of C t is similar to that of a foil with a single flexible joint at λ = 0.2 indicating that the effect of the combination of the two joints was not destructive to the thrust production capacity. All the more so,for larger flexibilities, C t is larger than what was achieved by any of the single jointed foil configurations due to the constructive combination effect. In the same figure, in blue is shown C ′ t where thrust is normalized by the trailing edge velocity rather than the swimming velocity. Similar to the case of a single spring, C ′ t drops with increasing flexibility. p in blue. is curve shows a fast drop in the power consumption with flexibility before the resonance (which is typical of high flexion ratio foils in single spring configurations), where at Π k ≥ 0.25 power coefficient is already below what was achieved by most of the single spring configurations (figure ). C ′ p continues to drop even more so a er resonance. In such a way, the two spring configuration maintains high propulsive efficiency across a wide range of flexibilities spanning on the both sides of resonance. Overall, the results of this section supports our earlier hypothesis.

. CONCLUSIONS

It has been shown that the unsteady propulsive performance of flexible foils with a single torsional spring hinge is not only a function of their effective flexibility but also their bending pa erns. Across all flexibilities tested here, increasing flexion ratio was beneficial to the efficiency while diminishing the thrust production. We showed that the combined effect of the flexibility and the flexion ratio can result in propulsive efficiencies as large as % or more for a purely pitching foil.

is is more than % larger than the propulsive efficiency of a rigid foil with the same leading edge kinematics. Additionally, flexible foils with two flexible joints were examined to probe whether multiple flexible hinges could be used to a enuate the trade-off between thrust and efficiency to achieve fast and efficient swimming simultaneously. e flexibility of the joints was determined such that the effective flexibility was constant along the chord. We found that this combination of flexible joints have a constructive effect on the propulsive performance of a pitching foil with regard to both thrust and efficiency across a wide range of flexibilities.

Finally, it is important to note that the fluid-structure model used in this study is subject to several assumptions. e fluid model, for instance, does not account for viscous effects such as the separation that may occur at the leading edge or along the deforming body, especially when the solid deformation is large. ese effects can potentially influence the propulsive performance especially with regard to the power consumption.

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Figure .

 . Figure . Schematic of the model for a single spring.

Figure .

 . Figure . Propulsive efficiency as a function of (a) number of time steps and (b) number of body panels.

Figure .

 . Figure . Analytical solutions for thrust and power coefficient as a function of reduced frequency, for two different non-dimensional spring stiffnesses, are shown with solid lines. ese solutions are taken from [ ]. Closed circles are the solutions calculated by the present numerical method.

  c) Calculate the location of the neutral axis of the foil in the fluid solver via known values of the leading element and passive elements. (d) Calculate the position and velocity of the fluid panels on the foil surface. (e) Calculate fluid forces and moments.

Figure

  Figure shows propulsive efficiency as a function of number time steps within an oscillation period, N t , and number of body panels, N p . e leading edge kinematics and the St are set to the same values as the main case studies. λ and Π k are set to . and . respectively. Π k = 0.3 marks the resonance frequency of the flow-structure system. It is evident that η converges to the discretization independent solutions as the number of body panels and time steps increases. is performance matrices change less than % when N panel = 700 and N time = 800.

Figure .

 . Figure . (a) Trailing edge amplitude, (b) thrust coefficient, (c) power coefficient, and (d) efficiency as a function of Π k .

Figure .

 . Figure . Variation of thrust (a) and power (b) coefficients, C ′ t and C ′ p , defined by equation with Π k . Contours of C ′ t and C ′ p in λ -Π k plane. Do ed lines are the contours of propulsive efficiency.

Figure .

 . Figure . (a) Trailing edge amplitude, (b) thrust coefficient, (c) power coefficient, and (d) efficiency as a function of Π k for two flexible hinge configuration.

Figure

  Figure a shows the trailing edge amplitude as a function of Π k . A foil with two flexible joints has two resonances, which only one is captured in this figure. e resonance occurs at Π k of 0.35 which is slightly larger than the resonance Π k for each individual flexible joint.is could be due to

Figure .

 . Figure . Change in the relative orientation of the solid elements versus time, within one cycle. Black line shows the prescribed pitching motion of the leading edge. Blue and red lines, respectively, show the deflection angles of the second and the third element. e deflection angles are measured relative to the preceding element. e angles are shown for three different Π k values of . , . , and . .

[]

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ACKNOWLEDGMENTS is work was funded by the Office of Naval Research under Program Director Dr B. Brizzolara, MURI Grant Number N ---.


Xiao Kang

Alan Palazzolo
email: palazzolo@tamu.edu

Test Rig Development for the Magnetic Bearing's Backup Bearing Contact Forces Measurement during Rotor Drops

Keywords:

This paper introduces a new catcher bearing (CB) test rig. Compared with the CB test rig described in the previous published literatures, firstly, this test rig eliminates the influence of the motor and coupling on the rotor drop process by implementing a magnetic coupling. The magnetic coupling can make it possible to mechanically pull back the drive motor axially, separating it from the rotating assembly to prevent its influence on the rotor drop. Secondly, the test rig enables the rotor drop contact force measurement in both the horizontal and vertical direction. The implementation of a 3 axis piezoelectric load cell eliminates the measurement error caused by the friction forces on the contact surfaces between the CB housing and the single axis load cells that applied by the former researchers [1,2,9] for the rotor drop force measurement. The 10K Hz sampling frequency of the data acquisition (DAQ) system and the high stiffness of the load cell can also guarantee the accuracy of the force measurement. Thirdly, the tachometers are installed to not only measure the spin speed of the rotor but the spin speed of the catcher bearing inner race. The developed test rig can provide the rotor drop test data to validate the mathematical model of the rotor drop event. Additionally, this test rig is also an essential apparatus for the future experimental research on both the catcher bearing fatigue life estimation, and the catcher bearing damper devices development.

INTRODUCTION

The magnetic bearing (MB) has been widely used in the turbomachinery industry. It has the advantages such as a low friction, controllable stiffness and damping [START_REF] Schweitzer | Magnetic Bearings Therory, Design, and Application to Roating machinery[END_REF]. The backup bearing, or catcher bearing (CB), is a crucial part in the magnetic bearing system. It can support the rotor when the magnetic bearing is shut down or malfunctioning due to power loss or other technical failures. When the magnetic bearing suddenly loses power, the rotor with high rotational speed will drop onto the catcher bearing. The large contact force and friction force may impair the catcher bearing or cause large vibrations such as reverse whirls, which may even bring harm to the entire magnetic bearing systems. It is important to both theoretically and experimentally investigate the rotor drop behavior to find better ways to accurately predict the catcher bearing fatigue life and, additionally, prevent large vibrations and contact forces during rotor drops.

The relevant literature [START_REF] Schweitzer | Magnetic Bearings Therory, Design, and Application to Roating machinery[END_REF][START_REF] Fumagalli | Modelling and measurement analysis of the contact interaction between a high speed rotor and its stator[END_REF][START_REF] Helfert | Rotorabsturze in Walzlager -Experimentelle Untersuchungen des Rotor-Fanglager-Kontakts[END_REF][START_REF] Gelin | Dynamic behavior of flexible rotors with active magnetic bearings on safety auxiliary bearings[END_REF][START_REF] Ishii | Transient Response Technique Applied to Active Magnetic Bearing Machinery During Rotor Drop[END_REF] contains a number of papers that consider the vibration response of a rotating shaft assembly supported by MBs during a MB power outage incident resulting in the sudden drop of the shaft onto the backup (catcher) bearings. Some of these papers compared the predicted and measured vibration displacements of the shaft near the catcher bearings [START_REF] Schweitzer | Magnetic Bearings Therory, Design, and Application to Roating machinery[END_REF][START_REF] Fumagalli | Modelling and measurement analysis of the contact interaction between a high speed rotor and its stator[END_REF][START_REF] Helfert | Rotorabsturze in Walzlager -Experimentelle Untersuchungen des Rotor-Fanglager-Kontakts[END_REF]. Their comparison of the displacements provides a good means to improve the prediction of displacement motion. Some literatures have investigated the ways to predict the fatigue life of the catcher bearing. Sun et al. [START_REF] Sun | Auxiliary Bearing Life Prediction Using Hertzian Contact Bearing Model[END_REF] used the Lundberg-Palmgren formula to estimate the fatigue life of the CB. Lee et al. [START_REF] Lee | Catcher bearing life Zoom in View Zoom in View prediction using a rainflow counting approach[END_REF] expanded the fatigue life calculation of the catcher bearing based on the rain flow counting method together with the Miner's rule and the material's torsional S-N curves. Some researchers also investigated other kinds of catcher bearing. Lahriri et al. [START_REF] Lahriri | Experimental quantification of dynamic forces and shaft motion in two different types of backup bearings under several contact conditions[END_REF] numerically and experimentally investigates the pin type catcher bearing. Swanson et al [START_REF] Swanson | Test Results And Numerical Simulation Of AMB Rotor Drop[END_REF] and Kang et al [START_REF] Kang | Dynamic and Thermal Analysis when Rotor Drop on to the Sleeve Type Catcher Bearings[END_REF] studied the transient response when a rotor drops onto the sleeve type catcher bearing. Kang et al [START_REF] Kang | Dynamic and Thermal Analysis when Rotor Drop on to the Sleeve Type Catcher Bearings[END_REF] built the 2D dynamic thermal coupled CB model, which enabled the 2D temperature and Von-Mises stress distribution of the CB during rotor drops. However, catcher bearing life is more directly related to the contact forces exerted on it by the journal during a drop event [START_REF] Schweitzer | Magnetic Bearings Therory, Design, and Application to Roating machinery[END_REF]. Only a very small number of papers have experimentally measured the impact and friction forces during a rotor drop event. Fumagalli et al. [START_REF] Fumagalli | Modelling and measurement analysis of the contact interaction between a high speed rotor and its stator[END_REF] obtained the contact force based on the motion of the catcher bearing housing (elastic ring) that is supported by 4 elastic springs. However, their contact forces were obtained indirectly and the calculation accuracy was highly sensitive to the accuracy of the system identification of the housing-spring system. Additionally, the friction force on the contact surfaces between the elastic spring and the CB housing also influenced the accuracy of the contact force measurement, especially for the force in horizontal direction, since the friction force between the rotor and the CB is relatively small compared to the normal contact force. Lahriri et al [START_REF] Lahriri | Experimental quantification of dynamic forces and shaft motion in two different types of backup bearings under several contact conditions[END_REF] measured the impact force between the rotor and two types of backup bearings. However, the rotor-bearing contacts in their tests was resulted from the large vibration under resonance conditions or from forced impact excitation, but not from a free drop onto the catcher bearings. Unlike a true rotor drop the rotor's drop behavior was influenced by the supporting bearings and motor coupling. Saket et al. [START_REF] Fawas Y Saket | Touchdown Bearing Contact Forces in Magnetic Bearing Systems[END_REF] installed a strain gauge type load measurement system in their catcher bearing system; however, the drop tests were not conducted and only the load cell calibration results were provided.

It can be seen that in the aforementioned papers, the measurement accuracy of the rotor drop contact forces is either influenced by the motor, coupling or the friction force between the catcher bearing housing and the load sensor. To eliminate those influences, the present paper introduces a new catcher bearing test rig, which implements a 3-axis piezo electronic load cell to measure the rotor drop contact force. The 3-axis load cell can capture the contact force in both the horizontal and the vertical directions, so the friction force between the load cell and the CB housing will no longer influence the accuracy of the rotor drop force measurements. Additionally, a magnetic coupling is implemented to connect the drive motor and the rotor. There is an air gap between the two pads of the magnetic coupling as shown in Figure 1. Therefore, before the rotor drops onto the CB, the motor can be pulled back axially and there will be no force and torque transmission between the two pads of the magnetic coupling, so the motor and coupling will no longer influence the rotor drop events. Moreover, the vibration of the shaft, the spin speed of the rotor and the catcher bearing inner race can also be monitored. Furthermore, the paper experimentally investigated the influence of rotor dropping with different spin speed, especially of the rotor drops with the spin speed higher than its critical speed. The test rig enables the investigations of the catcher bearing fatigue life and the catcher bearing damper devices in the future.

TEST RIG DESCRIPTION

Figure 1 shows a diagram of the catcher bearing (CB) test rig when the rotor drops onto the CB. The photo of the real catcher bearing test rig is shown in Figure 3. In this rig, the rotor is initially supported by a tapered roller bearing and a self-aligning bearing. The drop test is initiated when the drive motor is mechanically pulled axially backwards, separating it from the rotating assembly and in this process the gap in the magnetic coupling that connects the motor and rig rotor is suddenly thrust open. Next, the outer race of the tapered roller bearing is pushed forward by a solenoid-spring system, opening a clearance in the bearing to let the rotor drop freely onto the catcher bearing, where the tapered roller bearing and the self aligning bearing are shown in Figure 2. From Figure 2. it can be seen that the outer race of the tapered roller bearing can be separated from its inner race and rollers. The inner race of the self aligning bearing can rotate freely so it can act as a pivot during the rotor drop and has limited influence on the rotor drop process.

Figure 3. Catcher bearing test rig

A disk with balancing holes is installed on the non drive end of the rotor as shown in Figure 3. A 3-axis piezoelectric load cell is installed under the catcher bearing housing to measure the impact forces on the catcher bearing during rotor drops. The high stiffness of the load cell and its support and the 10 kHz DAQ sampling rate yield highly accurate measurements in both the horizontal and vertical directions. Here the duplex pair of the angular contact ball bearings is working as the catcher bearing. The data acquisition system is shown in Figure 4.

Rotor vibration measurement

Tachometer for the spin speed of the CB inner race catcher bearing inner race. The Labview based data acquisition software is used to monitor and save the measured vibration and force information.

Rotor free drop test

Because the drop test is initiated by separating the inner and outer race of the tapered roller bearing to create the clearance, it is essential to make sure that during rotor drops, the radial clearance in the tapered roller bearing is always larger than the displacement of the rotor. Then the rotor can drop freely without the influence of the tapered roller bearing. Thus, the rotor free drop test is conducted as shown in Figure 5. From Figure 6 it can be seen that when the tapered roller bearing outer race is pushed away, the radial clearance in the tapered roller bearing is alwasys larger than the vertical displacement of the rotor. Therefore, the rotor drop process will not be influenced by the tapered roller bearing. The rotor can drop freely.

Load cell calibration for the impact force measurment

The 3-axis piezoelectric load cell has been precisely calibrated by the manufacturer for the attached static and harmonic dynamic load. To make sure the 3-axis piezoelectric load cell can also effectively capture the impact force, vertical impact tests are applied. As shown in Figure 7, a spring with the stiffness of 1.926e5N/m (obtained from the manufacturer) is fixed on the load cell by a pin. A steel cube with the mass of 544g is dropped on to the spring with different heights (1 inch, 2 inches and 3 inches).

TEST RESULTS

In this rotor drop test, the initial rotor drop spin speed is 2444 RPM which is larger than the rotor's first critical speed (1500 RPM). Due to the limitation of the transmission rate of the current NI DAQ board, the sampling frequency for both the vibration and force information is 3300Hz. The rotor geometry information is shown as Figure 11. Figure 14 shows that the 3 axes piezoelectric load cells can effectively capture the contact forces in both the horizontal and vertical directions simultaneously. The first few hits have higher contact forces and those contact forces are about 5 to 6 times larger than the static load acting on the CB, which is about 60N based on the initial test. Then the vertical contact forces vibrate around 60N which is caused by the rotor's vibrations. The spin speed of both the rotor and the catcher bearing inner race (CBIR) are measured as Figure 15. Figure 15 shows that the rotational speed of the rotor starts to decay when the rotor contacts the CB. It takes about 0.2s for the CBIR to reach a similar rotational speed as the rotor, while the rotational speed of the CB is slightly lower than the rotor. This is because the CBIR has a larger radius than the rotor.

INFLUENCE OF THE ROTATIONAL SPEED

Rotor drop tests with different spin speeds are also conducted. The rotor orbits that with different spin speeds can be found in Figure 16. The tested three spin speeds are 593.6RPM, 1139 RPM and 2444 RPM, while the first critical speed of the rotor is about 1500 RPM according to the initial test. The goal it to observe the influence of rotor dropping with the spin speed higher than the rotor's critical speed and how the rotor drop spin speed influence the rotor drop behavior. To better observe how the rotor's drop spin speed influences the rotor drop behavior, here the time span is chosen as 0-25s. The rotor spin speeds and contact forces during 0-25s is shown as Figure 17. Note the first critical speed of the rotor is about 1500RPM, so from Figure 17 we can also observe the rotor transient response when it passes through its critical speed.

From Figure 17 (a), it can be seen that when rotor drop spin speeds get higher, it will take longer time to let the rotor stop spinning. From Figure 17 (b), it can be seen that for the first few hits, higher drop spin speeds do not results in higher contact force. This is because the contact forces for the first few hits are highly related to the initial drop conditions, while from the Figure 17 (d) and (f), they show for the rest hits, higher rotational speeds will result in higher contact forces. This may be caused by the residual imbalance of the rotor. Although the rotor is balanced for these cases, there is still some minor residual imbalance (within 5e-5 kg.m). The spin speed contributes much to the imbalance force of the rotor and causes higher contact forces. 17 (c) that when rotor passes its critical speed, the contact forces is greatly increased which is due to the large vibration caused by the resonance. So the rotor critical speed is also needed to be considered for the catcher bearing design.

CONCLUSION

A catcher bearing test rig is developed which can measure the rotor's vibration, catcher bearing contact force and the spin speeds of the catcher bearing inner race.

For the influence of rotor drops with different spin speeds, the results show that for the first hit, high spin speed doesn't necessarily cause higher contact force, because the contact force for the first few hits are highly related to the initial drop conditions. While for the rest hits, higher rotational speeds will result in higher contact forces. It may be caused by the residue imbalance of the rotor. Though the rotor is balanced for these cases, there may still be some residual imbalance. The spin speed will contribute much to the imbalance force of the rotor and finally results higher contact forces.

Additionally, when rotor passes its critical speed, the contact forces are greatly increases which is due to the large vibration caused by the resonance. Therefore, the rotor critical speed is also need to be considered for the catcher bearing design.

Future tests with varying surface roughness and other parameters such as rotor imbalance and catcher bearing clearance will be performed to investigate the approaches to mitigate the reverse whirl.

Future tests will also include different types of dampers to investigate their effect in preventing the reverse whirl and reducing the contact forces.

Figure 1 .

 1 Figure 1. Scheme plot of the way to let rotor drop onto the CB

Figure 4 .

 4 Data acquisition system of the Catcher Bearing test rig. (a) Bently eddy current sensors (b) NI USB DAQ board (c) Load cell signal conditioner Two eddy current sensors (Bently Nevada 3300XL probe) are placed near the catcher bearing location, measuring the displacement of the rotor during rotor drop. Because the sensors are fixed onto the housing, the measured rotor displacements are relative to the housing. Two eddy current sensors (Bently Nevada 3300XL probe) are working as tachometers to measure the spin speeds of the rotor and the

Figure 5 .C

 5 Figure 5. Rotor free drop test One Bently sensor (Bently Nevada 3300XL probe) measures the axial displacement of the tapered roller bearing outer race (TRBOR). Another Bently sensor (Bently Nevada 3300XL probe) measures the vertical displacement of the rotor near the catcher bearing location. The radial clearance in the tapered roller bearing can be obtained by the equation (1). The TRBOR's axial displacement, radial clearance and rotor's vertical displacement are also shown in the Figure 1.tan( ) TB TPOR TP CA

Figure 6 .

 6 Figure 6. Rotor vertical displacement and the radial clearance between the tapered roller bearing inner and outer race

Figure 7

 7 Figure 7 Impact force calibration test There are more than 15 tests are made for each height to verify the tests' repeatability. The measured forces and the predicted forces are compared as following. When the drop height is 1inch, the prediction and measurement are in Figure 8, where Figure 8 a is the comparison of the force amplitude among all the tests and the prediction.Figure 8 b and c are the transient force response of the prediction and the highlighted test.

Figure 8 b

 8 and c are the transient force response of the prediction and the highlighted test.

Figure 11 .Figure 12 .Figure 13 .Figure 14 .

 11121314 Figure 11. Geometry of the test rotorThe measured rotor orbit at the sensor's location is shown in Figure12.

  Figure 15. Measured rotational speed of the rotor and CBIR

5 X

 5 Direction (mm) Y Direction (mm) (a)594.6 RPM test (b) 1139 RPM test (c) 2444 RPM test Figure 16. Rotor orbit with different spin speeds

  (a) Spin speeds of rotors and catcher bearing inner races

ACKNOWLEDGMENTS

This research was supported by the Turbomachinery Research Consortium (TRC) at Texas A&M University. Special thanks to Mr. Erwin Thomas, Mr. Xiaojun li, Mr. Xiaomeng Tong and Mr.

Stephen Farris for their assistance in the construction of the rig and the DAQ system..

In Figure 9, the predicted force is similar to 70% of the drop cases (with 10% differences). A few cases (20%) have the contact forces 100N (30%) larger than the predicted value. The transient force response of the highlighted case is compared with the predicted transient response. The results show the duration of the forces are both around 0.007s, while the duration of the measured force is a little bit smaller. When the drop height increases to 3 inches, the results are shown in Figure 10.

From Figure 10 (a), the predicted force is similar to 75% of the drop cases (with 10% differences). A few cases (18%) have the contact forces 100N (40%) larger than the predicted value.

The transient force response of the highlighted case is compared with the predicted transient response. The results show the time duration of the forces is around 0.007s. While there are several impacts for the measured force. Thus, with different drop heights, 70% of the measured force amplitudes are within 10% of the predicted force. Therefore, the load cell is reliable for the impact force measurement.