VIBRATIONAL ANALYSIS OF A BALL BEARING Rohini Kesavan ASEN 5519: Fluid-Structure Interaction
OVERVIEW Introduction Structural Defect Induced Vibration Structural Model of the outer ring Finite Element Model of the outer ring Results Comparison Conclusion Future Work
INTRODUCTION The project presents an analysis of the vibrational behavior of a deep groove ball bearing with a structurally integrated force sensor The miniaturized force sensor, accommodated within a slot on the bearing s outer ring provides on-line conditioning monitoring capability to the bearing Analytical and finite element models are developed to predict the sensor output due to the bearing dynamic load and rotational speed variation
SENSOR-INTEGRATED SMART BALLBEARING
GEOMETRY OF A DEEP GROOVE BALL BEARING D-Outer Diameter d b -Ball Diameter d m -Pitch Diameter w-raceway width L-Slot length y-limiting angle d -Bore Diameter
Load distribution on a rolling element bearing is given by 1 q ( ψ ) = qmax [1 (1 cosψ )] 2ε where n= 3/2 for roller bearings n=10/9 for ball bearings q max = Z 5F r cosα F r Applied radial Load Z Number of rolling elements a- Mounted contact angle n
STRUCTURAL MODEL OF THE OUTER RING A piezoelectric sensor is embedded into a slot cut through the outer ring to monitor load and vibration within the bearing structure The piezoelectric sensor is modeled as a spring with stiffness k that is related to its material composition Clamped boundary conditions are considered appropriate since the ends of the beam are solidly connected to the surrounding bearing structure
LOADS ON MODIFIED SECTION OF THE OUTER RING
SIMPLIFIED MODEL OF THE BEARING OUTER RING
The bearing load q(ψ ) is determined by Eq. in slide 6 and the location a where the load is applied is related to by the expression 1 a = ( d m d b )sin( ψ 2 when a<l/2, the deflection of the sensor is given by δ a L / 2 = 4qa 2 (4a 192EI 3 when a>l/2, the sensor deflection is given by δ a> L / 2 = 4q( L a) 192EI γ 33 A I ) 3L) kl ( L 4a) 3 kl 2 where k= γ 33 - Elastic Modulus of the piezoelectric material.
This translates into a load on the sensor as R s = γ 33 The electrical charge produced by the sensor is l Q = d 33 R s Using a charge amplifier with a gain G, the voltage output produced by the embedded sensor is V= GQ = Aδ Gd γ 33 33 This voltage output is a direct measure of the outer ring deflection due to a specific bearing load. Conversely, from the sensor voltage output, the load applied to the bearing can be determined to identify overloading conditions Aδ l
BEAM MODEL
BENDING MOMENT DIAGRAM
SHEAR FORCE DIAGRAM
MODE SHAPES 1
MODE SHAPE 2
Variation of amplitude against excitation frequency when an harmonic load is applied
FINITE ELEMENT MODEL OF THE OUTER RING Technically the complete bearing structure can be modeled using a complex three-dimensional FE model. However, by observing the nature of the boundary conditions and loads on the outer ring, it was determined that the FE model could be simplified by using symmetry of the system Loads on the bearing structure are applied to the outer ring through small ellipsoidal contact areas between the rolling elements and the outer ring groove
Assuming a pure radial force, the resulting Hertzian stress distribution is located at the base of the groove Since the stress distribution is symmetric about the plane which divides the outer ring through the base of the groove, the strain normal to the plane e x, is zero The FE model was constructed using a four-node quadrilateral plain strain element.
) (1 ) (1 )] )(1 (1 ) )(1 (1 ) )(1 (1 ) )(1 (1 [ 4 1 ) (1 ) (1 )] )(1 (1 ) )(1 (1 ) )(1 (1 ) )(1 (1 [ 4 1 2 2 2 1 2 2 2 1 t v s v t s v t s v t s v t s v v t u s u t s u t s u t s u t s u u L K J I L K J I = = where u and v are the displacements along z and y directions.
Finite Element Model of the modified outer ring
GEOMETRIC MODEL OF THE OUTER RING
OUTER RING WITH BOUNDARY CONDITIONS
MODE SHAPE 1
MODE SHAPE 2
MODE SHAPE 3
FREQUENCY VS. DISPLACEMENT PLOT
FREQUENCY VS. VOLTAGE PLOT
SCHEMATICS OF THE BEARING TEST BED
Inner ring of the bearing is mounted on a shaft and secured with an end cap. The outer ring is secured with a flat ring With the shaft mounted in the chuck of a lathe a threaded rod is inserted through a frame, screwed into the top of the housing and secured with a nut Turning a nut at the top of the rod compressed a calibrated spring and applied a radial load to the bearing The free end of the shaft is supported with a live center, and the lathe was used to spin the shaft at various rotational speeds while the housing remained fixed Data from the embedded sensor was recorded using a computer which was equipped with a data acquisition system
PREDICTED AND EXPERIMENTAL SENSOR OUTPUTS
The difference between the predicted sensor output and experimental output is due to two reasons. 1,Unlike the sensor output predicted by the FE model, the experimental data did not reach exactly the zero output line due to vibrational noise, which was observed from the machine environment during the experiments. 2, In the FE model, the housing was assumed to be infinitely rigid. However, the experimental housing has a certain degree of flexibility since it was made of aluminum which is an elastic material.
CONCLUSION The structural integration of a load sensor into the outer ring of a rolling element bearing provides an effective means for assessing the time varying load conditions within the bearing structure The project emphasizes the importance of combined time and frequency domain analysis of the sensor signal to accurately assess the conditions of the bearing
FUTURE WORK Comparing analytical and FE results with experimental data Improving on the beam model by adding more elements Developing advanced and efficient signal processing techniques using wavelet transformation and neural-fuzzy networks Developing monitoring systems to provide early defect warning capabilities to a wide range of rolling element bearings
References Brian T. Holm, Robert X. Gao, Vibrational Analysis of a Sensor-Integrated Ball bearing, Vol.122, October 2000, ASME Brown, P.J., Condition Monitoring of Rolling Element Bearing, Noise Control Vibrat. Insul.8, No.2, pp 41-44 Berggren, J.C.1988, Diagnosing Faults in Rolling Element Bearings, Part I: Assessing Bearing condition, Vibrations, 4, No. 1, pp 5-13 Tandon N., Nakra B.C, 1990, Defect Detection in Rolling Element Bearings by Acoustic Emission Method, J. Acoustic Emiss.9, No.1, pp 25-28