Vibrational Analysis of a Ball Bearing

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 bearings outer ring, provides online condition monitoring capability to the bearing. Analytical and finite element models are developed to predict the sensor output due to bearing dynamic load and rotational speed variation. Good agreement was found between the model predicted sensor outputs and the analytical results. Introduction Rolling elements are widely used as low friction joints between rotating machine components. Since the rotational motion is often a significant function of the overall system, such as wheels on a train, rollers in a paper mill, or a rotor on a helicopter, proper functioning of a bearing over its designed life cycle is of vital importance to ensure product quality, prevent machine damage or even loss of human life. A new technique has been developed that involves the direct structural integration of sensing elements and telemetric data communication electronics into the bearing environment to provide “self-diagnostic” capabilities. During bearing operation, the sensing elements continuously measure the real-time fluctuations of the bearing load variations. If pre determined threshold values representing critical bearing operational parameters are exceeded, an alarm will be triggered which is then transmitted out of the integrated sensor by a wireless link to the machine control system for corresponding actions. The major advantage of this technique is the close vicinity of the fault detection mechanism to the fault source. The resulting shorter signal transmission path leads to quicker fault localization. In addition the sensors will be less vulnerable to background noise and distortion of fault characteristics. Structural Defect Induced Vibration A commercially available deep groove ball bearing is analyzed. Figure 1 Geometry of a deep groove ball bearing The load distribution on a rolling element bearing is given by (1) where qmax-Maximum load ψ-Limiting angle ε Load distribution factor n= 3/2 for roller bearings n=10/9 for ball bearings In a bearing with nominal diametral clearance, qmax can be approximated as, (2) where Fr – Applied radial Load Z – Number of rolling elements α Mounted contact angle Structural Model of the Outer Ring To monitor load and vibration within the bearing structure, a piezoelectric sensor is embedded into a slot cut through the outer ring. The sensor has solid contact with both the top of the slot and the bearing housing. Each time a rolling element passes over the slot, the sensor generates an electrical charge output that is proportional to the load applied to the bearing Fr. Since the outer ring is structurally supported by the bearing housing, it can be assumed as rigid. The piezoelectric sensor is modeled as a spring with stiffness k that is related to its material composition. The section of the bearing outer ring where the slot is cut can be modeled as a beam of varying cross-section, with a spring support at the midpoint. Since the ends of the beam are solidly connected to the surrounding bearing structure, which is directly supported by a rigid housing, clamped boundary conditions are considered appropriate. Figure 2 Simplified model of the bearing outer ring n q q )] cos 1 ( 2 1 1 [ ) ( max ψ ε ψ − − = α cos 5 max Z F q r = A static analysis of the beam was done in ANSYS using beam and spring elements. The necessary boundary conditions are applied to the model. A modal analysis using Subspace Iteration method is done and the natural frequencies of the system are noted. A harmonic analysis using Mode Superposition method is done and a graph between displacement and frequency is plotted. The peak amplitude and frequency is noted. The results are verified using the following analytical equations. The bearing load is determined and the location ‘a’ where the load is applied is related to by the expression (3) when aL/2, the sensor deflection is given by