Stator Coupling Model Analysis By Johan Ihsan Mahmood Motion Control Products Division, Avago Technologies White Paper Abstract In this study, finite element analysis was used to optimize the design of a stator coupling in order to increase its natural frequency. Through design iterations and analysis, it was discovered that by increasing the center hole diameter of the coupling, as well as introducing stiffening ribs at the flexible part of the structure, natural frequency will increase from its original design of 714Hz to an optimized design of 1611Hz. 1. Introduction Coupling is a device mainly used to connect two shafts in order to transfer power or rotation. In motion control applications, a rotor coupling is used to connect encoders to the motor shafts (rotor part), whereas a stator coupling is used to connect encoders to the motor housing (stator part). A rotor coupling rotates together with the motor shaft, whereas a stator coupling remains static during the operation of the motor. Proper coupling design is crucial for an encoder to function well in detecting rotational speed and position of the shaft. Since motors may rotate at any rotational speed or frequency, the coupling used with the encoder should be designed with high natural frequency, above the operating frequency range of the motor, to avoid resonance in the system. Resonance is a condition where the motor s operating frequency coincides with the coupling s natural frequency. When this happens, the coupling may oscillate at high amplitude and, as this amplitude builds up, premature failure may occur. The main objective of conducting this study is to determine the natural frequency of the coupling using a finite element analysis. The design of the coupling can be optimized and fine-tuned to achieve high natural frequency before the coupling is fabricated. This will avoid unnecessary design modifications and will save development time and cost. 1.1. Rotor and Stator Coupling A rotor coupling is typically constructed with materials such as an elastomer, a polymer or metal with helical (spring-like) design (Figure 1a) for flexibility. This coupling assists with the radial and axial alignment of shafts. It also has to be able to withstand enormous torque resulting from angular acceleration of the motor. A stator coupling (Figure 1b) is normally constructed of steel and designed to be flexible in order to tolerate axial or radial shaft misalignment. A stator coupling carries only a static load from the encoder mass, and very low torque results from the friction in the bearing. To Shaft Figure 1a. Rotor Coupling (helical design) To Shaft Figure 1b. Stator Coupling
A typical configuration showing how a rotor and stator coupling is assembled in a system is shown in Figures 2a and b. In order to understand how the coupling interacts in the mechanical system, a free body diagram is constructed (Figures 3a and b). As the rotor coupling rotates together with the shaft, the problem is analyzed as a torsional degree of freedom system. The mass moment of inertia of the motor is very large compared to the mass moment of inertia of the encoder (I). The motor can be considered to be stationary (fixed) and the shaft, coupling and encoder are connected in series in the torsional system. The shaft of the motor has a torsional stiffness of (Kt1) connected to the encoder shaft with torsional stiffness of (Kt3) via the rotor coupling with torsional stiffness of (Kt2). If the torsional stiffness of both shafts (Kt1 & Kt3) is significantly larger compared to the rotor coupling s torsional stiffness (Kt2), natural frequency is then determined by the least stiff part of the system, i.e., the rotor coupling s torsional stiffness (Kt2). Encoder Encoder Shaft Rotor coupling Stator coupling Motor Motor Figure 2a. Rotor Coupling Figure 2b. Stator Coupling I (encoder) θ (rotational) y (vertical) Kt3 (shaft) Kt2 (rotor coupling) M1 (encoder) x (horizontal) Kt1 (shaft) K1 (shaft) K2 (stator coupling) Figure 3a. Rotor Coupling Figure 3b. Stator Coupling Natural frequency for a torsional system can be estimated using the formula below (Eqn.1). It can be shown that as the coupling stiffness (Kt) increases, natural frequency (Fn) also increases, and as moment inertia or rotor (I) increases, natural frequency (Fn) decreases. Due to its torsional degree of freedom, a rotor coupling has the disadvantage of having a lower natural frequency compared to a stator coupling. Fn = 1 2 π Kt I (Eqn.1) Fn = Natural Frequency (Hz) Kt = Torsional Stiffness Coupling (Nm/rad) I = Moment Inertia of Rotor (kgm2) 2
In the case of stator coupling, the problem can be simplified as a spring-mass vibrating system. The system consists of an encoder mass (M1) with a spring from the shaft with stiffness (K1) and a combined effective spring from the stator coupling with stiffness (K2). The motor mass is considered to be very large compared to the encoder s mass and is assumed to be stationary. Only the encoder s mass is taken into consideration in this analysis. Due to the complex shapes of the stator coupling which may vibrate with multiple degrees of freedom, computational analysis using a finite element method is used to solve for natural frequencies and mode shapes. Stator coupling is preferred for highly dynamic applications because of its stability and high natural frequency. For optimum performance, the stator coupling must be as stiff as possible to achieve high natural frequency, but at the same time flexible enough to accommodate axial and radial shaft misalignment. 2. Finite Element Modeling Finite element analysis is conducted in the design and development of the stator coupling using ABAQUS software. The initial design of the stator coupling is shown in Figure 4. It has a physical outline measuring 50x50 mm and a height of 10 mm. The top part of the coupling is attached to the encoder and the base is attached to the motor housing. The coupling is made of stainless steel with a 0.2 mm thickness. Figure 4. Stator coupling (initial design) The finite element model is shown in Figure 5. The bearing base is modeled completely, but the encoder is modeled with a simplified shape. The encoder s mass is assumed to be a lumped mass to simplify the analysis, and it still retains the same mass of the actual encoder. By doing this, we are able to eliminate any local insignificant mode in the encoder that does not contribute to the global mode of vibrations. The Lanczos Eigensolver algorithm is used to solve for natural frequencies and mode shapes. Encoder (simplified) Shaft Bearing Base Stator Coupling Figure 5. Stator coupling finite element model 3
3. Natural Frequency and Mode Shape Results Results of the lowest natural frequencies for the stator coupling are shown in Figure 6: 714Hz, 987Hz, and 1011Hz. In order for the coupling to be used in highly dynamic applications, further modifications must be done to increase the coupling s stiffness to increase its natural frequency. 1200 Natural Frequency (Hz) 1000 800 600 400 200 714 987 1011 0 1 2 3 Mode Figure 6. Results of the three lowest natural frequencies The corresponding mode shapes are shown with their contour plot in Figure 7. Mode shapes represent how a structure would behave under resonance. This helps to pinpoint the critical area in the structure that needs to have its rigidity increased. In Mode 1, the center hole of the metal plate vibrates vertically in the Z axis. This indicates that design modification is required in the center area to increase the structure s rigidity. In Mode 2, the oscillation of the coupling is in the Y axis, and in Mode 3, the oscillation is in the X axis. Mode 1 at 714 Hz Mode 2 at 987 Hz Mode 3 at 1011 Hz Figure 7. Mode shapes for stator coupling (original design) 4
4. Coupling Design Optimization for High Natural Frequency The design of the coupling is then optimized to achieve higher natural frequency. First, the effect of the center hole of the coupling is investigated. When the center hole of the coupling is removed, natural frequency dropped to 661Hz (Figure 8b). When the center hole diameter is increased, the natural frequency increases. A natural frequency of 1133Hz was recorded when the hole diameter was 30 mm. Mode shapes from the first natural frequency of the design iteration are shown below: Fn = 714 Hz er Figure 8a. Original design, hole dia. = 16mm Figure 8b. Center hole removed Fn = 661 Hz Fn = 925 Hz Figure 8c. Center hole dia. = 24mm Fn = 1133Hz Figure 8d. Center hole dia. = 30mm Next, stiffening ribs surrounding the circumference of the hole are added to the final iteration of the coupling design. As a result the natural frequency increases to 1611Hz (Figure 9). Since the design of the stiffening ribs surrounding the hole is not feasible for manufacturing purposes, modification is still required. Fn = 1611 Hz Figure 9. Center hole dia. = 30 mm with stiffening rib, Fn = 1611 Hz 5
Further modification to the coupling s design is shown in Figure 10a. Natural frequency achieved in this design, which had center stiffening ribs, is still quite low at 891Hz. In order to overcome this, the coupling is further modified to add two center stiffening ribs and four side stiffening ribs (Figure 10b). This successfully returned the natural frequency of the stator coupling to 1611Hz. The final design of the coupling is shown in Figure 11 and the corresponding mode shapes are shown in Figure12. Figure 10a. Center stiffening rib only Fn = 891 Hz Fn = 1611 Hz Figure 10b. Center stiffening rib and 4 sides Figure 11. Final design of the stator coupling Mode 1 at 1611 Hz Mode 2 at 1677 Hz Mode 3 at 1780 Hz Figure 12. Mode shapes for final design of the stator coupling 6
2100 Natural Frequency (Hz) 1800 1500 1200 900 600 300 0 1611 714 1677 1780 987 1011 1 2 3 Mode Original Design Optimized Design Figure 13. Natural frequency comparison between original design and optimized design 5. Conclusion Based on the study conducted, it was discovered that finite element analysis can be used to optimize the design of a stator coupling in order to increase its natural frequency (Figure 13). Increasing the size of the coupling s center hole, as well as introducing stiffening ribs at the flexible part of the structure, increased the coupling s natural frequency. The stator coupling s natural frequency was increased from its original design of 714Hz to an optimized design of 1611Hz. 6. References 1. J.C. Wachel, Fred R. Szenasi, Analysis of Torsional Vibrations in Rotating Machinery, Proceedings of 22 nd Turbomachinery Symposium, 1993. 2. Jon R. Mancuso, Couplings and Joints, Design Selection and Applications, Second Edition, New York, Marcel and Dekker, 1999. 3. Rao Singresu.S, Mechanical Vibrations, Third Edition, Addison Wesley, 1995. 4. ABAQUS, User s Manual, ABAQUS Inc. Version 6.7, 2007. For product information and a complete list of distributors, please go to our web site: www.avagotech.com Avago, Avago Technologies, and the A logo are trademarks of Avago Technologies Limited in the United States and other countries. Data subject to change. Copyright 2008-2010 Avago Technologies Limited. All rights reserved. AV02-1075EN - March 8, 2010