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Torsional Vibration Testing of a Non-ferromagnetic Shaft by Magnetostrictive Transducers Yoon Young Kim, Seung Hyun Cho, Soon Woo Han, and Chan Il Park School of Mechanical and Aerospace Engineering, Seoul National University, Shillim-dong San 56-1, Kwanak-gu, Seoul, 151-742, Korea ABSTRACT Torsional vibration is an important vibration mode in cylindrical structures such as shafts and pipes. However, the modal testing of torsional vibrations of a cylindrical structure is not an easy task because of the lack of proper transducers. This work presents a new torsional vibration transducer based on the magnetostrictive principle and its application to torsional modal testing. The transducer is so designed as to generate/measure torsional vibrations excluding other vibration modes such as longitudinal and bending vibrations. The transducer is composed of ferromagnetic patches bonded to a test structure, permanent magnets, and a solenoid. Because only patches and magnets are bonded to a structure, torsional vibrations are generated and measured wirelessly by a solenoid encircling a test structure. Cost-effectiveness is another advantage of the proposed transducer. To check the performance of the proposed method, the torsional natural frequencies and mode shapes on a hollow aluminum shaft were extracted. The results obtained by the proposed transducer agreed favorably with theoretical results. 1. INTRODUCTION Cylindrical structures such as drive shafts, structural beams, and pipes have been used at various industrial fields. These structures may undergo longitudinal, bending, and torsional vibrations. Among various vibration modes of a cylindrical structure, torsional vibration critically affects its stability whenever it is put in rotational motion. Therefore, torsional modal testing may be needed to understand its torsional characteristics but the lack of proper transducers makes it difficult to perform torsional modal testing. For accurate torsional modal testing, it is necessary to generate and measure only a torsional mode excluding other modes. The major difficulty in torsional modal testing is the generation of torsional vibration so that torsional modal testing has not been much reported regardless of the practical importance of the testing. To generate torsional vibration in a cylinder, some existing methods using an electric motor [1], a piezoelectric transducer [2,3], a hydraulic transducer [4], and eddy current [5] may be employed. If an electric motor is used, a single impulse signal is difficult to produce. Furthermore, the motor must be attached only to the end of a cylinder. A piezoelectric transducer does not seem to have sufficient power to generate torsional vibration. However, torsional vibration is relatively easy to measure if a strain gauge or a torsional laser vibrometer [1] is used. This work presents a new torsional modal testing method that employs magnetostrictive transducers for the generation and measurement of torsional vibration. Thin magnetostrictive patches, permanent magnets and a solenoid comprise the proposed transducer [6]. It is capable of generating and measuring nearly pure torsional vibration. When this transducer is used, excitation and sensing positions can be anywhere in a test specimen. Furthermore, the modal testing by the proposed method is useful in a relatively higher frequency range reaching tens of kilohertz. For torsional modal testing experiments, a hollow aluminum shaft with free supports was used. Two transducers, one serving as an actuator and the other a sensor, were installed and the impulse-type testing was conducted. From the testing, the eigenfrequencies and mode shapes were obtained. 2. MAGNETOSTRICTIVE TORSIONAL VIBRATION TRANSDUCER The working principle of the present torsional vibration transducer is magnetostriction [7], which refers to the coupling phenomenon between magnetic field and mechanical deformation. Ferromagnetic material deforms mechanically under external magnetic field by magnetostriction. On the other hand, the deformed ferromagnetic body induces magnetic field. When a ferromagnetic body under static magnetic field experiences dynamic magnetic field in the perpendicular direction to the static field, the body undergoes shear deformation [8]. Inversely, a shear-deformed body under static magnetic field induces magnetic field in the perpendicular direction to the static field. We use this principle to generate and measure torsional vibrations in a cylinder. Figure 1 shows the proposed transducer, which is composed of two magnetostrictive patches made of ferromagnetic material, two bias permanent magnets, and a solenoid. The patches should be bonded tightly with epoxy or adhesive tape and magnets are placed in a gap between the patches. Then, a solenoid encircling the patches and the magnets is installed. The permanent magnets supply a static magnetic field to the patches as illustrated by arrows in Fig. 1(b). The solenoid supplies current into an actuating transducer. It detects induced magnetic field by torsional vibration of the test specimen in a sensing transducer. The applied and detected (a) (b) Fig. 1 Magnetostrictive patch transducer [6] for torsional vibrations modal testing; (a) schematics, (b) the flow direction of the static bias magnetic field. dynamic magnetic fields by the solenoid are parallel to the z axis of Fig. 1(a). When the transducer is used as an actuator, the input excitation current ( ie ) to the solenoid induces a dynamic magnetic field in the patch along the z direction. Then, the magnetic field generates shear deformation in the patch, which is transferred to a test specimen and generates torsional moment ( mt ). Assuming a linear relation between the patch deformation and the excitation magnetic field, the resulting torsional moment can be expressed as mt ( z, t ) ∝ ie (t ) . (1) When the transducer is used as a sensor, the twist rate ( α , twist angle per unit length) of the specimen causes deformation in a magnetostrictive patch. The patch deformation induces magnetic field which is picked up by a solenoid through the Faraday-Lenz law. In this case, the voltage difference ( vs ) measured by the solenoid can be expressed as vs ∝ ∂ α ( z, t ) . ∂t (2) 3. TORSIONAL MODAL TESTING EXPERIMENT The proposed transducer was applied for torsional vibration testing of a hollow aluminum shaft with free supports. The n th torsional natural frequency and mode shape of twist angle ( φ ) for a cylinder (length: L ) with free supports are known to be expressed as [9] fn = nπ GJ , 2L I (3) ⎛ nπ ⎝ L φn ( z ) = cos ⎜ ⎞ z⎟. ⎠ (4) where G , J , and I denote the shear modulus of elasticity, the polar moment of inertia of the cross sectional area, and the mass polar moment of inertia per unit length, respectively. As shown in Fig. 2 (a), two transducers, one as an actuator and the other as a sensor, were installed on an aluminum shaft (length: 1 m, outer radius: 25 mm, thickness: 1 mm). Using Eqs. (1) and (2), one can see that the (a) (b) Fig. 2 (a) Experimental setup for torsional vibration testing with the present transducers. (b) Actuating and sensing points on the test specimen. (a) (b) Fig. 3 (a) Upper: the input exciting current by the actuating transducer at Pt. 6. Lower: the output voltage by the sensing transducer at Pt. 14. (b) Frequency response function for the actuation point at Pt. 6 and the sensing at Pt. 14. Fig. 4 The torsional mode shapes – circles: experimental results, lines: theoretical results. input current to the actuating transducer is proportional to the actuating torsional moment of the specimen. One can also see that the output voltage from the sensing transducer is proportional to the twist rate. Figure 3(a) shows excitation current and output voltage. Using these signals, the frequency response function (FRF) relating the torsional strain to the applied torsional moment can be expressed as ∂ Φ (ω ) Vs (ω ) ∂ . ∝ jω z FRF = I e (ω ) M t (ω ) (5) Figure 3(b) shows the frequency response function obtained from the measured signals in Fig. 3(a). The frequency response functions were calculated by using the current signals of the actuating transducers at different location and the output voltage of the sensing transducer at Pt. 14. By using the peak-picking method Table I. The torsional natural frequencies theoretically calculated and experimentally acquired with the present method. Mode number 1st 2nd 3rd 4th 5th Theoretical Results (Hz) 1554 3108 4662 6216 7770 Experimental Results (Hz) 1536 3077 4654 6120 7750 Error 1.2 % 1.0 % 0.2 % 1.5 % 0.3 % [10], the torsional natural frequencies listed in Table 1 and the torsional mode shapes shown in Fig. 4 were obtained. The torsional modal parameters obtained by the proposed method agree well with the theoretical results. The good correction clearly demonstrates the validity of the proposed torsional vibration testing method. 4. CONCLUSIONS This investigation proposes a new method to conduct torsional modal testing of a cylindrical structure. The present method employs two magnetostrictive transducers to generate and measure torsional vibrations. Using the proposed method, torsional eigenfrequencies and mode shapes agreeing well with the theoretical results were obtained. Several advantages of the proposed method, such as wireless sensing/actuation, arbitrary selection of test locations in a test specimen, a large frequency band of application, were demonstrated by actual experimental tests. More studies such as transducer calibration are needed for further development of the proposed method. ACKNOWLEDGEMENTS This research was supported by the National Creative Research Initiatives Program (Korea Science and Technology Foundation grant No. 2006-033) contracted through the Institute of Advanced Machinery Design at Seoul National University. REFERENCES [1] Drew, S. J. and Stone, B. J., Torsional damping measurement for a gear box, Mechanical Systems and Signal Processing, Vol. 19, pp.1096-1106, 2005 [2] Sasida, T. and Kenjo, T., An Introduction to Ultrasonic Motors, Oxford University Press, Oxford, 1993 [3] Kim, J. O. and Kwon, O. S., Vibration characteristics of piezoelectric torsional transducers, Journal of Sound and Vibration, Vol. 264, pp.453-473, 2003 [4] http://www.corad.com [5] Simpson, H. M. and Pearson, J., Simple apparatus for measuring the dynamic shear modulus of cylindrical specimens, Reviews in Scientific Instrumentation, Vol. 50, pp.418-420, 1979 [6] Kim, Y. Y., Cho, S. H., Han, S. W., and Park, C. I., Apparatus and method for generating and sensing torsional vibrations using magnetostriction, Patent Pending to US, 2006 [7] Borzorth, R. M., Ferromagnetism, Piscataway, N. J. IEEE Press., 1993 [8] Thompson, R. B., Generation of horizontally polarized shear waves in ferromagnetic materials using magnetostrictively coupled meander-coil electromagnetic transducers, Applied Physics Letters, Vol. 34, pp.175177, 1978 [9] Meirovitch, L., Elements of Vibration Analysis, 2nd ed., McGraw-Hill, Singapore, 1986 [10] Ewins, D. J., Modal Testing: Theory and Practice, Research Studies Press, Letchworth, 1984