Download Torsional Vibration Testing of a Non

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
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[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