DESIGN AND OPTIMIZATION OF A COMPACT HIGH-FREQUENCY ELECTROMAGNETIC SHAKER

Transcription

1 11 th International Conference on Engineering Vibration Ljubljana, Slovenia, 7 10 September 2015 DESIGN AND OPTIMIZATION OF A COMPACT HIGH-FREQUENCY ELECTROMAGNETIC SHAKER Leonardo Bertini 1, Paolo Neri* 1, Ciro Santus 1 1 University of Pisa, Department of Civil and Industrial Engineering - DICI. Largo L. Lazzarino 2, Pisa, Italy leonardo.bertini@ing.unipi.it paolo.neri@for.unipi.it ciro.santus@ing.unipi.it Keywords: Electromagnetic shaker, Stinger, Bladed wheels, Modal analysis, Harmonic response analysis. Abstract. Dynamic characterization and resonances prediction of mechanical structures is a crucial issue in industry. Performing experimental modal analysis and harmonic response requires an excitation source able to operate in a wide frequency range, depending on the analyzed structure. In the present paper, the optimization and the design of an electromagnetic shaker is presented. This device has been engineered for a test bench to investigate the vibrational dynamics of centrifugal compressor bladed wheels. A really compact solution is needed since the final test bench provides up to 20 shakers on the circumference hoop, one for each blade, and the excitation frequency ranges from 1 to 10 khz. Different stinger solutions are proposed and compared in the paper, in order to monitor the stinger effects on the dynamic response of the analyzed structure. The investigated solutions are: a beam stinger (diameter 1 mm), a wire stinger (diameter 0.2 mm), and a ball stinger (diameter 3 mm) which was tested with two different contact solutions. Experimental tests were performed on a shaker prototype to verify the vibrational loads applied by the device and to compare the different stinger solutions. 217

2 1 INTRODUCTION Experimental Modal Analysis (EMA) and Harmonic Response Analysis (HRA) are crucial for structure dynamic characterization. Any time that a component is excited by a cyclic load, a deep knowledge of its vibrational response is needed in order to avoid resonance excitations that could imply efficiency drop or even induce fatigue fracture failures. High rotational speed bladed wheels, such as turbines and compressors, are remarkable examples of mechanical structures that experience vibrations with high frequency harmonics. The blades interact with the fluid coming from stator vanes [1, 2, 3], so that high frequency excitation happens at the rotational speed regimes of the machine. Indeed, EMA tests have been proposed for this kind of components [4, 5] with single or double shaker configurations. On the other hand, HRA needs a more complex setup because a high number of excitation sources is needed to properly reproduce the operational conditions, in principle one exciter is required for each blade. Since the number of blades can be considerable, the exciter lateral dimension issue implies a really critical constraint for this application. Contact-less solutions are available in literature, however they are limited to low load intensity and low maximum frequency [6], while for the present investigation the frequency values are in the range 1 10 khz, therefore electromagnetic shakers were considered. Several commercial solutions are available, but they do not guarantee to cover the whole frequency range or they do not satisfy the dimensional limitations, for this reason a custom electromagnetic shaker was designed. A static copper coil was wired and axially aligned with a moving permanent magnet, which was then connected to the target structure. Analytical and numerical models were implemented for the geometry optimization, in order to achieve relevant force values, still within the dimensional limitations and the required frequency range. The main drawback of this solution is the connection between the blades and the exciters, which can influence the dynamic component response. In order to prevent any inertia effect caused by the exciter moving part (i.e. the permanent magnet and the load cell) several stinger configurations were analyzed. The aim was to select a stinger transferring the force along the exciter direction only, filtering out all the other forces and torques, and minimizing the difference between the force, as measured by the load cell, and the actual force (and torque) acting on the tested structure. Four different configurations were compared: rigid connection, beam stinger, wire stinger, and ball stinger. A test bench was developed to study the effect of each stinger kind, with respect to the measured vibration, by comparing the Frequency Response Functions (FRFs). 2 ELECTROMAGNETIC SHAKER DESIGN The required severe design specifications for the electromagnetic exciter are summarized in Table 1. Quantity Value Frequency range 1 10 khz Maximum radius 17 mm Peak force (at 10 khz) 0.1N Table 1: Specifications for the electromagnetic excitation device. The basic design configuration was chosen, with a coil wired on a support (fixed part) which is axially aligned with a floating permanent magnet (moving part). The permanent magnet al- 218

3 lowed to have a high force density just with a limited volume of the moving part. Neodymium magnets were chosen for this application, since they can be found in a great variety of dimensions and at reasonable costs. The use of permanent magnets also allowed to build simple analytical/numerical models of the device, which made possible an optimization of the geometry sizes. All the structural parts of the device were manufactured in aluminum alloy, being nonferromagnetic, to have the generated magnetic field not influenced. Other non-ferromagnetic and non-metal materials, such as plexiglass, were also considered in order to limit the eddy current losses. However, the machining of those materials would have introduced some limitations on the possible manufacturing geometries, thus they were not chosen at the present research stage. As a consequence, the eddy current issue implies a higher power supply and then some overheating to be monitored. The connection between the fixed and the moving parts was finally designed. As reported below, the specification was to provide an almost free (or largely compliant) support along the axial direction while a quite stiffer support along the radial direction. 2.1 Analytical and numerical models Several different approaches are available in literature for solenoid-permanent magnet interaction modeling [7]; among them two analytical models have been focused in this research. The first, and simpler, uses the Biot-Savart law which is aimed at the determination of the flux density B on the axes of a current loop. The solenoid was modeled as a cylinder, having the coil winded between r 1 and r 2 (inner and outer solenoid radii). A cylindrical portion of the coil was initially considered, having a radial extension of dr and z as axial coordinate. Each dz segment was supposed to contain a number of coils equal to N/(L(r 2 r 1 ))dzdr, being N the total number of coils and L the total solenoid length. The intensity of B at any coordinate z can then be computed by integrating the contribution of each portion da = dzdr, from L/2 to L/2 and from r 1 to r 2, being the origin of the reference system positioned at the center of the solenoid. The value of B was computed along the solenoid axes, assuming that it remains constant along the radius with a variation along the z coordinate only. This hypothesis introduces an error which decreases as the solenoid length L increases with respect to its radius. The permanent magnet was modeled as a cylinder having an height h and a radius R m and a constant magnetization M. Thus, it was possible to compute the magnetic field in the solenoid [8] and then obtain the force acting on the magnet [7]: B 1 =(z + L/2)ln r 2 + r2 2 +(z + L/2)2 (1) r 1 + r1 2 +(z + L/2)2 B 2 =(z L/2)ln r 2 + r2 2 +(z L/2)2 r 1 + r1 2 +(z L/2)2 B z (z)= B(z) = μ 0IN 2L(r 2 r 1 ) (B 1 B 2 ) F(Z)= πr2 mm (B z (Z + h/2) B z (Z h/2)) μ 0 where μ 0 is the vacuum permeability, Z represents the distance between the center of the solenoid and the center of the permanent magnet and I is the current circulating in the solenoid. 219

4 The second considered model was the so called shell method. The solenoid was modeled as a series of thin cylinders in which the current density is I/N r, where N r is the number of radial windings. Also the permanent magnet was modeled as a thin cylindrical surface. The current density circulating on this surface was related to the value of M. The total force acting on the permanent magnet could then be obtained by summing the contributions of all the thin solenoid surfaces F s : F(Z)= 1 N r N r F s (R m,r n,h,l,z) (2) n=1 where r n represents the radial distance between the axes of the permanent magnet and the n th thin surface. F s formulation can be found in [9, 10] and it involves complete elliptic integrals of the first, second and third kinds. These two models were implemented on an electronic sheet and very similar results were obtained in terms of force estimation, Figure 1. An experimental validation (described below) has been performed and the results are also reported in the figure Biot-Savart model Shell model Exp. results Z F (N) r 1 r 2 R m h 0.1 L Z (mm) Figure 1: Analytical models comparison with L/r 1 = 5 and successful experimental validation. The models were then used for identifying the geometrical parameters of the solenoid and optimize the force intensity. Since some dimensions were fixed (see Table 1), the models were really useful to choose the total length L of the solenoid along with the permanent magnet dimensions. The optimized chosen parameter values are reported in Table Moving part support The support of the moving part is a crucial issue in the design. Depending on the chosen stinger, the exciter should work with no axial preload (beam stinger), tensile preload (wire stinger) or compression preload (ball stinger). The connection between the fixed and the moving parts of the shaker needs to carry the desired static preload, and it should be almost insensitive to high frequency excitation. In order to achieve this result, an elastic membrane was chosen. During the assembly, a rubber membrane disk was glued to the outer diameter of the fixed body so that the shaker moving part was supported as (almost) free-free [11] along the axial direction. The radial stiffness was significantly higher then axial stiffness providing ad adequate static support to the moving part of the shaker for any mounting orientation. Nevertheless, 220

5 Quantity Value r 1 10mm r 2 15mm L 50mm Wire diam. 1.2mm N 150 h 30mm 5mm R m Table 2: Electromagnetic shaker dimensions. this setup introduced a new constraint on the shaker sizing. Since the support stiffness highly depends on the unglued portion of the membrane, the clearance between the inner diameter of the coil and the outer diameter of the permanent magnet can not be too small. Figure 2 shows a schematic section view of the elastic support. The permanent magnet was implemented as separate cylinders, which are attached to each other (and to the membrane) by means of the reciprocal magnetic forces. A very simple and reliable connection between the moving part and the elastic support was therefore obtained without any fastener that would be difficult to include in such a compact design. In addition, this elastic support solution allowed for the application of the static preload both tensile or compression. Glued region Magnetic cylinder Mounting thread Elastic membrane Figure 2: Elastic support section view. 3 SHAKER TESTS The described models allowed to determine the device dimensions to satisfy the Table 1 specifications. The force as calculated with the proposed models is the interaction between the permanent magnet and the external coil and this is equal to the actual force acting on the structure only under static conditions. A static test was performed on a shaker prototype, Figure 3 (a), and a successful validation of the analytical models, described above, is reported in Fig- 221

6 ure 1. When the shaker is powered with a time varying signal, the difference between the force acting on the structure and that applied to the permanent magnet equals the product between mass and acceleration of the shaker moving part. This is the reason why dynamic tests were initially performed on a shaker prototype, to check if the peak force specification was satisfied at the highest operational frequency of 10 khz, which implies the highest acceleration. A simplified test setup was arranged with a fixed support to hold the shaker and a rigid structure as constraint, indeed, a high stiffness target reduces the moving mass acceleration. The target object was a steel beam having a cross section of mm held in fixed conditions, obviously remarkably stiffer than the parts of the shaker. The moving part was then connected to the beam through a piezoelectric load cell using a magnetic disk, Figure 3 (b). In order to decouple the shaker behavior from other effects, no stinger was introduced. The device was controlled through a LMS SCADAS function generator in terms of excitation amplitude and frequency. A Behringer EP2000 amplifier was used as a power supply for the device. The shaker voltage and current values were measured along with the obtained force, to check both electric and mechanical device properties. The supply current was limited to 10 A, since the copper wire used for the coil had a diameter of 1.2 mm, thus it could not undertake such high current levels for more then few minutes. Anyway, this limit was reached only at higher frequency (about 10 A were needed to produce 0.14 N at 10 khz), so that the overheating of the device was found to be a potential warning only for long tests at very high frequency. Since the actual test conditions do not require long time excitation, the result was considered acceptable. Solenoid Magnetic disk Load cell Permanent magnet (a) (b) Figure 3: Shaker tests: (a) static test and (b) dynamic test. 4 STINGERS COMPARISON After the achievement of the design specifications, the attention was focused to the characterization of the different stinger solutions. An ideal stinger should transmit just the load component along the chosen direction, filtering out all the other five components (i.e. 2 forces and 3 torques). A specific test bench was prepared to prove the effectiveness of the different stingers, Figure 4. Two shakers were mounted on a cantilever beam along the two transver- 222

7 Horizontal shaker Specimen Tested stinger Vertical shaker Laser Vibrometer Figure 4: Test bench setup for stinger characterization. SPHERICAL SLOT CONICAL SLOT Figure 5: Ball stinger setup. sal orthogonal directions. A load cell was mounted at the end of the vertical shaker, and one among the investigated stingers was used to connect this load cell with the beam along the vertical direction. The moving part of the other shaker was then attached to the vertical load cell, along the horizontal direction, through another load cell and a beam stinger. A laser Doppler vibrometer was used to measure the oscillation velocity of the cantilever beam (horizontal direction). Four different connection systems were compared in this study. Since some of them needed a compressive or a tensile preload, the vertical shaker was used as a support to provide this preload. Only the horizontal stinger was then excited during the test and the FRF between the force and the horizontal velocity was calculated. Stiffer stingers would lead the FRF to be not-zero while more compliant stingers should guarantee a good filtering of the horizontal force component, giving a really low FRF. Firstly, the vertical load cell was glued to the specimen using no stinger, then the tests were performed with: a beam stinger (no preload, φ = 1 mm, l = 10 mm), a ball stinger (compression preload, φ = 3 mm) and a wire stinger (tensile preload, φ = 0.2 mm, l = 7 mm). Two different supports were considered for keeping the ball in its 223

8 stable equilibrium, either with a conical or with a spherical slot. Figure 5 shows a picture of the ball stinger mounting. All the stingers were screwed on the load cell and glued to the specimen. On the contrary, the ball stinger does not involve any gluing to connect the exciter to the structure, and this implies higher reliability and also reduces the mounting time. Figure 6 shows the comparison between the obtained FRFs. As can be seen, all the FRFs show a peak around 5500 Hz, which is a resonance frequency of the excited specimen. The glue, the beam stinger, the wire stinger and the ball stinger with the conical slot also show one or more other peaks at frequencies lower then 4000 Hz. On the other hand, the ball stinger with the spherical slot provided the lowest FRF in the whole studied frequency range, showing just one peak in proximity of 5500 Hz. For this reason the ball with the spherical slot was chosen as the best stinger solution at least in this range of load level and frequency. Ampl (m s 1 N 1 ) Specimen natural mode FRF X direction Glue Beam Wire Sphere conical slot Sphere spherical slot Freq (Hz) Figure 6: Test results, FRF along the X direction. 5 CONCLUSIONS The paper presents the design and the optimization of a customized electromagnetic shaker. This device should operate in a wide frequency range, with very small lateral dimensions. A solenoid with permanent magnet configuration was chosen, proving to meet all the design specification prescribed. Several different stinger configurations were also compared: glue, beam, wire, and ball using two different mounting interfaces about the latter. The ball stinger with spherical slot support was found to be the more effective by means of transverse load filtering, and also the more convenient in terms of mounting time and reliability. 224

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