Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers)

Transcription

1 A omparison between ylindrial and ross-shaped magneti vibration isolators : ideal and pratial van Casteren, D.T.E.H.; Paulides, J.J.H.; Lomonova, E. Published in: Arhives of Eletrial Engineering DOI: /aee Published: 05/12/2015 Doument Version Publisher s PDF, also known as Version of Reord (inludes final page, issue and volume numbers) Please hek the doument version of this publiation: A submitted manusript is the author's version of the artile upon submission and before peer-review. There an be important differenes between the submitted version and the offiial published version of reord. People interested in the researh are advised to ontat the author for the final version of the publiation, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publiation after peer review. The final published version features the final layout of the paper inluding the volume, issue and page numbers. Link to publiation Citation for published version (APA): van Casteren, D. T. E. H., Paulides, J. J. H., & Lomonova, E. A. (2015). A omparison between ylindrial and ross-shaped magneti vibration isolators : ideal and pratial. Arhives of Eletrial Engineering, 64(4), DOI: /aee General rights Copyright and moral rights for the publiations made aessible in the publi portal are retained by the authors and/or other opyright owners and it is a ondition of aessing publiations that users reognise and abide by the legal requirements assoiated with these rights. Users may download and print one opy of any publiation from the publi portal for the purpose of private study or researh. You may not further distribute the material or use it for any profit-making ativity or ommerial gain You may freely distribute the URL identifying the publiation in the publi portal? Take down poliy If you believe that this doument breahes opyright please ontat us providing details, and we will remove aess to the work immediately and investigate your laim. Download date: 12. Ot. 2018

2 ARCHIVES OF ELECTRICAL ENGINEERING DOI /aee VOL. 64(4), pp (2015) A omparison between ylindrial and ross-shaped magneti vibration isolators: ideal and pratial D.T.E.H. VAN CASTEREN, J.J.H. PAULIDES, E.A. LOMONOVA Eindhoven University of Tehnology Eletromehanis and Power Eletronis group Den Doleh 2, 5600MB Eindhoven, The Netherlands d.t.e.h.v.asteren@tue.nl (Reeived: , revised: ) Abstrat: In this paper a ross-shaped isolator onsisting of uboidal magnets and a ylindrial isolator are ompared by resonane frequeny to volume ratio and shape. Both isolators are apable of obtaining a low resonane frequeny, i.e Hz and 0.01 Hz for the ross and ylinder, respetively. The volume of both isolators is omparable, only the shape is different, resulting in a tall struture with a small footprint for the ross and a flat with a large diameter ylindrial struture. A sensitivity analysis shows that due to the large amount of magnets, the ross-shaped isolator is less sensitive to manufaturing toleranes. Key words: Gravity ompensation, magneti, vibration isolation. 1. Introdution For preision mahines the influene of floor vibrations and aousti disturbanes beome ever more ritial. In these high-preision systems, therefore, the demands on vibration isolation systems inrease. Currently, the most mature tehnique for vibration isolation is air bearings augmented by linear atuators. These air bearings an, however, not work in a vau um, whih is required for an extreme ultraviolet lithography system, for instane. To ope with the requirements, researh is dediated to magneti vibration isolation systems whih have the advantages of being ontatless. Therefore, no lubriants are required, making it vauum ompatible. A magneti vibration isolation system onsists of a magneti spring (gravity ompensator) whih generates a vertial fore to levitate the mass, while maintaining a low stiffness to isolate the mass from floor vibrations. Sine a magneti spring is always unstable, linear atuators are present to stabilize the system. Magneti vibration isolators an ome in many shapes, suh as ylindrial [1], onial [2], planar (vertial and horizontal) [3], et. In this paper a omparison is made between two pratial isolators. The first isolator whih is disussed onsists of uboidal magnets plaed in a ross [4], whih an be seen in Figure 1(a). The seond isolator uses ylindrial magnets [5].

3 594 D.T.E.H. van Casteren, J.J.H. Paulides, E.A. Lomonova Arh. Elet. Eng. The translator onsists of two axially magnetized magnets and the stator magnet is magnetized radially, as an be seen in Figure 1(b). An important harateristi is the resonane or isolation frequeny. This determines from whih frequeny the isolator starts to redue the vibrations from the floor. In general, this frequeny is desired to be as small as possible in order to redue as muh vibration as possible. Sine the volume of the isolator is also important, in this paper, two isolator types are ompared by volume and resonane frequeny [6]. Another important harateristi is the manufaturability of these isolators. The alulations assume idential magnets, this is, however, not the ase in reality and small deviations on size and magnetization are always present. As in [7] the influenes of these deviations are shown by means of a sensitivity analysis. Fig. 1. Magneti struture of the (a) ross-shaped, and (b) ylindrial isolator To ompare these isolators, first the alulation of fore, stiffness, and resonane frequeny of a gravity ompensator is explained. Seond, the omparison is made for a fixed vertial fore. Third, the optimal designs are subjeted to a sensitivity analysis. Finally, onlusions are made. 2. Charateristi alulation A gravity ompensator has four important harateristis, namely vertial output fore, stiffness, resonane frequeny, and volume. To alulate these harateristis, first the magneti flux density is alulated. Sine the gravity ompensators are in free spae and only onsist of permanent magnets, the harge model is ideal (under the assumption that the relative permeability is unity). If the magnetization vetor M is onfined to the magnet volume V with surfae S, and falls abruptly to zero outside this volume, the magneti field, B, is alulated using µ 0 r r r r B = ρ dv + σ d S, 3 3 4π V r r (1.1) S r r

4 Vol. 64 (2015) A omparison between ylindrial and ross-shaped magneti vibration isolators 595 where r and rn are the observation and soure point, respetively [8]. The volume and surfae harge density, ρ and σ, respetively, are alulated by ρ = M, (1.2) M n, σ = (1.3) where M is the magnetization vetor. Note that it is assumed that the relative permeability, μ r, of the magnets is unity. In ase of low values of μ r (1,,1.1) the magnets an be modeled by saling the harge density with [9]. 2 µ + 1 r The fore between 2 magnets is alulated using the Lorentz fore equation. The fore on magnet 1, F 1, is alulated by F = ρ B dv + σ B d S, V S where ρ 1 and σ 1 are the volume and surfae harge density of magnet 1, respetively, and B 2 is the magneti flux density generated by magnet 2. The stiffness, K, an be alulated diretly from the fore equation by (1.4) K = J( F ), (1.5) where J is the Jaobian. Using the vertial fore, F z, and stiffness matrix, K, the resonane frequeny matrix, f r, of the vibration isolator is alulated by 1 Kg f r =, (1.6) π F 2 z where the gravitation onstant is denoted by g. Note that negative stiffness results in a virtual, imaginary resonane frequeny. In the results this is denoted as a negative resonane frequeny. In the ase of uboidal magnets with homogeneous magnetization, there is only a surfae harge density and the fore an be solved with analytial equations [10]. This results in an analytial expression for the stiffness and resonane frequeny. For ylindrial magnets this is not the ase. The axial magnetized ylinder also onsists of surfae harges, but the radial magnet also onsists of volume harges [11]. The field, however, an only be alulated semi-analytially. Therefore, the fore, stiffness, and resonane frequeny are alulated numerially. To estimate the volume of the isolators the smallest possible ube and ylinder is used in whih the ross-shaped and ylindrial isolator, respetively, fit.

5 596 D.T.E.H. van Casteren, J.J.H. Paulides, E.A. Lomonova Arh. Elet. Eng. 3. Comparison To attempt a fair omparison between the two strutures, several design parameters are fixed. First of all, the airgap, l g, between the translator and stator magnets is set to 4 mm, and the saled remanent magnetization is 1.28 T. Furthermore, the vertial output fore of the isolator should be around 3 kn Parameter variation In Figure 2 the side views of the isolators are given. For the ross-shaped isolator, Figure 2a, the magnets on both the translator and stator are hosen to have equal width w m, height, h m, and depth, d m. The distane between the stator magnets is given by w i and the offset of the translator is denoted by h off. The range in whih these parameters are varied are listed in Table 1. Fig. 2: Cross-setion of the (a) ross-shaped, and (b) ylindrial isolator In Figure 2b the ross-setion of the ylindrial isolator is given. As with the ross-shaped isolator, the height, h m, and width, w m, of both the stator and translator magnet are hosen equal. The other parameters whih are varied, are the inner radius, R in, and the gap between the translator magnets, h g. The range of these parameters are also listed in Table 1. Table 1. The variation in parameters for the omparison Cross-shaped Cylindrial Parameter Range Parameter Range w m w m 6 12 mm mm w i 0 15 mm R in 5 90 mm d m mm h g mm h m h m mm mm 3.2. Cross-shaped isolator To obtain a fore output of 3 kn, first the fore was alulated for a large amount of parameter ombinations. In the ase of the ross-shaped isolator the parameters, w m, w i, d m, h m, are

6 Vol. 64 (2015) A omparison between ylindrial and ross-shaped magneti vibration isolators 597 modeled by 4, 4, 21, 16 steps, respetively. Initially h off. was hosen to be 1/2 h m. In Figure 3 the alulated fore is shown for various values of d m and h m, while w m and w i are fixed at 6 mm and 5 mm, respetively. Sine the parameter sweep has oarse steps, to obtain the orret output fore, it was hosen to fit the urve using a polynomial approximation. In this ase, the parameter h m is fixed and the polynomial is used to approximate the fore for a muh smaller step in d m. This leads to the blak dots shown in Figure 3. Fig. 3. Fore of different ombinations when w m = 6 mm and w i = 5 mm of the ross-shaped isolator It an be seen that for the small values of h m an inrease in h m signifiantly dereases the d m required to obtain an output fore of 3 kn. For larger values (>40 mm), an inrease in height does not alter the fore level. In Figure 4 the differene in fore ompared to the 3 kn goal is shown for various values of h m and different movements. It an be seen that a large fore variation ours for small values of h m Note that to obtain this result the offset of the ross-shaped isolator was not kept equal, but shifted slightly to obtain a symmetri urve. Fig. 4. Fore of optimal ombination for various movements of the ross-shaped isolator This symmetry an also be seen in the results of the stiffness and the resonane frequeny, Fig. 5a and Fig. 5b, respetively. Here it an also be seen that the large fore variation for small h m, results in a large stiffness and resonane frequeny. These values derease when the height of the magnets inreases.

7 598 D.T.E.H. van Casteren, J.J.H. Paulides, E.A. Lomonova Arh. Elet. Eng. Fig. 5. (a) Stiffness and (b) resonane frequeny of optimal ombination for various movements of the ross-shaped isolator The results of the other set of ombinations are shown in Fig. 6, where the mean of the absolute resonane frequeny is plotted in ombination with the volume of the total system. It shows that while inreasing the height of the magnets redues the resonane frequeny, it does inrease the volume of the isolator. Furthermore, it an be seen that inreasing the inner thikness, w i, results in a large volume, but not in a derease in resonane frequeny. In this ase the optimal width of the magnets is around 8 mm. These lines an, therefore, be used as the optimal region for eah w i Cylindrial isolator Fig. 6. Volume to resonane frequeny plot of the ross-shaped isolator For the ylindrial isolator the same steps were taken as with the ross-shaped isolator to obtain a fore of 3 kn. The parameters paired in this ase were R in and w m. An inrease in R in resulted in a derease in w m. Using these pairs, the resonane frequeny is alulated for the other two parameters, h g and h m. In Figure 7 and Figure 8 the mean resonane frequeny versus the oupied volume is shown, when movements of 1 mm till 0 mm are taken into aount. Due to the symmetry the absolute value is equal to the ase when 1 mm to +1 mm is hosen. However, now also a differene in the sign of the steepness is visible, hene the positive and negative resonane frequenies.

8 Vol. 64 (2015) A omparison between ylindrial and ross-shaped magneti vibration isolators 599 Fig. 7. Volume to resonane frequeny plot of the ylindrial isolator for h = 30 mm and 40 mm m Fig. 8. Volume to resonane frequeny plot of the ylindrial isolator for h = 20 mm m For the ross-shaped isolator the results of the various parameters in one single figure. For the ylindrial struture, this is not possible, sine the shape varies more. Therefore, the results have been divided into two figures. In Fig. 7 the results are shown of magnets with h m is 30 mm and 40 mm for 3 different gaps, h g is 1.5 mm, 3.0 mm, and 4.5 mm, repsetively. The 6.0 mm gap is omitted, sine it is in the same trend and otherwise would make the figure unreadable. The same holds for h m = 50 mm. For h m = 20 mm the results are different and they are shown in Figure 9. It an be seen that the lines of the different gaps now ross eah other. For both figures it an, however, be seen that for eah ombination there is an optimum volume Differenes Looking at the two different strutures, it an be seen that the shape of the volume, differ greatly for the two strutures. For the ross-shaped struture a larger height is better for low resonane frequenies. The ylindrial isolator, on the other hand, is mainly flat and has a large surfae area, S. To show the differene between the two isolator types the optimal isolators

9 600 D.T.E.H. van Casteren, J.J.H. Paulides, E.A. Lomonova Arh. Elet. Eng. are hosen. In order to obtain an optimal isolator, a trade-off is made between the resonane frequeny and the volume. In the ase of the ross-shaped isolator, the topology is hosen whih has, w i = 0 mm, w i = 8 mm, h m = 38 mm, d m = 26 mm as parameters. The optimal ylindrial struture has h g = 1.5 mm, h m = 30 mm, R in = 10 mm, d m = 43.5 mm as parameters. In Table 2 the results are shown for the two isolators. It also shows that the volume of the magnets, V mag, is signifiantly smaller for the ross-shaped struture. Isolator Table 2. Summary of the optimal results f r [Hz] V [dm 3 ] S [dm 2 ] V mag [dm 3 ] Cross Cylindrial Sensitivity analysis In the previous setion, the optimal sizes of the isolators were determined using a parametri searh. In reality, however, it is not possible to exatly math these values due to manu faturing toleranes. To see how sensitive eah topology is for manufaturing toleranes, a sen sitivity analysis is onduted. For this analysis a 1000 different ombinations are mo deled, whih all are within the tolerane speifiation. First, the impat of tolerane on the dimension of the magnet is analyzed and seond, the influene of magnetization variations of the magnet is analyzed. In order to obtain appropriate results, the vertial offset of the topology is hosen per struture. The zero-position is hosen with onsideration for minimizing the resonane frequeny Magnet dimension When manufaturing magnets, there are always toleranes on the dimensions of the magnets. A standard value is 0.1 mm. Lower toleranes are possible, however, this omes at the ost of extra expenses. For the ross-shaped topology, this means that the dimensions of the magnet an beome w m = 8 ± 0.1 mm, h m = 38 ± 0.1 mm, and d m = 26 ± 0.1 mm. In Fig. 9 the result is shown of the sensitivity analysis. It an be seen that the vertial fore varies (±1%) between the different strutures. The resonane frequeny also varies about 1%. This is due to the large amount of magnets, whih averages the error. Resulting in a small deviation ompared to the ideal ase. For the ylindrial topology, the dimensions of the magnets are h g = 1.5 ± 0.1 mm, h m = 1.5 ± 0.1 mm, R in = 10 ± 0.1mm, d m = 43.5 ± 0.1 mm. The result of the sensitivity analysis is shown in Figure 10. Sine the variation in size is relatively small, the variation in fore is equal to the ross-shaped topology. Due to the small amount of magnets, i.e. 3 magnets, the resonane frequeny is more sensitive to size variations, resulting in larger variations (+0.2 Hz).

10 Vol. 64 (2015) A omparison between ylindrial and ross-shaped magneti vibration isolators 601 Fig. 9. The resonane frequeny versus vertial fore for 1000 different possible ross-shaped topologies whih satisfy the tolerane on the dimension of the magnets. Fig. 10. The resonane frequeny versus vertial fore for 1000 different possible ylindrial topologies whih satisfy the tolerane on the dimension of the magnets Magnetization Besides the dimensions of the magnets, the magnetization of the magnet is also subjeted to toleranes on i.e. magnitude and diretion. In this paper, it is assumed that the diretion is ideal and the remanent magnetization has a typial value of 1.32 T and a minimal value of 1.28 T. The relative permeability is onsidered In Figure 11 and Figure 12 the results of the sensitivity analysis are shown for the rossshaped and ylindrial topology, respetively. One more, it an be seen that the ross-shaped struture has a small variation in resonane frequeny (< 0.3%). It illustrates, however, that the variations do not reah the fore obtained with the typial magnetizations. This is due to the large amount of magnets, i.e. 64 in total. Sine the magnetization is randomly distributed between 1.28 T and 1.32 T, the average magnetization is equal to approximately the ideal ase for eah variation and the results are, therefore, entered around this point. It is, however, possible to be in between the typial and minimal ase. The ylindrial struture has more variation in resonane frequeny (+0.2 Hz) and the output fore overs the entire range from minimal to typial. This is again due to the small amount of magnets.

11 602 D.T.E.H. van Casteren, J.J.H. Paulides, E.A. Lomonova Arh. Elet. Eng. Fig. 11. The resonane frequeny versus vertial fore for 1000 different possible ross-shaped topologies whih satisfy the tolerane on the magnitude of the remanent magnetization of the magnets Fig. 12. The resonane frequeny versus vertial fore for 1000 different possible ross-shaped topologies whih satisfy the tolerane on the magnitude of the remanent magnetization of the magnets Up till now, the ylindrial struture was modeled using an ideal radially magnetized outer magnet. In pratie, however, this magnet is usually reated by ombining multiple smaller magnets with a parallel magnetization, as an be seen in Figure 13. Inreasing the amount of magnets inreases the resemblane with the radially magnetized magnet as an be seen in Figure 14, whih shows the results of 2 to 20 magnets. It an be seen that using parallel magnets only influenes the vertial output fore, where the resonane frequeny does not hange. This is due to the magneti field reated by the inner magnets being very onstant in the vertial diretion, suh that movements in the vertial diretions do not result in large fore variations. Hene a low resonane frequeny is obtained. Fig. 13. Radially magnetized ylindrial magnets, (a) ideal and (b) pratial

12 Vol. 64 (2015) A omparison between ylindrial and ross-shaped magneti vibration isolators 603 Fig. 14. The resonane frequeny versus vertial fore when the radial magnet is replaed by multiple parallel magnets 5. Conlusions In this paper two types of vibration isolators are ompared, namely a ross-shaped and a ylindrial struture. These are ompared by volume and resonane frequeny. The results have shown that both strutures are able to obtain a very low resonane frequeny, < 0.2 Hz. At these values the ross-shaped isolator has a volume of 2.4 dm 3 and the ylinder 2.1 dm 3. The main differene, however, is the shape of the volume. In the ase of the ross, a small ground surfae is ombined with a large height. For the ylinder, this is the opposite, as it has a large ground surfae and a small height. So in most ases the available room in the system will determine the best solution for the vibration isolation appliation. If there is, however, plenty of room, other fators ould be manufaturability and osts. On the dimensions of the magnets, toleranes are always present. In this paper it was hosen to do a sensitivity analysis for a 0.1 mm tolerane on its sizes. Due to the large amount of magnets in the ross-shaped topology, i.e. 64 in total, the global variation in output fore and resonane frequeny is small, <1% and <0.004 Hz, respetively. The ylindrial struture uses only 3 magnets, resulting in larger deviations in resonane frequeny of about 0.2 Hz. The same an be seen when the influene of variations of the remanent magnetization is verified. For both analyses, however, the absolute value of the resonane frequeny is still lower than the ross-shaped topology. Furthermore, the amount of magneti material in the ylindrial struture is muh larger, whih means its osts are higher ompared to the ross-shaped struture. If this is important, the ross-shaped isolator is the best topology. Referenes [1] Deng R., Saathof R., Spronk J.W. et al., Integrated 6-dof lorentz atuator with gravity ompensator for preision positioning. the 22nd International Conferene on Magnetially Levitated Systems and Linear Drives, Rio de Janeiro (2014). [2] van Casteren D.T.E.H., Paulides J.J.H., Janssen J.L.G., Lomonova E.A., Analytial fore, stiffness, and resonane frequeny alulations of a magneti vibration isolator for a miro balane. IEEE Transations on Industry Appliations 51(1): (2015).

13 604 D.T.E.H. van Casteren, J.J.H. Paulides, E.A. Lomonova Arh. Elet. Eng. [3] Janssen J.L.G., Paulides J.J.H., Lomonova E.A., Study of Magneti Gravity Compensator Topo logies Using an Abstration in the Analytial Interation Equations. Progress In Eletro magnetis Researh 128: (2012). [4] Janssen J.L.G., Gysen B.L.J., Paulides J.J.H., Lomonova E.A., Advaned Eletromagneti Modeling applied to Anti-Vibration Systems for High Preision and Automotive Appliations. International Compumag Soiety Newsletter 19(1): 3-16 (2012). [5] de Weerdt R.E.M.L., Dams J.A.A.T., Magneti atuator under piezoeletri ontrol European Patent, EP A2 (2004). [6] van Casteren D.T.E.H., Paulides J.J.H., Lomonova E.A., Gravity Compensation with Cylindrial or Cross-Shaped Magneti Vibration Isolators. Proeedings of the 10th International Symposium on Linear Drives for Industry Appliations, Aahen, pp 1-4 (2015). [7] van Casteren D.T.E.H., Pluk K.J.W., Paulides J.J.H., Lomonova E.A., Modeling the effets of magne tization variations on a permanent magnet based levitation or vibration isolation system. Applied Mehanis and Materials : (2013). [8] Furlani, E.P., Permanent Magnet and Eletromehanial Devies: Materials, Analysis, and Appliations. Aademi press (2001). [9] van Casteren D.T.E.H., Paulides J.J.H., Lomonova E.A., 3-D Numerial Surfae Charge Model inlud ing Relative Permeability: the General Theory. IEEE Transations on Magnetis, 50(11): (2014). [10] Allag H., Yonnet J.P., Latrehe M.E.H., 3D Analytial Calulation of Fores between Linear Halbah-Type Permanent-Magnet Arrays, Advaned Eletromehanial Motion Systems & Eletri Drives Joint Symposium, Lille, pp. 1-6 (2009). [11] Ravaud R., Lemarquand G., Lemarquand V., Depollier C., Permanent magnet ouplings: Field and torque three-dimensional expressions based on the oulombian model, IEEE Transations on Magnetis, 45(4): (2009).