Electrostatic sensor modeling for torque measurements

Transcription

1 Adv. Radio Sci., 15, 55 60, Author(s) This work is distributed under the Creative Commons Attribution 3.0 License. Eectrostatic sensor modeing for torque measurements Michał Mika 1, Mirjam Dannert 1, Fei Mett 3, Harr Weber 2, Wofgang Mathis 2, and Udo Nackenhorst 1 1 Institut für Numerische Mechanik, Leibni Universität Hannover, Appestraße 9A, Hannover, German 2 Institut für Theoretische Eektrotechnik, Leibni Universität Hannover, Appestraße 9A, Hannover, German 3 Leibni Universität Hannover, Wefengarten 1, Hannover, German Correspondence to: Michał Mika (mika@stud.uni-hannover.de) Received: 28 December 2016 Accepted: 9 Apri 2017 Pubished: 21 September 2017 Abstract. Torque oad measurements pa an important part in various engineering appications, as for automotive industr, in which the drive torque of a motor has to be determined. A wide used measuring method are strain gauges. A thin feibe foi, which supports a metaic pattern, is gued to the surface of the object the torque is being appied to. In case of a deformation due to the torque oad, the change in the eectrica resistance is measured. With the combination of constitutive equations the appied torque oad is determined b the change of eectrica resistance. The creep of the gue and the foi materia, together with the temperature and humidit dependence, ma become an obstace for some appications (Kapraov and Fesenko, 1984). Thus, there have been optica and magnetica, as we as capacitive sensors introduced (Turner, 1988; Woffenbutte and Foerster, 1990). This paper discusses the genera idea behind an eectrostatic capacitive sensor based on a simpe draft of an eempar measurement setup. For better understanding an own eectrostatica, geometrica and mechanica mode of this setup has been deveoped. 1 Introduction The state of the art method for the measuring of mechanica torque oads are strain gauges. These are thin feibe fois gued to the surfaces of an object, that change the eectrica resistance due to deformation. Their appication for engineering purposes is cheap, simpe and we estabished. However, the measurement accurac depends on the humidit and the surrounding temperature. Aso, the creep of the foi and the gue contributes to the measurement error (Kapraov and Fesenko, 1984). To address these probems an eectrostatic capacitive sensor is being proposed. Simiar concepts have been aread introduced in the works of Turner (1988) and Woffenbutte and Foerster (1990). 2 Measuring setup and operation principe As shown in Fig. 1, the sensor consists of a cantiever beam with a circuar cross-section over an idea conducting pate. A thin, eastic eectret foi with a constant surface charge densit is attached to the bottom side of the beam. In comparison to the eectret the pate is assumed to have much bigger dimensions. The beam itsef and its mounting are made out of poethene. Due to the permittivit of poethene, ε pet 1, these can be ignored in the eectrica modeing. The charge eads to an eectrostatic fied E and the reated eectric dispacement fied D. This induces an eectric charge on the pate, which can be measured. As a resut of the torque oad M t, the beam deforms, which eads to a changed position of the eectret reative to the height of the pate. Thus, the induced charge on the pate changes. This change can be associated with the equivaent torque oad. In the foowing the reation between the torque oad and the measured charge is being discussed, b modeing the sstem in Matab and performing a numerica anasis for different torque oads. First, the probem is being approached b anaing the mechanica behavior of the beam. 3 Mode geometr and torsiona mechanics In order to mode the behavior of an eectrostatic fied, it is important to understand the mode geometr and to define the mechanica assumptions for a beam with a torque oad M t at the end of the beam. The beam deformation dictates the deformation of the eectret, therefore it describes the dispace- Pubished b Copernicus Pubications on behaf of the URSI Landesausschuss in der Bundesrepubik Deutschand e.v.

2 56 M. Mika et a.: Eectrostatic sensor modeing for torque measurements Poethene cantiever beam (a) M t eectret h Idea conducting pate r # # M t Figure 1. Measuring setup consisting of the beam, eectret and the idea conducting pate. ment of the eectric charge in the space, when a torque oad is appied. For the St. Vénant torsion theor (Sabó, 2001) the fundamenta assumption is a constant twist rate ϑ over the ength of the beam. Furthermore, there is no distortion of the cross-sections in and direction. The cross-sections rotate as rigid bodies around the beam ais. Aso, the warping of the cross-section is aowed, though it remains equa over the ength of the beam. Furthermore, the eectret is being ignored in this modeing, because it is thin and eastic enough, not to contribute to the overa stiffness of the beam. From the mathematica perspective the cross-sections have to be a connected space with a boundar R. The shear stress component σ t is tangent to the boundar R. Assuming the shear ange γ 1 as shown in Fig. 2a, it is possibe to describe the twist ange at as ϑ = ϑ. (1) As shown in Fig. 2b, the motion of a point P depends on the twist ange ϑ as foows + v = r cos(β + ϑ ), + w = r sin(β + ϑ ). (2) Using the trigonometric identities and assuming sma deformations, the inear approimation of motion of P in reation to the twist ange ϑ at the end of the beam is described b + v = r cosβ rϑ sinβ = ϑ (3) + w = r sinβ + rϑ cosβ = + ϑ. (4) The dispacements are v = ϑ, w = ϑ. (5) Based on the St. Vénant assumption of constant cross-section warping, the deformation aong the beam ais can be formuated as u(,,) = ϑ ϕ(,) [0;]. (6) Leaving the warping function ϕ(, ) unknown for the time being, it is eas to see, that the warping of the cross-section for sma twist rates ϑ aso remains sma. (b) + w # P ( + v, + w) + v r P(, ) Figure 2. Torsiona deformation of the beam. Figures adapted from Sabó (2001). (a) Twist of the beam aong the -ais. (b) Movement of a point P (,) in the cross-section pane as a resut of torsiona beam deformation. 4 Epicit formuation of the point motion Foregoing the derivation of the St. Vénant torsion theor, the torsion rate at the end is defined as D = ϑ = M t GJ t, (7) where G is the shear moduus and the J t describes the torsiona second moment of area (Sabó, 2001), which for circuar beam cross-sections equas J t = r 2 da = 1 2 πr4. (8) A Using the described geometr and the St. Vénant torsion theor, it is possibe to formuate a simpe geometrica mode of the sma deformations as shown in Fig. 3 Ɣ r cos (ϕ + ϑ ) Ɣ = Ɣ = Ɣ r sin (ϕ + ϑ ) (9) [ ] where ϕ 118 π; 13 8 π and [0; ]. Adv. Radio Sci., 15, 55 60,

3 M. Mika et a.: Eectrostatic sensor modeing for torque measurements 57 condition rot(grad( )) = 0 is satisfied, therefore the potentia fieds are cur-free. Furthermore, the divergence of the eectric dispacement fied is used to derive an eiptic partia differentia equation div D = div ( ε gradϕ) = ε div(gradϕ) = ϱ(r). (16) Thus, the eectrostatic fied can be described b a Poisson equation of the fied potentia ϕ Figure 3. Simpe geometrica mode of the charge hoding partices. For a inear approimation the twist rate foows from Eqs. (1), (7) and (8). 5 Mathematica modeing of the eectrostatic fied The presence of a static charge hoding partice resuts in a phsica eectrostatic fied (Küpfmüer et a., 2013). The reation of the charge densit ϱ(r) to the eectric dispacement fied D is described b div D = ϱ(r). (10) Furthermore, based on the eperiments b Couomb and Cavendish, the eectrostatic fied E is defined as the force on the charge hoding partice normaied b the hoden charge q, E = F q. (11) Observations of D ead in first order to a inear isotropic constitutive equation D = εe. (12) with the constant permittivit ε. Based on the eperiments, it is known that the forces on the partice are centra forces. Thus, the cur of the described fieds has to be equa ero, rot F = q rot E = εq rot D = 0. (13) Therefore, the phsica eectrostatic fied can be mathematica described b div D = ϱ(r), D = εe, rot E = 0. (14) Foowing these equations, a scaar eectrostatic fied potentia ϕ : R 3 R is being defined as E =: grad ϕ. (15) This is possibe without oss of generait, because of the cur-free characteristic of the described fieds. In genera, the ϕ = 2 ϕ ϕ ϕ 2 = ϱ(r) ε. (17) 6 Kirchhoff s Integra Theorem In the case of homogeneous potentia fied probems, in which the on boundar condition is postuated b the decrease of the fied over the distance and eventua vanishing at an infinite distance, there are known anatic soutions to the eiptic partia differentia, e.g. Kirchhoff s integra theorem based on Greens identities (Küpfmüer et a., 2013), ϕ(r) = 1 4πε V ϱ( r) d V. (18) r r Considering the measuring setup in Fig. 4, the Kirchhoff s integra can not be appied. The pate beneath the eectret postuates an inhomogeneous boundar condition. However, there is a possibiit to substitute the current setup with an aternative setup just for the computation purpose. The first point to consider, is the fact that the potentia ϕ at the pate has a constant vaue. This is a consequence of the assumption of the eectrostatic fied theor. The static characteristic of the eectrons sha not be vioated, therefore no potentia differences in the idea conducting pate are aowed. Furthermore, because of the fact that the charge infuence decreases with the distance to the charge itsef, the potentia at the pate in a certain sufficient distance from the charge equas ero. The pate is assumed as infinite arge, therefore the potentia at the pate equas ero. This observation aows to aternate the current setup, so there are ecusive homogeneous boundar conditions and the Kirchhoff s integra can be appied. The idea is, to mirror the eectret at the pane of the pate and assume the mirrored charge q to be the opposite charge to q as shown in Fig. 5. The infuence of the phsica eectret charge q and the introduced artificia opposite charge q cance each other resuting in a constant ero potentia at the pane of the pate. Doing so, the boundar condition of a constant fied potentia at the pane of the pate is impicit satisfied and the pate as an inhomogeneous boundar condition can be eft out. Thus, we obtain an aternated setup with homogeneous boundar conditions and the Kirchhoff s integra can be appied (Küpfmüer et a., 2013, p. 157). Adv. Radio Sci., 15, 55 60, 2017

4 58 M. Mika et a.: Eectrostatic sensor modeing for torque measurements Γ; q Ã; q ϕ = const Figure 4. Inhomogeneous eectrostatic fied probem to sove, with a constant potentia boundar condition at the pate. Γ; q Γ; Q; A Figure 6. Reguar discretiation of the eectret. 8 Induced charge on the pane Γ; q = q ϕ = const Figure 5. Homogeneous eectrostatic fied probem, with a constant potentia at the pane described b the pate. Furthermore, the eectret consists of a thin foi, so it is possibe to simpif the integration in Eq. (18) to ϕ(r) = 1 ϱ( r) d Ã. (19) 4πε r r à 7 Approimation of the anatic soution Athough, an anatic soution for the sensor mode has been derived, there sti eists the probem of integrating the charge over a compe geometr. For the modeing purpose it is easier to use a discretiation of the eectret and transate the integration into a sum. For the discretiation, the eectret is divided into reguar sma part-areas à with a part-charge q as shown in Fig. 6. The integration in Eq. (19) is transated into a sum ϕ(r) = q à 1 4πε r r à and is appied for (20) ϕ(,) =const [ h;+ ). (21) This restriction to the soution is based on the fact, that in reait the space beow the idea conducting pate is shieded from the infuence of the eectric charge b the pate itsef. As described in Sect. 2 the induced charge Q on the pane depends on the torque oad M t. This induced charge can be cacuated using the resuts in Sect. 6 for the potentia ϕ b apping the divergence theorem Q = divd dv = D n da (22) V and using the constitutive equation A D = εgradϕ. (23) Assuming on a static eectric fied, the fied in the immediate proimit of the pate is perpendicuar to the idea conducting pate and the gradient at the pate is defined b ϕ(,) = h ϕ(,) = h+ h D = ε im e. (24) h 0 h Simiar to the discretiation of the eectret, it is possibe to sum over the sma part-areas of the idea conducting pane as foows Q = ÃDe Ã. (25) Athough, the idea conducting pate is assumed to be infinite, it is accurate enough, to sum over a finite area. The areas further awa from the eectric charge do not contribute to the overa induced charge on the pate, because the potentia fied decreases over the distance. 9 Computation resuts In Fig. 7 the potentia is shown in two different heights over the eectret. It becomes apparent that the potentia reaches its major vaues, where it is coser to the eectret. Figure 8 represents the charge densit on the idea conducting pate. Adv. Radio Sci., 15, 55 60,

5 M. Mika et a.: Eectrostatic sensor modeing for torque measurements 59 (a) (b) [ V ] [ V ] Figure 7. Potentia panes ϕ(,) =const in two different heights. (a) = h h. (b) = h h. Figure 8. Charge densit σ (,) as the charge per area. The quantit of the induced charge is cacuated as described in Sect. 8. This charge can be measured and depends on the torque oad, which is appied to the beam. Therefore, a characteristic curve of the sensor is derived. For this purpose, the charge for different torque oads is simuated and recored. Fina, the characteristic curve is approimated b a ponomia function of 2nd order. The resuts are shown in Fig Concusions As one can see in Fig. 9, the maimum induced charge occurs, when there is no torque oad being appied to the beam. At that configuration the eectret is cosest to the pate. B increasing the oad, the eectret is being moved further awa from the idea conducting pate and one can observe a decrease in the induced charge on the pate. The numerica anasis of the measuring setup aso shows, that it is on possibe to concude the absoute vaue of the torque oad from the Figure 9. The characteristic curve of the sensor fitted b a ponomia function of 2nd order Q p (M t ). charge induced on the idea conducting pate. This resut is in ine with the smmetr of the sstem. Furthermore, an epected sensitivit deficienc is being observed. This is bound to the sma quasi-permanent charge an eectret hods and the distance to the idea conducting pate. Interesting, the ponomia function of 2nd order Q p (M t ) fitted b the method of east squares fits the characteristic curve we. Introducing an objective error metric as a median error ni=1 Q i (M i ) Q p (M i ) 2 e Q = (26) n where Q i (M i ) is the measured charge for an appied torque M i and Q p (M i ) is the vaue of the ponomia function at that point; assuming there are n different probed torque Adv. Radio Sci., 15, 55 60, 2017

6 60 M. Mika et a.: Eectrostatic sensor modeing for torque measurements oads, for the characteristic in Fig. 9 the error adds up to As the dimension of the charge is of order , it is reasonabe to deduce a quadratic trend of the sensor, when inear assumptions are met. 11 Summar In this contribution an anasis of an eectrostatic sensor for torque oad measurements is presented. Due to the necessit of describing the position of the charge in the phsica space, a geometrica and mechanica mode based on the St. Vénant torsion theor is described. In order to sove the we known Poisson equation for this probem, the Kirchhoff s integra theorem and an aternated mode are used. Further, a discretiation is used to sove the integra and aso to cacuate the induced charge numerica. As shown, the proposed sensor has on a ow sensitivit. Considering the dimensions and the sensitivit issues of the measuring setup, the practica appication is not optima. Nevertheess, in the further research an improvement of the sensitivit wi be considered. The pubication of this artice was funded b the open-access fund of Leibni Universität Hannover. Edited b: J. Anders Reviewed b: two anonmous referees References Kapraov, V. M. and Fesenko, N. I.: Accurac in measuring mechanica stresses (strains) with resistance strain gauges, Strength Mater., 16, 1209, Küpfmüer, K., Mathis, W., and Reibiger, A.: Theoretische Eektrotechnik: Eine Einführung, Springerverag Berin Heideberg, Sabó, I.: Höhere Technische Mechanik: nach Voresungen, Springerverag Berin Heideberg, Turner, J.: The deveopment of a thick-fim non-contact shaft torque sensor for automotive appications, Department of Mechanica Engineering, Universit of Southampton, Highfied, Southampton, UK, Woffenbutte, R. F. and Foerster, J. A.: Noncontact Capacitive Torque Sensor For Use on a Rotating Ae, IEEE T. Instrum. Meas., 39, , Data avaiabiit. No data sets were used in this artice. Competing interests. The authors decare that the have no confict of interest. Adv. Radio Sci., 15, 55 60,