sensors Article

Analysis on Node Position of Imperfect Resonators for Cylindrical Shell Gyroscopes Zidan Wang, Yulie Wu *, Xiang Xi, Yongmeng Zhang and Xuezhong Wu * College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073, China; [email protected] (Z.W.); [email protected] (X.X.); [email protected] (Y.Z.) * Correspondence: [email protected] (Y.W.); [email protected] (X.W.); Tel.: +86-0731-8457-4958 (Y.W.) Academic Editor: Francesco De Leonardis Received: 8 June 2016; Accepted: 25 July 2016; Published: 30 July 2016

Abstract: For cylindrical shell gyroscopes, node position of their operating eigenmodes has an important influence on the gyroscopes’ performance. It is considered that the nodes are equally separated from each other by 90˝ when the resonator vibrates in the standing wave eigenmode. However, we found that, due to manufacturing errors and trimming, the nodes may not be equally distributed. This paper mainly analyzes the influences of unbalanced masses on the cylindrical resonators’ node position, by using FEM simulation and experimental measurement. Keywords: node position; imperfect resonator; trimming method

1. Introduction As a kind of solid-state wave gyroscope, the cylindrical shell vibrating gyroscope can be used to measure the angular velocity of a rotating body based on the inertia effect of the standing wave in two vibration modes of the axial symmetric resonator [1], which have advantages such as small size, high operation accuracy, low cost, low power consumption, good shock resistance, and long life [2]. The advantages offered by the cylindrical shell vibrating gyroscope have always been of great interest to many researchers, institutes and companies. The Innalabs Holding’s Coriolis Vibration Gyroscope (CVG) and the Watson Industries’ Vibrating Structure Gyroscope (VSG) are typical cylinder shell vibrating gyroscopes [3,4]. Besides, Marconi Avionics has developed the START vibrating gyroscope [5]. The working cylindrical resonator vibrating in its second-order mode has a maximum vibration amplitude at four antinodes and a zero vibration amplitude at four nodes. For a perfect resonator, the nodes are equally distributed 90˝ from each other [6]. However, various kinds of cylindrical shell gyroscopes with different structures will have inhomogeneous errors after manufacturing, leading to the instability of vibratory modes and node positions [7]. Experts home and abroad have studied different kinds of inhomogeneous errors and the resultant quality imperfection. Fox at the University of Nottingham researched the vibratory theory and mode trimming principles of ring and cylindrical gyroscopes and proved that by adjusting part masses and rigidity, the frequency split can be reduced [8,9]. Zhbanov revealed the influence of the movability of the resonator center on the operation of the hemispherical gyroscope [10]. Dag Kristiansen studied the nonlinear oscillations and the problem of superharmonic resonances [11]. Dzhashitov in Russia conducted a theoretical analysis on the model of thermo-elastic stress-strain states and revealed the inner relationship between temperature and stress states [12]. At present, papers researching the characteristics of nodes in cylindrical resonators are quite few, in spite of those studies on the errors of imperfect resonators. In fact, it is necessary to study the characteristics of nodes, for they are closely related to the zero-bias of the gyroscope. This paper will develop an in-depth study in this field, revealing how mass, rigidity, and geometric errors affect the Sensors 2016, 16, 1206; doi:10.3390/s16081206

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localization of nodes, and providing helpful instructions for manufacture, trimming and accuracy control of high quality resonators. 2. Structure and Working Principle Sensors 2016, 16, 1206 A typical Sensors 2016, 16, 1206resonant

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shell of the cylindrical gyro is shown in Figure 1. The structure comprises 2 of 11 a rigid substrate, a round bottom, a cylindrical suspension and a thick resonant ring. Eight piezoelectric electrodes are glued theproviding bottom. The vibration of the cylindrical wall is excited and and then converted localization ofofnodes, and helpful instructions for trimming localization nodes,on and providing helpful instructions formanufacture, manufacture, trimming andaccuracy accuracy to a voltage signal viaresonators. the electrodes. control ofofhigh quality control high quality resonators. The working principle of the resonator is well understood. Figure 2 describes the vibration Structure and Working Principle 2.2. Structure Working Principle pattern of aand perfect resonator. There is a pair of operating modes known as the drive mode and sense mode, which are separated each other by 45°. The operating have the same natural typical resonant shelloffrom ofthe thecylindrical cylindrical gyro shown Figuremodes Thestructure structure comprises AAtypical resonant shell gyro isisshown ininFigure 1.1.The comprises aa frequency and the mode shape in the form of a standing wave has four antinodes and four nodes. rigid substrate, a round bottom, a cylindrical suspension and a thick resonant ring. Eight piezoelectric rigid substrate, a round bottom, a cylindrical suspension and a thick resonant ring. Eight piezoelectric The drive are mode of the shell is usedThe to generate standing vibration the exciting electrodes glued onthe thebottom. bottom. vibrationthe thecylindrical cylindrical wallisalong isexcited excited andthen thenorientation. converted electrodes are glued on The vibration ofofthe wall and converted When the gyroscope rotates, the sense mode can be detected by the electrodes due to the Coriolis voltagesignal signalvia viathe theelectrodes. electrodes. totoaavoltage effect. The working principle of the resonator is well understood. Figure 2 describes the vibration pattern of a perfect resonator. There is a pair of operatingSubstrate modes known as the drive mode and sense mode, which are separated from each other by 45°. The operating modes have the same natural Piezoelectric Bottom frequency and the modeelectrodes shape in the form of a standing wave has four plate antinodes and four nodes. The drive mode of the shell is used to generate the standing vibration along the exciting orientation. When the gyroscope rotates, the sense mode can be detected by theSuspension electrodes due to the Coriolis effect. Resonant Substrate ring Piezoelectric Bottom plate electrodes

Suspension Resonant ring Figure 1. Structure of the cylindrical resonator. Figure 1. Structure of the cylindrical resonator.

The working principle of the resonator is well understood. Figure 2 describes the vibration pattern Antinode Axis Sense mode Antinode Axis Drive mode of a perfect resonator. There is a pair of operating modes known as the drive mode and sense mode, which are separated from each other by 45˝ . The operating modes have the same natural frequency Node Axis Node Axis and the mode shape in the form of a standing wave has four antinodes and four nodes. The drive Node mode of the shell is used to generate the standing vibration along the exciting orientation. When the Structure of the by cylindrical resonator. Antinode gyroscope rotates, the sense Figure mode 1. can be detected the electrodes due to the Coriolis effect. Antinode Antinode Axis

Drive mode

Antinode Axis

Sense mode

Node Axis

Node NodeAxis

Node

Electrode

Antinode

(a)

Electrode

Antinode

(b)

Node Figure 2. Mode shape of a perfect resonator. (a) Drive mode; (b) Sense mode.

3. ANSYS Simulation on Node Position Electrode

Electrode

According to the preceding description of a perfect resonator, the antinodes and nodes are 45° apart from each other. However, the mode shape of an imperfect resonator is related to the complex (a) (b) Figure Figure2.2.Mode Modeshape shapeofofa aperfect perfectresonator. resonator.(a) (a)Drive Drivemode; mode;(b) (b)Sense Sensemode. mode.

3. ANSYS Simulation on Node Position According to the preceding description of a perfect resonator, the antinodes and nodes are 45° apart from each other. However, the mode shape of an imperfect resonator is related to the complex

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3. ANSYS Simulation on Node Position 3 of˝ 11 preceding description of a perfect resonator, the antinodes and nodes are 45 apart from each other. However, the mode shape of an imperfect resonator is related to the complex theory shell dynamics and numerical solutions difficult obtain. a result, the FEM software theory ofof shell dynamics and numerical solutions areare difficult to to obtain. AsAs a result, the FEM software ANSYS is employed to analyze the node localization of imperfect resonators. It is worth mentioning ANSYS is employed to analyze the node localization of imperfect resonators. It is worth mentioning that FEM can induce artificial frequency split a perfect resonator with a general mesh that FEM can induce anan artificial frequency split ofof 0.20.2 HzHz toto a perfect resonator with a general mesh size used our research. However, this paper mainly discusses the conditions with large frequency size used inin our research. However, this paper mainly discusses the conditions with large frequency splits caused by distinct geometric errors. So the artificial frequency split of a perfect resonator splits caused by distinct geometric errors. So the artificial frequency split of a perfect resonator is small is small enough to be neglected. enough to be neglected. Sensors 2016, 16, 1206 According to the

3.1. Finite Element Modeling Resonator 3.1. Finite Element Modeling of of thethe Resonator The geometry parameters listed in Table are used to build the model (shown in Figure 3),the and The geometry parameters listed in Table 1 are1 used to build the model (shown in Figure 3), and the material properties are listed in Table 2. Further, the element type SOLID186 is chosen. Then, material properties are listed in Table 2. Further, the element type SOLID186 is chosen. Then, different different kinds of imperfections exerted on theinFE model in order to reveal their on the kinds of imperfections are exerted are on the FE model order to reveal their influence oninfluence the operating operating eigenmodes the node position. means of modal analysis, the displacement andof the eigenmodes and the nodeand position. By means of By modal analysis, the displacement and the azimuth ˝ azimuth of every point on a circle with an interval of 1° are extracted. The position with minimum every point on a circle with an interval of 1 are extracted. The position with minimum displacement to be where the nodes are located. is displacement considered to is beconsidered where the nodes are located.

T1

R1

H1

R0

H2 H3 (a)

(b)

Figure Thegeometric geometricmodel modelofofthethe resonator.(a)(a)The Thepartial partialsectional sectionalmodel; model;(b)(b)The The entire Figure 3. 3.The resonator. entire geometric model. geometric model. Table 1. Material properties of the cylindrical resonator. Table 1. Material properties of the cylindrical resonator. Parameter Parameter Young′s modulus E Poisson′sEratio μ Young1 s modulus Density Poisson1 s ratio µ ρ Density ρ

Value Value 210 GPa 0.3 210 GPa 7800 0.3kg/m3 7800 kg/m3

Table 2. Structure parameters of the resonator. Table 2. Structure parameters of the resonator. Parameter Value Height of resonant ring H1 8 mm Parameter Value Height of suspension ring H2 10 mm Height of resonant H1 Radius ofring substrate R0 48 mm HeightInternal of suspension ring H 10 mm 2 diameter of resonator R1 12 mm Radius of substrate R 4 mm 0 Thickness of resonant ring T1 1 mm Internal diameter R1 ring T2 12 mm Thicknessofofresonator suspension 0.3 mm Thickness of resonant ring T1 1 mm Thickness of bottom H3 0.3 mm Thickness of suspension ring T2 0.3 mm Thickness of bottom H3 0.3 mm

3.2. Influence of Mass Variation on Node Position

The mass variation relative to a perfect resonator is a common imperfection. The mass distribution may not be homogeneous after machining because of the limited machining accuracy or the inhomogeneity of the material density. To simplify this condition, rectangular mass is added on the original model before simulative modal analysis.

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3.2. Influence of Mass Variation on Node Position The mass variation relative to a perfect resonator is a common imperfection. The mass distribution may not be homogeneous after machining because of the limited machining accuracy or the inhomogeneity of the material density. To simplify this condition, rectangular mass is added on Sensors 16, 1206model before simulative modal analysis. 4 of 11 the2016, original Sensors 2016, 16, 1206 3.2.1. Influence of Single Added Mass 3.2.1. Influence of Single Added Mass

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isAs shown Figure 4, when a 2.8 amm 2.2 mm 1 mm mass is mass addedison the top 3.2.1.As Influence of in Single Mass is shown in Added Figure 4, when 2.8 ×mm ˆ 2.2× mm ˆrectangular 1 mm rectangular added on the of the resonator ring (notring on (not the internal or external surface to ensure that it does not increase the top of the resonator on the internal or external surface to ensure that it does not increase As is shown in Figure 4, when a 2.8 mm × 2.2 mm × 1 mm rectangular mass is added on the top radial too much), the frequency split reaches 30 Hz.30Furthermore, the four nodes are all therigidity radial rigidity too on much), the frequency splitsurface reaches Hz. Furthermore, theincrease four nodes of the resonator ring (not the internal or external to ensure that it does not the are all shifting towards the added mass. shifting towards the added mass. radial rigidity too much), the frequency split reaches 30 Hz. Furthermore, the four nodes are all shifting towards the added mass.

(a) (a)

(b) (b)

Figure 4. Resonator condition with with single added mass.mass. (a) The model applied in simulation; Figure 4. Resonator condition single added (a)geometric The geometric model applied in simulation; Figure 4. Resonator condition with single added mass. (a) The geometric model applied in simulation; (b) The node position in this case. (b) The node position in this case. (b) The node position in this case.

3.2.2.3.2.2. Influence of Double Added Mass Influence of Double Added 3.2.2. Influence of Double Added MassMass In this which is shown in Figure 5, when two rectangular masses are added on the In case, this which is shown in Figure 5, when two rectangular are on added onresonator the resonator In this case,case, which is shown in Figure 5, when two rectangular massesmasses are added the resonator ring,ring, the the actual node position has moved slightly away from the ideal position. However, even actual node position has movedslightly slightlyaway awayfrom fromthe the ideal ideal position. position. However, However, even ring, the actual node position has moved even though though the frequency split caused by the mass has reached 60 Hz, the angular error of the nodes the frequency splitsplit caused by the mass hashas reached 60 60 Hz,Hz, thethe angular error though the frequency caused by the mass reached angular errorofofthe thenodes nodesremains remains within merely 0.12° (shown in Figure 6 and Table 3). Since the frequency split of our ˝ withinwithin merely 0.12 (shown in Figure 6 and Table Since 3). theSince frequency split of our resonators remains merely 0.12° (shown in Figure 6 and3).Table the frequency split of our can be resonators canwithin be controlled within 5 Hz after precision manufacture, asplit 60 Hz frequency split controlled 5 Hz after precision Hz frequency resonators can be controlled within 5 Hz manufacture, after precisiona 60 manufacture, a 60 Hzrepresents frequency quite split a large represents quite a large mass disturbance. In summary, we can conclude that mass mass disturbance. In summary, we can conclude that simple disturbance a simple limited influence represents quite a large mass disturbance. In summary, we mass can conclude thathassimple mass disturbance has limited influence influenceon onthe thenode nodeposition. position. disturbance has aaposition. limited on the node

(a) (a)

(b)(b)

Figure 5. 5. Resonator Resonator condition with masses. (a)(a) The geometric model applied in simulation; Figure condition withtwo twoadded added masses. The model applied in simulation; Figure 5. Resonator condition with two added masses. (a)geometric The geometric model applied in simulation; (b) The The node node position in this case. (b) position in this case. (b) The node position in this case.

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0.15 0.15 0.12 0.12 0.09 angular angular 0.09 variation variation 0.06 0.06 (°) (°) 0.03 0.03 00

00

20 20

40 40 frequency split frequency split(Hz) (Hz)

60 60

80 80

Figure Figure 6. The relationship between angular variation and frequency split. Figure6. 6.The Therelationship relationshipbetween betweenangular angularvariation variationand andfrequency frequencysplit. split. Table Table 3. Frequency splits and node position variations due to mass disturbance. Table3. 3.Frequency Frequencysplits splitsand andnode nodeposition positionvariations variationsdue dueto tomass massdisturbance. disturbance. Side of aa11mm Thick Node 3) SideLength Length ofLength mmof Thick NodePosition Position Volume Frequency Split 3) Side a 1 mm Thick Volume Frequency Volume(mm (mm Frequency Split(Hz) (Hz) Node Position Square Block (mm) Variation (° )) 3 ˝ Square Block (mm) Variation (° Square Block (mm) (mm ) Split (Hz) Variation ( ) 00 00 00 00 0 0 0 0 0.072 1.8 3.24 34.8 1.8 3.24 34.8 0.072 1.8 3.24 34.8 0.072 2.2 4.84 51.4 0.098 2.2 4.84 51.4 0.098 2.2 4.84 51.4 0.098 0.122 2.5 6.25 65.8 2.5 6.25 65.8 2.5 6.25 65.8 0.122 0.122

3.3. 3.3.Influence InfluenceofofRigidity RigidityVariation Variationon onNode NodePosition Position 3.3. Influence of Rigidity Variation on Node Position Trimming Trimmingisisaakey keyprocess processto toeliminate eliminatemanufacture manufactureerror errorand andfrequency frequencysplit. split.ItItisisconsidered considered Trimmingaistiny a key process tomaterial eliminate manufacture error and frequency split. variation It is considered that trimming quantity of off the resonator will lead to the obvious that trimming a tiny quantity of material off the resonator will lead to the obvious variationof ofthe the that trimming a tiny quantity of material off the resonator will lead to the obvious variation of the natural frequency and mode shape [13] as a result of the rigidity change in this process. natural frequency and mode shape [13] as a result of the rigidity change in this process. natural frequencyrigidity and mode shape [13] as atrimmed result of the rigidityischange in this process.in Figure 7). Firstly, Firstly, the the rigidity condition condition of of the the trimmed resonator resonator is illustrated illustrated (shown (shown in Figure 7). Firstly, the rigidity condition of the trimmed resonator is illustrated (shownradial in Figure 7). Suppose Supposethat thataaresonator resonatorisistrimmed trimmedand andsome someof ofthe theresonator resonatorrings ringsare are‘cut’, ‘cut’,and andthe the radialrigidity rigidity Suppose that a resonator is trimmed and some of the resonator rings are ‘cut’, and the radial rigidity will vary as the trimming groove position changes. will vary as the trimming groove position changes. will vary as the trimming groove position changes.

Imperfect Imperfect section section

Groove Groove

Perfect Perfect section section

Figure Figure7. 7.The Therigidity rigiditycondition. condition. Figure 7. The rigidity condition.

Afterwards, Afterwards,the thecalculation calculationpresents presentsthat thataa2.2 2.2mm mm××1.2 1.2mm mmtrimming trimminggroove groovecauses causesaarigidity rigidity Afterwards, the calculation presents that a 2.2 mm ˆ 1.2 mm trimming groove causes agrooves rigidity variation by 8.8%, while its mass has merely changed by 0.26%, which reveals that trimming variation by 8.8%, while its mass has merely changed by 0.26%, which reveals that trimming grooves variation by 8.8%, whileinitsrigidity mass has merely changed byin0.26%, which reveals that trimming grooves cause causeaalarger largervariation variation in rigiditythan thanin inmass mass(shown (shown inFigure Figure8). 8). cause a larger variation in rigidity than in mass (shown in Figure 8). 3.3.1. Influence of Single Trimming Groove on Node Position From Figure 9, since the trimming groove lacks position constraint, the actual nodes are 1 pulled1 to the groove, resulting in the relative position change of nodes in the same side. Also, the angular

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1.06 × 10 1.0 × 10

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9.6 × 10 9.6 × 10

6

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5

9.25 × 105 9.2 ×depth. 10 variation is proportional to the groove It can reach 0.86˝ with a 2.2 mm deepˆ 1.2 mm wide 5 8.8considerable × 105 8.8of×the 10 trimming groove. In addition, in view angular variation values in Figure 10,6we can Sensors 2016, 16, 1206 of 11 5 5 8.4 × 10 8.4 × 10 come to a conclusion that groove structure has a great influence on the node position. 8.0 × 10 8.05 (N/m) × 105 (N/m) 1.0 × 106 9.6 × 105 9.2 × 105 8.8 × 105 8.4 × 105 ) (N/m) 8.0(º ×) 105 (º FigureFigure 8. The8. rigidity condition of a trimmed resonator. The rigidity condition of a trimmed resonator.

3.3.1. 3.3.1. Influence of Single Trimming Groove on Node Position Influence of Single Trimming Groove on Node Position FromFrom Figure 9, since the trimming groove lacks position constraint, the actual nodesnodes are ′pulled′ Figure 9, since the trimming groove lacks position constraint, the actual are ′pulled′ to thetogroove, resulting in theinrelative position change of nodes in thein same angular the groove, resulting the relative position change of nodes the side. sameAlso, side. the Also, the angular variation is proportional to thetogroove depth.depth. It canItreach 2.2 mm 1.2 mm )0.86° with variation is proportional the groove can(º reach 0.86°awith a 2.2deep× mm deep× 1.2wide mm wide trimming groove. In addition, in view of the considerable angular variation values in Figure 10, we10, we trimming groove. In addition, in view of the considerable angular variation values in Figure Figure8. 8.The Therigidity rigidity condition of trimmed resonator. can come to a conclusion that groove structure has acondition great on the node Figure of aainfluence trimmed resonator. can come to a conclusion that groove structure has a influence great on the position. node position. 3.3.1. Influence of Single Trimming Groove on Node Position From Figure 9, since the trimming groove lacks position constraint, the actual nodes are ′pulled′ to the groove, resulting in the relative position change of nodes in the same side. Also, the angular variation is proportional to the groove depth. It can reach 0.86° with a 2.2 mm deep× 1.2 mm wide trimming groove. In addition, in view of the considerable angular variation values in Figure 10, we can come to a conclusion that groove structure has a great influence on the node position.

(a)

(b)

(a)

(b)

FigureFigure 9. Resonator condition with one groove in simulation. (a) The(a)geometric model model appliedapplied in 9. Resonator Resonator condition in in Figure 9. condition with withone onegroove grooveininsimulation. simulation. (a)The Thegeometric geometric model applied simulation; (b) The node position in this case. simulation; (b) The node position simulation; position in in this thiscase. case.

1

1

0.8

0.8

0.6

0.6

(a)

(b)

angular Figure 9. Resonator angularcondition with one groove in simulation. (a) The geometric model applied in variation 0.4 0.4 variation simulation; (b) The node position in this case. (°)

(°)0.2 0

0.2 1 0 0 0.8 0 0.6

5

10 15 20 5 10 15 frequency split (Hz) frequency split (Hz)

20

25

25

Figure 10. angular The relationship between angular variation and frequency split. Figure 10. relationship split. Figure 10. The The0.4 relationshipbetween betweenangular angularvariation variationand andfrequency frequency split. variation

(°) 0.2 3.3.2. Influence of Multiple Trimming Groove on Node Position 0 In the interest of a deeper-level on the 0 exploration 5 10 effect of 15the groove 20 structure, 25 the conditions of multiple trimming grooves are considered below.frequency split (Hz) Figure 10. The relationship between angular variation and frequency split.

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3.3.2. Influence of Multiple Trimming Groove on Node Position In the interest of a deeper-level exploration on the effect of the groove structure, the conditions of multiple trimming grooves are considered below. Sensors 2016, 16, 1206

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Firstly, two 2.2 mm deep × 1.2 mm wide trimming grooves are applied on the resonator ring at a 90° angle. The consequent mode pattern is shown in Figure 11a, where the two nodes on the Firstly, two mm deep ˆ 1.2 mm are applied on the resonator at a symmetry axis2.2 remain unchanged whilewide two trimming side nodesgrooves shift towards the groove. The variablering trend 90˝ofangle. The consequent mode pattern is shown in Figure 11a, where the two nodes on the symmetry the nodes is similar to the former condition with one groove. axis remain unchanged while two side nodes shift towards the applied groove. on The variable trend nodes Similarly, when two 2.2-mm-deep trimming grooves are the resonator ringofatthe a 180° is similar to the former condition with 0.77°, one groove. angle, the angular variation becomes which is a rather big angle (shown in Figure 11b).

(a)

(b)

Figure 11. Simulation results with two grooves. (a) The node position with two grooves at a 90° angle; Figure 11. Simulation results with two grooves. (a) The node position with two grooves at a 90˝ angle; (b) The node position with two grooves at a 180° angle. (b) The node position with two grooves at a 180˝ angle.

However, when four evenly distributed trimming grooves are applied on the resonator ring, the Similarly, when two 2.2-mm-deep trimming grooves are isapplied ontothe ring at a angular variation is merely 0.01°, even though the groove depth increased 1.5resonator mm. The angular ˝ angle, the angular variation becomes 0.77˝ , which is a rather big angle (shown in Figure 11b). 180variation is usually considered to be zero in this condition. However, since the depths of the grooves four different evenly distributed trimming grooves area distinct appliednode on the resonator ring, areHowever, not exactlywhen the same, trimmed resonators actually have position. ˝ the angular variation is merely 0.01 , even though the groove depth is increased to 1.5 mm. The angular 3.4. Influence of Roundness ErrortoonbeNode variation is usually considered zeroPosition in this condition. However, since the depths of the grooves are not Roundness exactly the same, different trimmed resonators actually have a distinct node position. (the difference between the maximum outer diameter and the minimum outer diameter) is a significant property for evaluating the manufacture accuracy of the resonator. 3.4. Influence of Roundness Error on Node Position Apart from the groove structures, the shape of the resonator ring will also cause node position Roundness (the difference between the nodes maximum outer diameter theofminimum outer variation. In this case, the distribution of the depends on the specificand shape the resonator. An example of an elliptical resonator is given outthe in manufacture this paper. Asaccuracy is shownof inthe Figure 12, the nodes diameter) is a significant property for evaluating resonator. of Apart the resonator have a trend of approaching theof long The larger the ellipticity (the deviation from the groove structures, the shape theaxis. resonator ring will also cause node position from theInperfect circular to the elliptic the greater In variation. this case, the form distribution of theform) nodesis,depends on the the angular specific variation shape of becomes. the resonator. addition, 10 mm roundness leads to about a 0.06° angular error. Considering that precision An example of an elliptical resonator is given out in this paper. As is shown in Figure 12, the nodes of thearesonator shall be guaranteed at axis. the micron scale,the theellipticity node error(the caused by shape themachining resonator on have trend of approaching the long The larger deviation from is rather small. theerror perfect circular form to the elliptic form) is, the greater the angular variation becomes. In addition, In summary, in view of our machining condition, the node position variationmachining is more due to ˝ angular 10 mm roundness leads to about a 0.06 error. Considering that precision on Sensors 2016, 16, 1206 8 of 11 the trimming grooves than roundness error. resonator shall be guaranteed at the micron scale, the node error caused by shape error is rather small. 0.07 0.06 0.05 Angular 0.04 variation 0.03 (º) 0.02 0.01 0 -0.01 0

2

4

6

8

10

12

Roundness (µm) (a)

(b)

Figure 12. Simulation condition with roundness error. (a) The node position in this case; (b) The

Figure 12. Simulation condition with roundness error. (a) The node position in this case; relationship between angular variation and roundness. (b) The relationship between angular variation and roundness.

3.5. Analysis on Rigidity Variation and Angular Error Caused by Digging Trimming Removing mass near the mid-line of the resonator ring leads to a limited change of the radial rigidity. With the removing method shown in Figure 13, the rigidity merely varies by 0.18%, while the rigidity change caused by trimming grooves can reach 2%.

Roundness (µm) (b)

(a)

12. Simulation condition with roundness error. (a) The node position in this case; (b) The8 of 11 SensorsFigure 2016, 16, 1206 relationship between angular variation and roundness.

In summary, in view of ourand machining condition, position variation is more due to 3.5. Analysis on Rigidity Variation Angular Error Causedthe by node Digging Trimming trimming grooves than roundness error. Removing mass near the mid-line of the resonator ring leads to a limited change of the radial rigidity. With removing method shown in Figure 13, by theDigging rigidityTrimming merely varies by 0.18%, while 3.5. Analysis onthe Rigidity Variation and Angular Error Caused the rigidity change caused by trimming grooves can reach 2%. Removing mass near the mid-line of the resonator ring leads to a limited change of the radial Digging trimming is a kind of mass trimming. It causes less node position error. A single dug rigidity. With the removing method shown in Figure 13, the rigidity merely varies by 0.18%, while the pit can eliminate a 3 Hz frequency split, bringing a 0.01° angular variation. rigidity change caused by trimming grooves can reach 2%. Diameter: 0.75mm,Depth:1mm

Figure 13. Influence of roundness error. error.

4. Experiment Digging trimming is a kind of mass trimming. It causes less node position error. A single dug pit can eliminate Hz frequency split, bringing 0.01˝ angular variation. In order ato3 verify the simulation results,a acoustic experiments are implemented to detect the node position in specific imperfection cases. 4. Experiment 4.1. Test Equipments and System In order to verify the simulation results, acoustic experiments are implemented to detect the node position specifictest imperfection cases. An in acoustic system is set up to carry out the experiments, and it is shown in Figure 14. In order to acquire the accurate angular position of the four nodes, the output signals in circumferential 4.1. Test Equipments and System positions should be measured. Firstly, piezoelectrodes are used to drive the resonator fixed on the An acoustic system is set up to carry out theeigenfrequency experiments, and it is shown in Figure 14. turntable and the test driving frequency is stabilized at the of the drive mode. Secondly, In order to acquire the accurate angular position of the four nodes, the output signals in circumferential the acoustic signals are detected by a microphone at places every 0.2°, while the location with minimal positions should be measured. Firstly, used toposition drive the fixed on the output can be considered as a node. Wepiezoelectrodes can change theare detecting byresonator rotating the turntable turntable thestep driving frequency is stabilized the eigenfrequency mode. Secondly, controlledand by the motor controller. At last, theatangle between nodes of canthe bedrive counted from the step the acoustic signals are detected by a microphone at places every 0.2˝ , while the location with minimal motor controller screen. output can be considered as a node. We can change the detecting position by rotating the turntable controlled by the step motor controller. At last, the angle between nodes can be counted from the step Sensors 2016, 16, 1206 9 of 11 motor controller screen. Resonator in a cap

Microphone fixed on the cap

Multimeter

Step motor controller

DC stabilized power supply

Turntable

Figure 14. Experiment system. Figure 14. Experiment system.

4.2. Influence of Single Groove on Node Position Figure 15 describes the experimental results for this condition. As a 1.2-mm-deep groove is cut out, the angle between node 1 and node 2 near the groove is about 88.86°, much smaller than 90°. Meanwhile, the two angles between node 1 and node 3, and node 2 and node 4 become obviously

Step motor controller

Multimeter

Turntable

Figure 14. Experiment system.

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Groove on Node Position

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Figure 15 describes the experimental results for this condition. As a 1.2-mm-deep groove is cut out, the angle between nodeon1 Node and node 2 near the groove is about 88.86°, much smaller than 90°. 4.2. Influence of Single Groove Position Turntable Meanwhile, the two angles between node 1 and node 3, and node 2 and node 4 become obviously Figure describes the experimental results for this condition. As a 1.2-mm-deep groove is cut larger than 15 before, reaching 90.63° and 90.86°, respectively. Figure 14. Experiment system. out, the angle between node 1 and node 2 near the is about 88.86˝between , much smaller thannode 90˝ . Furthermore, when the groove is trimmed to 1.8groove mm deep, the angle node 1 and Meanwhile, the twoasangles node 1 and node and are node 2 and node 4 become obviously 2 becomes 88.45°,between thePosition angles on the two3,sides 90.84° and 91.33°. Thus, we can see 4.2. Influenceasofsmall Single Groove onand Node ˝ and ˝ , respectively. larger than before, reaching 90.63 90.86 that the deeper the groove is, the larger the angular variation becomes. Figure 15 describes the experimental results for this condition. As a 1.2-mm-deep groove is cut out, the angle between node 1 and node 2 near the groove is about 88.86°, much smaller than 90°. Meanwhile, the two angles between node 1 and node 3, and node 2 and node 4 become obviously larger than before, reaching 90.63° and 90.86°, respectively. Furthermore, when the groove is trimmed to 1.8 mm deep, the angle between node 1 and node 2 becomes as small as 88.45°, and the angles on the two sides are 90.84° and 91.33°. Thus, we can see that the deeper the groove is, the larger the angular variation becomes.

(a)

(b)

Figure 15. 15. Resonator with oneone groove in experiment. (a) The(a)tested (b) The Figure Resonatorcondition condition with groove in experiment. The resonator; tested resonator; experiment result of nodeofposition. (b) The experiment result node position.

4.3. Influence of Two Grooves on Node Position Furthermore, when the groove is trimmed to 1.8 mm deep, the angle between node 1 and node 2 ˝ , and ˝ and Twoas1.2-mm-wide and 1.8-mm-deep grooves cut out 91.33 at 0° ˝and 180°we (shown in becomes small as 88.45 the angles rectangular on the two sides are are 90.84 . Thus, can see (a) (b) Figure 16a). From the statistic data we can see that the angle between node 1 and node 2 is 88.28° that the deeper the groove is, the larger the angular variation becomes. while the opposite anglecondition between with nodeone 3 and nodein4 experiment. is 88.23° (shown intested Figureresonator; 16b). Figure 15. Resonator groove (a) The (b) The 4.3. Influence of Two Grooves on Node Position experiment result of node position.

Two 1.2-mm-wide and 1.8-mm-deep rectangular grooves are cut out at 0˝ and 180˝ (shown in 4.3. Influence of Twothe Grooves on Node Figure 16a). From statistic data Position we can see that the angle between node 1 and node 2 is 88.28˝ whileTwo the opposite angleand between node 3 and node 4 is 88.23˝ (shown in Figure 16b). 1.2-mm-wide 1.8-mm-deep

(a) Figure 16. Resonator condition with two grooves in experiment. (b) The experiment result of node position.

(b) (a) The tested resonator;

4.4. Influence of Added Mass on Node Position When a 2.8 mm ˆ 2.2 mm ˆ 1 mm small mass is glued on the top of the resonator edge (shown in Figure 17a), all four nodes are moved towards the position of the mass (shown in Figure 17b). In addition, when a larger mass is added, the shift of the nodes becomes more apparent. Therefore, the experiment results can well fit the simulation.

4.4. Influence of Added Mass on Node Position When a 2.8 mm × 2.2 mm × 1 mm small mass is glued on the top of the resonator edge (shown in Figure 17a), all four nodes are moved towards the position of the mass (shown in Figure 17b). In addition, a larger mass is added, the shift of the nodes becomes more apparent. Therefore, the Sensors 2016,when 16, 1206 10 of 11 experiment results can well fit the simulation.

(a)

(b)

Figure condition with one one added massmass in experiment. (a) The(a) tested (b) The Figure 17. 17.Resonator Resonator condition with added in experiment. Theresonator; tested resonator; experiment result of node position. (b) The experiment result of node position.

5. 5. Conclusions Conclusions In In this this paper, paper, the the influence influence principle principle of of different different errors errors on on node node position position is is studied. studied. Simulation Simulation results results show show that that mass mass disturbance disturbance and and roundness roundness error error exert exert little little influence influence on on the the node node position position while rigidityvariation variationcaused caused trimming grooves exerts a significant influence on the node while rigidity byby trimming grooves exerts a significant influence on the node position. position. Then vibration experiments on resonators with specific imperfections are implemented to Then vibration experiments on resonators with specific imperfections are implemented to validate the validate the mentioned situations, showing that simulated and measured results are in close mentioned situations, showing that simulated and measured results are in close agreement. agreement. Therefore, in order to avoid rigidity failure, mass trimming is recommended. Considering that Therefore, in order to avoid rigidity failure, mass trimming is recommended. Considering that the node position is relative to the zero bias stability, the present work is useful for performance the node position is relative to the zero bias stability, the present work is useful for performance improvement of the resonator. improvement of the resonator. Acknowledgments: The authors would like to thank to the Laboratory of Microsystems, National University Acknowledgments: The authors would like to thank to the Laboratory of Microsystems, National University of of Defense Technology, China, for technical support and access to equipment. This work was supported by the Defense China, for technical support andNo. access to equipment. This work was supported by the NationalTechnology, Natural Science Foundation of China (Grant 51335011 and 51275522). National Natural Science Foundation of China (Grant No. 51335011 and 51275522). Author Contributions: Yulie Wu and Xuezhong Wu conceived and designed the experiments. Xiang Xi and Yongmeng Zhang performed and Wu simulations. Zidan andthe Xiang Xi analyzedXiang the data and Author Contributions: Yulie the Wuexperiments and Xuezhong conceived and Wang designed experiments. Xi and wrote the paper. Besides, YulieWu and Xuezhong Wu offered sound advices on the language. Yongmeng Zhang performed the experiments and simulations. Zidan Wang and Xiang Xi analyzed the data and wrote the paper. Besides, and Xuezhong Wuofoffered sound advices on the language. Conflicts of Interest: TheYulieWu authors declare no conflict interest.

Conflicts of Interest: The authors declare no conflict of interest.

References

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Analysis on Node Position of Imperfect Resonators for Cylindrical Shell Gyroscopes.

For cylindrical shell gyroscopes, node position of their operating eigenmodes has an important influence on the gyroscopes' performance. It is conside...
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