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Dynamic characteristics of a beam angular-rate sensor
ARTICLE IN PRESS
International Journal of Mechanical Sciences 48 (2006) 11–20 www.elsevier.com/locate/ijmecsci
Dynamic characteristics of a beam angular-rate sensor Jongwon Seoka,, Henry A. Scartonb a
School of Mechanical Engineering, College of Engineering, Chung-Ang University 221, Heukseok-Dong, Dongjak-Gu, Seoul 156-756, Korea Laboratory for Noise and Vibration Control Research, Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590, USA
b
Received 12 October 2004; accepted 10 September 2005
Abstract This paper gives a bandwidth and sensitivity analysis for a vibrating square beam angular-rate sensor. A skew-symmetric gyroscopic inertia matrix results from rotating about an axis orthogonal to the plane of a forced-vibration elastic system giving unique characteristics permitting measurement of its rotation rate. The quality factor has a huge influence on the forced response at the resonant frequencies governing the dynamic operating range in its use as an angular-rate sensor. Fourier-type solution for the cantilever beam with electrostatic excitation reveals characteristics analogous to those occurring for more complex systems. The deflection-dependent nature of the electrostatic excitation induces the parametric excitation into the system, yielding a Hill-type equation. Although it causes the behavior to be more complex, linear sensitivity curves can be obtained, provided that the displacement of the beam is small compared with the gap in static equilibrium. Finally, it is seen that the system provides the applicability of the rotating beam for use as an angularrate sensor within a certain small range of parametric excitation. Design rules are suggested by observing that sensor sensitivity decreases with increasing sensor bandwidth. r 2005 Elsevier Ltd. All rights reserved. Keywords: Vibratory gyroscope; MEMS; Rotating cantilever beam; Electrostatic and parametric excitation; Hill-type equation; Fourier-type solution
1. Introduction Micromachined vibratory angular-rate sensors with high sensitivity and broad bandwidth are highly desirable for inertial sensing of rotational motion of a moving body. For the development of such sensor systems, mathematical analysis of the transverse vibration of a rotating body for measuring the angular velocity has drawn the attention of many researchers. Conservation of momentum induces a vibrational component perpendicular to its in-plane reference vibration excited by an electrical means when rotated about its out-of-plane axis. Since the amplitude of this rotation-induced vibration is proportional to the outof-plane angular velocity, the angular rate of the rotating body can be detected by measuring this amplitude electrically. This rotation-induced out-of-plane motion is seen in a spherical pendulum [1] or in the more Corresponding author. Tel.:+82 2 820 5729; fax: +82 2 814 9476.
E-mail addresses: [email protected] (J. Seok), [email protected] (H.A. Scarton). 0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.09.015
sophisticated transversally vibrating beam [2,3] rotating about a vertical axis passing through the support, or in vertically mounted transversally vibrating beam rotating about an axis perpendicular to its axial direction [4]. The equations for the vibration of the spinning beam have been mathematically formulated by expanding the eigenfunctions of the non-rotating boom attached to a satellite [2], by using modal analysis for a rotating shaft [3] and for a non-uniform Rayleigh beam [5], and by using Galerkin’s method for a rotating uniform Rayleigh beam subject to a force traveling at a constant speed along the axial direction [6]. However, most of the previous works are concerned mainly in the suppression of the undesirable (to those researchers) Coriolis-induced vibration. During the past few decades, the rotating beam with piezoelectric material has been conceived as a gyroscopically induced angular-rate sensor appearing in the shape of a parallelepiped [7] or tuning fork [8,9]. The parallelepiped beam gyroscope with piezoelectric films bonded to elastic structures [10], in which it was found that the higher Q factor gives higher but narrower peaks, requiring better
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
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Nomenclature A Bk b
body frame operators for boundary conditions width (or height) of a square cross-sectioned beam cDI structural damping coefficient ðnÞ C ðnÞ complex Fourier coefficients of the nth mode k ; Dk E Young’s modulus of the beam F forcing vector F^ x external force per unit length in the x direction f^x external force per unit length in the x direction normalized by EI ð¼ F^ x =EIÞ f^x0 magnitude of the normalized external force per unit length in the x direction ð¼ b e V 20 =ð4EI umax ÞÞ G, K, M, L matrix operators I identity matrix I area moment of inertia of a square crosssectioned beam L Lagrangian l length of a square cross-sectioned beam l~ dimensionless length of a square cross-sectioned beam normalized by s^ n ð¼ s^ n lÞ N inertial frame p~ n ; q~ n generalized coordinates of the nth mode normalized with the modal function of straight beam ð¼ ½pn ; qn T =Fn Þ Q quality factor Rn amplitude ration of the nth modal function of straight beam
control in tuning the device into resonant conditions. More recently, a vibratory gyroscope with bimorph plates arranged in the shape of one-half of a tuning fork composed of two cross-jointed bimorph plates connected to an annular sector plate [11,12] have also been analyzed. More detailed historical survey on these microscale vibrating angular-sensor systems is made and described in Ref. [13]. Electrostatic excitation plays a key role in inducing an oscillatory motion into a micromachined system. The complete field equation of electrostatics coupled to that
umax
equilibrium spacing between the beam and the substrate V0 magnitude of the applied voltage w,w1,w2 complex state vectors x, y, z coordinate axes in a body frame X, Y, Z coordinate axes in an inertial frame z~ dimensionless coordinate in z direction ð¼ s^ n zÞ Greek letters a g
wn d dW dmn e Ln1;2 [m,n]T [mn,nn]T O O On o on0 ~ o o ¯n Fn r s^ n tn z¯ n wn
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 beam parameter ð¼ rb2 =EI Þ Kelvin–Voigt structural damping coefficient ð¼ cD =EÞ dimensionless external force of the nth mode in the x direction variation operator virtual work Kronecker delta function electric permittivity of the spacing medium (F/m) eigenvalues normalized by on0 ð¼ 1 On Þ eigenvector of the rotating beam eigenvector of the rotating beam corresponding to the nth eigenvalue rotation rate (rad/s) dimensionless rotation rate normalized by on0 dimensionless rotation rate normalized by on0 excitation frequency (rad/s) undamped natural frequency of the nth mode in the absence of angular velocity ð¼ s^ n =a2 Þ apparent excitation frequency ð92oÞ dimensionless apparent excitation frequency ~ n0 Þ normalized by on0 ð¼ o=o nth modal function of straight beam mass density (kg/m3) modal parameter of straight beams (m1) dimensionless time normalized by 1=on0 (¼ on0 t) damping ratio associated with on0 (¼ gon0 =2) dimensionless forcing parameter ð¼ 2f^x0 = ðs^ 4n umax ÞÞ
of the moving body should be solved to describe the motion of the body. However, a simplification of the field equation of electrostatics can be done to yield electrostatic forces per unit length bounded by, and acting straightly on, the moving body if the dimensions of the electroded substrate are well matched with those of the body, (as is the case of this work). With further simplifications, Branebjerg and Gravensen [14] solved the electrostatic pressure acting on a membrane actuator and computed the correction functions for these simplified formulations. More recently, Wang [15] analyzed a micromachined cantilever beam with
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
two electrostatic actuators acting on the tip of the beam for damping the vibrations. He calculated feedback control input to suppress the vibrations deriving the stability conditions using the total energy equation. The main objective of this work is to present the salient aspects of these Coriolis-induced angular rate proportional effects in as simple a form as possible; the results can then be used as an aid in designing more complex angular-rate sensor embodiments [11,12]. The difficulties caused by the skew-symmetric gyroscopic matrix operator are resolved by using non-self-adjoint theory [16,17]. In this case, the uncoupled boundary conditions in the two orthogonal directions suggest the more convenient Ritz-type approach employed in Ref. [3]. The final form of model composed of a linear cantileverbeam gyroscope with electrostatic excitation will be presented. The displacement-dependent behavior of excitation is treated as a parametric excitation, which can be linearized with proper assumptions in electrostatic forcing function and finally yields a Hill-type equation [18]. The solution procedure of this Hill-type equation using a finite Fourier expansion will also be presented. 2. Mathematical model of a cantilever-beam angular-rate gyroscope 2.1. Model description and governing equations of the rotating beam The angular-rate sensor considered is a long, slender and square cross-sectional cantilever beam (length/width O(10)) exerted by an external force along an axis perpendicular to a lateral surface of the beam (see Fig. 1). For the system considered here, the deflection and the slope of the beam are small and the wavelength is compared to the beam thickness, large enough to satisfy the assumptions of the classical Euler–Bernoulli beam theory.1 Furthermore, although the system under consideration is in microscale, the fundamental framework of this paper is that the system is operated in an ideal environment, e.g. in vacuum condition, so that all the other microscopic effects such as surface tension, air damping, etc. are ignored. Consider two reference frames; a fixed frame N with coordinate axes X ; Y ; Z and a moving frame A with coordinate axes x; y; z. Here the frame A, with the actuating direction being in the x direction and the sensing direction along the y-axis, is rotating with a constant rotation rate O about the longitudinal z- (or Z-) axis. Eventually, an electrostatic forcing element and structural damping will be added, but for the moment, only the gyroscopic effects are included. With the ratio of Young’s modulus to the shear modulus 83 for a typical elastic body, the correction for rotary inertia and shear together is about 2% if the wave length is ten times larger than the thickness of the beam of rectangular cross-section [19]. 1
Sensing Direction
13
y x
Fˆx
Y
Actuating Direction
Frame A Frame N
[u,v]T z
Z X Ω Fig. 1. Plan view of a rotating beam with coordinate systems.
The Lagrangian function L of the beam in the absence of superposed angular velocity takes the form, Z 1 2 2 _ þ O2 ðu2 þ v2 Þg rb fu_ þ v_2 þ 2Oðu_v vuÞ L¼ 2 l 1 2 2 EIðu00 þ v00 Þ dz, ð1Þ 2 where E is Young’s modulus, I is the area moment of inertia for both the x- and y-axes, respectively, r is the density of the beam, uðz; tÞ and vðz; tÞ represents the displacement along the x- and y-axes, respectively, l is the total length of the beam, superscript dot ( ) means the partial differentiation with respect to time t, and superscript prime (0 ) means the partial differentiation with respect to z. Here, the kinetic energy associated with the rigid-body rotation about z-axis is simply ignored because the rotation rate O interested in this work will be assumed to be small compared with the lowest natural frequency of the system. With the Lagrangian given in Eq. (1) and the virtual work by the external force per unit length F^ x applied to the beam in the x direction, Z F^ x du dz, dW ¼ (2) l
the extended Hamilton’s principle [20], Z t Z t d L dt þ dW dt ¼ 0, t0
(3)
t0
yields the governing differential equations of the undamped rotating beam (damped case is considered in Section 2.3): uð4Þ þ a4 ðu€ 2O_v O2 uÞ ¼ f^x , vð4Þ þ a4 ð€v þ 2Ou_ O2 vÞ ¼ 0,
ð4Þ
and boundary conditions: ½u00 du0 l0 ¼ ½u000 dul0 ¼ ½v00 dv0 l0 ¼ ½v000 dvl0 ¼ 0, (5) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 where the beam parameter a ¼ rb =EI , b is the width (and/or height) of the beam, f^x ¼ F^ x =EI and the superscript number in parenthesis stands for the order of the derivative with respect to z. Note that although the
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
14
boundary conditions (5) naturally arise and trivially satisfy the conditions on the boundary on which the mechanical displacement or slope are prescribed,2 and thus are not useful in this case, these difficulties can be removed if we modify the extended Hamilton’s principle by introducing Lagrange multipliers [20]. Writing the second order in time Eq. (4) into its equivalent first-order vector form gives the vector state equation, _ ¼ Lw þ F, Mw where " M¼ 2 K¼4
0 I
G
1 q4 a4 qz4
" ;
L¼
I
0
0
K
2
ðÞO
#
" ;
G¼ 3
0 4
1 q a4 qz4
0
f ðzÞ ¼ A 1 sin d z þ A2 cos d z þ A3 sinh d z þ A 4 cosh d z.
ðÞO
2
0
2O
2O
0
ð14Þ
The boundary conditions for the cantilever beam fixed at z ¼ 0 and free at z ¼ l are
#
mð0; tÞ ¼ nð0; tÞ ¼ m0 ð0; tÞ ¼ n0 ð0; tÞ ¼ 0,
,
m00 ðl; tÞ ¼ n00 ðl; tÞ ¼ m000 ðl; tÞ ¼ n000 ðl; tÞ ¼ 0,
5,
F ¼ ½0; 0; f^x ; 0T ,
_ v_; u; vT ; w ¼ ½u;
ð7Þ
and 0 and I are a 2 2 zero matrix and identity matrix, respectively. At the end points of this cantilevered beam, the two homogeneous boundary conditions reduce to Bk wðz; tÞ ¼ 0;
where
(6)
#
I
Using the Euler formula eia ¼ cos a þ i sin a, the solutions of Eq. (10) can be denoted as " # " þ # mðzÞ f ðzÞ þ f ðzÞ ¼ , (13) nðzÞ ifþ ðzÞ þ if ðzÞ
k ¼ 1; 2.
(8)
Note that matrix G is skew-symmetric, while matrix K is symmetric. The non-symmetry of matrix G makes the problem non-self-adjoint [16,17], for which the detailed modal analysis is performed and explained in Appendix A.
ð15Þ
where the prime denotes the differentiation with respect to the spatial coordinate. Note that the boundary conditions in Eq. (15) are obtained by modifying Eq. (3) to treat the constraint-type conditions (vanishing deflection and slope at the fixed boundary in this work) as natural ones using the undetermined Lagrange multipliers [20]. It should be also noted that m and n are coupled in the differential equations but not in the boundary conditions. The solution functions Eq. (13) satisfy the boundary conditions provided, " # cos d l þ cosh d l sin d l þ sinh d l sin d l þ sinh d l cos d l þ cosh d l " ( )# " # A A1 0 3 ¼ , or A2 A4 0
ð16Þ
2.2. Eigenvalues and eigenfunctions of the rotating beam
which yield the following two independent transcendental equations:
Seeing that w ¼ ½lm; ln; m; nT elt , the homogeneous part of Eq. (6) yields
cos d l cosh d l þ 1 ¼ 0.
mð4Þ =a4 þ ðl2 O2 Þm 2Oln ¼ 0, nð4Þ =a4 þ ðl2 O2 Þn þ 2Olm ¼ 0,
ð9Þ
(10)
ð‘Þ
s ¼ dþ ; dþ i; s ¼ d ; d i; k ¼ 1 . . . 4, ‘ ¼ 5 . . . 8,
ð11Þ
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ a li O;
pffiffiffiffiffiffiffi i ¼ 1
with the amplitude ratios
(12)
On such a boundary, the variations of the displacement or slope simply vanish as seen in Eq. (5), which does not yield constraint-type conditions on such a boundary with the current formulation.
ð19Þ
where s^ n l ¼ 1:8751; 4:6941; . . . ;
and the superscripts 7 correspond to the associated 7 signs. 2
(18)
Rn ¼ AðnÞ =A1ðnÞ 2 ¼ ðsin s^ n l þ sinh s^ n lÞ=ðcos s^ n l þ cosh s^ n lÞ, ¼ ðcos s^ n l þ cosh s^ n lÞ=ðsin s^ n l sinh s^ n lÞ,
with the result ðkÞ
Eq. (17) has the same form as the characteristic equation of the vibrating cantilever beam and gives the well-known results [21], ^ nl d nl ¼ s
Eq. (9) can be solved classically by assuming ^ n^ T esz ½m; nT ¼ ½m;
(17)
n ¼ 1; 2; . . . ; 1.
For the fundamental natural frequency, which is to be the apparent excitation frequency in this work, the geometric limitation for the application of the classical beam theory becomes 2p l 10 (20) s^ 1 l b
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
The natural frequencies and mode shapes can be expressed as # " # " fn ðzÞ mn ðzÞ for oðÞ ¼ ðÞs^ 2n =a2 O, (21) ¼ n ifn ðzÞ n n ðzÞ where the superscripts 7 and (7) correspond to the associated 7 and (7) signs, respectively, and fn ðzÞ ¼ Rn ðcos s^ n z cosh s^ n zÞ þ sin s^ n z sinh s^ n z.
(22)
2.3. Forced vibration analysis of a damped beam system with electrostatic excitation In order to make the rotating beam solution more realistic, damping and electrostatic excitation in the actuator (x) direction are added. For this purpose consider the isotropic cantilever beam with electrostatic excitation as shown in Fig. 2. It is assumed that very thin electrodes are attached to the bottom of the beam and the top of the substrate to induce the electrostatic force. Coordinate axes x; y; z are rigidly attached to the rotating body mode, and Coriolis-induced movement in the y direction is caused by the electrostatic excitation in x direction (refer to Fig. 1 for the plan view). Introducing the Kelvin–Voigt damping [22], the system of coupled differential Eq. (4) now yields
v
þ g_v
ð4Þ
2
4
þ a ð€v þ 2Ou_ O vÞ ¼ 0,
Rigid Wall z
y x Substrate
~ ¼ 2o and dropping Letting the apparent frequency o the symbol
ð23Þ
where g ¼ cD =E, and cDI is the structural damping coefficient. When the excitation is done by the electrostatic force with an alternating voltage <½V 0 eiot and the time varying capacitance due to the motion of the beam, the forcing term in the right-hand side of Eq. (23) may be expressed in the following brief form: ( ) 2 2iot beV e 0 f^x ¼ < , (24) 2EIu2max ð1 u=umax Þ2
Beam
where b is the width of the beam, and e and umax means the electricity permittivity of the spacing medium and the equilibrium spacing between the beam and the substrate, respectively. Assuming small defection compared with the equilibrium spacing, u umax , Eq. (24) can be linearized as beV 20 u 2iot f^x < 1 þ 1 þ e . (25) 2umax 4EIu2max
beV 20 f^x0 ¼ . 4EIumax
uð4Þ þ gu_ ð4Þ þ a4 ðu€ 2Ov_ O2 uÞ ¼ f^x , ð4Þ
Actuating Direction
15
Sensing Direction
Fig. 2. A cantilever beam with electrostatic excitation in the x direction.
(27)
Defining dimensionless quantities: on0 ¼ s^ n =a2 ;
~ n0 , On ¼ O=on0 ; o ¯ n ¼ o=o ~ z~ ¼ s^ n z; l ¼ s^ n l,
tn ¼ on0 t; w ¼ 2f^ =ðs^ 4 umax Þ; n
x0
n
u~ ¼ us^ 4n =f^x0 ;
v~ ¼ vs^ 4n =f^x0 ,
Eq. (26) takes the form " # " #" # 0 2On u_~ u€~ þ 2On 0 v_~ v€~ 2 2 3" # On þ wn ðeio¯ n tn þ 1Þ 0 u~ 5 4 2 v~ 0 On 2 3 2 3 " # ð4Þ gon0 u_~ =s^ 4n u~ ð4Þ =s^ 4n eio¯ n tn þ 1 5¼ þ 4 ð4Þ 4 5 þ 4 , ð4Þ v~ =s^ n 0 gon0 v_~ =s^ 4n
ð28Þ
ð29Þ
where n can be any integer, and the meaning of the superscript dot and the superscript prime is changed to denote the differentiation with respect to the dimensionless time tn and dimensionless coordinate z~, respectively. This non-dimensionalization has been done for notational convenience in the derivation of the following equations. Even though the inclusion of the damping changes not only the differential equations but also the boundary conditions on which the moments or the shear forces are prescribed, hence its eigenstates, addition of small damping
ARTICLE IN PRESS 16
J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
would make small corrections on the eigenmodes. Instead of sacrificing small accuracy due to the corrections, big advantages can be obtained mathematically by adopting the eigenmodes of the undamped case and making use of the orthogonality of the eigenfunctions of the stationary cantilever beam by assuming the Ritz-series-type solutions for u~ and v~ of the form: ( ) X 1 fpm ðtm Þ þ qm ðtm Þgfm ðs^ m zÞ u~ ¼ < . (30) ifpm ðtm Þ þ qm ðtm Þgfm ðs^ m zÞ v~ m¼1 Integrating Eq. (29) over its domain after inserting the assumed solution in Eq. (30) multiplied by fn ðs^ n zÞ, the equation can be expressed in the form 2 3 " #" _ # p~ n p€~ 2ðz¯ n þ On iÞ 0 4 n5þ q€~ n 0 2ðz¯ n On iÞ q_~ n 2 3 2 1 On wn ðeio¯ n tn þ 1Þ=2 wn ðeio¯ n tn þ 1Þ=2 5 þ4 2 wn ðeio¯ n tn þ 1Þ=2 1 On wn ðeio¯ n tn þ 1Þ=2 3 " # 2 io¯ n tn p~ n þ 1Þ=2 ðe 5, ð31Þ ¼ 4 io¯ t q~ n ðe n n þ 1Þ=2
o ¯ n0 ffi Ln1;2 .
Z l~ Z l~ gon0 ; Fn ¼ z¯ n ¼ fn ð~zÞd~z= f2n ð~zÞd~z, 2 0 0 ½p~ n ; q~ n T ¼ ½pn ; qn T =Fn .
ðnÞ where C ðnÞ k and Dk are complex Fourier coefficients to be determined. Substituting Eq. (38) into Eq. (31) yields 1 X k¼1 ¯ n tn ¯ n tn iko iko wn DðnÞ =2 C ðnÞ k e k e ðnÞ iðkþ1Þo ¯ n tn wn ðC ðnÞ =2 ðeio¯ n tn þ 1Þ=2 ¼ 0 k þ Dk Þe
ð32Þ
(34)
Note that the Eq. (31) represents the time-dependent coupled differential equation with both parametric and external excitation yielding a Hill-type equation [18]. It has been well known that this Hill-type equation has significant regions of instability for undamped cases [23]. For the damped case, the characteristic determinant of Eq. (31),
ðnÞ iðkþ1Þo ¯ n tn wn ðC ðnÞ =2 ðeio¯ n tn þ 1Þ=2 ¼ 0 k þ Dk Þe
2
ðnÞ f2‘ðz¯ n i þ On Þ ‘2 o ¯ 2n þ 1 On wn =2gD‘ w wn ðnÞ ðC n C ðnÞ þ DðnÞ ‘ ‘1 Þ ðd1‘ þ d0‘ Þ=2 ¼ 0, 2 2 ‘1 ‘ ¼ 0; 1; 2; 1. ð40bÞ
2
(36)
ð39bÞ
Integrating Eqs. (39a) and (39b) over the fundamental period after multiplying it by ei‘on tn gives the relationship between the coefficients w ðnÞ 2 ðnÞ f2‘ðz¯ n i On Þ ‘2 o ¯ 2n þ 1 On wn =2gC ‘ n D‘ 2 wn ðnÞ ðnÞ ðC ‘1 þ D‘1 Þ ðd1‘ þ d0‘ Þ=2 ¼ 0, 2 ‘ ¼ 0; 1; 2; 1, ð40aÞ
2ðz¯ n On iÞln þ 1 On wn ðeio¯ n tn
ln3;4 ffi ðz¯ n iÞLn2 ,
for the case of wn 1 and z¯ n On 1.
2
½f2kðz¯ n i þ On Þ k2 o ¯ 2n þ 1 On wn =2g
wn ðeio¯ n tn þ 1Þ=2
yields
Ln1;2 ¼ 1 On
ð39aÞ
¯ n tn ¯ n tn iko iko wn C ðnÞ =2 DðnÞ k e k e
S ¼ jv~n j=ju~ n j.
where
2
½f2kðz¯ n i On Þ k2 o ¯ 2n þ 1 On wn =2g
k¼1
For the future usage, the dimensionless time-dependent displacements and the sensitivity are respectively defined as
½u~ n v~n T ¼ < p~ n ðtn Þ þ q~ n ðtn Þ T o , ð33Þ iðp~ n ðtn Þ þ q~ n ðtn ÞÞ
2
¯
2ðzn þ On iÞln þ 1 On wn ðeio¯ n tn þ 1Þ=2
wn ðeio¯ n tn þ 1Þ=2
(37)
Even though the exact solution of Eq. (31) is hard to obtain, Fourier-series-type solution can be obtained quite accurately by assuming the solution in the form: 2 3 " # ¯ n tn iko 1 X C ðnÞ p~ n k e 4 5, (38) ¼ ðnÞ iko ¯ n tn q~ n D e k k¼1
1 X
where
ln1;2 ffi ðz¯ n iÞLn1 ;
Note that the real parts of the Eq. (36) become positive at On ¼ 1 which indicates instability and the imaginary parts represent that the resonance occurs when the excitation frequency closely matches with either natural frequency, say,
¼0
þ 1Þ=2
(35)
Observing Eqs. (40a) and (40b), it is not difficult to prove the necessary convergence of the coefficients for large ‘. Finally, Eqs. (40a) and (40b) can yield 2ð2‘max þ 1Þ equations with 2ð2‘max þ 1Þ unknown coefficients with ðnÞ sufficient accuracy of coefficients C ðnÞ for ‘ and D‘ ‘ ¼ ‘max . Due to the harmonic components, the waves are no longer pure tones. Hence, to calculate the sensitivity, the amplitudes of both directions will be defined as
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
the components of dimensionless generalized coordinates, p~ n and q~ n , and of dimensionless apparent excitation frequency o ¯ n.
3. Discussion of results Eq. (36) shows that there are two distinct natural frequencies for the undamped case i.e.,1 On . With increasing dimensionless rotation rate On the lower natural frequency ðLn1 Þ decreases, while the upper natural frequency ðLn2 Þ increases linearly in both cases. Due to this rotation-dependent nature of the eigenvalues, it is not possible to drive this system at its natural frequencies even though the resonant frequencies are agreed to be the best driving frequencies from the energy and sensitivity point of view. The best alternative may be on the natural frequency
17
of the system at rest since this is a fixed frequency close to the actual natural frequencies known in advance. When a resonance occurs, the behavior of the vibrating system is mostly governed by the resonant denominator, and hence, the damping ratio of the resonant frequency. Under this circumstance, the Q factor [24] can be defined as 2Q ¼ 1=z¯ n and the Q factor plays a very important role in the sensitivity since the peak resonant response of the forced vibration is mainly governed by the Q factor in this damping dominant region. To investigate the influence of the damping effect on the forced response in detail, a computer simulation was performed first for small parametric excitation ðwn ¼ 104 Þ and two cases of Q factors— either Q ¼ 100 or 1000. Fig. 3(a) represents the dimensionless forced response of the system measured in the rotating coordinate system in the actuating direction and Fig. 3(b) in the sensing
Fig. 3. Steady-state sensor displacement response of the beam angular-rate gyroscope ðwn ¼ 104 Þ with Q ¼ 100 (a) for actuating direction, (b) for sensing direction, with Q ¼ 1000 (c) for actuating direction, (d) for sensing direction.
Fig. 4. Comparisons of sensitivities of the beam angular-rate gyroscope ðwn ¼ 104 Þ: (a) Q ¼ 100 and (b) Q ¼ 1000.
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
18
direction with Q of 100; while, Figs. 3 (c) and (d) shows the same relationships with Q factor of 1000. Note that for both directions in the second case, there are two distinct resonant peaks outside the very near range of zero angular frequency. On the other hand, when the Q factor is 100, the range of the single resonant peak becomes much broader although the responses for both directions become much smaller. This figure obviously illustrates the pros and cons of using the materials having low-quality factors. The sensitivity plot, which is the figure of the amplitude ratio of the sensing directional displacement vs. the actuating directional displacement under operational condition of small parametric excitation ðwn 1Þ, clearly shows the linear relationship against the frequency ratio, which can be seen in Figs. 4(a) and (b) for Q ¼ 100 and 1000, respectively, and in Fig. 5 for o ¯ n ¼ 1 and several different Q factors. Without any external control means, the resonant peak in which the system is stationary may govern the strength of the exciting voltage since the allowed maximum amplitude may also be restrained by this
20 Q=20 Q=40 Q=60 Q=80 Q=100
18 16 14
S×100
12 10 8 6 4 2 0 0
0.2
0.4
0.6
0.8
1
Ωn×1000 Fig. 5. Comparisons of percentage snsitivities with different Q factors ðo ¯ n ¼ 1; wn ¼ 104 Þ.
response. In this respect, high Q factor is preferable for narrow operation range while the low Q factor is for the wider operating range. Hence, the range of the single resonant peak should be one of the strongest design factors to make use of the resonant characteristics as much as possible. It is well known [19] that the linear classical Euler–Bernoulli beam theory can be applicable within the framework of the configuration considered in this work; the deflection and the slope of the beam are small and the wave length is sufficiently large compared to the beam thickness. Here the main measure is the size of the beam length compared to the other dimensions of the beam. Concerning the electrostatic excitation, the main measure is the size of the gap distance, between the beam and the substrate, which should be kept small to apply this forcing means effectively [14]. Even in the range of the gap distance that the linearized forcing function is quite valid, the characteristics of parametric excitation may cause undesirable behavior for the application of the system under consideration to be an angular-rate sensor. Thus it is of importance to investigate the effects of the gap distance on the behavior of this vibratory beam system. Fig. 6 shows the dimensionless orbital shapes of the system for fixed Q factors and rotation rate observed in a rotating coordinate system when the parametric excitation is relatively small ðwn ¼ 104 Þ. Here, the vertical axis stands for the dimensionless exciting frequency, while the horizontal plane includes the dimensionless displacements of the beam along the x- and y-axes. The trend of the smooth change of the orbital shape for the lower Q factor and the sharp change of its shape for the higher Q factor simply duplicates the explanation previously made. However, a distinct tilt of the orbital shape for larger values of wn can be observed in Fig. 7. Here the parametric excitation is relatively large ðwn ¼ 102 Þ compared to the previous case. This figure shows that the mean plane of vibration deviates severely from its static equilibrium position and strongly implies that the system would eventually be unstable if it exceeds a certain limit. The stability is generally handled in the parametric excitation problem; however, this will not be performed here because the system cannot be excited
Fig. 6. Orbital plots of the beam angular-rate gyroscope ðwn ¼ 104 ;
On ¼ 0:01Þ: (a) Q ¼ 100 and (b) Q ¼ 1000.
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
Fig. 7. Orbital plots of the beam angular-rate gyroscope ðwn ¼ 102 ;
only parametrically without any isolated external forcing function. Note that the linear theory adopted in this work does not support its validity if the parametric excitation exceeds a certain limit because the maximum deviation of this parametric excitation induced by the truncation of the second order is jdevjmax ¼ j3u=2umax j and mathematically proper linearization of the forcing function was based on the assumption of small higher-order terms, i.e., jdevjmax 1. Since the system equation is still linear associated with the displacement for small deflection, the response of the fundamental frequency can be used to explain its dynamic characteristics, and clearly exhibits its linear relationship with the rotation rate (see Figs. 4 and 5). 4. Conclusions Problems related with the vibratory gyroscope have been formulated and solved analytically for a special case. The forced response of the rotating beam was analyzed. The effect of the Q factor on the system time response and on the sensitivity was closely investigated for two cases of the Q factors. The system with relatively smaller Q factor yields smaller sensitivity while simultaneously giving a broader range in the response plateau before branching into two distinct resonant peaks, resulting in a broader measuring range of rotation rates. The additional parametric excitation due to the deflection-dependent nature of the electrostatic excitation causes the behavior to be more complex. However, within a certain small range of parametric excitation, the linear sensitivity curve could be obtained, operationally proving the applicability of the rotating beam as an angular-rate sensor. Appendix A. Modal analysis of the non-self-adjoint system The non-symmetry of the Coriolis matrix requires the implementation of the non-self-adjoint theory. The inner product of two complex state vectors w1,w2 defined as hw1 ; w2 i ¼ hu_ 1 ; u_ 2 i þ h_v1 ; v_2 i þ hu1 ; u2 i þ hv1 ; v2 i,
(A.1)
19
On ¼ 0:01Þ: (a) Q ¼ 100 and (b) Q ¼ 1000.
will be used with the following biorthogonal pair [16,17] lm Mnm ðzÞ ¼ Lnm ðzÞ; m ¼ 1; 2; . . . , l¯ n M n n ðzÞ ¼ L n n ðzÞ; n ¼ 1; 2; . . . ,
ðA:2Þ
where M and L are the adjoint operators of M and L, nm and n n are the eigenvectors of the principal and its adjoint system, respectively, lm is the mth eigenvalue, and the over bar operator l¯ n denotes the complex conjugate of ln . Here the inner product of two vectors a and b was defined as n Z X ha; bi ¼ b¯ i ai dG, (A.3) i¼1
G
where G is the domain in which the vectors a ¼ ½a1 ; a2 ; . . . ; an T ; b ¼ ½b1 ; b2 ; . . . ; bn T are defined. Note that while the differential matrix operator L is self-adjoint with the same boundary conditions as the principle system, the mass-matrix operator M is non-self-adjoint due to the skew-symmetric nature of G, that is to say, M ¼ MT ;
L ¼ L.
(A.4)
Taking the inner product of Eq. (A.2) with the various eigenfunctions, and using Eq. (5), gives lm Mnm ; n n ¼ Lnm ; n n , l¯ n nm ; M n n ¼ nm ; L n n , ðA:5Þ which with subtraction yields ðlm ln Þ Mnm ; n n ¼ 0.
(A.6)
For distinct eigenvalues lmaln Eq. (A.6) represents the biorthogonality condition Mnm ; n n ¼ 0; man, which with proper normalization, leads to Mn ; n ¼ dmn , m n Lnm ; nn ¼ lm dmn , ðA:7Þ where dmn is the Kronecker delta function. To obtain the relations between the two systems, let the eigenfunctions be nm ¼ ½lm mm ; lm nm ; mm ; nm T , n ¼ ½l¯ n m ; l¯ n n ; m ; n T . n
n
n
n
n
ðA:8Þ
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
20
Substituting Eq. (A.8) into Eq. (A.2) with the use of Eq. (7) gives " #" # lm 2O mm lm 2O lm nm " 1 0 #" m # 4 m 1 q 2 ¼ 4 4ðÞþO , ðA:9Þ a qz nm 0 1 " ln
ln
2O
#"
n¯ n
#
m¯ n 2O ln " 1 1 q4 2 ¼ 4 4ðÞþO a qz 0
0 1
#"
n¯ n m¯ n
# .
ðA:10Þ
Comparing Eq. (A.9) with (A.10) implies the linear relationship between the two systems ½m¯ n ; n¯ n T ¼ K¯ n ½nn ; mn T ,
(A.11)
where Kn is a constant determined by using the biorthogonality conditions with the result 1 i Kn ¼ h R l Rl 2 2 2 O 0 ðmn nn Þdz þ 2ln 0 mn nn dz or ln i. Kn ¼ h Rl Rl 2 2 2 ðln þ O Þ 0 mn nn dz a14 0 m00n n00n dz
ðA:12Þ
References [1] Foucault L. Travaux scientifiques de Foucault, Paris, 1878. [2] Meirovitch L, Silverberg LM. Control of non-self-adjoint distributedparameter systems. Journal of Optimization Theory and Application 1985;47:77–90. [3] Bauer HF. Vibration of a rotating uniform beam, Part I: orientation in the axis of rotation. Journal of Sound and Vibration 1980;72(2):177–89. [4] Cai G-P, Hong J-Z, Yang SX. Model study and active control of a rotating flexible cantilever beam. International Journal of Mechanical Sciences 2004;46(6):871–89. [5] Lee CW, Katz R, Ulsoy AG, Scott RA. Modal analysis of a distributed parameter rotating shaft. Journal of Sound and Vibration 1988;122(1):119–30.
[6] Huang Y-M, Lee C-Y. Dynamics of a rotating Rayleigh beam subject to a repetitively traveling force. International Journal of Mechanical Sciences 1998;40(8):779–92. [7] Gates WD. Vibrating angular rate sensor may threaten the gyroscope. Electronics 1968;41(10):130–4. [8] Ulitko IA. Mathematical theory of the fork-type wave gyroscope. In: Proceedings of the IEEE international frequency control symposium 1995:786–93. [9] So¨derkvist J. Design of a solid-state gyroscope sensor made of quartz. Sensors and Actuators 1990;A21–A23:293–6. [10] Fang HY, Yang JS. Vibration analysis of a rotating elastic beam with piezoelectric films as an angular rate sensor. In: Proceedings of the IEEE international frequency control symposium and PDA exhibition 2001:507–13. [11] Jongwon Seok, Tiersten HF, Scarton HA. An analysis of a vibratory angular-rate gyroscope using polarized piezoceramic bimorph plates, Part 1: derivation of variational equations in the absence of angular velocity. Journal of Sound and Vibration 2005;280(1–2):263–87. [12] Jongwon Seok, Tiersten HF, Scarton HA. An analysis of a vibratory angular-rate gyroscope using polarized piezoceramic bimorph plates, Part 2: solution procedure for the gyroscope with superposed angular velocity. Journal of Sound and Vibration 2005;280(1–2):289–310. [13] So¨derkvist J. Micromachined gyroscopes. Sensors and Actuators 1994;43:65–71. [14] Branebjerg J, Gravesen P. A new electrostatic actuator providing improved stroke length and force. In: Proceedings of IEEE micro electro mechanical systems 1992:6–11. [15] Wang PKC. Feedback control of vibrations in a micromachined cantilever beam with electrostatic actuators. Journal of Sound and Vibration 1998;213(3):537–50. [16] Albers WF, Scarton HA. Solution of the end problem of a liquidfilled cylindrical acoustic waveguide using a biorthogonality principle. American Society of Mechanical Engineers Transactions, Journal of Vibration and Acoustics 1992;114:425–31. [17] Ginsberg JH. Mechanical and structural vibration: theory and application. New York: Wiley; 2001. [18] Bolotin VV. The dynamic stability of elastic systems. San Francisco: Holden-Day Inc.; 1964. [19] Timoshenko S, Young DH. Vibration problems in engineering. New York: D. Van Nostrand Company; 1955. [20] Tiersten HF. Linear piezoelectric plate vibrations. New York: Plenum Press; 1969. [21] Blevins RD. Formulas for natural frequency and mode shape. New York: Van Nostrand Reinhold; 1979. [22] Banks HT, Wang Y, Inman DJ. Bending and shear damping in beams: frequency domain estimation techniques. Journal of Vibration and Acoustics 1994;116:188–97. [23] Meirovitch L. Methods of Analytical Dynamics. New York: McGraw-Hill Book Company; 1970. [24] Thomson WT, Dahleh MD. Theory of vibration with applications. Englewood Cliffs, NJ: Prentice-Hall Inc.; 1998.
International Journal of Mechanical Sciences 48 (2006) 11–20 www.elsevier.com/locate/ijmecsci
Dynamic characteristics of a beam angular-rate sensor Jongwon Seoka,, Henry A. Scartonb a
School of Mechanical Engineering, College of Engineering, Chung-Ang University 221, Heukseok-Dong, Dongjak-Gu, Seoul 156-756, Korea Laboratory for Noise and Vibration Control Research, Department of Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590, USA
b
Received 12 October 2004; accepted 10 September 2005
Abstract This paper gives a bandwidth and sensitivity analysis for a vibrating square beam angular-rate sensor. A skew-symmetric gyroscopic inertia matrix results from rotating about an axis orthogonal to the plane of a forced-vibration elastic system giving unique characteristics permitting measurement of its rotation rate. The quality factor has a huge influence on the forced response at the resonant frequencies governing the dynamic operating range in its use as an angular-rate sensor. Fourier-type solution for the cantilever beam with electrostatic excitation reveals characteristics analogous to those occurring for more complex systems. The deflection-dependent nature of the electrostatic excitation induces the parametric excitation into the system, yielding a Hill-type equation. Although it causes the behavior to be more complex, linear sensitivity curves can be obtained, provided that the displacement of the beam is small compared with the gap in static equilibrium. Finally, it is seen that the system provides the applicability of the rotating beam for use as an angularrate sensor within a certain small range of parametric excitation. Design rules are suggested by observing that sensor sensitivity decreases with increasing sensor bandwidth. r 2005 Elsevier Ltd. All rights reserved. Keywords: Vibratory gyroscope; MEMS; Rotating cantilever beam; Electrostatic and parametric excitation; Hill-type equation; Fourier-type solution
1. Introduction Micromachined vibratory angular-rate sensors with high sensitivity and broad bandwidth are highly desirable for inertial sensing of rotational motion of a moving body. For the development of such sensor systems, mathematical analysis of the transverse vibration of a rotating body for measuring the angular velocity has drawn the attention of many researchers. Conservation of momentum induces a vibrational component perpendicular to its in-plane reference vibration excited by an electrical means when rotated about its out-of-plane axis. Since the amplitude of this rotation-induced vibration is proportional to the outof-plane angular velocity, the angular rate of the rotating body can be detected by measuring this amplitude electrically. This rotation-induced out-of-plane motion is seen in a spherical pendulum [1] or in the more Corresponding author. Tel.:+82 2 820 5729; fax: +82 2 814 9476.
E-mail addresses: [email protected] (J. Seok), [email protected] (H.A. Scarton). 0020-7403/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2005.09.015
sophisticated transversally vibrating beam [2,3] rotating about a vertical axis passing through the support, or in vertically mounted transversally vibrating beam rotating about an axis perpendicular to its axial direction [4]. The equations for the vibration of the spinning beam have been mathematically formulated by expanding the eigenfunctions of the non-rotating boom attached to a satellite [2], by using modal analysis for a rotating shaft [3] and for a non-uniform Rayleigh beam [5], and by using Galerkin’s method for a rotating uniform Rayleigh beam subject to a force traveling at a constant speed along the axial direction [6]. However, most of the previous works are concerned mainly in the suppression of the undesirable (to those researchers) Coriolis-induced vibration. During the past few decades, the rotating beam with piezoelectric material has been conceived as a gyroscopically induced angular-rate sensor appearing in the shape of a parallelepiped [7] or tuning fork [8,9]. The parallelepiped beam gyroscope with piezoelectric films bonded to elastic structures [10], in which it was found that the higher Q factor gives higher but narrower peaks, requiring better
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
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Nomenclature A Bk b
body frame operators for boundary conditions width (or height) of a square cross-sectioned beam cDI structural damping coefficient ðnÞ C ðnÞ complex Fourier coefficients of the nth mode k ; Dk E Young’s modulus of the beam F forcing vector F^ x external force per unit length in the x direction f^x external force per unit length in the x direction normalized by EI ð¼ F^ x =EIÞ f^x0 magnitude of the normalized external force per unit length in the x direction ð¼ b e V 20 =ð4EI umax ÞÞ G, K, M, L matrix operators I identity matrix I area moment of inertia of a square crosssectioned beam L Lagrangian l length of a square cross-sectioned beam l~ dimensionless length of a square cross-sectioned beam normalized by s^ n ð¼ s^ n lÞ N inertial frame p~ n ; q~ n generalized coordinates of the nth mode normalized with the modal function of straight beam ð¼ ½pn ; qn T =Fn Þ Q quality factor Rn amplitude ration of the nth modal function of straight beam
control in tuning the device into resonant conditions. More recently, a vibratory gyroscope with bimorph plates arranged in the shape of one-half of a tuning fork composed of two cross-jointed bimorph plates connected to an annular sector plate [11,12] have also been analyzed. More detailed historical survey on these microscale vibrating angular-sensor systems is made and described in Ref. [13]. Electrostatic excitation plays a key role in inducing an oscillatory motion into a micromachined system. The complete field equation of electrostatics coupled to that
umax
equilibrium spacing between the beam and the substrate V0 magnitude of the applied voltage w,w1,w2 complex state vectors x, y, z coordinate axes in a body frame X, Y, Z coordinate axes in an inertial frame z~ dimensionless coordinate in z direction ð¼ s^ n zÞ Greek letters a g
wn d dW dmn e Ln1;2 [m,n]T [mn,nn]T O O On o on0 ~ o o ¯n Fn r s^ n tn z¯ n wn
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 beam parameter ð¼ rb2 =EI Þ Kelvin–Voigt structural damping coefficient ð¼ cD =EÞ dimensionless external force of the nth mode in the x direction variation operator virtual work Kronecker delta function electric permittivity of the spacing medium (F/m) eigenvalues normalized by on0 ð¼ 1 On Þ eigenvector of the rotating beam eigenvector of the rotating beam corresponding to the nth eigenvalue rotation rate (rad/s) dimensionless rotation rate normalized by on0 dimensionless rotation rate normalized by on0 excitation frequency (rad/s) undamped natural frequency of the nth mode in the absence of angular velocity ð¼ s^ n =a2 Þ apparent excitation frequency ð92oÞ dimensionless apparent excitation frequency ~ n0 Þ normalized by on0 ð¼ o=o nth modal function of straight beam mass density (kg/m3) modal parameter of straight beams (m1) dimensionless time normalized by 1=on0 (¼ on0 t) damping ratio associated with on0 (¼ gon0 =2) dimensionless forcing parameter ð¼ 2f^x0 = ðs^ 4n umax ÞÞ
of the moving body should be solved to describe the motion of the body. However, a simplification of the field equation of electrostatics can be done to yield electrostatic forces per unit length bounded by, and acting straightly on, the moving body if the dimensions of the electroded substrate are well matched with those of the body, (as is the case of this work). With further simplifications, Branebjerg and Gravensen [14] solved the electrostatic pressure acting on a membrane actuator and computed the correction functions for these simplified formulations. More recently, Wang [15] analyzed a micromachined cantilever beam with
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
two electrostatic actuators acting on the tip of the beam for damping the vibrations. He calculated feedback control input to suppress the vibrations deriving the stability conditions using the total energy equation. The main objective of this work is to present the salient aspects of these Coriolis-induced angular rate proportional effects in as simple a form as possible; the results can then be used as an aid in designing more complex angular-rate sensor embodiments [11,12]. The difficulties caused by the skew-symmetric gyroscopic matrix operator are resolved by using non-self-adjoint theory [16,17]. In this case, the uncoupled boundary conditions in the two orthogonal directions suggest the more convenient Ritz-type approach employed in Ref. [3]. The final form of model composed of a linear cantileverbeam gyroscope with electrostatic excitation will be presented. The displacement-dependent behavior of excitation is treated as a parametric excitation, which can be linearized with proper assumptions in electrostatic forcing function and finally yields a Hill-type equation [18]. The solution procedure of this Hill-type equation using a finite Fourier expansion will also be presented. 2. Mathematical model of a cantilever-beam angular-rate gyroscope 2.1. Model description and governing equations of the rotating beam The angular-rate sensor considered is a long, slender and square cross-sectional cantilever beam (length/width O(10)) exerted by an external force along an axis perpendicular to a lateral surface of the beam (see Fig. 1). For the system considered here, the deflection and the slope of the beam are small and the wavelength is compared to the beam thickness, large enough to satisfy the assumptions of the classical Euler–Bernoulli beam theory.1 Furthermore, although the system under consideration is in microscale, the fundamental framework of this paper is that the system is operated in an ideal environment, e.g. in vacuum condition, so that all the other microscopic effects such as surface tension, air damping, etc. are ignored. Consider two reference frames; a fixed frame N with coordinate axes X ; Y ; Z and a moving frame A with coordinate axes x; y; z. Here the frame A, with the actuating direction being in the x direction and the sensing direction along the y-axis, is rotating with a constant rotation rate O about the longitudinal z- (or Z-) axis. Eventually, an electrostatic forcing element and structural damping will be added, but for the moment, only the gyroscopic effects are included. With the ratio of Young’s modulus to the shear modulus 83 for a typical elastic body, the correction for rotary inertia and shear together is about 2% if the wave length is ten times larger than the thickness of the beam of rectangular cross-section [19]. 1
Sensing Direction
13
y x
Fˆx
Y
Actuating Direction
Frame A Frame N
[u,v]T z
Z X Ω Fig. 1. Plan view of a rotating beam with coordinate systems.
The Lagrangian function L of the beam in the absence of superposed angular velocity takes the form, Z 1 2 2 _ þ O2 ðu2 þ v2 Þg rb fu_ þ v_2 þ 2Oðu_v vuÞ L¼ 2 l 1 2 2 EIðu00 þ v00 Þ dz, ð1Þ 2 where E is Young’s modulus, I is the area moment of inertia for both the x- and y-axes, respectively, r is the density of the beam, uðz; tÞ and vðz; tÞ represents the displacement along the x- and y-axes, respectively, l is the total length of the beam, superscript dot ( ) means the partial differentiation with respect to time t, and superscript prime (0 ) means the partial differentiation with respect to z. Here, the kinetic energy associated with the rigid-body rotation about z-axis is simply ignored because the rotation rate O interested in this work will be assumed to be small compared with the lowest natural frequency of the system. With the Lagrangian given in Eq. (1) and the virtual work by the external force per unit length F^ x applied to the beam in the x direction, Z F^ x du dz, dW ¼ (2) l
the extended Hamilton’s principle [20], Z t Z t d L dt þ dW dt ¼ 0, t0
(3)
t0
yields the governing differential equations of the undamped rotating beam (damped case is considered in Section 2.3): uð4Þ þ a4 ðu€ 2O_v O2 uÞ ¼ f^x , vð4Þ þ a4 ð€v þ 2Ou_ O2 vÞ ¼ 0,
ð4Þ
and boundary conditions: ½u00 du0 l0 ¼ ½u000 dul0 ¼ ½v00 dv0 l0 ¼ ½v000 dvl0 ¼ 0, (5) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 where the beam parameter a ¼ rb =EI , b is the width (and/or height) of the beam, f^x ¼ F^ x =EI and the superscript number in parenthesis stands for the order of the derivative with respect to z. Note that although the
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14
boundary conditions (5) naturally arise and trivially satisfy the conditions on the boundary on which the mechanical displacement or slope are prescribed,2 and thus are not useful in this case, these difficulties can be removed if we modify the extended Hamilton’s principle by introducing Lagrange multipliers [20]. Writing the second order in time Eq. (4) into its equivalent first-order vector form gives the vector state equation, _ ¼ Lw þ F, Mw where " M¼ 2 K¼4
0 I
G
1 q4 a4 qz4
" ;
L¼
I
0
0
K
2
ðÞO
#
" ;
G¼ 3
0 4
1 q a4 qz4
0
f ðzÞ ¼ A 1 sin d z þ A2 cos d z þ A3 sinh d z þ A 4 cosh d z.
ðÞO
2
0
2O
2O
0
ð14Þ
The boundary conditions for the cantilever beam fixed at z ¼ 0 and free at z ¼ l are
#
mð0; tÞ ¼ nð0; tÞ ¼ m0 ð0; tÞ ¼ n0 ð0; tÞ ¼ 0,
,
m00 ðl; tÞ ¼ n00 ðl; tÞ ¼ m000 ðl; tÞ ¼ n000 ðl; tÞ ¼ 0,
5,
F ¼ ½0; 0; f^x ; 0T ,
_ v_; u; vT ; w ¼ ½u;
ð7Þ
and 0 and I are a 2 2 zero matrix and identity matrix, respectively. At the end points of this cantilevered beam, the two homogeneous boundary conditions reduce to Bk wðz; tÞ ¼ 0;
where
(6)
#
I
Using the Euler formula eia ¼ cos a þ i sin a, the solutions of Eq. (10) can be denoted as " # " þ # mðzÞ f ðzÞ þ f ðzÞ ¼ , (13) nðzÞ ifþ ðzÞ þ if ðzÞ
k ¼ 1; 2.
(8)
Note that matrix G is skew-symmetric, while matrix K is symmetric. The non-symmetry of matrix G makes the problem non-self-adjoint [16,17], for which the detailed modal analysis is performed and explained in Appendix A.
ð15Þ
where the prime denotes the differentiation with respect to the spatial coordinate. Note that the boundary conditions in Eq. (15) are obtained by modifying Eq. (3) to treat the constraint-type conditions (vanishing deflection and slope at the fixed boundary in this work) as natural ones using the undetermined Lagrange multipliers [20]. It should be also noted that m and n are coupled in the differential equations but not in the boundary conditions. The solution functions Eq. (13) satisfy the boundary conditions provided, " # cos d l þ cosh d l sin d l þ sinh d l sin d l þ sinh d l cos d l þ cosh d l " ( )# " # A A1 0 3 ¼ , or A2 A4 0
ð16Þ
2.2. Eigenvalues and eigenfunctions of the rotating beam
which yield the following two independent transcendental equations:
Seeing that w ¼ ½lm; ln; m; nT elt , the homogeneous part of Eq. (6) yields
cos d l cosh d l þ 1 ¼ 0.
mð4Þ =a4 þ ðl2 O2 Þm 2Oln ¼ 0, nð4Þ =a4 þ ðl2 O2 Þn þ 2Olm ¼ 0,
ð9Þ
(10)
ð‘Þ
s ¼ dþ ; dþ i; s ¼ d ; d i; k ¼ 1 . . . 4, ‘ ¼ 5 . . . 8,
ð11Þ
where pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ a li O;
pffiffiffiffiffiffiffi i ¼ 1
with the amplitude ratios
(12)
On such a boundary, the variations of the displacement or slope simply vanish as seen in Eq. (5), which does not yield constraint-type conditions on such a boundary with the current formulation.
ð19Þ
where s^ n l ¼ 1:8751; 4:6941; . . . ;
and the superscripts 7 correspond to the associated 7 signs. 2
(18)
Rn ¼ AðnÞ =A1ðnÞ 2 ¼ ðsin s^ n l þ sinh s^ n lÞ=ðcos s^ n l þ cosh s^ n lÞ, ¼ ðcos s^ n l þ cosh s^ n lÞ=ðsin s^ n l sinh s^ n lÞ,
with the result ðkÞ
Eq. (17) has the same form as the characteristic equation of the vibrating cantilever beam and gives the well-known results [21], ^ nl d nl ¼ s
Eq. (9) can be solved classically by assuming ^ n^ T esz ½m; nT ¼ ½m;
(17)
n ¼ 1; 2; . . . ; 1.
For the fundamental natural frequency, which is to be the apparent excitation frequency in this work, the geometric limitation for the application of the classical beam theory becomes 2p l 10 (20) s^ 1 l b
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
The natural frequencies and mode shapes can be expressed as # " # " fn ðzÞ mn ðzÞ for oðÞ ¼ ðÞs^ 2n =a2 O, (21) ¼ n ifn ðzÞ n n ðzÞ where the superscripts 7 and (7) correspond to the associated 7 and (7) signs, respectively, and fn ðzÞ ¼ Rn ðcos s^ n z cosh s^ n zÞ þ sin s^ n z sinh s^ n z.
(22)
2.3. Forced vibration analysis of a damped beam system with electrostatic excitation In order to make the rotating beam solution more realistic, damping and electrostatic excitation in the actuator (x) direction are added. For this purpose consider the isotropic cantilever beam with electrostatic excitation as shown in Fig. 2. It is assumed that very thin electrodes are attached to the bottom of the beam and the top of the substrate to induce the electrostatic force. Coordinate axes x; y; z are rigidly attached to the rotating body mode, and Coriolis-induced movement in the y direction is caused by the electrostatic excitation in x direction (refer to Fig. 1 for the plan view). Introducing the Kelvin–Voigt damping [22], the system of coupled differential Eq. (4) now yields
v
þ g_v
ð4Þ
2
4
þ a ð€v þ 2Ou_ O vÞ ¼ 0,
Rigid Wall z
y x Substrate
~ ¼ 2o and dropping Letting the apparent frequency o the symbol
ð23Þ
where g ¼ cD =E, and cDI is the structural damping coefficient. When the excitation is done by the electrostatic force with an alternating voltage <½V 0 eiot and the time varying capacitance due to the motion of the beam, the forcing term in the right-hand side of Eq. (23) may be expressed in the following brief form: ( ) 2 2iot beV e 0 f^x ¼ < , (24) 2EIu2max ð1 u=umax Þ2
Beam
where b is the width of the beam, and e and umax means the electricity permittivity of the spacing medium and the equilibrium spacing between the beam and the substrate, respectively. Assuming small defection compared with the equilibrium spacing, u umax , Eq. (24) can be linearized as beV 20 u 2iot f^x < 1 þ 1 þ e . (25) 2umax 4EIu2max
beV 20 f^x0 ¼ . 4EIumax
uð4Þ þ gu_ ð4Þ þ a4 ðu€ 2Ov_ O2 uÞ ¼ f^x , ð4Þ
Actuating Direction
15
Sensing Direction
Fig. 2. A cantilever beam with electrostatic excitation in the x direction.
(27)
Defining dimensionless quantities: on0 ¼ s^ n =a2 ;
~ n0 , On ¼ O=on0 ; o ¯ n ¼ o=o ~ z~ ¼ s^ n z; l ¼ s^ n l,
tn ¼ on0 t; w ¼ 2f^ =ðs^ 4 umax Þ; n
x0
n
u~ ¼ us^ 4n =f^x0 ;
v~ ¼ vs^ 4n =f^x0 ,
Eq. (26) takes the form " # " #" # 0 2On u_~ u€~ þ 2On 0 v_~ v€~ 2 2 3" # On þ wn ðeio¯ n tn þ 1Þ 0 u~ 5 4 2 v~ 0 On 2 3 2 3 " # ð4Þ gon0 u_~ =s^ 4n u~ ð4Þ =s^ 4n eio¯ n tn þ 1 5¼ þ 4 ð4Þ 4 5 þ 4 , ð4Þ v~ =s^ n 0 gon0 v_~ =s^ 4n
ð28Þ
ð29Þ
where n can be any integer, and the meaning of the superscript dot and the superscript prime is changed to denote the differentiation with respect to the dimensionless time tn and dimensionless coordinate z~, respectively. This non-dimensionalization has been done for notational convenience in the derivation of the following equations. Even though the inclusion of the damping changes not only the differential equations but also the boundary conditions on which the moments or the shear forces are prescribed, hence its eigenstates, addition of small damping
ARTICLE IN PRESS 16
J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
would make small corrections on the eigenmodes. Instead of sacrificing small accuracy due to the corrections, big advantages can be obtained mathematically by adopting the eigenmodes of the undamped case and making use of the orthogonality of the eigenfunctions of the stationary cantilever beam by assuming the Ritz-series-type solutions for u~ and v~ of the form: ( ) X 1 fpm ðtm Þ þ qm ðtm Þgfm ðs^ m zÞ u~ ¼ < . (30) ifpm ðtm Þ þ qm ðtm Þgfm ðs^ m zÞ v~ m¼1 Integrating Eq. (29) over its domain after inserting the assumed solution in Eq. (30) multiplied by fn ðs^ n zÞ, the equation can be expressed in the form 2 3 " #" _ # p~ n p€~ 2ðz¯ n þ On iÞ 0 4 n5þ q€~ n 0 2ðz¯ n On iÞ q_~ n 2 3 2 1 On wn ðeio¯ n tn þ 1Þ=2 wn ðeio¯ n tn þ 1Þ=2 5 þ4 2 wn ðeio¯ n tn þ 1Þ=2 1 On wn ðeio¯ n tn þ 1Þ=2 3 " # 2 io¯ n tn p~ n þ 1Þ=2 ðe 5, ð31Þ ¼ 4 io¯ t q~ n ðe n n þ 1Þ=2
o ¯ n0 ffi Ln1;2 .
Z l~ Z l~ gon0 ; Fn ¼ z¯ n ¼ fn ð~zÞd~z= f2n ð~zÞd~z, 2 0 0 ½p~ n ; q~ n T ¼ ½pn ; qn T =Fn .
ðnÞ where C ðnÞ k and Dk are complex Fourier coefficients to be determined. Substituting Eq. (38) into Eq. (31) yields 1 X k¼1 ¯ n tn ¯ n tn iko iko wn DðnÞ =2 C ðnÞ k e k e ðnÞ iðkþ1Þo ¯ n tn wn ðC ðnÞ =2 ðeio¯ n tn þ 1Þ=2 ¼ 0 k þ Dk Þe
ð32Þ
(34)
Note that the Eq. (31) represents the time-dependent coupled differential equation with both parametric and external excitation yielding a Hill-type equation [18]. It has been well known that this Hill-type equation has significant regions of instability for undamped cases [23]. For the damped case, the characteristic determinant of Eq. (31),
ðnÞ iðkþ1Þo ¯ n tn wn ðC ðnÞ =2 ðeio¯ n tn þ 1Þ=2 ¼ 0 k þ Dk Þe
2
ðnÞ f2‘ðz¯ n i þ On Þ ‘2 o ¯ 2n þ 1 On wn =2gD‘ w wn ðnÞ ðC n C ðnÞ þ DðnÞ ‘ ‘1 Þ ðd1‘ þ d0‘ Þ=2 ¼ 0, 2 2 ‘1 ‘ ¼ 0; 1; 2; 1. ð40bÞ
2
(36)
ð39bÞ
Integrating Eqs. (39a) and (39b) over the fundamental period after multiplying it by ei‘on tn gives the relationship between the coefficients w ðnÞ 2 ðnÞ f2‘ðz¯ n i On Þ ‘2 o ¯ 2n þ 1 On wn =2gC ‘ n D‘ 2 wn ðnÞ ðnÞ ðC ‘1 þ D‘1 Þ ðd1‘ þ d0‘ Þ=2 ¼ 0, 2 ‘ ¼ 0; 1; 2; 1, ð40aÞ
2ðz¯ n On iÞln þ 1 On wn ðeio¯ n tn
ln3;4 ffi ðz¯ n iÞLn2 ,
for the case of wn 1 and z¯ n On 1.
2
½f2kðz¯ n i þ On Þ k2 o ¯ 2n þ 1 On wn =2g
wn ðeio¯ n tn þ 1Þ=2
yields
Ln1;2 ¼ 1 On
ð39aÞ
¯ n tn ¯ n tn iko iko wn C ðnÞ =2 DðnÞ k e k e
S ¼ jv~n j=ju~ n j.
where
2
½f2kðz¯ n i On Þ k2 o ¯ 2n þ 1 On wn =2g
k¼1
For the future usage, the dimensionless time-dependent displacements and the sensitivity are respectively defined as
½u~ n v~n T ¼ < p~ n ðtn Þ þ q~ n ðtn Þ T o , ð33Þ iðp~ n ðtn Þ þ q~ n ðtn ÞÞ
2
¯
2ðzn þ On iÞln þ 1 On wn ðeio¯ n tn þ 1Þ=2
wn ðeio¯ n tn þ 1Þ=2
(37)
Even though the exact solution of Eq. (31) is hard to obtain, Fourier-series-type solution can be obtained quite accurately by assuming the solution in the form: 2 3 " # ¯ n tn iko 1 X C ðnÞ p~ n k e 4 5, (38) ¼ ðnÞ iko ¯ n tn q~ n D e k k¼1
1 X
where
ln1;2 ffi ðz¯ n iÞLn1 ;
Note that the real parts of the Eq. (36) become positive at On ¼ 1 which indicates instability and the imaginary parts represent that the resonance occurs when the excitation frequency closely matches with either natural frequency, say,
¼0
þ 1Þ=2
(35)
Observing Eqs. (40a) and (40b), it is not difficult to prove the necessary convergence of the coefficients for large ‘. Finally, Eqs. (40a) and (40b) can yield 2ð2‘max þ 1Þ equations with 2ð2‘max þ 1Þ unknown coefficients with ðnÞ sufficient accuracy of coefficients C ðnÞ for ‘ and D‘ ‘ ¼ ‘max . Due to the harmonic components, the waves are no longer pure tones. Hence, to calculate the sensitivity, the amplitudes of both directions will be defined as
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
the components of dimensionless generalized coordinates, p~ n and q~ n , and of dimensionless apparent excitation frequency o ¯ n.
3. Discussion of results Eq. (36) shows that there are two distinct natural frequencies for the undamped case i.e.,1 On . With increasing dimensionless rotation rate On the lower natural frequency ðLn1 Þ decreases, while the upper natural frequency ðLn2 Þ increases linearly in both cases. Due to this rotation-dependent nature of the eigenvalues, it is not possible to drive this system at its natural frequencies even though the resonant frequencies are agreed to be the best driving frequencies from the energy and sensitivity point of view. The best alternative may be on the natural frequency
17
of the system at rest since this is a fixed frequency close to the actual natural frequencies known in advance. When a resonance occurs, the behavior of the vibrating system is mostly governed by the resonant denominator, and hence, the damping ratio of the resonant frequency. Under this circumstance, the Q factor [24] can be defined as 2Q ¼ 1=z¯ n and the Q factor plays a very important role in the sensitivity since the peak resonant response of the forced vibration is mainly governed by the Q factor in this damping dominant region. To investigate the influence of the damping effect on the forced response in detail, a computer simulation was performed first for small parametric excitation ðwn ¼ 104 Þ and two cases of Q factors— either Q ¼ 100 or 1000. Fig. 3(a) represents the dimensionless forced response of the system measured in the rotating coordinate system in the actuating direction and Fig. 3(b) in the sensing
Fig. 3. Steady-state sensor displacement response of the beam angular-rate gyroscope ðwn ¼ 104 Þ with Q ¼ 100 (a) for actuating direction, (b) for sensing direction, with Q ¼ 1000 (c) for actuating direction, (d) for sensing direction.
Fig. 4. Comparisons of sensitivities of the beam angular-rate gyroscope ðwn ¼ 104 Þ: (a) Q ¼ 100 and (b) Q ¼ 1000.
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
18
direction with Q of 100; while, Figs. 3 (c) and (d) shows the same relationships with Q factor of 1000. Note that for both directions in the second case, there are two distinct resonant peaks outside the very near range of zero angular frequency. On the other hand, when the Q factor is 100, the range of the single resonant peak becomes much broader although the responses for both directions become much smaller. This figure obviously illustrates the pros and cons of using the materials having low-quality factors. The sensitivity plot, which is the figure of the amplitude ratio of the sensing directional displacement vs. the actuating directional displacement under operational condition of small parametric excitation ðwn 1Þ, clearly shows the linear relationship against the frequency ratio, which can be seen in Figs. 4(a) and (b) for Q ¼ 100 and 1000, respectively, and in Fig. 5 for o ¯ n ¼ 1 and several different Q factors. Without any external control means, the resonant peak in which the system is stationary may govern the strength of the exciting voltage since the allowed maximum amplitude may also be restrained by this
20 Q=20 Q=40 Q=60 Q=80 Q=100
18 16 14
S×100
12 10 8 6 4 2 0 0
0.2
0.4
0.6
0.8
1
Ωn×1000 Fig. 5. Comparisons of percentage snsitivities with different Q factors ðo ¯ n ¼ 1; wn ¼ 104 Þ.
response. In this respect, high Q factor is preferable for narrow operation range while the low Q factor is for the wider operating range. Hence, the range of the single resonant peak should be one of the strongest design factors to make use of the resonant characteristics as much as possible. It is well known [19] that the linear classical Euler–Bernoulli beam theory can be applicable within the framework of the configuration considered in this work; the deflection and the slope of the beam are small and the wave length is sufficiently large compared to the beam thickness. Here the main measure is the size of the beam length compared to the other dimensions of the beam. Concerning the electrostatic excitation, the main measure is the size of the gap distance, between the beam and the substrate, which should be kept small to apply this forcing means effectively [14]. Even in the range of the gap distance that the linearized forcing function is quite valid, the characteristics of parametric excitation may cause undesirable behavior for the application of the system under consideration to be an angular-rate sensor. Thus it is of importance to investigate the effects of the gap distance on the behavior of this vibratory beam system. Fig. 6 shows the dimensionless orbital shapes of the system for fixed Q factors and rotation rate observed in a rotating coordinate system when the parametric excitation is relatively small ðwn ¼ 104 Þ. Here, the vertical axis stands for the dimensionless exciting frequency, while the horizontal plane includes the dimensionless displacements of the beam along the x- and y-axes. The trend of the smooth change of the orbital shape for the lower Q factor and the sharp change of its shape for the higher Q factor simply duplicates the explanation previously made. However, a distinct tilt of the orbital shape for larger values of wn can be observed in Fig. 7. Here the parametric excitation is relatively large ðwn ¼ 102 Þ compared to the previous case. This figure shows that the mean plane of vibration deviates severely from its static equilibrium position and strongly implies that the system would eventually be unstable if it exceeds a certain limit. The stability is generally handled in the parametric excitation problem; however, this will not be performed here because the system cannot be excited
Fig. 6. Orbital plots of the beam angular-rate gyroscope ðwn ¼ 104 ;
On ¼ 0:01Þ: (a) Q ¼ 100 and (b) Q ¼ 1000.
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
Fig. 7. Orbital plots of the beam angular-rate gyroscope ðwn ¼ 102 ;
only parametrically without any isolated external forcing function. Note that the linear theory adopted in this work does not support its validity if the parametric excitation exceeds a certain limit because the maximum deviation of this parametric excitation induced by the truncation of the second order is jdevjmax ¼ j3u=2umax j and mathematically proper linearization of the forcing function was based on the assumption of small higher-order terms, i.e., jdevjmax 1. Since the system equation is still linear associated with the displacement for small deflection, the response of the fundamental frequency can be used to explain its dynamic characteristics, and clearly exhibits its linear relationship with the rotation rate (see Figs. 4 and 5). 4. Conclusions Problems related with the vibratory gyroscope have been formulated and solved analytically for a special case. The forced response of the rotating beam was analyzed. The effect of the Q factor on the system time response and on the sensitivity was closely investigated for two cases of the Q factors. The system with relatively smaller Q factor yields smaller sensitivity while simultaneously giving a broader range in the response plateau before branching into two distinct resonant peaks, resulting in a broader measuring range of rotation rates. The additional parametric excitation due to the deflection-dependent nature of the electrostatic excitation causes the behavior to be more complex. However, within a certain small range of parametric excitation, the linear sensitivity curve could be obtained, operationally proving the applicability of the rotating beam as an angular-rate sensor. Appendix A. Modal analysis of the non-self-adjoint system The non-symmetry of the Coriolis matrix requires the implementation of the non-self-adjoint theory. The inner product of two complex state vectors w1,w2 defined as hw1 ; w2 i ¼ hu_ 1 ; u_ 2 i þ h_v1 ; v_2 i þ hu1 ; u2 i þ hv1 ; v2 i,
(A.1)
19
On ¼ 0:01Þ: (a) Q ¼ 100 and (b) Q ¼ 1000.
will be used with the following biorthogonal pair [16,17] lm Mnm ðzÞ ¼ Lnm ðzÞ; m ¼ 1; 2; . . . , l¯ n M n n ðzÞ ¼ L n n ðzÞ; n ¼ 1; 2; . . . ,
ðA:2Þ
where M and L are the adjoint operators of M and L, nm and n n are the eigenvectors of the principal and its adjoint system, respectively, lm is the mth eigenvalue, and the over bar operator l¯ n denotes the complex conjugate of ln . Here the inner product of two vectors a and b was defined as n Z X ha; bi ¼ b¯ i ai dG, (A.3) i¼1
G
where G is the domain in which the vectors a ¼ ½a1 ; a2 ; . . . ; an T ; b ¼ ½b1 ; b2 ; . . . ; bn T are defined. Note that while the differential matrix operator L is self-adjoint with the same boundary conditions as the principle system, the mass-matrix operator M is non-self-adjoint due to the skew-symmetric nature of G, that is to say, M ¼ MT ;
L ¼ L.
(A.4)
Taking the inner product of Eq. (A.2) with the various eigenfunctions, and using Eq. (5), gives lm Mnm ; n n ¼ Lnm ; n n , l¯ n nm ; M n n ¼ nm ; L n n , ðA:5Þ which with subtraction yields ðlm ln Þ Mnm ; n n ¼ 0.
(A.6)
For distinct eigenvalues lmaln Eq. (A.6) represents the biorthogonality condition Mnm ; n n ¼ 0; man, which with proper normalization, leads to Mn ; n ¼ dmn , m n Lnm ; nn ¼ lm dmn , ðA:7Þ where dmn is the Kronecker delta function. To obtain the relations between the two systems, let the eigenfunctions be nm ¼ ½lm mm ; lm nm ; mm ; nm T , n ¼ ½l¯ n m ; l¯ n n ; m ; n T . n
n
n
n
n
ðA:8Þ
ARTICLE IN PRESS J. Seok, H.A. Scarton / International Journal of Mechanical Sciences 48 (2006) 11–20
20
Substituting Eq. (A.8) into Eq. (A.2) with the use of Eq. (7) gives " #" # lm 2O mm lm 2O lm nm " 1 0 #" m # 4 m 1 q 2 ¼ 4 4ðÞþO , ðA:9Þ a qz nm 0 1 " ln
ln
2O
#"
n¯ n
#
m¯ n 2O ln " 1 1 q4 2 ¼ 4 4ðÞþO a qz 0
0 1
#"
n¯ n m¯ n
# .
ðA:10Þ
Comparing Eq. (A.9) with (A.10) implies the linear relationship between the two systems ½m¯ n ; n¯ n T ¼ K¯ n ½nn ; mn T ,
(A.11)
where Kn is a constant determined by using the biorthogonality conditions with the result 1 i Kn ¼ h R l Rl 2 2 2 O 0 ðmn nn Þdz þ 2ln 0 mn nn dz or ln i. Kn ¼ h Rl Rl 2 2 2 ðln þ O Þ 0 mn nn dz a14 0 m00n n00n dz
ðA:12Þ
References [1] Foucault L. Travaux scientifiques de Foucault, Paris, 1878. [2] Meirovitch L, Silverberg LM. Control of non-self-adjoint distributedparameter systems. Journal of Optimization Theory and Application 1985;47:77–90. [3] Bauer HF. Vibration of a rotating uniform beam, Part I: orientation in the axis of rotation. Journal of Sound and Vibration 1980;72(2):177–89. [4] Cai G-P, Hong J-Z, Yang SX. Model study and active control of a rotating flexible cantilever beam. International Journal of Mechanical Sciences 2004;46(6):871–89. [5] Lee CW, Katz R, Ulsoy AG, Scott RA. Modal analysis of a distributed parameter rotating shaft. Journal of Sound and Vibration 1988;122(1):119–30.
[6] Huang Y-M, Lee C-Y. Dynamics of a rotating Rayleigh beam subject to a repetitively traveling force. International Journal of Mechanical Sciences 1998;40(8):779–92. [7] Gates WD. Vibrating angular rate sensor may threaten the gyroscope. Electronics 1968;41(10):130–4. [8] Ulitko IA. Mathematical theory of the fork-type wave gyroscope. In: Proceedings of the IEEE international frequency control symposium 1995:786–93. [9] So¨derkvist J. Design of a solid-state gyroscope sensor made of quartz. Sensors and Actuators 1990;A21–A23:293–6. [10] Fang HY, Yang JS. Vibration analysis of a rotating elastic beam with piezoelectric films as an angular rate sensor. In: Proceedings of the IEEE international frequency control symposium and PDA exhibition 2001:507–13. [11] Jongwon Seok, Tiersten HF, Scarton HA. An analysis of a vibratory angular-rate gyroscope using polarized piezoceramic bimorph plates, Part 1: derivation of variational equations in the absence of angular velocity. Journal of Sound and Vibration 2005;280(1–2):263–87. [12] Jongwon Seok, Tiersten HF, Scarton HA. An analysis of a vibratory angular-rate gyroscope using polarized piezoceramic bimorph plates, Part 2: solution procedure for the gyroscope with superposed angular velocity. Journal of Sound and Vibration 2005;280(1–2):289–310. [13] So¨derkvist J. Micromachined gyroscopes. Sensors and Actuators 1994;43:65–71. [14] Branebjerg J, Gravesen P. A new electrostatic actuator providing improved stroke length and force. In: Proceedings of IEEE micro electro mechanical systems 1992:6–11. [15] Wang PKC. Feedback control of vibrations in a micromachined cantilever beam with electrostatic actuators. Journal of Sound and Vibration 1998;213(3):537–50. [16] Albers WF, Scarton HA. Solution of the end problem of a liquidfilled cylindrical acoustic waveguide using a biorthogonality principle. American Society of Mechanical Engineers Transactions, Journal of Vibration and Acoustics 1992;114:425–31. [17] Ginsberg JH. Mechanical and structural vibration: theory and application. New York: Wiley; 2001. [18] Bolotin VV. The dynamic stability of elastic systems. San Francisco: Holden-Day Inc.; 1964. [19] Timoshenko S, Young DH. Vibration problems in engineering. New York: D. Van Nostrand Company; 1955. [20] Tiersten HF. Linear piezoelectric plate vibrations. New York: Plenum Press; 1969. [21] Blevins RD. Formulas for natural frequency and mode shape. New York: Van Nostrand Reinhold; 1979. [22] Banks HT, Wang Y, Inman DJ. Bending and shear damping in beams: frequency domain estimation techniques. Journal of Vibration and Acoustics 1994;116:188–97. [23] Meirovitch L. Methods of Analytical Dynamics. New York: McGraw-Hill Book Company; 1970. [24] Thomson WT, Dahleh MD. Theory of vibration with applications. Englewood Cliffs, NJ: Prentice-Hall Inc.; 1998.