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0Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 2, pp. 231-257. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd240603
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70K28, 70K30, 70K40, 70K50, 74H10, 74H60
Mass Sensing by Symmetry Breaking and Mode Localization in a System of Parametrically Excited
Microbeam Resonators
N. V. Piskun, A.V. Lukin, I.A.Popov, L. V. Shtukin
In this work, we study the nonlinear dynamics of a mode-localized mass detector. A system of equations is obtained for two weakly coupled beam resonators with an alternating electric current flowing through them and taking into account the point mass on one of the resonators. The one-dimensional problem of thermal conductivity is solved, and a steady-state harmonic temperature distribution in the volume of the resonators is obtained. Using the method of multiple scales, a system of equations in slow variables is obtained, on the basis of which instability zones of parametric resonance, amplitude-frequency characteristics, as well as zones of attraction of various branches, are found. It is shown that in a completely symmetrical system (without a deposited particle), the effect of branching of the antiphase branch of the frequency response is observed, which leads to the existence of an oscillation regime with different amplitudes in a certain frequency range. In the presence of a deposited particle, this effect is enhanced, and the branching point and the ratio of the amplitudes of oscillations of the resonators depend on the mass of the deposited particle.
Keywords: nano- and microsystems, nano- and microelectromechanical systems (NEMS/MEMS), sensors of masses of deposited particles, gas detectors, gas concentration sensors, nonlinear dynamics, coupled-field problems, modal localization, weakly coupled systems
Received January 11, 2024 Accepted May 20, 2024
The research is partially funded by the Ministry of Science and Higher Education of the Russian Federation as part of the World-class Research Center program: Advanced Digital Technologies (contract No. 07515-2022-311 dated 20.04.2022).
Nadezhda V. Piskun mozhgova_nv@spbstu.ru Alexey V. Lukin lukin_av@spbstu.ru Ivan A. Popov popov_ia@spbstu.ru Lev V. Shtukin shtukin_lv@spbstu.ru
Peter the Great St. Petersburg Polytechnic University ul. Polytechnicheskaya 29, Saint Petersburg, 195251 Russia
1. Introduction
Microelectromechanical systems (MEMS) sensors developed over the past decades have demonstrated high output performance and at the same time low power consumption along with high reliability. Due to their small size and high sensitivity to changes in various physical quantities, MEMS sensors have a very wide range of potential applications, including inertial sensors (accelerometers [1], gyroscopes [2]), pressure sensors [3], gravimeters [4], flow velocity and flow velocity gradient sensors [5], displacement sensors, as well as mass sensors [6] and gas detectors [7].
Particle mass detectors and gas detectors are finding use in a relatively new field of MEMS applications, namely, biomedical systems [8]. Specific applications include sensors for diagnosing diseases by measuring the mass of small biological elements such as molecules, viruses and bacteria [9, 10], as well as sensors for monitoring the composition of water and air (mainly measuring the concentration of a particular gas) [11] to monitor air quality and detect potential exhaust leaks in factories.
Recently, various architectures and mechanisms for measuring ultra-low masses through mechanical vibrations have been explored. The most common and widely studied is the resonant mass detector, where adding mass to a single sensitive element changes the value of its natural frequency of oscillation [12-15]. The authors of [16] obtained in their work a simple closed-form expression that allows one to quickly calculate the position and mass of individual particles placed on a micro- or nanostring by measuring the shifts in the resonant frequency of the first two bending modes of vibration. Continuous measurement of mass to calculate the evaporation rate of microdroplets was implemented by the authors of the work [17] also based on changes in the natural frequency of oscillations. A number of works are devoted to the analytical and experimental study of mass detectors with a sensitive element in the form of a clamped-clamped or cantilever beam, in which oscillations are excited by electrostatic forces [12, 13, 18-20]. The use of the internal resonance effect to increase the sensitivity of a mass detector based on a shift in the natural frequency of oscillations is discussed in [14, 15, 21-23]. Thus, in [21] internal resonance between the fundamental mode and higher-order modes is observed in a piezoelectric polyvinylidene fluoride membrane as a resonant mass sensor. Higher-order modes draw energy from the fundamental mode and vibrate at a frequency that is an integer number of times the frequency of the fundamental mode. A more complex sensing element geometry using the same effect is shown in [22]. And the authors of [24] propose to use the frequency shift of the resonator to identify defects of complex shape that arose in the resonator, for example, during the manufacturing process.
A separate class of devices for measuring mass should be noted — microchannel resonators, in which the resonance characteristics of a coupled resonator-liquid-particle system are used for detection. Thus, in [25] the influence of several important factors, including particle density, fluid viscosity and vibration frequency, is studied and analyzed. The results show that the Reynolds number and the liquid density coefficient are two important parameters that affect the measurement accuracy and the occurrence of errors. The work [26] of the same group of authors shows that the displacement of a particle inside a microchannel resonator consists of two components: one is in phase with the acceleration of the resonator and is responsible for the effect of attached mass, and the other is antiphase and leads to a slight attenuation of oscillations. The authors of [27] proposed a mass detector layout capable of independently filtering out particles of the required size from the flow and then measuring their mass also by measuring the shift of the natural frequency of the resonator.
Along with classical resonant MEMS sensors, an intensively developing area is the study of the modal interaction between different degrees of freedom of nonlinear and weakly coupled oscillatory systems to apply the features of this interaction to high-precision measurement problems, including mass detection problems [28-30]. For example, in [31, 32] the authors propose to use the phenomenon of modal localization, which occurs in a system of two cantilever resonators in the presence of a weak mechanical connection between them, to detect mass. The same system, but under the influence of a viscous medium, is considered by the authors of [33], emphasizing that sensors operating in a viscous medium contribute to the development of new drugs and the identification of biomarkers of diseases. A gas detector for simultaneous detection of the presence and determination of the concentration of two gases, namely, carbon dioxide (CO2) and methane (CH4), was proposed in the work [10] also based on two weakly coupled cantilever resonators. The authors use machine learning methods in their work to predict the characteristics of gases based on the dynamic characteristics of coupled resonators.
A system of two clamped microresonators coupled by electrostatic forces is being studied by several groups of researchers [34-39]. In [34, 35], the output parameter of the sensor is the shift in the amplitude ratio of the resonators due to the effect of modal localization, and it is shown that the sensitivity of movements along the third natural mode of oscillation is higher than through the first. The work [37] considers resonators of different thicknesses for the presence of asymmetry in the system, which generates interaction between the second- and third-order eigenforms. Modal localization on the above modes results in a 20-fold increase in sensor sensitivity compared to first-order mode-localized mass sensors. The authors of [36] study the phenomena of loss of symmetry during the deposition of particles using the apparatus of nonlinear normal modes and propose to measure their mass based on tracking the transition between various localized modes, where the ratio of the oscillation amplitudes of two resonators changes sharply.
In addition, the works [38, 39] show that with parametric excitation of oscillations, the sensitivity of mass detectors can be increased by 1-2 orders of magnitude compared to the forced oscillation mode due to the operation of the sensor based on bifurcation tracking. The authors show that the sensitivity near the fork bifurcation point can be increased by three orders of magnitude, and the sensor exhibits excellent noise immunity over a certain damping range, which opens the possibility of avoiding the problem of insufficient reliability for bifurcation-based mass sensors. The authors of [40] also note that when operating in viscous media, positive velocity feedback is often used to compensate for the damping effect on the resonators, which introduces relatively high noise as the device size decreases. Therefore, the authors propose to use parametric resonance to detect frequency shift without feedback control.
Moving along the path of increasing the sensitivity of mass detectors, researchers came to study arrays of weakly coupled resonators, in which the modal localization also makes it possible to measure the ultra-low mass of deposited particles [41-45]. Thus, the authors of [46] develop a general model of a mass sensor consisting of N mechanically weakly coupled cantilever microresonators, the oscillations in which are excited by the electrostatic method. The work shows that the sensor can operate both in threshold mode based on bifurcation, and in continuous mode based on amplitude measurement.
Less classical architectures and mechanisms for excitation of oscillations and signal pickup of mass detectors also occur in the literature. Thus, in [47] a mass sensor with modal localization based on self-oscillating weakly coupled resonators with a temperature mechanism for excitation of oscillations and a piezoresistive signal pickup mechanism is described and experimentally tested. In [48], for simultaneous measurement of relative humidity and helium concentration in the medium, various modes of oscillation of a resonator are used, consisting of a clamped arched
beam connected to a bending beam, and a T-shaped moving mass. The authors believe that using different vibration modes of a carefully designed MEMS device is a promising approach and could open new horizons for the development of smart and highly selective gas sensors. The authors of [49] also use the multimode excitation method, but to amplify the amplitude signal of sensors with classical beam microresonators. The authors of [50] explore the rich potential of multimodal movements of asymmetrical arched microbeams with electrostatic excitation of oscillations to develop inertial gas sensors with higher sensitivity and signal-to-noise ratio. The occurrence of modal localization between the first symmetrical and first antisymmetrical modes of vibration of a beam with initial curvature allows the use of such a structure as a mass sensor based on the modal relationship between the capacitances belonging to the symmetrical and antisymmetrical modes.
Mathematical modeling and design of mechanical structures of the class under consideration requires the use of very complex geometrically and physically nonlinear models of elastic deformation of continuum systems. An extensive literature is devoted to this group of issues (see, for example, [51-54]).
The purpose of this work is to study the nonlinear dynamics of a micromechanical mass detector, the operating principle of which is based on the effect of modal localization between two mechanically weakly coupled beam resonators under parametric electrothermal excitation of oscillations.
2. Mathematical model
We consider a system of two microbeam resonators clamped on both sides (Fig. 1), mechanically coupled through a thin elastic bridge, the flexural rigidity of which is much lower than the flexural rigidity of the resonators. A particle of mass mp is deposited in the center of one of the beams, and the resonators have different thicknesses.
Fig. 1. Model of weakly coupled microresonators
The geometric parameters of the resonators are selected in such a way that the resonators
have equal lengths L1 = L2 = L and widths b1 = b2 = b, and to take into account the difference
h
in thicknesses, the parameter n% = — 1. It is assumed that h12 <C L and that the deflections
of the resonators w12 during oscillations are small but finite, so the model takes into account the midplane stretching. The resonators are made of an isotropic elastic material with density p and Young's modulus E. Parametric bending vibrations are excited by heating microbeam elements due to the flow of electric current J of harmonic type (2.1). The mechanism for excitation of oscillations is illustrated in Fig. 2. Due to the boundary conditions, a nonuniform temperature distribution along the length of the resonators is formed:
j=Ydç
+ VAC cos ut
Pe L
(2.1)
where VDC is the amplitude of the constant voltage component, VAC and u are the amplitude and frequency of the variable voltage component, pe is the resistivity beam material, and t is time.
Front view
Fig. 2. Illustration of the vibration excitation mechanism
To derive the equations of oscillations of such a system, Hamilton's principle is used:
pS r
d 2
w.
l \ d2
w
dt 2
+ ß-
dWi
dt
+ Eli
+ k{w1 - w2)6 [x - - ) = 0,
L
PS2
d2
w2 ^dW 2 +M
dt2
dt
2 + EI*™*
dx4
dx V dx
d 4 W1 IB4
dx V 1 dx
+
-n{w1 -w2)ô [x-^
= 0,
(2.2)
where x is the coordinate along the axis of the resonators, ^ is the damping coefficient in the medium around the resonators, 2 is the axial force acting on the resonators, n = 3^jf- is the coefficient of static bending stiffness of the bridge, E', I' and l' are the moment of inertia of the cross-section, the length of the bridge and Young's modulus of the bridge material, respectively, I1 2 is the moment of inertia of the cross section of the resonators, S1 2 is the cross-sectional area of the resonators, and S() is the Dirac delta function.
The mechanical boundary conditions are
w
= 0
dw
1,2
l>2lx=0 '
= 0,
(2.3)
x=0
w
1,2\x=L
= 0,
dw
1,2
dx
= 0.
x=L
The axial force NV1 2 is caused by the Joule effect due to heating of the resonators N,
as well as the effect of midplane stretching TV
midstr(1,2) ' N1,2 = NVth(1,2) + Nmidstr(1,2),
th(1,2) '
(2.4)
r
N
th(1,2)
ESh.
17
a(Tu (x) - To) dx,
N •
ES
L
(2.5)
1,2
midstr (1,2) 2L J \ Qx o
dw
i,2
where a is the coefficient of linear thermal expansion of the resonator material, T1 2 (x) is the temperature distribution along the axis of the resonators, and T0 is the initial temperature of resonators. The temperature distribution along the axis of the resonators is calculated by solving the one-dimensional problem of thermal conductivity (Section 3).
By switching to dimensionless parameters
w
1,2
«'1,2 x _ x t_ t
hi
V
t.
tscale
, L scale
^th —
N
scale
V
V
i scale 2
EL
scale
EI1 k
IpSjlA EI, '
K
K
, Kscale
scale
Eh
L3
№ scale ^scale 4
mp
, mscale
= pS1L = m
beam
№ scale L m scale
the equations for bending vibrations of resonators can be obtained in dimensionless form:
d4w1 d2w1 _ ( 1 \ d2w1 dw-, (' „ 1 ( dw1 \2 , \ d2w1 ^—r + -^-rr- + mnrmd ( x--—^ + a—1 - \ Nth + 6 / —- dx —~r+
dx4 dt2 non V 2 J dt2 rJt th \ dx dx2
+ k,{w1 - w2)6 ( x - - ) = 0,
dx
d2w-2 dx2
- K,{wl - w2)6 ( x - - ) = 0,
with boundary conditions
W1,2 lx=0 = 0,
dw
1,2
dx
dwU2
(2.6) (2.7)
^2U = 0,
x=0
x=1
0.
The subject of the study is the parametric analysis of the nonlinear dynamics of the system (2.6)-(2.7). An important preliminary stage of this problem is to find the temperature distribution in the volume of microbeam resonators, resulting from the flow of a harmonically varying electric current.
L
2
0
2
0
1
2
0
3. Finding the steady-state harmonic temperature distribution in the resonators
The influence of the bridge on the thermal state of the resonators is absent, since under the assumed boundary conditions (thermal insulation on the outer surface of the beams) the temperature fields do not depend on the shape of the beams. Thus, we consider the unsteady one-dimensional heat conduction equation along the beam axis x G [0, L], which has the form
d2T 1 dT 1 _ ^ . .
= --Q(x,t), (3.1)
dx2 a dt k
where a = is the thermal diffusivit.y, k is the thermal conductivity coefficient, p is density, c is the specific heat capacity resonator material, x is the coordinate along the resonator axis, T(x) is a function of temperature, T0 is the initial temperature of the resonators (the same for both beams), t is time, and Q(x, t) is the volumetric heat release in the resonator due to Joule heating.
Thermal boundary conditions are shown in Fig. 2. The volumetric heat generation due to the flow of current can be described as follows:
____ —s t2 t Vdc + VAC cos ut
Q(x,t) = J2pe, J = -^f-, (3.2)
peL
where J is the current strength, pe is the resistivity of the resonator material, VDC is the amplitude of the constant voltage component, VAC and u are the amplitude and frequency of the alternating voltage component, and L is the length of the resonators. Thus, the volumetric heat release can be written as
? _ (VDC + VAC cos ujT)2 _ V£c + \V\C '2VdcVac cos ujT+ \Vlccos2ujt
l! — „ r2 ~ „ r2 +
or
PeL2 PeL2 PeL2
Q(x, i) = Qs(x)+ Qd(x, t ), (3.3)
where Qs(x) and Qd(x, t ) are the static and dynamic components of volumetric heat release, respectively:
Qs(x) = v°c+Avlc, Qd(%> t) = 2VDCVAcCOSUJt + lVAccos2u)t^ (3 4)
pel peL
The static component Ts of the steady-state temperature distribution in the resonator is determined by the Eq. (3.5)
d2Ts(x) 1
~8— = -TM$), (3-5)
dx2 k
with boundary conditions
Ts\x=0 = T0, Ts\x=L = T0- (3-6)
The solution to Eq. (3.5) is
n V'2 4- il/2
Ts(x) = -g (X2 - Lx) + T0 = - D^p2L2AC (x2 ~ Lx) + T0. (3.7)
The steady harmonic temperature distribution in the beam is determined by the harmonic solution of the equation
a%(i, t, i «■„<?, % = - u)
dx2 a dt k
with boundary conditions
Td\x=0 = T0, Td\x=L = T0 ■ (3-9)
The steady-state harmonic temperature distribution Td in the resonator can be found in the frequency domain; for this, the problem is written in complex form as the sum of two harmonics Td1 and Td2 with frequencies w and 2w, respectively,
Td(x, w) = Tdi(x, w) + Td2(x, 2w), (3.10)
dT i, 2 2 1
~~ 2^1,2 — —7"Q(il,2) (3.H)
where
dx2 d1'2 k
peL2 '
The physical temperature distribution is represented as
Td(x, t, w) = Re{Td1 exp(iwi)} + Re{Td2exp(2iwi)} (3.12)
or
Td(x, t,w) = T^3™ sin(wX) + T^,™3 cos(wX) + T™™ sin(2wX) + Tamp,cos cos(2wX),
where the amplitudes of the sine and cosine harmonics of the harmonic temperature distribution are determined by the expressions
rramp,sin T cm \ rpamp,cos X
T d1,2 = " ím{Id1,2}, Td1,2 = Re{T d1,2} ■
The solution to the harmonic problem (3.10)-(3.11) using the Green's function can be represented as follows [55]:
L
Tdiaix, = f J Gxiiix\x'] u))Qdl 2{x', to) dx'. (3.13)
0
The Green's function GX 11(x\x'; w) under given boundary conditions is
g-a(2L—lx-x '|) _ g—a(2L—x—x') g—aQx—x '|) _ g—a(x+x')
Gxll{x\x'- u>) = 2aa(l-e~2^) + 2aa(l - e~2^) "
Thus, by solving the thermal problem, static and steady-state harmonic temperature distributions along the resonator axis were found, from which, according to Eq. (2.5), the axial force that modulates the stiffness of the system can be calculated, thereby exciting parametric bending vibrations.
4. Mechanical problem
To study the nonlinear dynamics of the system (2.6)-(2.7), taking into account the found expressions for the axial temperature force, the Galerkin method [56-58] is used with expansion in terms of the natural modes of bending vibrations of an unloaded resonator. In the expansion, only the first term corresponding to the first form of bending vibrations is retained, since the movement of the resonators is assumed precisely along the lower eigenmodes:
wi(x) = 0oi(x) + Qi(t)^i(x), w2(x) = 0O2(x) + q2(t)^i(x), (4.1)
where $0i,2(x) = 0, since the boundary conditions of the problem are homogeneous.
The expansion (4.1) is carried out in terms of orthogonal eigenfunctions of the system satisfying the following equation:
(4-2)
where w2nan i is the square of the dimensionless first natural frequency of bending vibrations.
Taking into account the series expansion in terms of eigenforms and boundary conditions, the system (2.6) takes the form
Ql + mnoa4>l Q) Ql + m + Ql + K0? (0 + Nth j(</4)2 dx J -
- «9201 (0+6 dl I J dx\ = 0,
' , (4.3)
(1 + n%)q2 + ßq2 + q2 ^(1 + n%fuj'2ond + n<j>\ (0 + Nth J(c^)2 tfej -
" (0 + 6(1 + n%)<?23 J(</>'i)2 dx = 0,
t
Using (2.5), (3.7), (3.10) and (3.12), the axial force Nth can be written as follows:
Nth = " N + N%? sin(nt) + NCOs cos(nt) + N%r sin(2^t) + N%? cos(2^t)), (4.4)
where
6aL2 {V2C + \VIC) h\kpe
JVS — 7.2 7, „ ) 7 — w 1 scale!
L L
T\T.sin _ -^2CiL f amp,sin t\tcos _ ^Q-Zv f rpamp^cos
(¿1,2 — fo 2 / (il,2 ttX' (il,2 — jl2 / (il,2 ttX"
1 ° 1 °
After substituting Eq. (4.4) into Eq. (4.3) the system of equations of motion of resonators will take the form
Qi + ( \ ) Qi + Mi + Qi
u,.
i+ «</>? (0 " Naji^f dx-
- / (0l)2 dx [NSn sin(nt) + N%8 cos(nt) + sin(2nt) + № cos(2nt)]
(1 + n%)(?2 + M2 + Q2
(1 + n%)3wLu + «01 Q) - (1 + n%)Ns J(0i)2 dx-
-(1 + n%) (0l )2 dx [NZ? sin(nt) + Ncd? cos(nt) + NJ2n sin(2^t) + cos(2^t)]
- Q) + 6(1 + n%)<?23 |J(0i)2 ctej = 0.
Notations are introduced for the frequencies of the two equations:
1
= + «0? Q) - n8 J ((/>;)2 dx
(4.5)
(4.6)
q2 =(1 + n%)2wnora,i +
(1+ n%)r 1 V 2
0? ~NS ji^fdx
where we can express the frequency of the second equation in terms of the frequency of the first, taking into account the smallness of the parameter n%, expanding the corresponding terms into a Taylor series:
(l + n7f = l + 2n%+ni+...
(1+ n%)
Keeping the first 2 terms in the expansion, we obtain
l
2 3
= 1 - n% + n% - n% + ...
Q2 =
or
+ «0? (^j - Ns J((/>;)2 dx + 2n%ujf20nJ - nn%<j>\ =
0 J
= fi? + 2n%u)lonJ - K,n%(p\ (4.7)
Q2 = Q2 + AQ2, AQ2 = 2- ««%</>? Q) • (4-8)
/
i
0
i
0
i
i
0
2
i
i
K
Taking into account the introduced notations and dividing the second equation in the system (4.5) by (1 + n%), we obtain the final form of the system of equations that describes the nonlinear dynamics of bending vibrations of a system of coupled resonators:
f hi + Ciqi + C2<ii + qi (Qi - C4 [NT sin(nt) + NCOs cos(nt) + N¡2? sin(2nt)+ +NCOs cos(2nt)]) - C5q2 + C7q\ = 0, q2 + C3<?2 + q2 ("2 + AQ2 - C4 [NSC sin(nt) + N%s cos(nt) + N¡2n sin(2nt)+ +NCT cos(2nt)]) - C6qi + C7q3 = 0,
where the parameters are introduced as follows:
' 1
1
Cl — mnon<t> 1 ( 2 J ' C'2 — C3-
C4 =
y4 - J (0i )2 dx, °
%
C5 -
> =
1 + n
%
,
C7 - 6
2
1 )2 dx
(4.9)
1
2
5. Asymptotic analysis
The terms in Eq. (4.9) are scaled as follows:
f qi + e2Ciqi + e2C2(i + qi (Qi - e2C4 [NsdT sin(nt) + N^ cos(nt) + N¿2" sin(2nt) +
+NC2os cos(2nt)]) - e2Cq + C7q\ = 0, (5
CÎ2 + e2C3(Ï2 + q2 (Qi + e2AQ2 - e2C4 [N^ sin(nt) + N£s cos(nt) + NT sin(2nt) + +NC20s cos(2nt)]) - e2C6(i + C7q2 = 0,
where e is a book-keeping small parameter.
The expansion using the method of multiple scales [59] will be carried out up to the third order inclusive. We consider a three-term expansion in the form
( qi(t, e) = e(qi)i(TQ, Ti, T2) + e2(qxUT0, Ti, T2) + e3(qi):i(T0, Ti, T2), . 2) \ q2(t, e) = e(q2)i(T0, Ti, T2) + e2(q2U^, Ti, T2) + e^UU, Ti, T2),
where Tn = ent.
The time derivatives are defined as
d d
- = D0+eD1+e2D2, Dn = —.
Substituting the expansion (5.2) into Eq. (5.1) and balancing by powers of the small parameter gives the following expressions:
D°2(qi)i + Qj(qi)i -0 ^ (qi)i - Ai exp(iQiT°) + cc, D2(q2)i + n2i(q2)i -0 ^ (q2)i - A2exp(iQiT°) + cc,
where A12 = Al2(Tl, T2) = ^a,12 exp(if312), where a,12 and f312 are the slowly varying amplitude and phase of two resonators, and cc are complex conjugate terms:
D2(9i)2 + ^1(91)2 = -2Do D1(q1 )1; Dq2(92)2 + ^(9o)o = -2Do D1(q2 )1,
O № :
(5.5)
„ / 2\ I "UV11/2 1 -"1V11/2 U"1VH/1> ,N
O(£ ): { ^^ , „2,^ ^ .w^ (5.4)
< D02(q1 )3 + Q2(q1)3 = -2D0D1 (91)2 - 2D0D2(91)1 - D2(91 )1 + 6-5(92)1 -
- 67(q1)3 - C1D0(91)1 - Co Do (91 ^ + C,(qx \ [N¡1? sin(nt) + +Ncd°s cos(nt) + Nj2n sin(2nt) + Ncd°s cos(2nt)],
Dq(9O )3 + ^1(92)3 = -2D0D1 (qo)o - 2 Do Do (92)1 - D2(qo )1 + C6(q1)1 -
- C7(9o)1 - C3D0(92)1 - AQ2(9o)1 + C4(qo)1 [N™ sin(nt)+ +NCd°S cos(nt) + N¡0? sin(2nt) + NC2S cos(2^t)].
The work will consider the primary parametric resonance, therefore,
2 r? = 2Q1 + ea r? = + y, (5.6)
where a is the frequency detuning parameter. After carrying out the further procedure of the method of multiple scales, we obtain the following system of equations in slow variables (5.7):
' 1 ^ = -2^- 2C5a2 sin ^ - ^ - C^Ng1 cos Xl - C^NfJ sin Xl,
1 ^ = -2Q1C3a2 + 'IC.a, sin ^ - ^ - cos - C4a2^T sinX2,
'111 a ir,(,2,-„>^ Al , ,02,
(h 5 2 2 2
- 2C4aNd2S cos %1 + 2CaN%? sin %1,
= -a - 4C5a2cos ^ - ^ + 3C7a; - 4QiC1a1- (5 7)
^^ 2 2 ( )
4Q1a,2—= -<7 + 3C7a'2 + 4a2AQ2 - 4C60,! cos ^ -
ar 2 2
- 2C4aoNdOS cos xo + 2C4aoNJ2n sin xo, where the modified phase variables x12 are related to ^12 as follows:
X1,o = 2^1,2 - ar. (5.8)
6. Results
This section presents the results of a study of the dynamics of the system based on the analytical solutions obtained. Table 1 presents the geometric and physical-mechanical parameters of resonators made of silicon. The selected numerical values of the system parameters represent typical values for micromechanical sensors [10, 32, 33, 35-37].
6.1. Heat transfer problem
Figure 3 shows the dependence of the temperature in the center of the beam on time and the distribution of temperature along the axis of the beam, obtained analytically (Eqs. (3.7), (3.12), (3.13)) and numerically when solving the problem of unsteady heat conduction in the COMSOL Multiphysics.
Table 1. System parameter values
Name of quantity Designation, numerical value and units of measurement
Length of resonators L = 1000 ¿tni
Width of resonators b = 50 ¿tin
Thickness of resonator 1 hi = 5 ¿tin
Length of bridge between resonators Lsm = 500 Atm
Width of the bridge between the resonators bsm = 10 A«m
The thickness of the bridge between the resonators hsm = 5 A«m
Quality factor Q = 100
Young's modulus E = 169 GPa
Density p = 2330 kg/m3
Thermal conductivity coefficient k = 156 W/(m • Iv)
CLTE a. = 2.56 • 10~6 1/K
Specific heat c= 713 J/(kg • Iv)
Resistivity pe = 1.14 - 10~4 Om- m
Initial temperature of the resonators T0 = 0 °c
There is a complete coincidence of the results with the exception of the transition process in the temperature versus time dependence, which is caused by zero initial conditions in the numerical solution, while the analytical solution gives a steady-state distribution. This leads to the conclusion that the thermal fields are described correctly.
Next, the effect of current frequency on the type of temperature distribution along the axis of the beam was studied; the corresponding dependence is presented in Fig. 4, where the temperature scaled to the average along the length is plotted along the vertical axis: Tavg =
L
= ifTd(x)dx.
0
A strong influence of the current frequency on the type of temperature distribution is visible. In this work, we consider the case of the primary parametric resonance, when the current frequency is close to twice the first natural frequency of bending vibrations of the beam, which corresponds to the purple curve in Fig. 4. In this case, the dependence of the dynamic temperature component has local maxima near the beam edges.
6.2. Instability zones of parametric resonance
When analyzing parametric oscillations, the first step of the study is to search for zones of instability of the resulting oscillations. In this work, two methods were used to construct diagrams of the Ince-Strutt type: the classical approach of the Floquet theory, as well as using the Routh-Hurwitz stability criterion (see Appendix A). Figure 5 shows the instability zones of parametric resonance with the length of the bridge between the resonators equal to Lsm = = 500 ¡m, in the absence of thickness differences n% = 0 and deposited particles mp = 0.
Oscillation zones in a system of weakly coupled resonators are a combination of two zones typical of a single resonator. The left zone, indicated by a blue dotted line, corresponds to the zone of in-phase oscillations, and the right zone, indicated by a red dotted line, corresponds to
(a)
(b)
Fig. 3. Comparison of analytical and numerical solutions to the heat conduction problem at VDC = = VAC = 0.5 V, w = 200 rad/s: (a) time dependence of the temperature in the center of the beam; (b) temperature distribution along the axis of the beam
x, m xlO 3
Fig. 4. Dependence of the type of temperature distribution along the beam axis on the current frequency, — the first frequency of bending vibrations of the beam
antiphase oscillations. These zones have "spouts", which, with decreasing damping (increasing the quality factor), tend to the values of the natural frequencies of in-phase oscillations (without taking into account the work of the elastic bridge) and antiphase oscillations (taking into account the bridge). The zones have overlapping areas, in which the type of oscillations will depend on the initial conditions. It is also worth noting that the solutions for the instability zones of parametric resonance obtained by the two methods coincide.
Figure 6 shows the instability zones of parametric resonance for different lengths of the elastic bridge between the resonators.
The zone of in-phase oscillations does not depend on the length of the elastic bridge and does not change in the graphs of Fig. 5. The zone of antiphase oscillations shifts to the right as the length decreases and therefore increases the stiffness of the bridge. Thus, in this system it is possible to customize the type of instability zones of parametric resonance by changing the length of the elastic bridge.
The influence of the value of the resonator thickness difference parameter n% on the instability zones of parametric resonance was also analyzed, as shown in Fig. 7.
If there is a difference in the thickness of the resonators of the order of 2-5 %, the instability zones of parametric resonance are shifted to the right relative to the original system with the same thicknesses, and the displacement of the zone of antiphase oscillations is more significant compared to the zone of in-phase oscillations. Due to the displacement of the zone of antiphase oscillations, the range of voltage values at which the instability zones of parametric resonance do not intersect increases.
Lsm = 300 jam
Lsro = 1000 /mi
Fig. 6. Zone of the main parametric resonance at different values of elastic coupling between the resonators. The hatching is the Floquet theory, the dotted line is the Routh - Hurwitz criterion, VDC = 0,
mp = ° n% = 0
1.4 1.3
> 1.2
6
^ 1.1
riv, =0.02
n„ =0.05
0.9
\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ x
■ \ \
\ \
\ \
\ s
" \ j \Jy
1.7 1.8 1.9
2.1 2.2
2.1 2.2
Fig. 7. Zone of the main parametric resonance at different values of the resonator thickness difference parameter. The hatching is the Floquet theory, the dotted line is the Routh - Hurwitz criterion, VDC = 0,
mp = 0, Lsm = 500 fim
6.3. Frequency response of the system
The previously found instability zones of parametric resonance make it possible to construct resonance characteristics and perform a direct analysis of the nature of these oscillations in a geometrically nonlinear formulation (Eq. (2.6)). To construct the frequency responses of the system, methods of bifurcation theory are used, namely, the method of continuation by parameter, which implies the continuation of a stable equilibrium position when one of the active parameters of the system changes (in this case, the frequency detuning parameter a). Numerical continuation is implemented using the MATCONT [60]. Figure 8 shows the amplitude-frequency response of the system at VAC = 0.94 V, that is, until the moment of intersection of the two instability zones of parametric resonance.
1.5 r
2.04
1.5
— In-phase (stable)
■ - - ■ In-phase (unstable)
— Antiphase (stable)
■ - - ■ Antiphase (unstable)
— Antiphase after BP 1 (stable) ...........Antiphase after BP 1 (unstable)
Antiphase after BP 2 (stable)
........... Antiphase after BP 2 (unstable)
• Bifurcation points
0.5 -
0
1.9 1.92 1.94 1.96
1.98
2.02 2.04
Fig. 8. Frequency response of the system in the absence of intersection of instability zones of parametric resonance, VDC = 0, mp = 0, n% = 0
The in-phase branch of the frequency response (blue line) is a classical amplitude-frequency diagram under parametric excitation with a characteristic "hard" nonlinearity. The branch of antiphase oscillations (red line) has a feature in the form of a "fork" branching point at a frequency value close to twice the natural frequency of the unloaded resonator. After this point, one stable branch of the frequency response of each resonator is divided into two branches with different amplitudes (orange and purple lines), and if one of the resonators oscillates with the amplitude of the upper branch, then the other oscillates with the amplitude of the lower branch. Thus, in a perfectly symmetric system, in a certain frequency range, the symmetry breaking effect is observed — the emergence of a regime of oscillations with different vibration amplitudes of the two beam resonators. In this regime, modal localization occurs between the two resonators — energy is localized on one of them, which performs motions of larger amplitude. Small perturbations introduced into the system can be measured by the change in the amplitude ratio of the resonators.
In addition, Hopf bifurcations are observed on the unstable branches of the antiphase regime in Fig. 8 (red dotted line). In this case, they are not of interest from the point of view of system analysis, since they occur on unstable branches.
Figure 9 shows the amplitude-frequency response of the system at VAC = 0.96 V in the presence of intersection of the two instability zones of parametric resonance.
The intersection of instability zones of parametric resonance in Fig. 5 leads to the intersection of the in-phase and antiphase branches in the amplitude-frequency response of the system. As
2 1.5 1
0.5 0
0.6 -6.-S8
iiifei
1.974 1.975 1.976 1.977 \s78f
: :«/-':
„ 7/ i 11 ... / J — J — J — J... . \.. :. ■ i . ■ ■ ■ ■ i ■ ■ ... . ^ . . . .
1.9
1.95
2.05
2.1
2 1.5 1
0.5
0
1.9
In-phase (stable) uo
----In-phase (unstable)
Antiphase (stable)
----Antiphase (unstable) _ ^ It r
Antiphase after BP 1 (stable) .............Antiphase after BP 1 (unstable) Antiphase after BP 2 (stable) .............Antiphase after BP 2 (unstable) • Bifurcation points
^ yA
/ ( ?!
...... ..,....,....,....]...................
1.95
2.05
2.1
Fig. 9. Frequency response of the system in the presence of intersection of instability zones of parametric
resonance, VDC = 0, mp = 0, n% = 0
can be seen from Fig. 9, part of the stable branch of antiphase oscillations, which is located "inside" the branch of in-phase oscillations, is unstable.
The following presents the frequency response of the resonators in the presence of a disturbance on one of them in the form of a deposited particle of mass mp = 10 pg. Figure 10 shows the frequency response before intersection of the two instability zones of parametric resonance, VAC = 0.94 V, and Fig. 11, after intersection of the zones, VAC = 0.96 V.
In the presence of a disturbance, the antiphase branch of the frequency response of the system breaks up into two nonintersecting branches, that is, the system also possesses an oscillation mode with different amplitudes, and the amplitude ratio, as well as the transition point from the mode with the same amplitudes to the mode with different ones, depend on the magnitude of the disturbance, in our case on the mass of the deposited particle. From this we can conclude that it is possible to measure the mass of deposited particles with a detector, the sensitive element of which will be a system of weakly coupled resonators.
The influence of the resonator thickness difference parameter on the type of frequency response was also studied; the results of the study are shown in Fig. 12.
From Fig. 12 it is clear that a difference in the thickness of the resonators of the order of 2 % significantly changes the frequency response of the system. First of all, both branches shift to the right in frequency; in addition, the branch of antiphase oscillations in one of the resonators acquires a strong slope to the right, while the same branch in the frequency response of the second resonator remains unchanged. With an increase in the thickness difference parameter of
1.5 r
2.04
a
a.
— In-phase (stable) ^ g---- In-phase (unstable)
— Antiphase 1 (stable) ---- Antiphase 1 (unstable)
— Antiphase 2 (stable)
........Antiphase 2 (unstable)
• Bifurcation points
0.5 -
1.9
___ h^S.....•
1.92
1.94
1.96
1.98
2.02
2.04
Fig. 10. Frequency response of the system in the presence of a deposited particle and the absence of intersection of instability zones of parametric resonance VDC = 0, mp = 10, n% = 0
2.15
— In-phase (stable)
— - ■ In-phase (unstable)
— Antiphase 1 (stable)
■ - ■ Antiphase 1 (unstable)
— Antiphase 2 (stable) ...... Antiphase 2 (unstable)
Bifurcation points
2.15
Fig. 11. Frequency response of the system in the presence of a deposited particle and the intersection of instability zones of parametric resonance VDC = 0, mp = 10 pg, n% = 0
n% = 0.02
n„ =0.05
1.5
a i
a.
«g 0.5 0
1.92 1.94 1.96 1.98 2 2.02 2.04 2.06 2.08
u "o
-In-phase (stable)
-Antiphase (stable)
----Antiphase (unstable)
----datai
2.05 2.1 2.15
1.92 1.94 1.96 1.98 2 2.02 2.04 2.06 2.08
-In-phase (stable)
----In-phase (unstable)
-Antiphase (stable)
----Antiphase (unstable)
2.05 2.1 2.15
Fig. 12. Frequency response of the system in the presence of different thicknesses of resonators VDC = 0, mp = 0, VAC = 0.94 V
the resonators to 5 %, the branch of in-phase oscillations also undergoes changes, namely, its tilt to the right is also observed.
6.4. Attraction zones of various frequency branches
In a certain frequency range, the frequency response of the system, both in the absence and in the presence of a disturbance in the form of a deposited particle, has several nontrivial stable solutions (branches). In order for the resonators to reach the desired oscillation mode corresponding to a certain branch of the frequency response, it is necessary to set specific initial conditions, which can be determined by the attraction zones shown in Fig. 13 for the frequency ratio value jj- = 2.005, amplitude of alternating voltage VAC = 0.94 V and the absence of a deposited particle mp = 0. The initial conditions for the velocity of both resonators for these zones are zero.
No oscillations In-phase Antiphase, q1>q2 Antiphase, ?2>îi No oscillations_In-phase Antiphase, q1 >q2 Antiphase, q2 > q1
Is.
-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
h Resonator 1
-1 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 0.8 1
Si Resonator 2
Fig. 13. Attraction zones,
2.005, VAC = 0.94 V, mp = 0
0
The attraction zones have a rather complex structure on the plane q1-q2, but they are completely symmetrical for two resonators: when resonator 1 is attracted to the upper branch of the frequency response and resonator 2 is attracted to the lower one. The zone for each resonator also has mirror symmetry along the diagonal q1 = q2. Figure 14 shows similar zones of attraction in the presence of a disturbance in the system mp = 10 pg.
No oscillations
In-phase Antiphase, q^ > </2 Antiphase, q2 > q%
No oscillations
In-phase Antiphase, q1 >q2 Antiphase, q2 > Qi
Resonator 1
Resonator 2
Fig. 14. Attraction zones, — = 2.005, VAC = 0.94 V, m = 10 pg
It can be seen that the attraction zones are resistant to disturbance in the form of a deposited particle; the influence of the nonzero mass mp on the attraction zones is barely noticeable. Thus, there is no need to define zones of attraction for each value of the parameter mp of interest.
7. Conclusion
In this work, a parametric study of the nonlinear dynamics of weakly coupled resonators under electrothermal excitation was performed. Using Hamilton's principle, a system of equations was obtained taking into account geometric nonlinearity and the presence of a disturbance in the form of a point mass. Next, a reduced order model was obtained using the Galerkin method. The method of multiple scales was used to obtain a system in slow variables, which was then studied by methods of bifurcation theory. It is shown that in a fully symmetric system in a certain frequency range the symmetry breaking effect is observed — the emergence of a mode with different oscillation amplitudes of two beam resonators caused by branching of a stable branch of antiphase oscillations on the amplitude-frequency characteristic of the system. In this mode of oscillations, modal localization between the two resonators occurs — energy is localized on one of them, which performs motions of larger amplitude. In the presence of a deposited particle, the appearance of an isolated branch is observed in the frequency response, which also corresponds to antiphase oscillations with different amplitudes. The attraction zones of various branches of the frequency response are obtained and the symmetry of these zones for two resonators is shown.
Appendix A
This appendix provides a brief description of the Floquet theory and the Routh-Hurwitz criterion used in the construction and analysis of zones of instability of parametric oscillations in Section 6.2. It is important to note that both methods for determining zones of instability are
applied to a system of equations without taking into account geometric nonlinearity — C7 = 0 in Eq. (4.9).
The main provisions of Floquet theory are described in [59, Section 11.2]. In this work, by numerically integrating the system of equations (4.9) at C7 = 0 over the oscillation period T with the initial conditions (A.1), the monodromy matrix was obtained (A.2):
(1) <i(0) = 1, q2(0) = 0, 4i(0) = 0, q2(0) = 0,
(2) qi(0)=0, q2(0) = 1, <i(0) = 0, ^(0) = 0,
(3) <i(0) = 0, <2(0) = 0, <i(0) = 1, <2(0) = 0,
(4) <i(0) = 0, <2(0) = 0, <i(0) = 0, <?2(0) = 1,
"?i(i) (T) <i(2) (T) <i(3)(T) <i(4) (T)"
<2(i) (T) <2(2) (T) <2(3) (T) <2(4) (T)
9i(i) (T) <i(2)(T) <i(3)(T) <7i(4) (T)
À(i) (T) <2(2) (T) <2(3) (T) <?2(4) (T)J
(A.1)
A =
(A.2)
The eigenvalues \i of the matrix A determine the nature of the oscillations of the original equation: when \Xi | ^ 1 the solution remains limited, when \AJ > 1 oscillations are building up. The zones of instability of parametric oscillations correspond to green areas in Figs. 5-7.
The Routh —Hurwitz criterion is applied to the system of equations in slow variables obtained after applying the method of multiple scales neglecting nonlinear terms:
a A ___
= "2iC5A2 - 2il1C2A1 - 2iQjC^ - iC^N^8etaT - CilA1N%ne%<TT,
r]A ___
= -2iC6A1 - 2Q1C3A2 + 2iA2Ml2 - iC4A2N%seiaT - C4A2N^eiaT.
dr
(A.3)
Next, a complex change of variables is carried out:
¿1,2 = (Br1>2 + iBth2 )e^/2 (A.4)
and the real and imaginary parts of Eqs. (A.3) are separated: dBr
= 2ii15iio- + 2BiC.nl - + 2B%C5 - BtC\Ncd°s - BrC4Ng,
dBt 2
= -2^Bra - 2BrCin2 - - 2BrC5 + B,C4N^ - BrC4Nc/2s,
dBr
= '2n,Bl2a - 2Bt2An2 + 2BtC6 - - BtC\N%s - BrC4N%n,
dBi 2
= -2n.B^a + 2BrAn2 - 2BrC6 - 23^, + BtC4N^ - BrC4N%°.
(A.5)
Thus, a system of four real-valued ordinary differential equations for the quantities B and Bi is obtained. The general solution of this system can be presented as
Br = br eYT, Bi = bi eYT, (A.6)
'1,2 '1,2 '1,2 '1,2 v '
where br12, bi^ ^ and 7 are the desired amplitude coefficients and characteristic number. Substituting the type of solution (A.6) into the system of equations (A.5) leads to a homogeneous system of linear algebraic equations for the amplitude coefficients br12,
f [2^62 + 4Ql7 + C4NT] bri + [-2Q?C! - + CANcdT] bh - 2C5b^ = 0, [2fil0- + 2Q?Ci + CNO] bri + [2QiC2 + 4Qi7 - C4NT] bH + 2C5br2 = 0, [2Q1C3 + 4Qi7 + C4NSn] bri + [2AQ2 - 2^1 * + C4NT] bi2 - 2C^ = 0, [2Qia - 2AQ2 + C4NdOs] br2 + [2QC + 4^7 - C4NT] bh + 2^ = 0,
the solvability condition of which has the form of an equation of degree 4 with respect to 7:
A(y) = a0Y4 + aij3 + a2j2 + a3 7 + a4 = 0, (A.8)
where the coefficients are
(A.7)
a0 — 1 a1 — C2 + C3i
a2 = (2Q,\C'l + 4QfC1cr + 2+ 8Q?C2C3 + 2Q?Cf +
8^1
-4QiAQ2a - C42 (NT)2 - C42 NT)2 + 2AQ4 + 4C5C6),
a3 = Î4QfCfC3 + 8Q?CiC3(T + 4Q2C|C3 + 4Q?C2C32 + 4Q?C2(t2+ 16Q2 V
+4Q2C3CT2 - 8QiC2AQ2a - C2C42 (NT)2 - C2C42 (N%T)2 + +4C2AQ4 + 4C5C6C2 - C3C42 (NC2s)2 - C3C42 (NT)2 + 4C5C6C^,
a4 = r (mnlcjc'i + 16Q?CiV - 32QfC2AQ2<r + 32fifC1Cf <7 + 32QfCia3-25604
-4qfCfo2 (Nf)2 - 4Q1 CC (NT)2 + 16Q4Ci2AQ4 - 64^4C1aq2a2+ +16QiC22C32 + 16Q4C22a2 + 16q1C32CT2 + 16q4O4 - 8Q?C1C42 (N0s)2 a--8Q?C1C42 (Nj2ra)2 a - 32QlC1C5C6o + 32QlC1AQ4o - 32^3C22AQ2o--32Q3AQ2o3 + 32Q2C1C5C6AQ2 - 4^2C22C42 (N0s)2 - 4Q2C22C42 (NT)2 +
+16Q2 C|aq4 + 32Q2 C2C3C5 C6 - 4Q2 c2c2 (NT)2 - 4Q2C3C2 (NT)2 -
-8Q?C42 (N^)2 a2 - 8Q?C42 (NT)2 a2 - 32Q?C5C6o2 + 16Q?AQ4o2+ +8Q1C42AQ2 (Ncd°s)2 a + 8Q1C42AQ2 (NT)2 a + 32Q1C5C6AQ2o + (N^)4 + +2C44 (NdOs)2 (NT)2 + C44 (NT)4 - 8C4CC6 (Nc2os)2 - 8C42C5C6 (NT)2 --4C42AQ4 (Ncos)2 - 4C42AQ4 (Ns2n)2 + 16C52C62.
The roots of this equation determine the nature of the dynamic response of the system under parametric excitation: the presence of a positive real part of the parameter 7 means the instability of the zero solution of the system of equations and the buildup of parametric oscillations. Thus, the boundaries of the parametric resonance regions in the space a, VAC can be found using the Routh-Hurwitz criterion:
All roots of the polynomial A(y) = 0 have negative real parts if and only if all n Hurwitz determinants are positive provided that a0 > 0 is positive:
A1 > 0, A2 > 0, A3 > 0, ..., an > 0. (A.9)
As a result of applying the above criteria, the boundaries of zones of instability of parametric oscillations were obtained, shown by red and blue dotted lines in Figs. 5-7.
Conflict of interest
The authors declare that they have no conflicts of interest.
Authors contributions
All authors participated in the article concept development, manuscript writing and discussing the results.
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