Proceedings of IMECE2008 2008 ASME International Mechanical Engineering Congress and Exposition
October 31-November 6, 2008, Boston, Massachusetts, USA
ABSTRACT
This work is concerned with the modeling, nonlinear dynamic analysis and control design of an electrostatically actuated clamped-clamped microbeam filter. The model accounts for the mid-plane stretching and nonlinear form of the electrostatic force actuated along the microbeam span. A reduced-order model is constructed, using the method of multiple scales, to examine the microsystem static and dynamics behaviors. To improve the microbeam behavior, a nonlinear feedback controller is proposed. The main control objective is to make it behave like commonly known one-degree-of-freedom self-excited oscillators, such as the van der Pol and Rayleigh oscillators, which depict attractive filtering features. We present a novel control design that regulates the pass band of the fixed-fixed microbeam and derive analytical expressions that approximate the nonlinear resonance frequencies and amplitudes of the periodic solutions when the microbeam is subjected to one-point and fully-distributed feedback forces.
1. INTRODUCTION
The evolution of microelectromechanical systems technology has led to a new generation of filters that have been a major focus in the litterature. Many studies discussed in details the design process of macroscopic mechanical filters made up with microbeams. Lin et al. [1] presented a mechanical filter based on polysilicon interdigitated comb resonators with a double-folded support structure. Bannon et al. [2] demonstrated filters composed of two doubly-clamped beams which are coupled mechanically by a soft mechanical spring. Johnson [3] discussed in details the design process of macroscopic mechanical filters. Bannon et al. [4] described the design process of a micromechanical filter comprised of two capacitive resonators made of doubly-clamped microbeams and coupled by a flexural one. Tilmans [5] studied the dynamic behavior of mechanical filters. Shaw et al. [6] analyzed the dynamics of MEMS
1
oscillators that act like frequency filters, where parametric resonance was used for frequency selection. Their design is appropriate for highly tunable microbeams, which offer minimal packaging constraints, low-power consumption, low damping, ease of parameter tuning, and relatively simple integration with electronics. Rhoads and co-workers [7-9] described a filter design based on the nonlinear response of parametrically-excited MEMS oscillators that have significant potential in many communications applications. They reviewed parametric resonance and discussed its relevance to MEMS and its potential use in filtering applications. They also modeled a single MEMS oscillator and analyzed its dynamic response. In addition, they presented a procedure of how to improve oscillator performance, specifically for filtering applications, and described one possible filter design that utilizes two tuned MEMS oscillators.
Motivated by the need to further improve the static and dynamic performances of microbeam filters, this paper considers the modeling, dynamic analysis, and control synthesis of an electrically-actuated microbeam. The control design is based on the microbeam reduced-order model, which is deduced using the method of multiple scales. Thus, the control design aims at altering the behavior of the microbeam so that it acts like one of the one-degree-of-freedom self-excited oscillators. A major contribution of this work is to provide a set of reliable analytical expressions that describe the system’s behavior. Such expressions make it possible to design feedback controllers for tuning oscillators and regulating the pass band of microbeam filters.
2. MODELING OF A CLAMPED-CLAMPED MICROBEAM FILTER
Here, we present a dynamic model for a clamped-clamped electrostatically actuated microbeam with a feedback input. This model exhibits the main characteristics that can be found in a large number of MEMS devices, which rely on electrostatic
actuation. In MEMS devices, one of the basic structures is the beam. This mechanical component and its extension, the plate, constitute the majority of MEMS sensors and actuators. Consequently, the first step to analyze the behavior of any device is to understand and model the dynamic characteristics of a beam. For this, we consider the clamped-clamped microbeam shown in Figure 1.
Figure 1: A clamped-clamped microbeam filter [10].
The nonlinear dynamics of an electrically actuated microbeam and associated boundary conditions can be described as follows [10 and 11]:
( )( )( )
( ) ( )
24 2
DC04 2 2
2 2
20
ˆˆ ˆ ˆˆˆ ˆˆ 2 ˆ
ˆ ˆˆˆ ˆˆ ˆ ˆ,ˆ ˆ2
l
V v tbw w wE I c Atx t d w
E A w wu x t N t dxl x x
ερ
+∂ ∂ ∂′ + + =∂∂ ∂ −
⎡ ⎤′ ∂ ∂⎛ ⎞+ + +⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠⎢ ⎥⎣ ⎦∫
( ) ( ) ( ) ( )ˆ ˆˆ ˆ0, ,ˆ ˆˆ ˆ0, , 0, 0
ˆ ˆw t w l t
w t w l tx x
∂ ∂= = = =
∂ ∂
(1)
where w is the deflection amplitude, ρ is the beam density, b and h are, respectively, the width and height of the beam section, l is the beam length, E is Young’s modulus, A bh= is the cross section area, 3 /12I bh= is the second moment of area, ( )ˆ ˆN t is
an axial force, ( )2/ 1E E ν′ = − is the modified Young modulus
of elasticity, ν is Poisson’s ratio, 0ε is the dielectric constant,
DCV and ( )ˆv t are the DC and AC voltages, respectively, and
( )ˆˆ ˆ,u x t is the distributed control input. For convenience, we recast Equations (1) in the nondimensional form.
( )
( )( )
( ) [ ]
12
10
2DC
2 2
'''' ' ''
, , 0,11
w cw w w d N w
V v tu x t x
w
α ξ
α
⎡ ⎤+ + = +⎢ ⎥
⎢ ⎥⎣ ⎦
⎡ ⎤+⎣ ⎦+ + ∈−
∫ (2)
( ) ( ) ( ) ( )0, 1, 0, ' 0, ' 1, 0w t w t w t w t= = = = (3) where
xxl
= wwd
=
'
4ˆ E It t
Alρ=
2
'
clcE I Aρ
=
2
1 6 dh
α ⎛ ⎞= ⎜ ⎟⎝ ⎠
40
2 ' 3 36l
E d hε
α =
4
'ˆlu u
E Id=
2
'ˆ lN N
E I=
y z
b
S t a t i o n a r y E l e c t r o d e
M i c r o b e a m V DC
( )ˆv t h
L
d
x
2
and the dot and prime denote the derivatives with respect to time and space, respectively. Next, we consider two types of feedback input: point and distributed.
3. A ONE-POINT FEEDBACK INPUT
Since the middle point of the microbeam experiences the largest amplitudes of vibration during motion, we choose the feedback input to be concentrated at this point; that is,
( ) ( ) ( ), 1 / 2u x t U t xδ= − (4)
In this study, we propose a feedback input that causes the microbeam to behave like the van der Pol or Rayleigh oscillator. We can do this by feeding back negative linear and positive nonlinear damping. Consequently, we propose to use the following form of ( )U t :
( ) 3 21 2 3
1 1 1 1, , , ,2 2 2 2
U t K w t K w t K w t w t⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(5)
where iK ( 1, 2, 3i = ) are positive feedback gains.
3.1. Approximate Solution
At this stage, it should be emphasized that the proposed control design aims at making a microbeam behave either as the van der Pol or the Rayleigh oscillator by adjusting the feedback gains. Here, the first is adopted for the control synthesis proposed in this paper. In other words, the microbeam filter can be tuned via feedback to meet certain specifications. For this, we apply the method of multiple scales [12-14] to determine a second-order uniform approximate solution to Equations (2)-(5). To this end, we seek the solution in the form
( ) ( ) ( )( ) ( )
( ) ( )1 0 2
2 32 0 2 3 0 2
, , ,
, ,
, , , , ...
s
s
w x t w x w x t
w x w x T T
w x T T w x T T
ε
ε
ε ε
= +
= +
+ + +
(6)
where nnT tε= . Moreover, we scale some of the variables as
2( )c O ε= ( ) 3AC cos( )v t V tε= Ω
( )21K O ε= ( )2 1K O= ( )3 1K O=
(7)
Substituting Equations (6) and (7) into Equations (2)-(5) and equating like powers of ε , we obtain Order 0ε : (static equation)
where /n nD T= ∂ ∂ . The boundary conditions for 1, 2, 3n = lead to:
( ) ( ) ( ) ( )0 1 0, 0 1 0n n n nw w w w′ ′= = = = (13)
The solution of Equation (9) is assumed to consist of only the directly excited mode. Accordingly, we express 1w as
( ) ( ) ( ) ( )0 01 0 2 2 2, , i T i Tw x T T A T e A T eω ω φ ξ−⎡ ⎤= +⎣ ⎦ (14)
where ( )2A T is a complex-valued function, the over bar denotes
the complex conjugate, and ω and ( )xφ are the natural frequency and corresponding eigenfunction of the directly excited mode, respectively. They are governed by the following boundary-value problem:
( )
( )( )
12
22 DC1 2
'''' '' , ''
22 , '' 0
1
s s
s ss
N w w
Vw w
w
φ φ α φ
αα ϕ φ ω φ
− − Γ
− Γ − − =−
( ) ( ) ( ) ( )0 1 0, 0 1 0φ φ φ φ′ ′= = = =
(15)
Substituting Equation (14) into Equation (10) yields the solution
( ) ( ) ( )( ) ( ) ( ) ( ) ( )
0
0
222 0 2 1 2
222 2 2 1 2
, ,
2
i T
i T
w x T T x A T e
x A T A T x A T e
ω
ω
ψ
ψ ψ −
=
+ + (16)
where 1ψ and 2ψ are the solutions of the following boundary-value problems:
( ) ( ) 1, 2i iM h xψ ωδ = 0 and 0 at 0 and 1, 1, 2j j x x jψ ψ ′= = = = =
(17)
δij is the Kronecker delta, the linear differential operator ( , )M ψ ω is defined by
( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( )
( ) ( )( )
2
1 1
322 DC
, '''' ''
, '' 2 , ''
2 / 1
s s s s
s
M x x x N x
w w x w x x w x
V x w x
ψ ω ψ ω ψ ψ
α ψ α ψ
α ψ
= − −
− Γ − Γ
− −
(18)
and
3
( ) ( )( ) ( )
42 22 DC
1 1
3 / 1
2 , '' , ''s
s s
h x V w
w w
α φ
α φ φ α φ φ
= −
+ Γ + Γ (19)
In order to describe the nearness of the excitation frequency Ω to the fundamental natural frequency ω , we introduce the detuning parameter σ defined by
2ω ε σΩ = + (20)
Substituting Equations (14), (16), and (20) into Equation (11), we obtain
( ) ( )( ) ( )( ) ( )( )
0
0
02
3 1
2 3 3 23 2
22 AC DC
1/ 2 2
( 3 ) ( 1 / 2)
2 / (1 )
i T
i T
i Ti Ts
w ic A iK A x i A x e
A A x K K i x A A x e
V V e e w CC NST
ω
ω
ωσ
ω ω δ ω φ
χ ω ω φ δ
α
′= − + − −
+ − − −
+ − + +
L
(21)
where A′ denotes the derivative of A with respect to 2T , CC indicates the complex conjugate of the preceding terms, NST stands for the terms that do not produce secular terms, and ( )xχ is defined by
( ) ( ) ( ) ( )( ) ( ) ( )
( ) ( )( ) ( )
1 1 1 1 2
1 1 1 2 1 1
52 31 2 2 DC
4 42 22 DC 1 2 DC 2
3 , 2 , 4 , ''
2 , 4 , '' 2 , ''
4 , '' 12 / 1
6 / 1 12 / 1
s s
s s
s s
s s
x w w
w w
w V w
V w V w
χ α φ φ α ψ α ψ φ
α φ ψ α φ ψ α φ ψ
α φ ψ α φ
α φψ α φψ
⎡ ⎤= Γ + Γ + Γ⎣ ⎦⎡ ⎤+ Γ + Γ + Γ⎣ ⎦
+ Γ + −
+ − + −
(22)
Multiplying the right-hand side of Equation (21) by ( ) 0i Tx e ωφ − and integrating the result from 0x = to 1x = yield the solvability condition
22 21 22 2 8 8 0i T
ei A i A i A A A A Fe σω ωμ ωμ α′− + − + + = (23)
where φ is normalized such that 1
2
0
1dxφ =∫ and
21 1
1 1 12 2 2
K cμ φ ⎛ ⎞= −⎜ ⎟⎝ ⎠ ( )2 4
2 3 21 138 2
K Kμ ω φ ⎛ ⎞= + ⎜ ⎟⎝ ⎠
1
0
18e dxα χφ= ∫
1
2 AC DC 2
0
2(1 )s
F V V dxwφα=
−∫
By inspecting Equation (23), we note that the linear feedback gain has been lumped with the inherent viscous damping in the system into an overall effective damping term. This is a direct result of the feedback law used, which calls for derivative feedback. In order that Equation (23) mimics either the van der Pol or Rayleigh oscillator we should have: 1 0μ > and 3 0μ > . Next, we express A in the polar form / 2iA ae β= , where
( )2a a T= and ( )2Tβ β= are real-valued functions, representing, respectively, the amplitude and phase of the response. Substituting for A in Equation (23), letting
2Tγ σ β= − , and separating the real and imaginary, we obtain the following modulation equations:
31 2
3
sin
1 cose
Fa a a
Fa a a
μ μ γω
γ σ α γω ω
′ = − +
′ = + +
(24)
(25)
Substituting Equations (14), (16), and (20) into Equation (6) and setting 1ε = , we obtain, to the second approximation, the following microbeam response to the external excitation:
( ) ( )
( ) ( )21 2
, ( ) cos( )1 cos 2( ) ...2
sw x t w x a x
a x t x
τ γ φ
ψ γ ψ
= + Ω −
⎡ ⎤+ Ω − + +⎣ ⎦ (26)
It follows from Equation (26) that periodic solutions correspond to constant a andγ ; that is, the fixed points ( )0 0,a γ of Equations (24) and (25). Thus, letting 0 and 0aγ ′ ′= = in Equations (24) and (25), we obtain
31 0 2 0 0
30 0 0
sin
1 cose
Fa a
Fa a
μ μ γω
σ α γω ω
⎧ − = −⎪⎪⎨⎪ + = −⎪⎩
(27)
(28)
Eliminating 0γ from Equations (27)-(28), we obtain the following frequency-response equation:
( )2 22 2 2 2 2
0 0 1 2 01/ eF a a aω σ α μ μω
⎡ ⎤⎛ ⎞= + + −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦
(29)
From Equation (29), we derive an expression of σ as function of 0a
( )2 22 2
0 1 2 0 02 20
1 0eFa a a
aσ α μ μ
ω ω= − ± − − ≠ (30)
The ± sign in Equation (30) depicts that σ is negative or positive. Figure 2 shows the shape of a typical frequency response curve of the microbeam with hardening-type behavior. In this curve, we distinguish three points: two bifurcation points A and B at which 0/ 0d daσ = and a maximum-amplitude point C at which 0 / 0da dσ = .
Figure 2: Typical frequency curve with two bifurcation points.
σ
0a Peak amplitude 0 / 0da dσ =
Bifurcation points 0/ 0d daσ = A
B C
4
The occurrence of the three points in the frequency response curve and their locations are key features in shaping the microbeam filter. Inspection of Equation (30) depicts that these locations are mainly functions of the feedback gains 1K , 2K and
3K , lumped into 1μ and 2μ , and the AC and DC amplitudes, lumped in F.
To this end, we determine expressions for the locations of these points. Taking the derivative of Equation (30) with respect to 0a and setting the resulting equation to zero yield
( )( )
4 2 2 22 0 1 2 0
22 6 2 8 20 0 1 2 0 0
2 /
2 / 0e
a a F
F a a a a
μ μ μ ω
α ω μ μω
⎡ ⎤± − −⎣ ⎦
= − − > (31)
whose roots gives the locations of the bifurcation points. By squaring both sides of Equation (31), its roots can be determined from the following equation:
( ) ( )( ) ( ) ( )
( ) ( )
2 2 2 2 6 2 2 2 52 2 1 2 2
22 2 2 2 4 2 2 2 31 2 2
2 421 2
4 / 8 /
4 / 4 / /
4 / / 0
e e
e e
X X
X F X
F X F
μ μ α ω μ μ μ α ω
μ μ α ω ω μ α ω
ω μ μ ω
+ − +
+ + + −
− + =
(32)
where 20X a= . Equation (32) has six roots, which are real and
complex conjugates. It can also be shown that the maximum amplitude points are the roots of the following equation:
( )22 2 2 20 1 2 0/ 0F a aω μ μ− − = (33)
or all solutions of Equation (33) are the union of solutions of the following two cubic polynomials.
( )( )
2 31 0 1 2 0 2 0 1 0
2 32 0 1 2 0 2 0 1 0
: / / 0
: / / 0
P F a a a a F
P F a a a a F
ω μ μ μ μ ω
ω μ μ μ μ ω
+ − = − − =
− − = − + =
(34)
To identify the total number of roots located on the right side of the complex plane, we use the Routh Hurwitz stability criterion. We found that the first polynomial has a unique real root on the right side for all values of 1μ , 2μ and F. It is given by
( )( )
2/32 33
0 1/32 33
12 9 81 12
18 9 81 12a
λ κ κ λ
κ κ λ
+ + −=
+ − (35)
( )
( )
21
12 4
2 3 2
2 42 3 2
1( )2 =4
3 (1/ 2)
8 3 (1/ 2)
K c
K K
F FK K
φμλ
μ ω φ
κωμ ω ω φ
−=
+
= =+
(36)
The other two roots, either all real roots or complex conjugates, are located on the left side. The second polynomial has two solutions, which can be real or complex conjugates depending on the sign of the following constant [15]:
Therefore, two types of frequency response curves of the microbeam are possible. Case 1: if 0J > , one real root of 1P and two complex conjugates of 2P are found, respectively, in the left and right halves of the complex plane. The shape of the frequency response curve looks as shown in Figure 3a. Case 2: if 0J ≤ , one real root of 1P and two real roots of 2P are found, respectively, on the right half of the complex plane, yielding the shape of the frequency response curve shown in Figure 3b.
(a)
(b)
Figure 3: Possible frequency-response curves. 4. A FULLY DISTRIBUTED FEEDBACK INPUT
We also consider the control of a microbeam filter using distributed-force feedback. In practical situations, the controller
5
is realized by depositing on the microbeam a piezoelectric patch, which stretches over at least 20% to 30% of the beam length, and thus, it cannot be approximated by a point load. For this, we propose adding the feedback signal to the signal used to actuate the beam via the electrode underneath the beam. In other words, we let the feedback be applied over the whole length of the beam. This is similar to the case of generating a DC and an AC field between the stationary electrode and microbeam. It is also possible to collocate the actuation force and feedback force at the same electrode. Therefore, let the distributed feedback input be given by
( ) ( ) ( ) ( ) ( )3 21 2 3, , , , ,u x t K w x t K w x t K w x t w x t= − − (38)
Again, we apply the method of multiple scales to approximate the solutions of Equations (2), (3) and (38). We take advantage of the definitions given in Equations (7), and we adopt the same procedure as for the case of one-point feedback. As a result, the modulation equations (27) and (28), of the one-point feedback, are modified as follows:
( ) 2 31 0 0 3 2 0
30 0 0
1 1 3ˆ sin 02 8 8
1cos 08
FK c a K K a G
F a S a
γ ωω
γ σω ω
⎧ ⎛ ⎞− + + − =⎜ ⎟⎝ ⎠⎨
+ + =⎩
(39)
1
2 AC DC 20
1 14
0 0
2 ,(1 )
,
sF V V d
w
G d S d
φα ξ
φ ξ φχ ξ
=−
= =
∫
∫ ∫ (40)
Then eliminating 0γ from the system (39), we obtain the following frequency-response equation:
Next, we evaluate numerically the parameters ω , φ , 1ψ , 2ψ , and sw associated with Equations (39-40) using the Differential Quadrature Method [16]. Once these parameters are computed, frequency-response curves can be generated. To describe the dynamic response of the microbeam, we need to determine the natural frequency ω , the effective excitation amplitude F, the effective nonlinearity of the system eα , and the effective viscous damping coefficients 1μ and 2μ . As a first step, Equation (8) is numerically integrated using DQM to determine the static deflection sw for a given DC voltage. Using the static
solution sw , we solve the boundary-value problem,
( ), 0M φ ω = , using DQM for the fundamental natural frequency ω and its corresponding eigenfunction φ . Next, we solve the two boundary-value problems in Equations (17) and
(18) to evaluate the functions 1ψ and 2ψ using DQM. Finally, we evaluate F, G and S from Equation (40).
We consider the microbeam shown in Figure 1 with the geometric and physical characteristics given in Table 1.
Table 1: Microbeam parameters
l (µm) b (µm) h (µm) d (µm) 510 100 1.5 1.18
N (N) c E' (GPa) 0ε (F/m) -41.561 10× 0.0239511 166 -128.854 10×
Figure 4 shows variation of the mid-point deflection of the microbeam maxw with the DC voltage, using 11 grid points in the DQM. The stable (lower) branch and the unstable (upper) branch meet at the static pull-in instability 4.81DCV = Volts and
max 0.42w = , resulting in the destruction of both branches in a saddle-node bifurcation. This static analysis shows that the MEMS filter should be designed to operate below this value, which serves as an upper bound of the stability limit of the microbeam.
Figure 4: The normalized static deflection curve of the electrostatically
actuated microbeam (Bifurcation Diagram).
Next, we study the dynamic behavior of the microbeam under an AC harmonic excitation near its fundamental frequency 1ω = 23.9. We assume a quality factor Q = 1000, which is related to the damping coefficient by 1ˆ /c Qω= .
We simulate the controlled frequency response of the microbeam using both one-point and distributed- feedback forces. Figure 5 shows the frequency-response curves for DCV = 1.5 Volt and
ACV = 0.05 Volt. The results are obtained by solving Equation (29) for the point feedback and Equation (41) for the distributed feedback. As expected, the amplitude becomes lower when the distributed actuator is used since all points along the microbeam axis are less compliant than the mid-span point where the point feedback is applied.
Stable branch Unstable branch
maxw
6
Figure 5: Frequency-response curves for VDC =1.5 Volt,
VAC = 0.05 Volt, K1=0.01, K2=1, and K3=0.
For 3 179.6K = , Figure 6 shows the resulting frequency-response curves of the controlled microbeam. This figure depicts a microbeam behavior that is similar to that of the van der Pol oscillator. This implies that it is possible to synthesize a set of feedback gains that makes the response characteristics of a microbeam resemble those of the van der Pol oscillator, which filters signals corresponding to the left and right of 0σ = .
Figure 6: Frequency-response curves when VDC =1 Volt,
VAC =0.01 Volt, K2=0, K3=179.6.
5. DESIGN OF AN ELECTRONIC CIRCUIT FOR THE IMPLEMENTATION OF THE FEEDBACK CONTROLLER
An analog circuit, using Analog Device AD633JN voltage multipliers, is proposed for the implementation of the feedback controller given by Equation (30). In order to detect the microbeam deflection and velocity, an electronic interface is necessary to out both of these variables. The position or velocity can be determined from the output of the circuit shown in Figure
Figure 7: Circuit for detecting the deflection and velocity of the clamped-clamped microbeam [17].
The circuit parameters are given by
7
01 2 ( )
AC C
d wε
= =−
0 11 2( )
AVi w
d wε
=−
0 12 2( )
AVi w
d wε
= −+
1 2 DCV V V− = (42)
In order to sense the position and velocity, parallel plate drives are used in a differential capacitance sense circuit [17]. With a constant DC voltage on the stationary combs, a current 3i is induced that is proportional to velocity. In Figure 7, Z is either an integrating capacitor IC or a transresistance amplifier resistor R, giving an output voltage proportional to the device position or velocity, respectively. In the limit that the deflection w is small compared to the gap ( w d ), then the output voltage with the integrating capacitor is proportional to deflection
02
2 DCout
I
AVV w
C dε
≈ (43)
With a resistor R, the output is proportional to velocity
02
2 DCout
AV RV w
dε
≈ (44)
The feedback control can be implemented using analog circuitry made of 6 analog devices AD633JN, shown in Figure 8.
Figure 8: Analog implementation of the feedback controller.
The AD633JN is a low cost multiplier comprising a translinear core, a buried Zener reference, and a unity gain connected output amplifier with an accessible summing node. Figure 9 shows the functional block diagram. The differential X and Y inputs are converted to differential currents by voltage-to-current converters. The product of these currents is generated by the multiplying core. A buried Zener reference provides an overall scale factor of 10 V. The sum of ( ) /10X Y Z× + is then applied to the output amplifier. The amplifier summing node Z allows the user to add two or more multiplier outputs, convert the output voltage to a current, and configure various analog computational functions.
Figure 9: Functional Block Diagram of an Analog Device AD633JN.
5. CONCLUSIONS We presented a novel control design that regulates the pass band of a microbeam filter whose principal component is an electrostatically actuated clamped-clamped microbeam. The feedback is primarily used to render the microbeam behave like the van der Pol oscillator or the Rayleigh oscillator whose dynamic and frequency response features were studies in the literature [14]. Using the method of multiple scales, we derived two nonlinear ordinary-differential equations that describe the modulation of the amplitude and phase of the response with time. These equations are used to approximate the nonlinear resonance frequencies and amplitudes of limit-cycle solutions in the presence of either a one-point or a fully distributed feedback force. We proposed an electronic circuitry made of six analog devices AD633JN for the implementation of the feedback controller.
6. REFERENCES
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