Shear dependent nonlinear vibration in a high quality factor
single crystal silicon micromechanical resonator
Supporting Materials
Modeling the dynamic behavior of the wine-glass (WG) mode square
plate resonator
Due to the symmetry in the WG mode shape, the planar principal strains (along
the x and y axes) can be transformed as pure shear strains along the x’ and y’ axes as
shown in Fig. S1 (45° from the in-plane principal strain directions).
Fig. S1: Definition of the coordinate systems used in the derivation of the model
Based on the linear model for WG mode microresonator [S1], we consider a new
coordinate system by rotating the original by 45 degree in counterclockwise direction
(illustrated in Fig. S1). The WG mode shape with the transformed coordinates is then
given as:
 2n 
sin
y  ,

2  L

(1)
 2n 
sin
x ,

2  L

(2)
x’-direction: U x, y  
1
y’-direction: V x, y  
1
where L is the edge length of the square resonator, n is a integer (n =1, 2, 3, …)
associated with the order of the mode shape. The mode shape equations are verified
by our finite element (FE) simulation in COMSOL multi-physics.
Since there is no normal strain in x’ and y’ directions, the equation of motion for each
element can be expressed as:
x’-direction: 
y’-direction: 
 2u 
t
2
 2 v
t
2


 xy
y 
 xy
x
 f x x' , y ' , t ,
(3)
 f y x' , y ' , t ,
(4)
where ρ is the material density, τ’x’y’ is the shear stress, fx’ and fy’ are the distributed
driving forces in x’ and y’ directions, respectively. u’ and v’ are the displacement
fields over time (t) in x’ and y’ directions, given as follows by applying the method of
separation of variables:
x’-direction: ux, y, t   U x, y  qt ,
(5)
y’-direction: vx, y, t   V x, y  qt ,
(6)
where q is the general displacement over time.
The displacement and shear strain relation is given as:
 xy 
u  v 

.
y  x 
(7)
To study the nonlinear mechanical property effect on the vibration, we consider
the nonlinear strain-stress relation including damping can be expressed as (up to
second order correction):


 xy  G0 1  G1 xy  G2 xy 2  xy    xy ,
(8)
where G0 is the linear shear modulus, G1 and G2 are first and second order correction
terms, respectively, ζ is the damping coefficient. The overdot represents the derivative
with respect to time.
Substitute (5), (6), (7) and (8) into (3) and (4), the partial differential equations
that govern the motion of each element are given as follows:
x’-direction:
  2U 
2
 2V  
  U  V  
  U  V   2

qt   G0 G1

 q t 
q t   G0


2







x

y

y

y

x

y   y  x 


 y 

3
  U  V   3

 q t   f x x' , y' , t .
 G0 G2

y   y  x 
U qt    

(9)
y’-direction:
2
  2U   2V  






q t   G0   U  V qt   G0 G1   U  V  q 2 t 
V qt    




2

x  y  x 
x  y  x 
 xy  x 
3
  U  V   3

 q t   f y x' , y' , t .
 G0 G2

x  y  x 
(10)
Since the displacements are small compared to resonator’s dimensions (i.e. L),
the nonlinearity due to large geometrical deformation can be ignored.
It is thus valid
to assume the change of mode shape due to nonlinearity in our system is negligible,
and the mode shape equations (1) and (2) derived in linear vibration are adopted.
Substitute (1) and (2) into (9) and (10), and apply the Galerkin method to
integrate (9) and (10) over the mode shape U’ and V’, respectively, the device in WG
mode
forced
vibration
can
be
modeled
as
an
single
degree-of-free
mass-damper-spring system and the lumped equation of motion can be expressed as
an asymmetric Duffing equation (both in x’ or y’ are the same):
M eff qt   Deff qt   K eff ,1qt   K eff ,2 qt 2  K eff ,3qt 3  F t 
Parameters
(annotation)
Meff (effective mass)
Deff (effective damping
coefficient)
Keff,1 (effective linear
spring constant)
(11)
Expression
Fundamental WG mode
(n=1)
hL2
hL2
4
4
n 2 2h
2
 2 h
n  G0 h
2
 G0 h
2
2
2
2
2
Keff,2 (effective
quadratic spring
constant)
Keff,3 (effective cubic
spring constant)
 n 
16 n 5  cosn sin 4 
G0 G1h
 2 

9L
9n 4 4 G0 G 2 h
8 L2
8L


U
64G0 G1h
9L
9 4 G0 G 2 h
2
 n
2 0 hL sin 2 
 2
2
ng
F (driving force)

DCU AC
cost 
2  0 hL
U DCU AC cost 
 g2
where ε0 is the permittivity of free space, ω is the angular velocity of the AC driving
signal, UDC and UAC are DC and AC driving voltage, respectively; L, h and g are the
edge length, thickness, and transducer gap distance of the device, respectively.
The parameter values for modeling the designed microresonator in this letter are
summarized in the Table I.
Table I: Modeling Parameters Summary
Parameters
ρ
L
h
g
ε0
G0
G1
G2
Q
Meff
Deff
Keff,1
Keff,2
Keff,3
Value
2330
800
25
3
8.85×10-12
81.7
0.2067
-1.4688
1.968×106
9.32×10-9
2.0169×10-7
1.0083×107
-1.1795×1010
-5.1385×1014
Units
kg/m3
μm
μm
μm
F/m
GPa
kg
N·s/m
N/m
N/m2
N/m3
References
[S1] M. Ziaei-Moayyed, D. Elata, E. P. Quévy and R. T. Howe, J. Micromech.
Microeng. 20, 115036 (2010).