![](https://csdnimg.cn/release/download_crawler_static/3458639/bg5.jpg)
Modeling of QCM Data
17
D. Johannsmann
Ways of modeling layer systems
mathematical
01="q" jj+1 N+1N
u
0
+,0
u
j
+,0
u
N+1
-,0
u
j
-,0
z
z
j,j+1
S
j,j+1
L
j
optical
quartz
1 2 liquid
i AZ
q
tan(k
q
h
q
)
i AZ
q
tan(k
q
h
q
)
−
i AZ
q
/sin(2k
q
h
q
)
1/(i
ω
C
0
)
Z
k
A
Z
L
Modeling of QCM Data
18
D. Johannsmann
Contents
1. General
2. Detection Schemes
3. Complex Resonance Frequencies and Overview of Modeling
4. Mathematical Description
5. Optical Description
6. Electrical Description and the Small Load Approximation
7. Layer Systems
8. Perturbation Analysis
9. Software Package QTM
10. Contact Mechanics
11. Nonlinearities
12. Conclusions
Modeling of QCM Data
19
D. Johannsmann
Mathematical description
01="q" jj+1 N+1N
u
0
+,0
u
j
+,0
u
N+1
-,0
u
j
-,0
z
z
j,j+1
S
j,j+1
L
j
()
()
,0 ,0
jj
,0
jj
,0
jj
- N layers, supporting two waves traveli
ng to the right ( ) and to the left ( ) eac
- wave equation holds
exp i
exp i
:wavenumber,
= : speed of so
j
j
jj
j
jjj
uu
uu kz
uu kz
k
cGZ
k
G
c
−+
−−
++
=−
=−
ρρ
ω
==ω =ω
ρ
und
: density
:shear modulus
Z= = acoustic impedance (materials property)
G
cG
ρ
ρρ
Modeling of QCM Data
20
D. Johannsmann
Mathematical description (2)
01="q" jj+1 N+1N
u
0
+,0
u
j
+,0
u
N+1
-,0
u
j
-,0
z
z
j,j+1
S
j,j+1
L
j
()() () ()
()() () ()
+- +
j j,j 1 j j,j 1 j+1 j,j 1 j 1 j,j 1
+- +
j j,j 1 j j,j 1 j+1 j,j 1 j 1 j,j 1
- Stress and displacement continuous at N+1 interfaces
homogenous linear system of 2N+2 equations for 2N+1 a
uz uz u z u z
uz uz u z u z
−
++ +++
−
++ +++
+= +
+= +
⇒
() ( )
() ( )
,0
0
,0
qq
q
,0
qq qq
q
,0
2
qqqq qq
For example
11 1 0
00
0
0 exp i exp i 1
0 exp i exp i 0
Set the determinant to zero .... .... .... .... and find the r
u
ZZ
u
kd kd
u
u
ZkdZ kd
+
+
−
−
−
−
=
−−
−−
- intransparent