Influence of shear and axial pretension on
the free-vibration behaviour of clampedclamped beam MEMS resonators
Hein van Beek
DCT 2008.127
Bachelor final project
Coordinators:
dr. ir. R.H.B. Fey
ir. R.M.C. Mestrom
Eindhoven University of Technology
Department of Mechanical Engineering
Dynamics and Control Technology Group
Eindhoven, September 14, 2008
Summary
Goal
A clamped-clamped beam MEMS resonator may be used as a time reference in the
near future. For accurate timekeeping, it is important that the frequency at which the
micro beam vibrates is very well known. Therefore a model needs to be developed
that can accurately predict this natural frequency. In this project is investigated if
shear and axial pretension need to be taken in account in such a model.
Method
In order to investigate the influence of shear and pretension on the mode shapes and
the natural frequencies of the clamped-clamped beam MEMS resonator, the project
has been divided into four different cases: cases with and without shear and with and
without pretension. An Euler beam model is used for the cases without shear and a
Timoshenko beam model is used for the cases with shear. Pretension is based on
temperature differences. From literature it is known, that shear starts to play a role
from a Height/Length ratio of around 1/10. Since the micro beam has this H/L ratio,
different H/L ratios are also investigated at all cases to investigate their influence.
Different calculation methods, both analytic as FEM based, are used to check for the
correctness of the results.
Results
Regarding the natural frequencies, the influence of shear has been found to be small at
low H/L ratios and big at higher H/L ratios, which is according to the expectation. The
influence of shear is also increasing at higher eigenmodes. The influence of pretension
has been found to be reasonably small at low H/L ratios and very small at higher H/L
ratios. The influence of pretension decreases at higher eigenmodes. Regarding the
mode shapes, it has been found that the mode shapes are independent of shear and
pretension. Numerical problems have been encountered in plotting the mode shapes of
the case with shear and with pretension.
Conclusions
The accuracy requirements in terms of calculating natural frequencies of the model
for both shear and pretension are very strict, in the order of ppm. The influence of
shear and pretension has been found to be small, but cannot be neglected. Therefore,
the model that predicts the frequency of the MEMS resonator should take both shear
and pretension into account. The analytic calculation method has been proven to be
the best. It is recommended for further research in this topic to solve the numerical
problems that have been encountered.
i
Table of contents
Summary
i
1
Introduction
1
2
Clamped-clamped beam MEMS resonator
2
3
Modelling
3.1
Model of the MEMS resonator
3.2
Modelling approach
3
3
3
4
Theoretical background
4.1
Euler-Bernoulli
4.1.1 Assumptions and differential equation
4.1.2 Mode shapes and natural frequencies
4.2
Timoshenko
4.2.1 Assumptions and differential equations
4.2.2 Mode shapes and natural frequencies
4.3
Pretension
4.3.1 Determining the range of pretensions
4.3.2 Effect of pretension on Euler model
4.3.3 Effect of pretension on Timoshenko model
4.4
Finite Element Methods
4.4.1 FEM using Matlab
4.4.2 FEM using Marc-Mentat
4
4
4
5
6
6
7
9
9
10
11
13
14
14
5
Analytical and numerical analysis
5.1
Euler without pretension
5.1.1 Case 1: Natural frequencies
5.1.2 Case 1: Mode shapes
5.2
Timoshenko without pretension
5.2.1 Case 2: Natural frequencies
5.2.2 Case 2: Mode shapes
5.3
Euler with pretension
5.3.1 Case 3: Natural frequencies
5.3.2 Case 3: Mode shapes
5.4
Timoshenko with pretension
5.4.1 Case 4: Natural frequencies
5.4.2 Case 4: Mode shapes
5.5
Comparison between the four cases
5.5.1 Comparison of the natural frequencies
5.5.2 Comparison of the mode shapes
16
16
16
17
18
18
19
20
20
22
23
23
24
27
27
30
6
Conclusions and recommendations
31
Bibliography
32
ii
Appendices
A
Nomenclature
A.1
Latin symbols
A.2
Greek symbols
33
33
34
B
Derivation of mode shape coefficients
B.1
Mode shape coefficients Timoshenko without pretension
B.2
Mode shape coefficients Timoshenko with pretension
35
35
36
C
Matlab files
C.1
Euler without pretension
C.2
Euler without pretension FEM
C.3
Timoshenko without pretension
C.4
Timoshenko without pretension FEM
C.5
Euler with compressive pretension
C.6
Euler with tensile pretension
C.7
Timoshenko with compressive pretension
38
38
39
40
42
44
45
47
iii
1
Introduction
A MEMS resonator is a small electro-mechanical device that can be used as a time
reference in the near future. MEMS stands for Micro-Electro-Mechanical System. The
resonator consists of a clamped-clamped micro beam that vibrates at very high
frequencies, along with some sensing and actuation circuitry. To obtain accurate
timekeeping, it is important that the frequency at which the beam vibrates is very well
known. Therefore, a model is needed that can accurately predict this frequency.
Since the resonator is operated in vacuum, the system is very weakly damped.
Measurements have also shown this. Therefore, the undamped free vibration is
analysed, i.e. the undamped natural frequencies and mode shapes are determined. Of
special interest is, whether shear and axial pretension affect the natural frequencies
and mode shapes such, that they should be taken in account in the model that is used
to predict the frequency of the MEMS resonator. The main goal of this Bachelor Final
Project is to find this out.
Therefore, the problem has been split up into four cases: with and without shear and
with and without pretension. An Euler beam model is used for the cases where shear
is neglected, a Timoshenko beam model is used for the cases where shear is taken in
account. Pretension is based on temperature differences that may occur in a MEMS
resonator during operation.
Different calculation methods are used for determining the natural frequencies and
mode shapes. Both FEM based methods and (semi-)analytic methods are used. This
allows for checking of the correctness of the results. Eventually, this may lead to the
conclusion that one method is preferred over the others.
In summary, the problem definition for this project is the following:
‘What is the influence of shear and axial pretension on the free-vibration behaviour
of clamped-clamped beam MEMS resonators?’
A short outline of this report is as follows. In chapter 2, more information about the
specific MEMS resonator under investigation can be found. Chapter 3 deals with the
modelling approach that is used in this project. In chapter 4, the theoretical
background on the different modelling approaches in this project is discussed. Chapter
5 deals with the analysis of the results of this project. In chapter 6 the conclusions and
recommendations can be found.
1
2
Clamped-clamped beam MEMS resonator
A MEMS (Micro-Electro-Mechanical System) resonator is a small device that
consists of a vibrating micro beam and some actuation and sensing circuitry, see
figure 2.1 and 2.2. It is proposed to be used as a timing device in for example a mobile
phone or a computer in the near future. The frequency at which the micro beam
vibrates will be used as a reference for the time that passes. To enable accurate
timekeeping it is important that the frequency at which the beam vibrates is very well
known. Even small deviations may lead to unacceptable differences in the time the
beam suggest and the real time. This causes the frequency tolerances of the MEMS
resonator to be very strict. An overview of the relevant frequency tolerances are given
in table 2.1, adopted from [8]. Furthermore, some physical properties of the MEMS
resonator are given in table 2.2.
Figure 2.1: MEMS resonator system.
Figure 2.2: Close-up of the micro beam.
Table 2.1: Overview of the frequency tolerances.
Property
static frequency @ 25° C
frequency change over temperature range (-20, 100) ° C
Tolerance
± 2 ppm
± 3 ppm
Table 2.2: Physical properties of the MEMS resonator.
Property
Material
Length L
Height H
Width B
Density ρ
Young’s modulus E
Poisson’s ratio ν
Thermal expansion coefficient α
Value
Silicon
44 µm
4 µm
1.4 µm
2330 kg/m3
131 GPa
0.28
4.2e-6 K-1
2
3
Modelling
3.1
Model of the MEMS resonator
A MEMS resonator’s most important part is a vibrating micro beam. The MEMS
resonator is modelled as a clamped-clamped beam with length L in x-direction, height
H in y-direction and width B in z-direction. This can also be seen in figure 3.1.
Figure 3.1: Model of a clamped-clamped beam MEMS resonator.
3.2
Modelling approach
Several different models are available for calculating mode shapes and natural
frequencies of vibrating beams. Two well-known models are the Euler-Bernoulli
beam model and the Timoshenko beam model. The biggest difference between these
models is that the Euler-Bernoulli model neglects shear and rotary inertia, while the
Timoshenko model takes these two in account. From literature, see for instance [1,2],
it is known that for a height / length ratio (H/L ratio) of around 1/10 and at higher
modes, shear and rotary inertia effects become important and lead to a significant
difference in natural frequencies. Since the dimensions of the beam are such that these
effects of shear and rotary inertia may play a role, natural frequencies and eigenmodes
are calculated both using the Euler-Bernoulli model and the Timoshenko model. This
is done for different H/L ratios, with L constant and H variable.
A further point of interest is pretension. Pretension can be caused by manufacturing
effects or by temperature differences. In this project, pretension is assumed to be
caused by temperature differences. From literature, see [5] or [6], it is known that
pretension affects the natural frequencies of vibrating beams, but it is not known
whether these effects are significant for micro beams. This will also be investigated in
this project, in order to find the effects of pretension on the natural frequencies and
mode shapes of the micro beam.
As a result, the following modelling approach is proposed, in which four different
cases will be investigated:
1. without shear and pretension, different H/L ratios
2. with shear and without pretension, different H/L ratios
3. without shear and with pretension, different H/L ratios, different pretensions
4. with shear and with pretension, different H/L ratios, different pretensions
The theoretical background on the different beam models will be described in the next
chapter.
3
4
Theoretical background
In this chapter the theoretical background used in this project will be discussed. First,
two commonly used beam theories, Euler-Bernoulli and Timoshenko, will be
explained regarding the used formulas and assumptions. In paragraph 4.3, the
influence of pretension for both the Euler-Bernoulli and the Timoshenko case will be
described. Finally, it will be described how the mode shapes and natural frequencies
for the two cases can be calculated using Finite Element Method (FEM), using both
Matlab and Marc-Mentat.
4.1
Euler-Bernoulli
In this project, two commonly used theories to describe the static and dynamic
behaviour of beams will be compared, namely Euler-Bernoulli beam theory and
Timoshenko beam theory. In this report, these theories will be referred to ‘Euler’ and
‘Timoshenko’.
4.1.1
Assumptions and differential equation
To describe the static and dynamic behaviour of beams, Euler-Bernoulli beam theory
uses the following assumptions:
1. The material of the beam is isotropic.
2. The material behaviour is linear elastic.
3. Rotary inertia is negligible.
4. Shear deformation is negligible.
Under these assumptions, the following partial differential equation of motion for
small deflections of an Euler beam can be written:
∂4 y
∂2 y
EI 4 + ρ A 2 = 0 ,
∂x
∂t
stiffness
(4.1)
mass inertia
where E is the Young’s modulus or modulus of elasticity [Pa], I is the second
moment of area [m4], y is de deflection of the beam [m], x is the coordinate along the
beam length [m], ρ is the density of the beam material [kg m-3], A is the cross section
area [m2] and t is the time [s]. When the beam vibrates in one of its natural modes, the
deflection y becomes (this is called separation of variables):
y ( x, t ) = Y ( x) cos(ω t ) ,
(4.2)
in which Y(x) is the modal displacement [m] and ω is the circular natural frequency
[rad/s]. This way, the differential equation becomes:
2
d 4Y
2 d Y
EI 4 − ρ Aω
= 0.
dx
dt 2
(4.3)
4
4.1.2
Mode shapes and natural frequencies
To determine the mode shapes and the natural frequencies of the beam, differential
equation (4.3) has to be solved. For a clamped-clamped beam, the boundary
conditions are as follows:
Y (0) = Y ( L) = 0 ,
(4.4)
and
dY (0) dY ( L)
=
=0.
dx
dx
(4.5)
The equations can be solved easiest by making them dimensionless. Therefore, a
dimensionless beam coordinate ξ [-] is defined as:
ξ=
x
.
L
(4.6)
By solving the differential equation, a characteristic equation for the natural frequency
and a general solution for the mode shapes can be obtained (the way to do this is
explained in more detail in the Timoshenko section, chapter 4.2 or can also be found
in Blevins [4]). For mode i, the general solution for the mode shapes can be shown to
be:
Yi (ξ ) = cosh(λi ξ ) − cos(λi ξ ) − σ i (sinh(λi ξ ) − sin(λi ξ )) ,
(4.7)
in which λi [-] is the solution of the characteristic or frequency equation,
cos(λ ) cosh(λ ) = 1 ,
(4.8)
and σi [-] a constant depending on λi:
σi =
cosh(λi ) − cos(λi )
.
sinh(λi ) − sin(λi )
(4.9)
Furthermore, the natural frequency fi [Hz] can be obtained by:
fi =
λi 2
2π L2
EI
,
ρA
(4.10)
where L is the length of the beam [m].
5
4.2
Timoshenko
The second model that has been used in this project is the Timoshenko model. It
differs from the Euler model in the fact that it takes shear and rotary inertia in
account.
4.2.1
Assumptions and differential equations
The assumptions in Timoshenko beam theory are the following:
1. The material of the beam is isotropic.
2. The material behaviour is linear elastic.
3. Rotary inertia is taken in account.
4. Shear deformation is taken in account.
So the Timoshenko model shares the isotropic and linear elastic material behaviour
assumptions with the Euler model and differs with respect to the inclusion of shear
and rotary inertia. These assumptions lead to a system of two coupled partial
differential equations, known as:
EI
∂ 2ψ
∂y
∂ 2ψ
+
AG
(
−
)
−
I
= 0,
κ
ψ
ρ
∂x
∂x 2
∂t 2
stiffness
2
ρA
shear
(4.11)
rotary inertia
2
∂ y
∂ y ∂ψ
− κ AG ( 2 −
) = 0,
2
∂x
∂t
∂x
mass inertia
(4.12)
shear
where ψ is the rotation of the beam [rad], G is the shear modulus [Pa] and κ is the
shear coefficient [-], which can be found to be for rectangular cross sections (see [4]):
κ=
10(1 + ν )
,
12 + 11ν
(3.13)
in which ν is the Poisson’s ratio [-]. A list of all symbols and their meaning can also
be found in appendix A. Above system of differential equations can be decoupled as
follows, adopted from reference [1]:
∂4 y
∂2 y
E
∂4 y
m2 R 2 ∂ 4 y
2
+
m
−
mR
(1
+
)
+
= 0,
κ G ∂x 2 ∂t 2 κ AG ∂t 4
∂x 4
∂t 2
∂ 4ψ
∂ 2ψ
E ∂ 4ψ
m2 R 2 ∂ 4ψ
EI 4 + m 2 − mR 2 (1 +
) 2 2 +
= 0,
κ G ∂x ∂t
κ AG ∂t 4
∂x
∂t
EI
(4.14)
(4.15)
where R is the radius of gyration [m],
R=
I
,
A
(4.16)
and m = ρA is the mass per length of the beam [kg/m]. According to reference [1], the
last term in (4.14) and (4.15) can be omitted due to its negligible contribution to yield:
6
∂4 y
∂2 y
E
∂4 y
2
+
m
−
mR
(1
+
)
= 0,
κ G ∂x 2 ∂t 2
∂x 4
∂t 2
∂ 4ψ
∂ 2ψ
E ∂ 4ψ
EI 4 + m 2 − mR 2 (1 +
)
= 0.
κ G ∂x 2 ∂t 2
∂x
∂t
EI
(4.17)
(4.18)
When the beam vibrates in one of its natural modes, the following equivalent forms
for the rotation ψ and the deflection y can be used:
y ( x, t ) = Y ( x) cos(ω t ) ,
ψ ( x, t ) = Ψ ( x) cos(ω t ) ,
(4.19)
(4.20)
where Ψ(x) is the modal rotation. The system of differential equation can be solved
easiest in dimensionless form, so the following dimensionless parameters are
introduced:
p2 =
ρ Aω 2 L4
EI
,
r=
R
,
L
b2 =
EI
.
κ AGL2
(4.21)
This way, the system of uncoupled differential equations becomes, as shown by
Abramovitch and Elishakoff in [1]:
2
d 4Y
2
2
2 d Y
+
p
(
r
+
b
)
− p 2Y = 0 ,
4
2
dξ
dξ
(4.22)
2
d 4Ψ
2
2
2 d Ψ
+
p
(
r
+
b
)
− p2Ψ = 0 .
4
2
dξ
dξ
(4.23)
4.2.2
Mode shapes and natural frequencies
The uncoupled set of differential equations, (4.22) and (4.23), have the following
general solutions for the transversal and rotational mode shape, see [1]:
Y (ξ ) = B1 cosh( ps1ξ ) + B2 sinh( ps1ξ ) + B3 cos( ps2ξ ) + B4 sin( ps2ξ ) ,
(4.24)
Ψ (ξ ) = C1 cosh( ps1ξ ) + C2 sinh( ps1ξ ) + C3 cos( ps2ξ ) + C4 sin( ps2ξ ) ,
(4.25)
where B1-4 and C1-4 are constants [-], depending on the mode of the beam and the
boundary conditions. The constants s1 and s2 [-] can be obtained from:
r 2 + b2 1
4
s1 = −
+
(r 2 + b 2 ) 2 + 2 ,
2
2
p
(4.26)
r 2 + b2 1
4
+
(r 2 + b 2 )2 + 2 ,
2
2
p
(4.27)
s2 = +
and finally, p can be obtained as the solution of the characteristic equation:
7
2 − 2 cosh( ps1 ) cos( ps2 ) +
p (3b 2 − r 2 + p 2 b 4 (b 2 + r 2 ))
sinh( ps1 )sin( ps2 ) = 0 .
1 + p 2b2 r 2
(4.28)
The constants B1-4 and C1-4 are not independent. Following the steps from reference
[1], so by substituting the general solutions, (4.24) and (4.25), into the differential
equations, (4.17) and (4.18), and taking in account (4.19) and (4.20), the following
relations between B1-4 and C1-4 can be found:
C1 =
p ( s12 + b 2 )
B2 ,
Ls1
C3 =
p ( s2 2 + b 2 )
− p ( s2 2 + b 2 )
B4 , C4 =
B3 .
Ls2
Ls2
C2 =
p ( s12 + b 2 )
B1 ,
Ls1
(4.29)
(4.30)
Since there is only interest in the transversal mode shape and not in the rotational
mode shape, above relations can be used in (4.24) and (4.25), together with the
boundary conditions:
Y (0) = Y (1) = 0 ,
Ψ (0) = Ψ (1) = 0 ,
(4.31)
(4.32)
to find expressions for B2-4 as a function of B1. Since there are four equations, one
would probably expect the ability to eliminate all four constants B1-4 in the general
equation for the mode shape. However, this cannot be done, for the simple fact that
one equation is already used to determine the characteristic (or frequency) equation.
The following expressions can be found for B2-4 as a function of B1 (a derivation of
the expressions can be found in appendix B.1):
 s2 2 − b 2  s1 


 sin( ps2 ) + sinh( ps1 )
s2  s12 + b 2 

B2 = B1
,
cos( ps2 ) − cosh( ps1 )
B3 = − B1 ,
 s12 + b 2  s2 


 sinh( ps1 ) + sin( ps2 )
s1  s2 2 − b2 

B4 = − B1
.
cos( ps2 ) − cosh( ps1 )
(4.33)
(4.34)
(4.35)
The value of B1 can be chosen arbitrarily, but for proper comparison between
different mode shapes, it is useful to choose a certain normalization. The
normalization chosen in this project is that the maximum value of the mode shape
equals 1, so:
max(Y ) = 1 .
(4.36)
Since p is already known from the characteristic equation, the natural frequencies fi
can be easily calculated from the definition of p:
8
fi =
p
2π L2
EI
.
ρA
(4.37)
Note that p is different for each eigenmode.
4.3
Pretension
From literature it is known (see for instance [2], [5] or [6]) that pretension affects the
mode shapes and the natural frequencies. It is unknown whether these effects are
significant for micro beams like in a MEMS resonator. This will be investigated in
this project.
4.3.1
Determining the range of pretensions
Pretension in the beam assumed to be caused by temperature differences. When
temperature increases or decreases compared to the reference temperature, the beam
material wants to expand or shrink. Due to the clamped-clamped boundary condition
the material however cannot expand or shrink freely. This results in tensile axial
pretension in case of a temperature decrease and compressive axial pretension in case
of a temperature increase. In Fenner [7] the following formula for calculating the axial
pretension can be found:
σ x = − Eα (T − Tref ) = − Eα∆T ,
(4.38)
where σx is the axial pretension [Pa], α is the thermal expansion coefficient [K-1], T is
the temperature [K], Tref is the reference temperature [K] and ∆T is the temperature
difference [K]. The reference temperature is set to room temperature, so 20° C. The
range of temperatures is set between -20° C and 100° C (the temperature range in
which the MEMS resonator should work properly, see table 2.1). This results in a
range of pretensions as listed in table 4.1. Note that a positive pretension indicates
tensile stress and a negative pretension indicates compressive stress.
Table 4.1: Pretensions at different temperatures.
T (°C) ∆T (°C) σx (MPa)
-20
-40
22
0
-20
11
20
0
0
40
20
-11
60
40
-22
80
60
-33
100
80
-44
9
4.3.2
Effect of pretension on Euler model
In both the Euler and Timoshenko case, pretension leads to an extra term in the
differential equation(s), namely:
±Nx
∂2 y
∂2 y
=
±
σ
A
,
x
∂x 2
∂x 2
(4.39)
where Nx is the axial force. The plus minus sign represents the difference between
tensile (-) and compressive (+) pretension. Thus, the partial differential equation in the
Euler case becomes:
EI
∂4 y
∂2 y
∂2 y
+
ρ
A
+
N
=0,
x
∂x 4
∂t 2
∂x 2
(4.40)
for compressive axial pretension and
EI
∂4 y
∂2 y
∂2 y
+
ρ
A
−
N
= 0,
x
∂x 4
∂t 2
∂x 2
(4.41)
in case of tensile axial pretension. These differential equations lead to a characteristic
equation for calculating the natural frequencies and a general solution of the
differential equation to calculate the mode shapes. These can be derived using exactly
the same procedure as has been done in the Timoshenko case without pretension
(section 4.2.2), although it becomes a little more complicated due to the extra term.
Here, only the resulting characteristic equation and general solution for the mode
shapes will be presented, for the derivation, see [5] for the compressive case and [6]
for the tensile case.
First, the following variables are defined, which help to represent the characteristic
equation and the general solution for the mode shapes in an easier way:
U=
N x L2
,
2 EI
Ω = ωL
ρA
EI
(4.42)
,
(4.43)
M = −U + U 2 + Ω 2 ,
(4.44)
N = +U + U 2 + Ω 2 ,
(4.45)
in which U is the relative axial force [-], Ω is the relative circular natural frequency [-]
and M and N are mode shape constants [-]. For compressive axially loaded beams, the
characteristic equation is as follows:
Ω − U sinh( M ) sin( N ) − Ω cosh( M ) cos( N ) = 0 .
(4.46)
The mode shapes for the compressive case can be found to be:
10
Y (ξ ) = sinh( M ξ ) +
M sin( N ) − N sinh( M )
M
(cosh( M ξ ) − cos( N ξ )) − sin( N ξ ) . (4.47)
N (cosh( M ) − cos( N ))
N
For tensile axially loaded beams, the characteristic equation is as follows:
Ω + U sinh( N ) sin( M ) − Ω cosh( N ) cos( M ) = 0 .
(4.48)
The mode shapes for the tensile case can be found to be:
Y (ξ ) = sinh( N ξ ) +
N sin( M ) − M sinh( N )
N
(cosh( N ξ ) − cos( M ξ )) − sin( M ξ ) . (4.49)
M (cosh( N ) − cos( M ))
M
It should be noted that both expressions for the mode shapes are not yet normalized.
The normalization used in this project is that the maximum of Y equals one. The
buckling load Nx,cr [N] can be expressed as:
N x ,cr =
4π 2 EI
.
L2
(4.50)
The buckling stress σx,cr [Pa] can be easily found by dividing the buckling load by the
cross-sectional area:
σ x ,cr =
N x ,cr
A
=
4π 2 EI
AL2
(4.51)
When the pretension is higher than the buckling stress, the beam will buckle and thus,
all formulas presented for eigenmodes and natural frequencies are not valid anymore.
In table 4.2 the maximum pretension is compared with the buckling stress for various
H/L ratios. It can be concluded that the pretension does not exceed or come close to
the buckling stress. Therefore, all presented formulas can be used.
Table 4.2: Buckling stress for various H/L ratios.
H/L ratio (-)
1/44
4/44
10/44
4.3.3
σx,cr (GPa)
-0.2226
-3.562
-22.26
σx,max (MPa)
-44
-44
-44
σx,max/σx,cr (%)
19.8
1.24
0.198
Effect of pretension on Timoshenko model
As mentioned in the Euler part about pretension, pretension affects the Timoshenko
model by introducing an extra term in the differential equations. Unfortunately, there
are only expressions available for Timoshenko beams under compressive axial loads
and not for tensile axial loads. Therefore, only compressive axial loads for the
Timoshenko model will be considered. The system of coupled differential equations
for Timoshenko beams under compressive axial loads is as follows:
11
N  ∂2 y
∂ψ 
ρ ∂2 y
+ 1 − x  2 −
=0,
κ G ∂t 2
∂x  κ GA  ∂x
∂ 2ψ
∂ 2ψ
 ∂y

EI 2 + κ GA  − ψ  − ρ I 2 = 0 .
∂x
∂t
 ∂x

−
(4.52)
(4.53)
Following the procedure explained in chapter 4.2 (see also [2]), expressions for the
mode shapes and natural frequencies can be obtained. First, some parameters are
defined:
I
EI
α = κ GA ,
r2 =
,
b2 =
,
(4.54)
2
AL
α L2
N x L2
ρ Aω 2 L4
k2 =
,
p2 =
,
(4.55)
Nx 
Nx 


EI 1 −
EI 1 −
α 
α 


in which α is the shear load [N], r is the relative radius of gyration [-], b is the stiffness
/ shear ratio [-], k is the relative axial load and p is the relative natural frequency [-].
Furthermore, s1 and s2 [-] are given by:
2
 1  2 2


N 
Nx  2 
1  
2
s1 = −  p 2  r 2 1 − x  + b 2  + k 2  +
+
b
+
k
+ 4 p 2 , (4.56)
 p  r 1 −



2  
α 
α 


 2   

2
 1  2 2


N 
Nx  2 
1  
s2 = +  p 2  r 2 1 − x  + b 2  + k 2  +
+ b  + k 2  + 4 p 2 . (4.57)
 p  r 1 −

2  
α 
α 


 2   

The characteristic or frequency equation (for determining p) can be found to be:

 

N 
k2 
p  p 2 b 4 − 1  r 2 1 − x  + b 2  + kb 4 + 4b 2 + 2 
α 
p 
 

2 − 2 cosh( s1 ) cos( s2 ) + 
sinh( s1 ) sin( s2 ) = 0 .
Nx  2 2
2 2 2 
1 + p b r 1 −
+b k
α 

(4.58)
(
)
The general solution for the transversal mode shape, obtained from [2], is as follows:
Y (ξ ) = B1 cosh( ps1ξ ) + B2 sinh( ps1ξ ) + B3 cos( ps2ξ ) + B4 sin( ps2ξ ) ,
(4.59)
in which the following expressions for the mode shape coefficients B2-4 can be found
as a function of B1 (see appendix B.2 for the derivation):
B2 = − B1
B3 = − B1 ,
M sinh( s1 ) + N sin( s2 )
,
M cosh( s1 ) − M cos( s2 )
(4.60)
(4.61)
12
B4 = B1
M ( M sinh( s1 ) + N sin( s2 ))
,
N ( M cosh( s1 ) − M cos( s2 ))
(4.62)
where M and N are given by:
Nx 

(4.63)
1 − α  ,


N 
s 2 − p2b2 
N= 2
1− x .
(4.64)

s2
α 

B1 can be obtained by choosing a normalization. In this project, the maximum of the
mode shape is set to one, so:
M =
s12 + p 2 b 2
s1
B1 =
1
.
max(Y )
(4.65)
The expression for the rotational mode shape is abandoned, since it is not of interest
in this project (although it is needed for determining the constants B2-4). Finally, the
buckling load for the Timoshenko case is given by:
N x ,cr
4π 2
.
= 2
L
4π 2
+
EI
α
(4.66)
For the most important H/L values the maximum axial pretension is compared to the
buckling stress (buckling load divided by area). It can be seen that the maximum
pretension does not exceed or come close to the buckling stress. Thus, the presented
formulas for calculating the natural frequencies and eigenmodes are valid.
Table 4.3: Buckling stress for various H/L ratios.
H/L ratio (-)
1/44
4/44
10/44
4.4
σx,cr (GPa)
-0.2215
-3.292
-14.72
σx,max (MPa)
-44
-44
-44
σx,max/σx,cr (%)
19.9
1.33
0.299
Finite Element Methods
Two different FE approaches are used in this project: one uses a self-written FEprogram in Matlab and the other uses the commercial FE software package MarcMentat. The self written FE program has the advantage that it is exactly known what
is done (in terms of formulas used etc.) to calculate mode shapes and natural
frequencies. Unfortunately, it is not capable of determining mode shapes and natural
frequencies when pretension is applied. Marc-Mentat has as advantage that it can be
used in case of pretension, but as disadvantage that it is a black box: some parameters
are put in, Marc-Mentat starts calculating and the end results pop up on the computer
screen. It is not exactly known how things are calculated. So if inaccurate or false
results are generated, it is unknown what causes them, which makes it difficult to
improve or compare results.
13
4.4.1
FEM using Matlab
With FEM, a structure (a beam in this case) is divided into a finite number of
elements. For each of these elements an element mass matrix Me and an element
stiffness matrix Ke can be defined. The element stiffness matrix is different for the
Euler and Timoshenko model, while the element mass matrix is the same for Euler
and Timoshenko. Expressions for these matrices can be found in [3]. The element
mass matrices and element stiffness matrices can be assembled to a system mass
matrix M and a system stiffness matrix K. By solving the eigenvalue problem:
[ K − M ω 2 ]u = 0 ,
(4.67)
natural frequencies and eigenmodes can be determined, where ω is the circular natural
frequency [rad/s] and u is the eigenmode. If the number of elements chosen is
sufficiently high, the exact solutions for mode shapes and natural frequencies will be
approximated. Matlab is used for assembling the element stiffness and mass matrices
into the system stiffness and mass matrices and for solving the eigenvalue problem. It
is not possible to use this FE method when pretension is applied, so it is only used for
the Euler and Timoshenko case without pretension.
4.4.2
FEM using Marc-Mentat
Marc-Mentat is also used as FE package for determining the mode shapes and natural
frequencies of the micro beam. For the case without pretension, calculating mode
shapes and natural frequencies is straightforward. The graphical user interface can be
used to draw a beam, to divide it into enough elements, to assign geometric and
material properties and the boundary conditions and, as a last step, to use a dynamic
modal load case to determine the mode shapes and the natural frequencies. For the
Euler case, the 2-node element 5 is used and for the Timoshenko case, the 3-node
element 45 is used (see Marc Mentat manual [9]).
However, assigning pretension to the beam in Marc-Mentat is not straightforward,
since it is not possible to assign stress to an element or putting forces on the beam
which should lead to the right pretensions. The method for introducing pretension in
the beam is by prescribing a displacement to one end of the beam. Here, the beam
starts with an initial length L0, different from the designed length L it should have
with the right pretension assigned:
L0 =
1
E
L=
L,
1+ ε
E +σx
(4.68)
where ε is the strain. As load case in Marc-Mentat, a displacement is prescribed as a
fixed value. The displacement and length L0 must be chosen such that, after
displacement, the beam has length L and the desired pretension. Therefore, the
displacement ∆x needs to be:
∆x =
ε
1+ ε
L=
σx
L.
E +σx
(4.69)
14
The other steps for calculating mode shapes and natural frequencies are the same as
for the case without pretension. Note that the load cases for the displacement and the
dynamic modal analysis must be arranged such that the modal analysis is done after
the displacement has been prescribed to the beam.
15
5
Analytical and numerical analysis
In the first four sections of this chapter the results of each of the four cases (Euler /
Timoshenko, with / without pretension, see also section 3.2) are presented. The results
are based on the theory, presented in the previous chapter. Next, in section 5.5, a
comparison is made between the four cases. The Matlab scripts that are used to
calculate the natural frequencies and eigenmodes can be found in appendix D.
5.1
Case 1: Euler without pretension
The first case is the case without shear and without pretension. For different H/L
ratios and for all three different calculation methods the mode shapes and natural
frequencies are presented.
5.1.1
Case 1: Natural frequencies
The first three natural frequencies for the case without shear and without pretension
are given in table 5.1. Three different calculation methods are used to calculate the
natural frequencies. First, a semi-analytic method is used. This method uses the
analytical equations of chapter 4. The method is called semi-analytic because it uses
the analytical equations, but they are solved using a numerical solver of Matlab. In the
table however, it is named analytic. The second calculation method used is the MarcMentat FE software package. In the table, this method is called Marc-Mentat. The
third method used is a self written FE program in Matlab. This method is called FEM
Matlab in the table. The method values in table 5.1 are calculated using the analytical
method. A comparison between the different calculation methods can be found in
tables 5.2-5.4 in paragraph 5.2.1.
Table 5.1: First three natural frequencies for Euler case without pretension (analytic).
H (µm)
1
2
3
4
5
6
7
8
9
10
L (µm)
44
44
44
44
44
44
44
44
44
44
f1 (MHz)
3.98118
7.96236
11.9435
15.9247
19.9059
23.8871
27.8682
31.8494
35.8306
39.8118
f2 (MHz)
10.9743
21.9485
32.9228
43.8971
54.8714
65.8456
76.8199
87.7942
98.7684
109.743
f3 (MHz)
21.5140
43.0279
64.5419
86.0558
107.570
129.084
150.598
172.112
193.626
215.140
From table 5.1 some conclusions can be drawn. Firstly, it can be seen that for higher
eigenmodes the frequency increases. For thicker beams (H higher) the natural
frequency also increases. The linearity of the Euler case also is remarkable: if H is
twice or ten times as large, the natural frequency also becomes twice or ten times as
large. The reason for this is that the height of the beam appears linearly in the natural
frequency equation (4.10):
fi =
λi 2
2π L2
λ2
λ2
EI
EBH 3
E
= i 2
= i 2H
.
ρ A 2π L 12 ρ BH 2π L
12 ρ
(5.1)
16
5.1.2
Case 1: Mode shapes
In figure 5.1 the mode shapes of H/L = 4/44 can be seen. There is hardly any
difference between the mode shapes for various H/L ratios. An example of this can be
seen in figure 5.2, where for H/L = 1/44, 4/44 and 10/44 the second mode is plotted.
1
0.8
0.6
0.4
Yi [-]
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
f1
f2
f3
0.2
0.4
0.6
0.8
1
0.8
1
ξ [-]
Figure 5.1: Mode shapes for H/L = 4/44.
1
0.8
0.6
0.4
Yi [-]
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
H/L = 1/44
H/L = 4/44
H/L = 10/44
0.2
0.4
0.6
ξ [-]
Figure 5.2: Second mode: comparison between different H/L ratios.
17
5.2
Case 2: Timoshenko without pretension
The second case is the case with shear and without pretension. For different H/L ratios
and for all three different calculation methods the mode shapes and natural
frequencies are presented.
5.2.1
Case 2: Natural frequencies
In tables 5.2-5.4 the natural frequencies for various calculation methods can be seen.
The natural frequencies are given for both the Euler (E) as the Timoshenko (T) case,
so they can be compared.
Table 5.2: Natural frequencies f1 for various calculation methods.
Beam dimensions
H (µm) L (µm)
1
44
2
44
3
44
4
44
5
44
6
44
7
44
8
44
9
44
10
44
Analytic
f1,E (MHz)
3.98118
7.96236
11.9435
15.9247
19.9059
23.8871
27.8682
31.8494
35.8306
39.8118
f1,T (MHz)
3.96741
7.35387
11.5864
15.1060
18.3716
21.3596
24.0628
26.4865
28.6451
30.5585
Marc-Mentat
f1,E (MHz) f1,T (MHz)
3.981
3.969
7.962
7.869
11.94
11.64
15.92
15.22
19.91
18.58
23.89
21.69
27.87
24.55
31.85
27.15
35.83
29.51
39.81
31.63
FEM Matlab
f1,E (MHz) f1,T (MHz)
3.98118
3.96845
7.96236
7.86194
11.9435
11.6122
15.9247
15.1630
19.9059
18.4734
23.8871
21.5185
27.8682
24.2883
31.8494
26.7849
35.8306
29.0194
39.8118
31.0092
Table 5.3: Natural frequencies f2 for various calculation methods.
Beam dimensions
H (µm) L (µm)
1
44
2
44
3
44
4
44
5
44
6
44
7
44
8
44
9
44
10
44
Analytic
f2,E (MHz)
10.9743
21.9485
32.9228
43.8971
54.8714
65.8456
76.8199
87.7942
98.7684
109.743
f2,T (MHz)
10.8868
21.2726
30.7621
39.1242
46.2905
52.3135
57.3145
61.4411
64.8399
67.6430
Marc-Mentat
f2,E (MHz) f2,T (MHz)
10.97
10.90
21.95
21.36
32.92
31.04
43.90
39.73
54.87
47.35
65.85
53.93
76.82
59.54
87.79
64.31
98.77
68.34
109.7
71.75
FEM Matlab
f2,E (MHz) f2,T (MHz)
10.9743
10.8975
21.9485
21.3523
32.9228
31.0043
43.8971
39.6263
54.8714
47.1309
65.8456
53.5425
76.8199
58.9569
87.7942
63.5031
98.7684
67.3163
109.743
70.5224
Table 5.4: Natural frequencies f3 for various calculation methods.
Beam dimensions
H (µm) L (µm)
1
44
2
44
3
44
4
44
5
44
6
44
7
44
8
44
9
44
10
44
Analytic
f3,E (MHz)
21.5140
43.0279
64.5419
86.0558
107.570
129.084
150.598
172.112
193.626
215.140
f3,T (MHz)
21.2135
40.7639
57.5661
71.2891
82.1550
90.6362
97.2402
102.409
106.492
109.754
Marc-Mentat
f3,E (MHz) f3,T (MHz)
21.51
21.25
43.03
41.05
64.54
58.45
86.06
73.11
107.6
85.16
129.1
94.95
150.6
102.9
172.1
109.3
193.6
114.6
215.1
119.0
FEM Matlab
f3,E (MHz) f3,T (MHz)
21.5140
21.2577
43.0279
41.0795
64.5419
58.4657
86.0558
73.0262
107.570
84.8683
129.084
94.3673
150.598
101.972
172.112
108.094
193.626
113.071
215.140
117.166
18
From tables 5.2-5.4 the following conclusions can be drawn. Firstly, it can be seen for
the Timoshenko case, just like the Euler case, that natural frequencies increase with
higher H/L ratios and higher modes. A difference between Euler and Timoshenko is
that the Euler frequencies are always higher than the corresponding Timoshenko
natural frequencies. These differences are larger at higher H/L ratios and higher
modes as can be seen from the tables, since shear has a greater influence at higher H/L
ratios and higher modes. The linearity in the Euler case is also lost at the Timoshenko
case. This is due the extra shear and rotary inertia terms in the Timoshenko partial
differential equations.
Furthermore, the various calculation methods can be compared for the Euler and
Timoshenko case. Considering the results of the Euler case for the various calculation
methods, it can be concluded that they match very well, since the first six numbers
exactly match. Therefore, the results of the Euler case seem to be reliable, since the
various calculation methods give identical results.
Considering the results for the Timoshenko case, it can be seen that the match
between the results for the various calculation methods is not very good; there are
differences between 5‰ and 6%. Differences are small at small H/L ratios and in the
first mode, differences become larger at higher H/L ratios and higher modes. Since all
three calculation methods give slightly different results, the results are a less reliable
than the Euler case, but since the overall trend in the results is clear, it is not a very
big problem. A possible explanation for this could be the use of a different shear
correction factor κ, see (3.13). There are multiple expressions for κ in use, see [4].
5.2.2
Case 2: Mode shapes
1
0.8
0.6
0.4
Yi [-]
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
f1
f2
f3
0.2
0.4
0.6
0.8
1
ξ [-]
Figure 5.3: Mode shapes for H/L = 4/44.
19
In figure 5.3 the Timoshenko mode shapes of H/L = 4/44 can be seen. When these are
compared to the Euler mode shapes (figure 5.1), it is hard to see any difference. Just
like the Euler case, mode shapes of various H/L ratios are very similar to each other.
Apparently, the mode shapes are independent of shear and H/L ratio.
5.3
Case 3: Euler with pretension
The third case is the case without shear and with pretension. For different pretensions
and different H/L ratios natural frequencies and mode shapes are given, calculated
using two different calculation methods, analytic and Marc-Mentat.
5.3.1
Case 3: Natural frequencies
In tables 5.5 – 5.7 the natural frequencies for various pretensions and three different
H/L ratios can be seen.
Table 5.5: Natural frequencies at H/L = 1/44 for various pretensions.
H/L = 1/44
σx (MPa)
22
11
0
-11
-22
-33
-44
Analytic
f1 (MHz)
4.16726
4.07539
3.98118
3.88445
3.78500
3.68261
3.57700
f2 (MHz)
11.2303
11.1031
10.9743
10.8438
10.7117
10.5778
10.4421
f3 (MHz)
21.7960
21.6555
21.5140
21.3715
21.2280
21.0836
20.9381
Marc-Mentat
f1 (MHz) f2 (MHz)
3.983
10.98
3.982
10.98
3.981
10.97
3.981
10.97
3.980
10.97
3.979
10.97
3.979
10.97
f3 (MHz)
21.52
21.52
21.51
21.51
21.51
21.50
21.50
Table 5.6: Natural frequencies at H/L = 4/44 for various pretensions.
H/L = 4/44
σx (MPa)
22
11
0
-11
-22
-33
-44
Analytic
f1 (MHz)
15.9724
15.9486
15.9247
15.9008
15.8769
15.8530
15.8290
f2 (MHz)
43.9618
43.9295
43.8971
43.8647
43.8322
43.7998
43.7673
f3 (MHz)
86.1268
86.0913
86.0558
86.0203
85.9848
85.9493
85.9137
Marc-Mentat
f1 (MHz) f2 (MHz)
15.93
43.91
15.93
43.90
15.92
43.90
15.92
43.89
15.92
43.88
15.92
43.87
15.91
43.87
f3 (MHz)
86.08
86.07
86.06
86.04
86.03
86.01
86.00
Table 5.7: Natural frequencies at H/L = 10/44 for various pretensions.
H/L = 10/44
σx (MPa)
22
11
0
-11
-22
-33
-44
Analytic
f1 (MHz)
39.8309
39.8213
39.8118
39.8022
39.7927
39.7831
39.7736
f2 (MHz)
109.769
109.756
109.743
109.730
109.717
109.704
109.691
f3 (MHz)
215.168
215.154
215.140
215.125
215.111
215.097
215.083
Marc-Mentat
f1 (MHz) f2 (MHz)
39.83
109.8
39.82
109.8
39.81
109.7
39.81
109.7
39.80
109.7
39.79
109.7
39.79
109.7
f3 (MHz)
215.2
215.2
215.1
215.1
215.1
215.0
215.0
In the tables, a positive σx denotes tensile stress and a negative σx denotes
compressive stress. Several things can be noticed about the results. Firstly, there can
be seen that when the pretension increases (going from negative stress to positive
stress), the natural frequency also increases. This is the case for all eigenmodes,
although it can be seen that for higher eigenmodes, this effect is less pronounced.
20
Therefore, the relative changes in natural frequency decrease at higher eigenmodes.
Furthermore, it can be noticed that these relative differences also decrease when the
H/L ratio increases. In order to see this, consider an illustrative example. When the
relative differences between a pretension of 22 MPa and -44 MPa are compared for
various H/L ratios and eigenmodes, the relative difference at H/L = 1/44 at first mode
is 14.2 %. The relative difference at the third mode is 3.94 %. The relative differences
at a H/L ratio of 10/44 can be found to be much smaller, namely 0.144 % at the first
mode and 0.040 % at the third mode (analytic results are used for calculation).
When the results for the two calculation methods (analytic and Marc-Mentat) are
compared, the following effect can be seen. Looking at the relative differences
between a pretension of 22 MPa and -44 MPa, at H/L = 1/44, at first mode, the
relative difference is 0.100 %. At the third mode, the relative difference is 0.930 %.
At H/L = 10/44, the relative differences are also 0.100 % at the first mode and 0.930
% at the third mode. Therefore, the results of Marc-Mentat seem to suggest that the
relative differences between a pretension of 22 MPa and -44 MPa are small
(compared to analytic) and almost independent of H/L ratio or eigenmode. Moreover,
Marc-Mentat suggests that at a certain pretension, the natural frequency varies
linearly with the H/L ratio, similar to the case without pretension (see section 5.1). For
example, at an H/L ratio of 1/44 and a pretension of 22 MPa, the first natural
frequency is 3.983 MHz. If the H/L ratio is ten times as large, so H/L = 10/44, the first
natural frequency is 39.83 MHz at 22 MPa, so also ten times as large. Therefore it is
likely that something is wrong with the Marc-Mentat results, since they are
inaccurate, especially at low H/L ratios.
The exact cause for this is unknown, but possible explanations are the following. The
first one has to do with the equations used by Marc-Mentat. Marc-Mentat may use
slightly different equations than the ones presented in chapter 4, containing more or
less terms. The manual of Marc-Mentat [9] does not contain any information about
the exact formulas that are used, so it is difficult to say more about this.
Numerical problems could be a second possible explanation. The dimensions of the
MEMS resonator beam are such that the numbers are in the range of the smallest
numbers Marc-Mentat can work with. This was experienced in the beginning of the
project. First, a 3D Euler element was used for the micro beam, which did not give
any results. Next, after the dimensions of the beam were slightly increased, results
were obtained.
Furthermore, it was found that the 2D Euler element was less vulnerable for this
effect, but it still may be affected by this kind of numerical problems. Concludingly, if
the dimensions of a beam are small enough, Marc-Mentat may not give good results.
This can be due to numerical errors that occur.
A final possible explanation is that the loading of the beam should be done in another
way. The proposed method (see section 4.2.2) may not be the right one. Still, the
pretension in Marc-Mentat is checked to have the right value and this has been found
to be the case.
To summarize, it is known that something in Marc-Mentat is wrong when natural
frequencies are calculated when a pretension is applied, but it is not known what the
cause is or how to improve it. This may be investigated further in future research.
21
5.3.2
Case 3: Mode shapes
1
0.8
0.6
0.4
Y [-]
0.2
0
-0.2
-0.4
f1
-0.6
f2
-0.8
-1
f3
0
0.2
0.4
0.6
0.8
1
ξ [-]
Figure 5.4: Mode shapes for H/L = 4/44 at a pretension of 22 MPa.
1
0.8
0.6
0.4
Y [-]
0.2
0
-0.2
-0.4
f1
-0.6
f2
-0.8
-1
f3
0
0.2
0.4
0.6
0.8
1
ξ [-]
Figure 5.5: Mode shapes for H/L = 4/44 at a pretension of -44 MPa.
22
In figure 5.4 the mode shapes are plotted for an H/L ratio of 4/44 at a pretension of 22
MPa. In figure 5.5 the mode shapes are plotted for the same H/L ratio, but at a
pretension of -44 MPa. When the figures are compared to each other and to the mode
shapes without pretension (figure 5.1), it is hard to see any difference. So, the mode
shapes are independent of the applied pretension. Furthermore, the mode shapes do
not differ for the various H/L ratios.
5.4
Case 4: Timoshenko with pretension
The fourth case is the case with shear and with pretension. For different pretensions
and three different H/L ratios, natural frequencies and mode shapes are given,
calculated using two different calculation methods, the analytical method and using
Marc-Mentat. Note that there are no analytical results at tensile pretensions, since
there is no analytical expression available for calculating eigenmodes and natural
frequencies (see also section 4.3.3).
5.4.1
Case 4: Natural frequencies
In tables 5.8 - 5.10 the natural frequencies for the Timoshenko case are given for
different H/L ratios and pretensions.
Table 5.8: Natural frequencies at H/L = 1/44 for various pretensions.
H/L = 1/44
σx (MPa)
22
11
0
-11
-22
-33
-44
Analytic
f1 (MHz)
3.96741
3.72087
3.45778
3.17408
2.86368
f2 (MHz)
10.8868
10.6668
10.4419
10.2119
9.97638
f3 (MHz)
21.2134
21.0076
20.7995
20.5894
20.3771
Marc-Mentat
f1 (MHz) f2 (MHz)
3.971
10.90
3.970
10.90
3.969
10.90
3.969
10.90
3.968
10.89
3.967
10.89
3.967
10.89
f3 (MHz)
21.26
21.26
21.25
21.25
21.25
21.24
21.24
Table 5.9: Natural frequencies at H/L = 4/44 for various pretensions.
H/L = 4/44
σx (MPa)
22
11
0
-11
-22
-33
-44
Analytic
f1 (MHz)
15.1060
15.0473
14.9884
14.9292
14.8699
f2 (MHz)
39.1242
39.0705
39.0170
38.9635
38.9099
f3 (MHz)
71.2891
71.2380
71.1873
71.1366
71.0860
Marc-Mentat
f1 (MHz) f2 (MHz)
15.22
39.74
15.22
39.74
15.22
39.73
15.21
39.73
15.21
39.72
15.21
39.71
15.21
39.71
f3 (MHz)
73.13
73.12
73.11
73.10
73.09
73.08
73.07
Table 5.10: Natural frequencies at H/L = 10/44 for various pretensions.
H/L = 10/44
σx (MPa)
22
11
0
-11
-22
-33
-44
Analytic
f1 (MHz)
30.5585
30.5353
30.5133
30.4915
30.4698
f2 (MHz)
67.6430
67.6193
67.5978
67.5766
67.5555
f3 (MHz)
109.754
109.729
109.707
109.684
109.662
Marc-Mentat
f1 (MHz) f2 (MHz)
31.64
71.77
31.63
71.76
31.63
71.75
31.63
71.74
31.62
71.74
31.62
71.73
31.61
71.72
f3 (MHz)
119.0
119.0
119.0
119.0
118.9
118.9
118.9
23
The tables of the Timoshenko case with pretension clearly show a similar trend as the
Euler case with pretension (tables 5.5-5.7). Firstly, it can be seen that when the
pretension increases (going from negative stress to positive stress), the natural
frequency also increases for both Timoshenko as Euler. This is true for all
eigenmodes, although it can be seen that for higher eigenmodes, the effect is lower.
Therefore, the relative changes in natural frequency decrease at higher eigenmodes.
Furthermore, it can be observed that these relative differences also decrease when the
H/L ratio increases.
When the results of the two calculation methods are compared, again the same
problem as in case 3 occurs. The results of Marc-Mentat suggest that the relative
differences between a pretension of 22 MPa and -44 MPa are small (compared to
analytic) and almost independent of H/L ratio or eigenmode. A similar explanation as
in section 5.3.1 holds.
5.4.2
Case 4: Mode shapes
1
0.8
0.6
0.4
Yi [-]
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
f1
f2
f3
0.2
0.4
0.6
0.8
1
ξ [-]
Figure 5.6: Mode shapes for H/L = 4/44 at a pretension of -44 MPa.
In figure 5.6 the mode shapes can be seen for H/L = 4/44 at a pretension of -44 MPa.
At first sight, the mode shapes may look exactly the same as all the others that have
been presented before. A close look to the right end of the graph (ξ = 1), however,
reveals that something is wrong: the mode shapes are not exactly equal to zero,
although this should be the case due to the applied boundary condition (Y(ξ = 1) = 0).
This effect occurs at every H/L ratio, every pretension and every eigenmode in the
Timoshenko case with pretension. However, this effect decreases at higher H/L ratios
or higher eigenmodes, as figures 5.6 and 5.7 show. When the pretension decreases
(for example from -44 MPa to -11 MPa), the mode shapes also are closer to zero at
24
the right end. This is showed in figure 5.8. Figure 5.9 clearly shows that a decrease of
the H/L ratio leads to a dramatic increase in the error that is made at the right side.
1
0.8
0.6
0.4
Yi [-]
0.2
0
-0.2
-0.4
-0.6
-0.8
f1
f2
f3
-1
0
0.2
0.4
0.6
0.8
1
ξ [-]
Figure 5.7: Mode shapes for H/L= 10/44 at a pretension of -44 MPa.
1
0.8
0.6
0.4
Yi [-]
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
f1
f2
f3
0.2
0.4
0.6
0.8
1
ξ [-]
Figure 5.8: Mode shapes for H/L = 4/44 at a pretension of -11 MPa.
25
1
0.8
0.6
0.4
Yi [-]
0.2
0
-0.2
-0.4
f1
-0.6
f2
-0.8
f3
-1
0
0.2
0.4
0.6
0.8
1
ξ [-]
Figure 5.9: Mode shapes for H/L = 1/44 at a pretension of -44 MPa.
Several things have been checked in order to determine the cause for the error at the
right side. Firstly, the pretension Matlab script has been run at a pretension of zero,
yielding exactly the same results as the script without pretension. Therefore, the
problem only occurs when a pretension is applied.
Secondly, it has been checked whether the applied pretension is close to or exceeds
the buckling stress. If this would be the case, strange results can be expected, since the
equations that have been used are not valid in that case. The results of this check have
been listed in table 5.11. As can be seen from the values in the table, the maximum
applied pretension σx of -44 MPa is not close to the buckling stress σx,cr. Therefore,
this is not the cause for the errors at the right side of the mode shape graphs.
Table 5.11: Buckling stress for various pretensions.
H/L ratio (-)
1/44
4/44
10/44
σx,cr (GPa)
-0.2215
-3.292
-14.72
σx,max (MPa)
-44
-44
-44
σx,max/σx,cr (%)
19.9
1.33
0.299
Lastly, it has been checked whether numerical errors occur in the mode shape
expressions. These mode shape expressions contain hyperbolic cosine, hyperbolic
sine, cosine and sine terms. Note that the hyperbolic cosine and hyperbolic sine equal:
cosh( x) =
e x + e− x
,
2
sinh( x) =
e x − e− x
.
2
(5.2)
These hyperbolic functions become very large if the argument x is large. Moreover,
cosh(x) ≈ sinh(x) if x is large. In the mode shapes, the cosh(x) terms have an opposite
26
sign as the sinh(x) terms. Therefore, at the right end of the mode shape graphs, where
the graphs should equal zero, a large positive cosh(x) term plus a large negative
sinh(x) term should equal exactly zero after small sin(x) and cos(x) terms are added.
This makes a perfect recipe for numerical errors. It would also explain why the error
is extremely small at the left of the graphs and large at the right end of the graphs. At
the right end of the graphs, the argument x is larger, so the sinh(x) and cosh(x) are
larger and closer to each other. Hence, the numerical error is larger.
However, the above explanation only holds if x is large. Therefore, the order of
magnitude of all the terms in the mode shape equation has been checked for the case
where the problems are worst (H/L = 1/44 and σx = -44 MPa). This leads to sin(x) and
cos(x) terms of around 1 and sinh(x) and cosh(x) terms of around 1000, with the first
four digits of sinh(x) and cosh(x) being the same. With these magnitudes some small
numerical errors could be expected in the mode shape calculation, but not to the
extent that has been found in figure 5.9.
Nevertheless, effort has been made to try to rewrite the mode shape expressions in
order for the errors to decrease. The mode shape expressions have been rewritten by
hand, without any result. Also, the mathematical package Maple has been used to
rewrite the equations. However, this has not given any improvements, either.
To summarize, the problems regarding the mode shape errors at the right side of the
graphs have not been solved. They are probably caused by numerical problems,
although that is not certain. It is recommended to investigate this problem further in
future research.
5.5
Comparison between the four cases
The goal of this section is to give an overview of the four cases. Therefore,
differences and similarities are presented. This will make it easier to draw
conclusions. First the differences and similarities of the natural frequencies are
discussed and secondly the differences and similarities of the mode shapes are given.
5.5.1
Comparison of the natural frequencies
In figure 5.10, a graph is plotted for the natural frequencies of both the Euler and
Timoshenko case without pretension as a function of H (values of analytic calculation
method are used). Since L is always kept the same (44 µm) in this project, the graphs
would be exactly the same as were they plotted as a function of H/L ratio. The figure
shows some important similarities and differences.
As a start, it can be seen that the natural frequencies increase if the H/L ratio
increases. This increase is bigger when the eigenmode is higher. This is true for both
Euler and Timoshenko. Difference is that, when the H/L ratio is increased, the Euler
natural frequencies increase linearly, while the Timoshenko natural frequencies
increase less than linearly. The reason for this is that the influence of shear is bigger at
higher H/L ratios. It can be seen that, for small H, the Timoshenko curves are tangent
to the Euler curves, while the differences grow larger at higher H/L ratios. Note
however, that the Timoshenko natural frequencies are always lower than the Euler
ones.
Also, natural frequencies are higher at higher modes. At higher modes the differences
between the Euler and Timoshenko natural frequencies are larger, since the effect of
shear plays a larger role at higher modes.
27
250
Euler
Timoshenko
200
f3
f [MHz]
150
100
f2
50
f1
0
1
2
3
4
5
6
H [µm]
7
8
9
10
Figure 5.10: Euler and Timoshenko natural frequencies for first three eigenmodes as a
function of H.
The results are in agreement with what can be found in literature (see [1] or [4]): until
a H/L ratio of around 1/10 the differences in natural frequency (at first eigenmode) are
small. At higher H/L ratios, shear and rotary inertia begin to play such a role, that
there can be quite some differences. At higher modes these effects start to play a role
at even lower H/L ratios. Figure 5.10 shows this nicely, since around H = 4,
differences between Euler and Timoshenko can be seen for mode 1. To be more
precise, the difference between Euler and Timoshenko is at that point 5.1 %. At higher
modes these differences can be seen earlier, just after H = 2 at second mode and just
before H = 2 at the third eigenmode. In summary, the differences between Euler and
Timoshenko at the first mode increase from 0.35 % at H/L = 1/44 to 23 % at H/L =
10/44. For the second, mode these differences increase from 0.80 % at H/L = 1/44 to
38 % at H/L = 10/44. For the third mode, these differences are 1.4 % and 49 %,
respectively. These differences have been calculated using the values from the
analytic calculation method.
Also, the influence of pretension on the Euler and Timoshenko model has been
investigated. Graphically, this can be seen in figure 5.11. In the figure the Euler and
Timoshenko natural frequencies are plotted as function of pretension for H/L = 1/44.
Note that the Timoshenko graphs only exist at compressive (negative) pretensions,
since there is no data for tensile pretensions (as explained earlier in section 4.3.3 and
5.4).
Several comments can be made about this figure regarding the influence of
pretension. At first, the influence may look small, since the slopes of the lines are
small. When the pretension increases (goes from negative to positive values), the
frequency increases slightly in both the Euler and the Timoshenko case. However, by
28
considering the numerical values, the differences in natural frequency caused by the
pretension are not very small at low modes and low H/L ratios. The difference for the
Euler case for the first mode (see tables 5.5 – 5.10) between a pretension of -44 and 0
MPa for an H/L ratio of 1/44 is 0.404 MHz. Since the frequency when no pretension
is applied is 3.98 MHz, this comes down to a relative difference of 10.2 %, which
cannot be considered small anymore. Timoshenko gives in this case a difference of
1.10 MHz, which makes a relative difference of 27.8 %, since the frequency when no
pretension is applied is 3.97 MHz. So apparently pretension has a much bigger effect
on the Timoshenko model than on the Euler model. This can also be seen directly
from the figure, since the Timoshenko line is steeper than the Euler line.
30
Euler
Timoshenko
25
f3
f [MHz]
20
15
f2
10
5
f1
0
-44
-33
-22
-11
σx [MPa]
0
11
22
Figure 5.11: Influence of pretension on Euler and Timoshenko natural frequencies for
H/L = 1/44.
Considering the influence of pretension on the natural frequencies of higher
eigenmodes, it can be seen that the differences in absolute value are about the same as
those of the first mode. Since the frequencies of the second and third modes are
higher, this translates to smaller relative differences. Again, the effect of pretension on
the Timoshenko model is bigger than on the Euler model.
Furthermore, the influence of pretension on Euler and Timoshenko has been
investigated for different H/L ratios. Here, for both Euler and Timoshenko,
differences in natural frequency for different pretensions decrease. Thus, relative
differences decrease even further, since the natural frequencies increase at higher H/L
ratios. For instance, relative differences are to 0.96 ‰ (first mode) and 0.26 ‰ (third
mode) for the Euler case and 2.9 ‰ and 0.84 ‰, respectively, for the Timoshenko
case with an H/L ratio of 10/44 (differences are for a pretension of -44 MPa relative to
the frequency at 0 MPa). Therefore, the influence of pretension is small at higher H/L
ratios.
29
5.5.2
Comparison of the mode shapes
The influence of H/L ratio, shear and pretension on the mode shapes has also been
investigated. It has been found that the changes in the mode shapes for different H/L
ratios, with or without shear (so Euler or Timoshenko model) and with or without
pretension are extremely small. Therefore, it can be stated that the mode shapes are
independent of these variables. All mode shapes therefore look like the ones that can
be seen in figure 5.12.
1
0.8
0.6
0.4
Yi [-]
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
f1
f2
f3
0.2
0.4
0.6
0.8
1
ξ [-]
Figure 5.12: Mode shapes for H/L = 4/44 for Euler with no pretension applied.
30
6
Conclusions and recommendations
In this project, the influence of shear and pretension on the natural frequencies and
mode shapes of a clamped-clamped beam MEMS resonator has been investigated.
When the influence of shear on the natural frequencies is considered, it has been
found that it depends on the Height/Length ratio of the MEMS resonator beam. At
low H/L ratios, in the first eigenmode, the differences between Euler and Timoshenko
are pretty small, in the order of several percents. At higher H/L ratios these
differences will increase up to 23 %. These differences will increase even further at
higher modes.
However, due to strict frequency tolerances for the MEMS resonator the model has to
predict the natural frequency very accurate (2 ppm). Since the differences between
Euler and Timoshenko, at all H/L ratios, are much larger, shear cannot be neglected.
Therefore, shear should be taken in account in the model that predicts the natural
frequencies of the MEMS resonator.
When the influence of pretension is considered, it has been found that this influence is
smaller than the influence of shear. The influence of pretension is found to be larger at
low H/L ratios than at higher H/L ratios. At higher modes the influence of pretension
is also smaller. It also has been found that the influence of pretension for the
Timoshenko beam is larger than for the Euler beam. The influence varies between
several promille till several percents. However, again the required accuracy is very
high (3 ppm), such that pretension should also be taken in account in the model that
predicts the natural frequencies of the MEMS resonator.
Considering the influence of shear and pretension on the mode shapes, it has been
found that differences in the mode shapes of the various cases are so small, that the
mode shapes can be considered to be independent of shear and pretension. For the
Timoshenko case with pretension, some numerical problems have occurred that have
not been solved yet.
Three different calculation methods to calculate the natural frequencies and mode
shapes have been used and have been compared. It has been found, for the Euler case,
that the results match very well. For the Timoshenko case, the results matched
reasonably well, although there are some small differences. For the cases with
pretension, Marc-Mentat gives some incorrect results which have not been solved.
Overall, the (semi-) analytic calculation method has been found to be the best. It is
easy to implement due to the simple geometry of the beam and it can deal with
pretension, while FEM with Matlab cannot. Furthermore, the use of exact formulas
gives a lot of insight, therefore, in case of problems or unexpected results, it is
straightforward to find and correct errors. Marc-Mentat does not have this advantage.
For a continuation of the research of this topic, it is recommended to investigate the
numerical problems that occur with the mode shapes in case of pretension.
Furthermore, it is recommended to further investigate the correct implementation of
pretension in Marc-Mentat.
31
Bibliography
[1] H. Abramovitch and I. Elishakoff. ‘Influence of shear deformation and rotary
inertia on vibration frequencies via Love’s equation,’ Journal of Sound and Vibration,
1990; 137(3): 516-522.
[2] H. Abramovitch. ‘Natural frequencies of Timoshenko beams under compressive
axial loads,’ Journal of Sound and Vibration, 1992; 157(1): 183-189.
[3] M.J. Beelen. ‘Validation of a method for error localization in finite element
models – Applied to a cantilever beam with a weld,’ Bachelor Final Project, DCT
internal report 2007.069, Technische Universiteit Eindhoven, 2007.
[4] R.D. Blevins. ‘Formulas for natural frequency and mode shape,’ Van Nostrand
Reinhold Company, 1979.
[5] A. Bokaian. ‘Natural frequencies of beams under compressive axial loads,’
Journal of Sound and Vibration, 1988; 126: 49-65.
[6] A. Bokaian. ‘Natural frequencies of beams under tensile axial loads,’ Journal of
Sound and Vibration, 1990; 142: 481-498.
[7] R.T. Fenner. ‘Mechanics of Solids,’ CRC Press, 1999: 113-115.
[8] R.M.C. Mestrom. ‘Microelectromechanical Oscillators – a literature survey,’
DCT internal report 2007.123, Technische Universiteit Eindhoven, 2007.
[9] MSC Software. ‘Marc 2005 r3 manual, Volume E: element library,’ 2005.
32
Appendices
A
Nomenclature
A.1
Latin symbols
Symbol
A
B
B1-4
b
C1-4
E
fi
G
H
I
k
L
L0
M
m
N
Nx
Nx,cr
p
R
r
s1,2
T
Tref
∆T
t
U
x
∆x
Y
y
y
z
Meaning
area of cross section
width
mode shape coefficient
stiffness / shear ratio
mode shape coefficient
Young’s modulus
natural frequency
shear modulus
height
second moment of area
relative axial load
length
initial length
mode shape coefficient
mass per length
mode shape coefficient
axial force
buckling load
relative natural frequency
radius of gyration
relative radius of gyration
mode shape coefficient
temperature
reference temperature
temperature difference
time
relative axial force
x-coordinate
displacement
modal deflection
y-coordinate
deflection
z-coordinate
Unit
[m]
[m]
[-]
[-]
[-]
[Pa]
[Hz]
[Pa]
[m]
[m4]
[-]
[m]
[m]
[-]
[kg/m]
[-]
[N]
[N]
[-]
[m]
[-]
[-]
[K]
[K]
[K]
[s]
[-]
[-]
[m]
[-]
[-]
[m]
[-]
33
A.2
Greek symbols
Symbol
α
α
ε
κ
λi
ν
ξ
ρ
σi
σx
σx,cr
Ψ
ψ
Ω
ω
Meaning
thermal expansion coefficient
shear load
strain
shear coefficient
solution of frequency equation
Poisson’s ratio
dimensionless beam coordinate
density
mode shape coefficient
axial pretension
critical buckling stress
modal rotation
rotation
relative circular natural frequency
circular natural frequency
Unit
[K-1]
[N]
[-]
[-]
[-]
[-]
[-]
[kg/m3]
[-]
[Pa]
[Pa]
[-]
[rad]
[-]
[rad/s]
34
B
Derivation of mode shape coefficients
B.1
Mode shape coefficients Timoshenko without pretension
The mode shape coefficients B2-4 as function of B1 can be found by inserting the
boundary conditions in the general solutions for the transversal and rotational mode
shapes. First, some helpful hyperbolic sine and cosine expressions are:
cosh( x) =
e x + e− x
,
2
d
(cosh( x)) = sinh( x) ,
dx
cosh(0) = 1 ,
sinh( x) =
e x − e− x
,
2
d
(sinh( x)) = cosh( x) ,
dx
sinh(0) = 0 .
(B.1)
(B.2)
(B.3)
The general solutions of the differential equations (4.24), (4.25) are:
Y (ξ ) = B1 cosh( ps1ξ ) + B2 sinh( ps1ξ ) + B3 cos( ps2ξ ) + B4 sin( ps2ξ ) ,
(B.4)
p ( s12 + b 2 )
p ( s12 + b 2 )
B2 cosh( ps1ξ ) +
B1 sinh( ps1ξ )
s1
s1
LΨ (ξ ) =
+
p ( s2 2 − b 2 )
p ( s2 2 − b 2 )
B4 cos( ps2ξ ) −
B3 sin( ps2ξ ) .
s2
s2
(B.5)
The boundary conditions are:
Y (0) = 0 ,
LΨ (0) = 0 ,
Y (1) = 0 ,
LΨ (1) = 0 .
(B.6)
(B.7)
From Y (0) = 0 , it follows that:
B1 + B3 = 0 ,
(B.8)
B3 = − B1 .
(B.9)
Applying boundary condition LΨ (0) = 0 gives:
p ( s12 + b 2 )
p ( s2 2 − b 2 )
B2 +
B4 = 0 ,
s1
s2
(B.10)
 s2 2 − b2   s1 
B2 = − 
B .
 2
2  4
 s2   s1 + b 
(B.11)
35
Applying LΨ (1) = 0 results in:
 s12 + b 2

s2 2 − b 2
pB1 
sinh( ps1 ) +
sin( ps2 ) 
s2
 s1

2
2
p ( s2 − b )
B4 (cos( ps2 ) − cosh( ps1 )) = 0 ,
+
s2
 s12 + b 2  s2 


 sinh( ps1 ) + sin( ps2 )
s1  s2 2 − b2 

B4 = − B1
.
cos( ps2 ) − cosh( ps1 )
(B.12)
(B.13)
Finally, substituting (B.13) in (B.11) gives the last unknown mode shape coefficient
B2 as a function of B1:
 s2 2 − b 2  s1 


 sin( ps2 ) + sinh( ps1 )
s2  s12 + b 2 

B2 = B1
.
cos( ps2 ) − cosh( ps1 )
B.2
(B.14)
Mode shape coefficients Timoshenko with pretension
The mode shape coefficients for the Timoshenko case with compressive axial
pretension can be found in exactly the same way as for the Timoshenko case without
pretension (section B.1). In fact, the expressions are very similar. First, two mode
shape constants M and N are defined, to help express the mode shape coefficients in
an easier way:
Nx 

1 − α  ,


2
2 2
N 
s −pb 
N= 2
1− x .

s2
α 

M =
s12 + p 2 b 2
s1
(B.15)
(B.16)
The general solutions for the transversal and rotational mode shape are:
Y (ξ ) = B1 cosh( s1ξ ) + B2 sinh( s1ξ ) + B3 cos( s2ξ ) + B4 sin( s2ξ ) ,
(B.17)
LΨ (ξ ) = MB2 cosh( s1ξ ) + MB1 sinh( s1ξ ) + NB4 cos( s2ξ ) − NB3 sin( s2ξ ) .
(B.18)
The boundary conditions are:
Y (0) = 0 ,
LΨ (0) = 0 ,
Y (1) = 0 ,
LΨ (1) = 0 .
(B.19)
(B.20)
From Y (0) = 0 , it follows that:
B1 + B3 = 0 ,
(B.21)
B3 = − B1 .
(B.22)
36
Applying boundary condition LΨ (0) = 0 gives:
MB2 + NB4 = 0 ,
M
B4 = − B2 .
N
(B.23)
(B.24)
Applying LΨ (1) = 0 results in:
MB2 cosh( s1 ) + MB1 sinh( s1 ) − MB2 cos( s2 ) + NB1 sin( s2 ) = 0 ,
B2 = − B1
M sinh( s1 ) + N sin( s2 )
.
M cosh( s1 ) − M cos( s2 )
(B.25)
(B.26)
Finally, substituting (B.26) in (B.24) gives the last unknown mode shape coefficient
B4 as a function of B1:
B4 = B1
M ( M sinh( s1 ) + N sin( s2 ))
.
N ( M cosh( s1 ) − M cos( s2 ))
(B.27)
37
C
Matlab files
C.1
Euler without pretension
% Calculate natural frequencies and eigenmodes without shear and
% without axial pretension
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
Variables
x = position in x-direction [m] (x = 0 at left end of the beam)
y = position in y-direction [m] (y = 0 at the middle of the beam)
z = position in z-direction [m] (z = 0 at the middle of the beam)
L = length of the beam [m]
H = height of the beam [m]
B = width of the beam [m]
I = second moment of area [m^4]
rho = density [kg.m^-3]
E = Young's modulus [Pa]
m = mass per meter [kg.m^-1]
nu = Poisson's ratio [-]
fi = natural frequency i (i = 1,2,3) [Hz]
labdai = solution of frequency equation of mode i [-]
sigmai = constant depending of labdai [-]
ytildei = modeshape at natural frequency i [-]
ksi = x/L = dimensionless beam coordinate [-]
n = number of eigenmodes and natural frequencies that will be
calculated
% Set some defaults
format long e;
clc;
clear all;
% Set some plotting defaults
set(0,'DefaultLineLineWidth',2)
set(0,'DefaultAxesFontSize',12)
% Values of various parameters
L = 44e-6;
H = 4e-6;
B = 1.4e-6;
rho = 2330;
E = 131e9;
nu = 0.28;
n = 3;
I = (1/12) * B * H^3;
m = B * H * rho;
labda = [ ]; % make vector for labdas at natural frequency i
% calculation of labdai
for i = 1 : n % calculate labdai for the first n natural frequencies
labdai = fzero(@(labda)cos(labda).*cosh(labda)-1,(2*i+1)*pi/2);
labda = [labda labdai];
end
sigma = [ ]; % make vector for the sigmas of natural frequency i
% Calculate sigmai
for i = 1 : n % calculate sigma for first n natural frequencies
sigmai = ( cosh(labda(1,i)) - cos(labda(1,i)) ) ./ ( sinh(labda(1,i)) sin(labda(1,i)) );
sigma = [sigma sigmai];
end
f = [ ]; % make vector with natural frequencies
% Calculate fi
for i = 1 : n % calculate first n natural frequencies
fi = labda(1,i).^2 ./ (2*pi*L.^2) .* sqrt((E*I)./m);
f = [f ; fi];
end
f % print the column of natural frequencies to the screen
% range of x-coordinate is 0 to L
x = 0:1e-8:L;
ksi = x / L;
ytilde = [ ]; % make a matrix with the values of ytilde
38
% Calculate ytildei
for i = 1 : n % calculate ytildei for all n natural frequencies
ytildei = cosh(labda(1,i)*ksi) - cos(labda(1,i)*ksi) - sigma(1,i)*(
sinh(labda(1,i)*ksi) - sin(labda(1,i)*ksi) );
ytildei = ytildei ./ max(abs(ytildei));
ytilde = [ytilde ; ytildei];
end
% Plotting of the results
figure
grid
hold on
plot(ksi,ytilde(1,:),'k')
plot(ksi,ytilde(2,:),'k--')
plot(ksi,ytilde(3,:),'k:')
legend('f_1','f_2','f_3','Location','SW')
xlabel('\xi [-]')
ylabel('Y_i [-]')
hold off
C.2
Euler without pretension FEM
% FEM calculation for clamped-clamped beam
% Euler beam elements are used
% Case: no shear, no pretension
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
Variables
m = number of elements [-]
n = number of nodes [-]
dof = degrees of freedom [-]
L = length of the beam [m]
B = width of the beam [m]
H = heigth of the beam [m]
K = system stiffness matrix [N/m]
Ke = element stiffness matrix [N/m]
M = system mass matrix [kg]
Me = element mass matrix [kg]
l = element length [m]
E = Young's modulus [Pa]
I = second moment of area [m^4]
A = area of cross section [m^2]
rho = density [kg/m^3]
k = number of natural frequencies that are calculated [-]
U = eigencolomn [-]
f = vector with natural frequencies [Hz]
% set some defaults
format long e; clc; close all;
% values of the different parameters
L = 44e-6;
B = 1.4e-6;
H = 4e-6;
m = 100;
n = m + 1;
dof = 2 * (n - 2);
l = L / m;
E = 1.31e11;
A = B * H;
I = 1/12 * B * H^3;
rho = 2330;
k = 3;
% determine element stiffness matrix
Ke = E * I / l^3 * [ 12
6*l -12
6*l
6*l 4*l^2 -6*l 2*l^2
-12 -6*l
12 -6*l
6*l 2*l^2 -6*l 4*l^2];
% Make empty matrix for system stiffness matrix
K = sparse(2*n,2*n);
% build system stiffness matrix from the element stiffness matrices
for i = 1:m
ni = (2*i-1):(2*i+2);
K(ni,ni) = K(ni,ni) + Ke;
39
end
% Partioning of K: the displacements and rotations of the nodes at
% both ends of the beam are zero. Hence, the first and the last two
% rows and colomns do not give a contribution. These will be
% eliminated from K.
K = K(3:2*n-2,3:2*n-2);
% determine element mass matrix
Me = rho * A * l / 420 * [ 156
22*l
54 -13*l
22*l 4*l^2 13*l -3*l^2
54
13*l
156 -22*l
-13*l -3*l^2 -22*l 4*l^2];
% Make an empty matrix for the system mass matrix
M = sparse(2*n,2*n);
% Build up the system mass matrix from the element mass matrices
for i = 1:m
ni = (2*i-1):(2*i+2);
M(ni,ni) = M(ni,ni) + Me;
end
% Partioning of M: idem as K
M = M(3:2*n-2,3:2*n-2);
% Natural frequencies and eigenmodes follow from the characteristic
% equation: [K - omega^2*M]*U = 0 or K*U = omega^2*M*U
[U,F] = eigs(K,M,k,'SM'); % calculate the solution of the equation
% calculate omega and sort in order of magnitude
[omega,iom] = sort(sqrt(diag(F)));
U = U(:,iom); % sort U in order of magnitude of the natural frequencies
%
%
%
%
%
U consists of the rotation and displacements of the nodes. Since
only the eigenmode as function of displacement is needed, are de
displacements extracted from vector U. Since the displacements are
positioned at uneven spots, are only the uneven terms of this
vector U needed.
% replace U with only the uneven terms of U
V = zeros(n-2,k);
for i = 1:k
V(:,i) = U(1:2:dof-1,i);
end
% Norm the eigencolomns to max(V) = 1
for i = 1:k
V(:,i) = V(:,i)/max(abs(V(:,i)));
end
% Change frequencies in rad/s to Hz
f = omega ./(2*pi)
% Plotting of the eigenmodes
% Add boundary conditions: displacements at both ends of the beam are % 0, so W =
extension of V with first and last element is 0
W = zeros(n,k);
W(2:n-1,1:k) = W(2:n-1,1:k) + V;
ksi = 0 : 1/m : 1;
figure
plot(ksi,W(:,1))
hold on;
plot(ksi,W(:,2))
plot(ksi,W(:,3))
C.3
Timoshenko without pretension
% Calculation of natural frequencies and eigenmodes with shear and
% without axial pretension
%
%
%
%
%
%
Variables
x = position in x-direction [m] (x = 0 at left end of the beam)
y = position in y-direction [m] (y = 0 at the middle of the beam)
z = position in z-direction [m] (z = 0 at the middle of the beam)
L = length of the beam [m]
H = height of the beam [m]
40
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
B = width of the beam [m]
I = second moment of area [m^4]
rho = density [kg.m^-3]
E = Young's modulus [Pa]
m = mass per meter [kg.m^-1]
nu = Poisson's ratio [-]
A = area of cross section [m^2]
fi = natural frequency i (i = 1,2,3) [Hz]
Yi = modeshape at natural frequentie i [-]
ksi = x/L = dimensionless beam coordinate [-]
omega = 2*pi*fi = circular natural frequency [rad/s]
p(i) = sqrt(m*omega^2*L^4/(E*I)) = relative natural frequency [-]
R = sqrt(I/A) = radius of gyration [m]
r = R / L [-]
G = shear modulus [Pa]
k = shear coefficient [-]
b = sqrt((E*I)/(k*A*G*L^2)) [-]
s1 = coefficient depending on r, b en p [-]
s2 = coefficient depending on r, b en p [-]
n = number of calculated natural frequencies [-]
B1 = normalisation constant for the modeshapes [-]
% set some defaults
format long e;
clc;
clear all;
close all;
set(0,'DefaultLineLineWidth',2)
set(0,'DefaultAxesFontSize',12)
% Values for various parameters
L = 44e-6;
H = 4e-6;
B = 1.4e-6;
rho = 2330;
E = 131e9;
nu = 0.28;
n = 3;
I = (1/12) * B * H^3;
m = B * H * rho;
A = H * B;
% calculate shear modulus G = E/(2*(1 + nu))
G = E/(2*(1 + nu));
% calculate k (formula valid for rectangular cross sections)
k = 10*(1 + nu)/(12 + 11*nu);
R = sqrt(I/A);
r = R / L;
b = sqrt((E*I)/(k*A*G*L^2));
p = [ ]; % make a vector for p(i)
% calculate p(i) as the solution of: 2 - 2*cosh(p*s1)*cos(p*s2)
% + p*((3*b^2 - r^2) + p^2*b^4*(b^2 + r^2)) / (1 + p^2*b^2*r^2)
% * sinh(p*s1)*sin(p*s2) = 0
for i = 1 : n
% The formula as function of i that determines the point around
% fzero tries to calculate the zero of the characteristic
% equation is as follows for various values of H:
% H = 1
39*i
% H = 2 t/m 7
30*i
% H = 8 t/m 10
20*i
px = fzero(@(p) 2 - 2.*cosh(p.*sqrt(-0.5.*(r.^2 + b.^2) + 0.5.*sqrt((r.^2 +
b.^2).^2 + 4./p.^2))).*cos(p.*sqrt(0.5*(r.^2 + b.^2) + 0.5.*sqrt((r.^2 + b.^2).^2 +
4./p.^2))) + p.*((3*b.^2 - r.^2) + p.^2.*b.^4.*(b.^2 + r.^2)) ./ (1 +
p.^2.*b.^2.*r.^2) .* sinh(p.*sqrt(-0.5.*(r.^2 + b.^2) + 0.5.*sqrt((r.^2 + b.^2)^2 +
4./p.^2))).*sin(p.*sqrt(0.5.*(r.^2 + b.^2) + 0.5.*sqrt((r.^2 + b.^2).^2 +
4./p.^2))),30*i);
p = [p px];
end
f = [ ]; % make vector for fi
% calculate fi
for i = 1 : n
fi = p(1,i) / (2*pi*L^2*sqrt(m / (E*I)));
f = [f ; fi];
end
f % print the natural frequencies on the screen
% range of x-coordinate is 0 to L
41
x = 0:1e-8:L;
ksi = x / L;
Y = [ ]; % make matrix for mode shapes Yi
for i = 1 : n
% coefficients s1 and s2 are dependent on p so are calculated for every p
s1 = sqrt(-0.5 * (r^2 + b^2) + 0.5 * sqrt((r^2 + b^2).^2 + 4/(p(1,i))^2));
s2 = sqrt(0.5 * (r^2 + b^2) + 0.5 * sqrt((r^2 + b^2).^2 + 4/(p(1,i))^2));
% calculate mode shape Yi
Yi = modeshape(ksi,p(1,i),s1,s2,b);
% calculate normalisation constant B1
B1 = 1 / max(abs(Yi));
Yi = B1 .* Yi;
Y = [Y ; Yi];
end
% Plotting the results
figure
grid
hold on
plot(ksi,Y(1,:),'k')
plot(ksi,Y(2,:),'k--')
plot(ksi,Y(3,:),'k:')
legend('f_1','f_2','f_3','Location','SW')
xlabel('\xi [-]')
ylabel('Y_i [-]')
hold off
C.4
Timoshenko without pretension FEM
% FEM calculation for clamped-clamped beam
% Timoshenko beam elements are used
% Case: with shear, without pretension
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
Variables
m = number of elements [-]
n = number of nodes [-]
dof = degrees of freedom [-]
L = length of the beam [m]
B = width of the beam [m]
H = heigth of the beam [m]
K = system stiffness matrix [N/m]
Ke = element stiffness matrix [N/m]
M = system mass matrix [kg]
Me = element mass matrix [kg]
l = element length [m]
E = Young's modulus [Pa]
I = second moment of area [m^4]
A = area of cross section [m^2]
rho = density [kg/m^3]
k = number of natural frequencies that are calculated [-]
U = eigencolomn [-]
f = vector with natural frequencies [Hz]
phi = correction factor for the influence of shear [-]
kappa = shear coefficient [-]
nu = poisson's ratio [-]
G = shear modulus [Pa]
% set some defaults
format long e; clc; close all;
% values of the different parameters
L = 44e-6;
B = 1.4e-6;
H = 4e-6;
m = 100;
n = m + 1;
dof = 2 * (n - 2);
l = L / m;
E = 1.31e11;
A = B * H;
I = 1/12 * B * H^3;
rho = 2330;
k = 3;
nu = 0.28;
42
kappa = 10*(1 + nu)/(12 + 11*nu);
G = E / (2*(1 + nu));
phi = 12 * E * I / (G * A * kappa * l^2);
% determine element stiffness matrix
Ke = E * I / (l^3*(1+phi)) * [ 12
6*l -12
6*l
6*l l^2*(4+phi) -6*l l^2*(2-phi)
-12
-6*l
12
-6*l
6*l l^2*(2-phi) -6*l l^2*(4+phi)];
% Make empty matrix for system stiffness matrix
K = sparse(2*n,2*n);
% build system stiffness matrix from the element stiffness matrices
for i = 1:m
ni = (2*i-1):(2*i+2);
K(ni,ni) = K(ni,ni) + Ke;
end
% Partioning of K: the displacements and rotations of the nodes at
% both ends of the beam are zero. Hence, the first and the last two
% rows and colomns do not give a contribution. These will be
% eliminated from K.
K = K(3:2*n-2,3:2*n-2);
% determine element mass matrix
Me = rho * A * l / 420 * [ 156
22*l
54 -13*l
22*l 4*l^2 13*l -3*l^2
54
13*l
156 -22*l
-13*l -3*l^2 -22*l 4*l^2];
% Make an empty matrix for the system mass matrix
M = sparse(2*n,2*n);
% Build up the system mass matrix from the element mass matrices
for i = 1:m
ni = (2*i-1):(2*i+2);
M(ni,ni) = M(ni,ni) + Me;
end
% Partioning of M: idem as K
M = M(3:2*n-2,3:2*n-2);
% Natural frequencies and eigenmodes follow from the characteristic
% equation: [K - omega^2*M]*U = 0 or K*U = omega^2*M*U
[U,F] = eigs(K,M,k,'SM'); % calculate the solution of the equation
% calculate omega and sort in order of magnitude
[omega,iom] = sort(sqrt(diag(F)));
U = U(:,iom); % sort U in order of magnitude of the natural frequencies
% U consists of the rotation and displacements of the nodes. Since
% only the eigenmode as function of displacement is needed, are the
% displacements extracted from vector U. Since the displacements are % positioned at
uneven spots, are only the uneven terms of this
% vector U needed.
% replace U with only the uneven terms of U
V = zeros(n-2,k);
for i = 1:k
V(:,i) = U(1:2:dof-1,i);
end
% Norm the eigencolomns to max(V) = 1
for i = 1:k
V(:,i) = V(:,i)/max(abs(V(:,i)));
end
% Change frequencies in rad/s to Hz
f = omega ./(2*pi)
% Plotting of the eigenmodes
% Add boundary conditions: displacements at both ends of the beam are % 0, so W =
extension of V with first and last element is 0
W = zeros(n,k);
W(2:n-1,1:k) = W(2:n-1,1:k) + V;
ksi = 0 : 1/m : 1;
figure
plot(ksi,W(:,1))
hold on;
plot(ksi,W(:,2))
43
plot(ksi,W(:,3))
C.5
Euler with compressive pretension
% Calculation of natural frequencies and eigenmodes without shear and % with
compressive axial pretension
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
Variables
x = position in x-direction [m] (x = 0 at left end of the beam)
y = position in y-direction [m] (y = 0 at the middle of the beam)
z = position in z-direction [m] (z = 0 at the middle of the beam)
L = length of the beam [m]
H = height of the beam [m]
B = width of the beam [m]
I = second moment of area [m^4]
rho = density [kg.m^-3]
E = Young's modulus [Pa]
A = area of cross section [m^2]
O, OMEGA = relative natural frequency [-]
a = sqrt((EI)/(rho*A) [m^2/s]
U = relative axial force [-]
cs = compressive pretension [Pa]
T = axial compressive force [N]
omega = natural frequency [rad/s]
M, N = mode shape coefficient [-]
ci (i=1,2,3,4) = mode shape coefficient [-]
k = number of natural frequencies [-]
ksi = dimensionless beam coordinate [-]
% set defaults
format long e;
clc;
clear all;
close all;
% values for various parameters
L = 44e-6;
H = 4e-6;
B = 1.4e-6;
rho = 2330;
E = 131e9;
nu = 0.28;
I = (1/12) * B * H^3;
A = H * B;
cs = 1e8;
T = cs * A;
a = sqrt((E*I) / (rho * A));
U = T*L^2 / (2*E*I);
k = 3;
% Solve the characteristic equation with O as unknown
OMEGA = [ ]; % make empty vector
for i = 1:k
Oi = fzero(@(O)O-U*sinh(sqrt(-U+sqrt(U^2+O^2)))*sin(sqrt(U+sqrt(U^2+O^2)))O*cosh(sqrt(-U+sqrt(U^2+O^2)))*cos(sqrt(U+sqrt(U^2+O^2))),((2*i+1)*0.5*pi)^2);
OMEGA = [OMEGA; Oi];
end
% calculate natural frequencies
f = OMEGA * a / (2 * pi * L^2)
% for calculating the mode shapes some coefficients have to be
% determined: first M and N, thereafter ci (i = 1,2,3,4) which are a
% function of M and N. These constants are different for each natural % frequency.
% Calculation of M en N
M = [ ]; % make empty vector for M
N = [ ]; % make empty vector for N
for i = 1:k
Mi = sqrt(-U + sqrt(U^2 + OMEGA(i,1)^2));
Ni = sqrt(U + sqrt(U^2 + OMEGA(i,1)^2));
M = [M ; Mi];
N = [N ; Ni];
end
44
% Calculation of ci (i = 1,2,3,4)
c1 = [ ];
c2 = [ ];
c3 = [ ];
c4 = [ ];
for i = 1:k
c1i = 1;
c2i = (M(i,1)*sin(N(i,1)) - N(i,1)*sinh(M(i,1))) / (N(i,1)*(cosh(M(i,1)) cos(N(i,1))));
c3i = -M(i,1) / N(i,1);
c4i = -c2i;
c1 = [c1 ; c1i];
c2 = [c2 ; c2i];
c3 = [c3 ; c3i];
c4 = [c4 ; c4i];
end
% calculation of ksi
ksi = 0 : 0.001 : 1;
% calculation of the mode shapes
Y = [ ]; % make empty matrix for the mode shapes
for i = 1 : k
Yi = c1(i,1) * sinh(M(i,1) * ksi) + c2(i,1) * cosh(M(i,1) * ksi) + c3(i,1) *
sin(N(i,1) * ksi) + c4(i,1) * cos(N(i,1) * ksi);
Y = [Y ; Yi];
end
% Normalization of the mode shapes with max(Y) = 1
for i = 1 : k
Y(i,:) = Y(i,:) ./ max(abs(Y(i,:)));
end
% Plotting of the results
figure
grid
hold on
for i = 1 : k
plot(ksi,Y(i,:))
end
C.6
Euler with tensile pretension
% Calculation of natural frequencies and eigenmodes without shear and
% with tensile axial pretension
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
Variables
x = position in x-direction [m] (x = 0 at left end of the beam)
y = position in y-direction [m] (y = 0 at the middle of the beam)
z = position in z-direction [m] (z = 0 at the middle of the beam)
L = length of the beam [m]
H = height of the beam [m]
B = width of the beam [m]
I = second moment of area [m^4]
rho = density [kg.m^-3]
E = Young's modulus [Pa]
A = area of cross section [m^2]
O, OMEGA = relative natural frequency [-]
a = sqrt((EI)/(rho*A) [m^2/s]
U = relative axial force [-]
ts = tensile pretension [Pa]
T = axial tensile force [N]
omega = natural frequency [rad/s]
M, N = mode shape coefficient [-]
ci (i=1,2,3,4) = mode schape coefficient [-]
k = number of natural frequencies [-]
ksi = dimensionless beam coordinate [-]
% set defaults
format long e;
clc;
clear all;
close all;
% values for various parameters
45
L = 44e-6;
H = 4e-6;
B = 1.4e-6;
rho = 2330;
E = 131e9;
nu = 0.28;
I = (1/12) * B * H^3;
A = H * B;
ts = 44e6;
T = ts * A;
a = sqrt((E*I) / (rho * A));
U = T*L^2 / (2*E*I);
k = 3;
% Solve the characteristic equation with O as unknown
OMEGA = [ ]; % make empty vector
for i = 1:k
Oi = fzero(@(O)O+U*sinh(sqrt(U+sqrt(U^2+O^2)))*sin(sqrt(-U+sqrt(U^2+O^2)))O*cosh(sqrt(U+sqrt(U^2+O^2)))*cos(sqrt(-U+sqrt(U^2+O^2))),((2*i+1)*0.5*pi)^2);
OMEGA = [OMEGA; Oi];
end
% calculate natural frequencies
f = OMEGA * a / (2 * pi * L^2)
% for calculating the mode shapes some coefficients have to be
% determined: first M and N, thereafter ci (i = 1,2,3,4) which are a
% function of M and N. These constants are different for each natural % frequency.
% Calculation of M en N
M = [ ]; % make empty vector for M
N = [ ]; % make empty vector for N
for i = 1:k
Mi = sqrt(U + sqrt(U^2 + OMEGA(i,1)^2));
Ni = sqrt(-U + sqrt(U^2 + OMEGA(i,1)^2));
M = [M ; Mi];
N = [N ; Ni];
end
% Calculation of ci (i = 1,2,3,4)
c1 = [ ];
c2 = [ ];
c3 = [ ];
c4 = [ ];
for i = 1:k
c1i = 1;
c2i = (M(i,1)*sin(N(i,1)) - N(i,1)*sinh(M(i,1))) / (N(i,1)*(cosh(M(i,1)) cos(N(i,1))));
c3i = -M(i,1) / N(i,1);
c4i = -c2i;
c1 = [c1 ; c1i];
c2 = [c2 ; c2i];
c3 = [c3 ; c3i];
c4 = [c4 ; c4i];
end
% calculation of ksi
ksi = 0 : 0.001 : 1;
% calculation of the mode shapes
Y = [ ]; % make empty matrix for the mode shapes
for i = 1 : k
Yi = c1(i,1) * sinh(M(i,1) * ksi) + c2(i,1) * cosh(M(i,1) * ksi) + c3(i,1) *
sin(N(i,1) * ksi) + c4(i,1) * cos(N(i,1) * ksi);
Y = [Y ; Yi];
end
% Normalization of the mode shapes with max(Y) = 1
for i = 1 : k
Y(i,:) = Y(i,:) ./ max(abs(Y(i,:)));
end
% Plotting of the results
figure
grid
hold on
for i = 1 : k
plot(ksi,Y(i,:))
46
end
C.7
Timoshenko with compressive pretension
% Calculation of natural frequencies and eigenmodes with shear and
% with compressive axial pretension
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
%
Variables
x = position in x-direction [m] (x = 0 at left end of the beam)
y = position in y-direction [m] (y = 0 at the middle of the beam)
z = position in z-direction [m] (z = 0 at the middle of the beam)
L = length of the beam [m]
H = height of the beam [m]
B = width of the beam [m]
I = second moment of area [m^4]
rho = density [kg.m^-3]
E = Young's modulus [Pa]
m = mass per meter [kg.m^-1]
nu = Poisson's ratio [-]
A = area of cross section [m^2]
fi = natural frequency i (i = 1,2,3) [Hz]
Yi = mode shape at natural frequency i [-]
ksi = x/L = dimensionless beam coordinate [-]
omega = 2*pi*fi = circular natural frequency [rad/s]
p(i) = sqrt(m*omega^2*L^4/(E*I(1-Nx/a))) = relative natural
frequency [-]
r = sqrt(I/A) = radius of gyration [m]
R = r / L = relative radius of gyration [-]
G = shear modulus [Pa]
kappa = shear coefficient [-]
s1 = mode shape coefficient [-]
s2 = mode shape coefficient [-]
n = number of calculated natural frequencies [-]
a = kappa * G * A = shear parameter [N]
b = sqrt(E * I / (a * L^2)) [-]
k = sqrt(Nx * L^2 / (E * I * (1 - Nx / a))) [-]
sigmax = axial compressive pretension [Pa]
Nx = axial compressive force [N]
beta = R^2 * (1 - Nx / a) + b^2 [-]
Nxc = critical buckling load
sigmaxc = critical buckling stress
% set defaults
format long e;
clc;
clear all;
close all;
set(0,'DefaultLineLineWidth',2)
set(0,'DefaultAxesFontSize',12)
% Values for different parameters
L = 44e-6;
H = 4e-6;
B = 1.4e-6;
rho = 2330;
E = 131e9;
nu = 0.28;
sigmax = 4.4e7;
n = 3;
I = (1/12) * B * H^3;
m = B * H * rho;
A = H * B;
Nx = sigmax * A;
G = E/(2*(1 + nu));
kappa = 10*(1 + nu)/(12 + 11*nu);
r = sqrt(I/A);
R = r / L;
a = kappa * G * A;
b = sqrt(E * I / (a * L^2));
k = sqrt(Nx * L^2 / (E * I * (1 - Nx / a)));
beta = R^2 * (1 - Nx / a) + b^2;
p = [ ]; % make empty vector for p(i)
% calculate p(i) from the characteristic equation
for i = 1 : n
47
% The formula as function of i that determines the point around
% fzero tries to calculate the zero of the characteristic
% equation is as follows for various values of H:
% H = 1
39*i
% H = 2 t/m 7
30*i
% H = 8 t/m 10
20*i
px = fzero(@(p) 2 - 2*cosh(sqrt(-0.5 * (p^2*(R^2*(1 - Nx / a) + b^2) + k^2) + 0.5
* sqrt((p^2*(R^2*(1 - Nx / a) + b^2) + k^2).^2 + 4*p^2))).*cos(sqrt(0.5 * (p^2*(R^2*(1
- Nx / a) + b^2) + k^2) + 0.5 * sqrt((p^2*(R^2*(1 - Nx / a) + b^2) + k^2).^2 +
4*p^2))) + p.*(p^2.*b^4*beta + k*b^4 + 4*b^2 - beta + k^2 ./ p^2) ./ (1 +
p.^2*b^2*R^2*(1 - Nx / a) + b^2*k^2) .* sinh(sqrt(-0.5 * (p^2*(R^2*(1 - Nx / a) + b^2)
+ k^2) + 0.5 * sqrt((p^2*(R^2*(1 - Nx / a) + b^2) + k^2).^2 + 4*p^2))).*sin(sqrt(0.5 *
(p^2*(R^2*(1 - Nx / a) + b^2) + k^2) + 0.5 * sqrt((p^2*(R^2*(1 - Nx / a) + b^2) +
k^2).^2 + 4*p^2))),30*i);
p = [p; px];
end
f = [ ]; % make empty vector for fi
% calculate fi from p
for i = 1 : n
fi = sqrt( ((p(i,1))^2*E*I*(1 - Nx / a)) / (pi^2*4*m*L^4) );
f = [f ; fi];
end
f % print f on the screen
% Calculate Buckling load
Nxc = 4*pi^2 / (L^2*((1/(E*I))+4*pi^2 / (L^2*a)));
sigmaxc = Nxc / A % print sigmaxc on the screen
% determine ksi
ksi = 0 : 0.001 : 1;
Y = [ ]; % make matrix for Yi
for i = 1 : n
% coefficients s1 and s2 are dependent on p so are calculated for % every p
s1 = sqrt(-0.5 * ((p(i,1))^2*(R^2*(1 - Nx / a) + b^2) + k^2) + 0.5 *
sqrt(((p(i,1))^2*(R^2*(1 - Nx / a) + b^2) + k^2).^2 + 4*(p(i,1))^2));
s2 = sqrt(0.5 * ((p(i,1))^2*(R^2*(1 - Nx / a) + b^2) + k^2) + 0.5 *
sqrt(((p(i,1))^2*(R^2*(1 - Nx / a) + b^2) + k^2).^2 + 4*(p(i,1))^2));
% calculation of mode shape constants M and N
M = (s1^2 + (p(i,1))^2*b^2) * (1 - Nx / a) / s1;
N = (s2^2 - (p(i,1))^2*b^2) * (1 - Nx / a) / s2;
% calculation of mode shape coefficients B2, B3 and B4
B2 = (-M*sinh(s1) - N*sin(s2)) / (M*cosh(s1) - M*cos(s2));
B3 = -1;
B4 = -M / N * B2;
% determine the mode shapes
Yi = 0.5*(1+ B2)*exp(s1*ksi) + 0.5*(1-B2)*exp(-s1*ksi) + B3*cos(s2*ksi) +
B4*sin(s2*ksi);
% Normalization: max(Yi) = 1
B1 = 1 / max(abs(Yi));
Yi = B1 * Yi;
Y = [Y ; Yi];
end
% Plotting of the results
figure
grid
hold on
plot(ksi,Y(1,:),'k')
plot(ksi,Y(2,:),'k--')
plot(ksi,Y(3,:),'k:')
xlabel('\xi [-]')
ylabel('Y_i [-]')
hold off
48