„Micromechatronics and MEMS” – Lab1 2014/2015 One of the most popular form of the MEMS’ design is the cantilever beam. Fig.1 shows the bimorph cantilever beam.

„Micromechatronics and MEMS” – Lab1 2014/2015 One of the most popular form of the MEMS’ design is the cantilever beam. Fig.1 shows the bimorph cantilever beam.

Fig. 1 Bimorph cantilever beam

The physical properties of the popular materials, which are widely used in the microsystem technology, are shown in Tab. 1.

Tab. 1 Properties of materials

Material Density

 [kg/m³]

Young’s modulus

E [GPa]

Thermal expansion coefficient

 [K^-1]

Thermal conductivity

 [W/(mK)]

Specific heat c [J/(kgK)]

SiO2 2500 73 4.0·10^-7 1.1 733

Si/Poly Si 2300 190 2.5·10^-6 160.0 700

Al 2700 70 2.3·10^-5 240.0 900

The stiffness

K

_z of the bimorph can be calculated by using the following expression:

3

12

_{n ny}

z z

z

F E I K   L



^{, (1.1)}

where

F

_z denotes the force applied on the cantilever tip,



_z is the tip displacement measured along the

F

_z direction. The effective Young’s modulus is given by E_n max



E E1, 2



, where indices 1 and 2 denote the material properties of the bimorph layers, according to Fig. 1. The effective moment of inertia of the bimorph cross-section is given by:

 

2 3

2

1

12

i i

ny i n i

i

I W t A h h



 

    

 



^{, (1.2)}

where Wi W E E



i n



is the effective width of the i-th layer;

A

_i

 W t

_{i i} denotes the effective area of the i-th layer cross section and

t

_i denotes the thickness of the i-th layer cross section.

h

_i is the coordinate of the center of gravity for the i-th layer, which is measured from the bottom edge of the cross section,

h

_n denotes the center of gravity for the bimorph cross-section:

2

1 2

1 i i i n

i i

A h h

A











^{. (1.3)}

The potential energy stored in the bimorph can be expressed by the equation:

1

m

2

z z

E  F 

. (1.4)

Density of the potential energy

w

_m is defined by:

m m

w E

 V

, (1.5)

where

V  2 WtL

denotes the volume of the bimorph. Mass of the bimorph is given by:

 

0 1 2

M     WtL. (1.6)

The bimorph, which is shown in Fig. 1, can be also considered as a resonator. The main parameter of the resonators is the eigenvalue



₀, which can be estimated by the 1-degree-of-freedom model:

0

0

K

z

  M

. (1.7)

The resonator can also work as a sensor. The relative change of the eigenvalue caused by the the change of the mass

 m

is given by:

0 0

0

1 1

1 m

M

  

  

. (1.8)

The bimorph can also work as an accelerometer. For the quasi-static acceleration, the sensitivity of the bimorph displacement x can be expressed by the following equation:

0 a

z

M S dx

da K

  , (1.9)

where a denotes the acceleration.

For the electrostatic actuation, the one electrode is the bottom side of the bimorph and the second one is located at a distance G from the bottom side of the cantilever beam, as is shown in Fig. 1. The maximum value of the electrostatic field energy in the air gap can be calculated by the following expression:

2 2

0

1

e

2

b

E   W GE

, (1.10)

where

 

₀

8.85 pF/m

is the vacuum dielectric permeability, W and G are the dimensions, as are shown in Fig.1;

E

_b denotes the breakdown electric field.

The electrostatic force is given by:

2 2 0

1

e

2

b

F   W E

, (1.11)

and the density of the electrostatic field energy

w

_e can be calculated by the (1.5), where instead of the mechanical energy, the electrostatic field energy must be substituted. The pressure on the bottom electrode is caused by the electrostatic force and it can be expressed by:

e e

e

p F

 A

, (1.12)

where the area of the electrode is equal to

A

_e

 W

².

The bimorph can also work as a thermal actuator. The difference between the thermal expansion coefficient values of the layers causes the bending of the beam. If T denotes the value of the temperature increment, the tip displacement



can be calculated from the following expression:

 

        ^{ }

2 2

2 1

2 2

3 1

3 1 1 1

L m T

t m mn m mn

      

     

 

, (1.13)

where

m  t t

_{1 2}

, n  E E

_{1 2}

, t   t

₁

t

₂;

t

_i denotes the i-th layer thickness. The thermal energy stored in the bimorph is then given by:



1 1 2 2



T l

E    c c V T , (1.14) where the density and the specific heat for the i-th layer are denoted by



_i

and c

_i, respectively; Vl

denotes the volume of the single layer.

The scaling issue can be considered by performing calculation of the mentioned physical quantities for a different values of the bimorph geometrical dimensions, keeping the ratio between them.

The skeleton of the m-function may have the following form:

function [out_1,out_2,..,out_n]=bimorph(L)

% L - the argument of the function 'bimorph' and the length of the bimorph

% out_1, out_2,..., out_n - outputs of the function 'bimorph'

% the SI system of units is recommended with basic units

% material data and geometric proportions ...

E1=...; % the Young modulus of the 1st layer E2=...; % the Young modulus of the 2nd layer ...

En=max([E1 E2]);

Iny=...; the expression for the moment of inertia

Kz=(12*E*Iny)./(L^3) % the stiffness of the bimorph out_1=Kz;

...