Coefficient about MEMS Planar Microspring
Li Hua, Shi Geng-chen
Beijing Institute of Technology, Beijing, 100081, China
Phone (010)81690204 E-Mail: lihua101@bit.edu.cn
Abstract: In this paper, the energy method in
mechanics of materials is used to establish the analytic model of a MEMS planar microspring, the form of which is “L”. The elastic coefficient formulas of other planar microsprings can also be obtained in the same way. The results of experiment and finite element analysis (FEA) simulation validate the correctness of the elastic coefficient formula. The stiffness of two planar microsprings with same size and material are compared and the variation law in some range is illustrated in theory. Key words: energy method; MEMS; planar microspring; elastic coefficient; variation law
I Introduction
Elastic microelement plays an important role in
MEMS device. It is an important part of microactuator, microaccelerometer, microgyroscope, and so on. Microspring is an elastic microelement widely used in MEMS-type mechanisms. In the process of designing and fabricating MEMS planar microspring, the elastic coefficient is a very important parameter, which directly determines the performance of planar microspring and whether it can work normally. The planar microsprings, which are fabricated using suitable microfabrication technology, are different from each other in shape, material, and so on. At present, the elastic coefficient is usually calculated by the simulation of finite element analysis (FEA) software or measured by experiment. But there is no special calculation formula for elastic coefficient, and this is very inconvenient to the design and the fabrication of planar microspring.
The calculation formulas of elastic coefficient about planar microspring are obtained with the Castigliano second theorem belonging to the energy
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method, which builds the theoretical foundation for deeply analyzing the various qualities of microspring.
II The Dduction of the Calculation Formulas of Elastic Coefficient
In the elastic range, the strain of the studied microspring is linear to its stress. According to the Castigliano second theorem, the displacement of the microspring in the point where the force acts is:
δ∂U
F(x)∂FN(x)M(x)∂M(x)i=∂F=i∫NlEA∂Fdx+i∫lEI∂Fdx+
i
∫T(x)∂T(x)
lGI (1)
p∂Fdx iWhereE is the modulus of elasticity of the microspring; A is the cross-sectional area of the microspring; Fis the external force acting on the microspring; M(x) is the bending moment acting on some cross-section of microspring; I is the second axial moment of area; Ip is the second polar moment of area of some cross-section; G is the shear modulus of the material; T(x) is the torsional moment acting on some cross-section.
If there is only a vertical planar force acting on the planar microspring, the moment of torque is zero. And the deformation produced by shear force can be neglected because the work of shear is much smaller than that of bending moment.
The microspring, the form of which is “L”, is composed of n basal units that are the same in configuration and material, so it is enough to choose a unit to analyze. Supposing the upper end of one unit is fixed and the lower end is applied by a
vertical force F, they are showed in figure 1. In the figure, b is the line width of microspring; d is the space length; h is the thickness; l is the half width. A unit of “L” microspring can be divided into six parts(①②③④⑤⑥)to be analyzed. The ①②③ parts are completely symmetrical with the ④⑤⑥ parts, so it only need to analyze the ①②③ parts.
FIG 1 The analytic model of a unit of “L” microspring
The ①part:
M1(x)=Fx (2)
∂M1(x)∂F
=x，（0≤x≤l）
(3) δl
M1(x)∂M1(x)Fl3
1=∫0EI∂Fdx=3EI
(4) The ②part:
M2(x)=Fl (5) ∂M2(x)
∂F
=l，（0≤x≤d+2b）
(6) δd+2bM2(x)∂M2
2
(x)2=∫0EI∂Fdx=Fl(d+2b)EI
(7) The deformation of the ③ part is δ3 and it is
equal to δ1. According to the symmetry, the whole
deformation of one unit of “L” microspring is:
δ=4δ1+2δ2. Following the Hooke’s law, the elastic coefficient of “L” microspring consisting of n
basic units is:
KFEb3hL=nδ=n[16l3+24(d+2b)l2] (8) The meanings of the parameters are shown in
FIG 1. The influence of structural parameters on elastic coefficient can be directly ascertained from the formula.
When the material of the microspring is nickel,
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the parameters are chosen as: h=300um, l=480um, d=100um, n=7. Some different
line widths b are chosen to calculate the elastic coefficient by the formula and the FEA software. The simulation result of FEA software is shown in FIG 2. The numeric values calculated by the formula and the FEA software are disposed by the curve fitting method, as shown in FIG 3.
FIG 2 The deformation nephogram of FEA simulation
FIG 3 The comparison of the results calculated by the formula and FEA software in the condition
of different line widths It can be observed from the FIG.3 that the
elastic coefficient of the microspring becomes large
with the increase of the line width b. The results
calculated by the formula are tally with the FEA
simulation results.
The “S” microspring is similar to the “L”
microspring in configuration, as shown in FIG 4. Its
elastic coefficient formula can also be calculated by
the Castigliano second theorem: b3KhES=n[16l3+12R(2l2π+8lR+R2
π)]
(9) The meanings of the parameters are shown in FIG 5.
are:h=300um,l=970um,b=80um,d=80um, n=10.
The curve of tensile force—extension is tested by a measurement system. In the elastic range, the tested data are disposed by the least square method, and the calculated result of the elastic coefficient is148N/m. The result calculated by the formula is
FIG 4 The scan photograph of the “S” microspring
FIG 5 The analyzed model of an “S” microspring unit
KS=154N/m. The results are almost equal in the elastic range and this validates the correctness of elastic coefficient formula about planar microsprings deduced by the energy method.
III The Comparison and Analyse of Elastic Coefficient of Two Microsprings
The same structural and material parameters are chosen to calculate the different elastic coefficients of “L” and “S” planar microsprings, and the results are shown as the Tab 1:
The “S” microspring is fabricated by UV−LIGA technology, whose material is nickel. And other parameters of the “S” microspring
The elastic coefficient of “L”
microspring(N/m) The elastic coefficient of “S”
microspring (N/m)
Tab 1 The comparison of the elastic coefficient of the two microspring in the same condition
208 651 1437 2625 4261 179 559 1234 2248 3627
According to Tab 1, it shows that the elastic coefficient of “S” microspring is smaller than that of the “L” microspring. The reason is that the stress and deformation are widely identical in the ①③④⑥ parts; in the ②、⑤ parts, the bending moment of “L” microspring isM2(x)=Fl, while the bending moment of “S” microspring is M2(x)=Fl+FRsinθ. So the deformation of “S” microspring is larger than that of the “L” microspring, and its elastic coefficient is smaller. In some range, it can be deduced that the elastic coefficient of planar microspring the form of which is arc is smaller than that of planer microspring the form of which is linear.
Castigliano second theorem belonging to energy method, are validated by the results of FEA simulation and the experiment data. The formulas can be used as the gist for designing and fabricating the two microsprings. The analytic method can also be applied to other MEMS microsprings.
2) In this paper, the calculation formulas and variation law are only applied in the elastic range. In the nonlinear condition, it needs more study.
References
[1] Yang BoYuan. Mechanics of material (I) [M]. Beijing: China Machine Press, 2002. PP 186-190
[2] Liu WeiGuo. Scientific calculation and MATLAB language[M]. Beijing: China Railway Press. 2000. PP 59-68
IV. Conclusion
1) The elastic coefficient formulas of planar microspring, which are calculated by the
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