Cantilever Rectangular Beam Type Micro-tweezers

From thod for this

case is given

Eqs. (4.7) and (4.8), the iteration formulation of the Newton-Raphson me by:

tion of a cantilever bea

adopted as the initial estimation for the iteration procedure described above.

4.2 Numerical

efficiency of the proposed DQM approach, this study first

simu onsidered in [9-11]

(Chen et al., and MacDonald et al 9 an Shi et al., 1995). The current simulation results are then compared with the results pres in th

In [11], the simplified rectangular beam type micro-tweezer with θ = shown in Fig. 4-1 0 BEM and FEM method. In the current example, the beam length It is noted that the linear solu m with a uniformly distributed load is

4.2.1 Cantilever Rectangular Beam Type Micro-tweezers

To verify the accuracy and

Analyses of Different Micro-tweezers

lates the uniform rectangular cantilever beam type micro-tweezers c ., 198 d

ented e relevant literature.

4.2.1.1 Simplified Micro-tweezers

was investigated using a hybrid

L is 200 mb µ , the Young’s modulus for the tungsten beam material is E =410 GPa, and the beam width is b=2.7 mµ . Furthermore, the thickness of the rectangular beam ish =2.9 m₀ µ and the initial gap is d₀ =2g₀ =4 mµ . Table 4-1 summarizes the details of the beam geometry.

Fig. 4-2 indicates that the gap between the two cantilever beams reduces gradually as the applied voltage is increased toward a critical voltage. Beyond this critical voltage, the two ca

beams are pulled sudd . It can be seen that a good agreement exists

betw the proposed DQM and the pull-in voltage of

78 V reported by Shi et al. in 1995 using the hybrid FEM-BED method [11]. The difference between the two sets of results is found to be less than 2%.

Calculation of Contact Forces

ntilever enly toward each other

een the pull-in voltage of 79.6 V calculated by

Once the applied voltage reaches the critical voltage, the tips of the micro-tweezers contact the object. The contact forces increase as the applied voltage is increased further. Since the assumption is made that the micro-tweezer arms are symmetr

along the centerline of the gap between the two arms.

This study uses a simple spring at the free end of the micro-beam to model the effect of the two arm tips contacting the gripped object. Hence, when the tips are in contact with the object, the beam is no longer cantilevered, but is considered to be spring-supported at the tip. This

mod the tip of the beam as a

boundary condition. This spring force

ical, the contact force will occur

ification can be accomplished by inserting the spring force at

f changes the final boundary condition in the original s

set of conditions (4.3). Accordingly, the shear force at the end of the beam is equal to the spring force, i.e.

( ) _s

EIy′′′ x = f (4.10)

The contact force can then be obtained by multiplying the stiffness m trix of the beam by ith this new form of tip support. Figure 4-3 compares the contact force results calculated by th ulation results predicted by the hybri

FEM method [1 that the tendencies of the two sets of results are similar and

o-tweezers without Silicon Dioxide Coating

References [9,10] fabricated and investigated CVD tungsten micro-tweezers, in which the two beams were rigidly attached at the fixed end with θ =0 5. ^°. The remaining dimensions were as shown in Table 4-1. The experimental results indicated that the pull-in voltage was 125 V 121.5 volts, while the result obtained using the hybrid FEM-BEM method is 112 V. Figure 4-4 presents a

parison of the tip deflection behaviors of the micro-beam as calculated by the current method and by the two reported methods. Meanwhile, Fig. 4-5 illustrates the variation in beam deflection with beam

micro-tweezers coated with a silicon dioxide film. The thickness of the silicon dioxide was 0.2 m

a the deflection calculated w

e DQM with the sim d BEM and

1]. It can be seen consistent.

4.2.1.2 Micr

[9,10]. By contrast, the proposed DQM approach predicts a pull-in voltage of

com

length at various applied voltages.

4.2.1.3 Micro-tweezers with Silicon Dioxide Coating

In order to prevent the occurrence of electric shorts when the two beams are pulled together, Chen et al., 1989 and MacDonald et al., 1989 [9,10] developed

µ and the beam length L _b was 200 mµ . Furthermore, the Young’s modulus for the tungsten beam material was E =410 Gpa, while that of the silicon dioxide was E =73 GPa. The true cross-sectional view of the _s composite beam is shown in Fig. 4-6a. In this study, a 2-D model is used for the composite beam,

and th sections remain plane” [11], as shown in Fig. 4-6b.

Hence, the effective EI of the composite beam is equal to the summation of its tungsten and e assumption is made that “plane

silicon dioxide components, i.e. _{, , ,}

2 2

1 SiO 2 tungsten 3 SiO

EI =EI +EI +EI . The result of the pull-in ltage ca lated by the DQM method is found to be 141.5 V, which compares to an

just nineteen sampling e beam and is capable of convergence with a tolerance of 10

vo lcu

experimental result of 150 volts [9,10] and a value of 157~160 volts calculated by the FEM-BEM hybrid method [11]. Significantly, the proposed DQM algorithm requires the use of

points along th

oposed DQM approach are in good ented in the literature. It is known that the critical value of the applied voltage which results in the pull-in effect is dependent upon the particular shape of the micro-beam. This phenomenon is studied and exploited in the following section.

-8 within four iterations. The convergence tendency of the tip deflections calculated using the DQM for different numbers of selected sampling points is shown in Fig. 4-7. As discussed previously, the locations of the sampling points along the beam are selected in accordance with the Chebyshev-Gauss-Lobatto distribution.

The comparisons presented above for the rectangular cantilever beam type micro-tweezers confirm that the simulation results obtained using the pr

agreement with the results pres