Mechanics of Microelectromechanical Systems - N.Lobontiu and E.Garcia Part 6 doc

138 Chapter 3 stiffnesses by utilizing the matrix transformation of Eq. (3.4), and the stiffness becomes: Equation (3.18) simplifies to Eq. (3.8) when which checks the validity of the generic model. Figure 3.7 plots the ratio of the stiffnesses that are given in Eqs. (3.8) and (3.18) in terms of the parameter and a parameter c (it has been assumed that the length and width are related as l = cw). It can be seen that the stiffness of the straight beam increases noticeably, compared to the stiffness of the inclined beam. When the inclination angle and the parameter c increase towards their upper limits in the selected ranges, the stiffness ratio reaches a local maximum. Figure 3.7 Inclined-to-straight beam-spring stiffness ratio 2.2 Bent Beam Suspensions A spring design which is formed of two compliant straight segments that are perpendicular can be utilized to enable the two-axis motion of a rigid, such as the one shown in Fig. 3.8, where four springs support the central mass symmetrically. While the body translates about one of the directions indicated in the figure, the spring leg that is directed perpendicularly to the motion direction will bend, whereas the other leg will be subject to axial extension/compression in addition to bending. Figure 3.9 indicates the geometry of a bent beam suspension (also called corner spring), where the two legs have different lengths. The boundary conditions are assumed to be fixed-free, as also indicated in the same figure. The main deformations of a bent beam spring are planar and they result from the two-dimensional motion 3. Microsuspensions 139 of the central mass. However, the bent beam is also sensitive to z-axis parasitic loading, generated by the weight of the central mass. As a consequence, in-plane stiffnesses about the x- and y-directions, as well as the out-of-the-plane stiffness about the z-direction will be derived for this microsuspension. Figure 3.8 Rigid body and four bent beam springs for planar motion Figure 3.9 Geometry of a bent beam microsuspension The in-plane deformations of the bent beam can be studied by applying the loads and and by calculating the corresponding tip displacements and through Castigliano’s displacement theorem. The strain energy collects contributions from bending and axial loading on the two segments, 1-2 and 2-3. The tip deformations can be related to the tip loads by means of a compliance matrix in the form: 140 Chapter 3 The terms of the compliance matrix are: The superscripts (1) and (2) refer to the first segment 1-2 and to the second one 2-3, respectively. The compliances of the right-hand side of Eqs. (3.20) through (3.25) are calculated for each of the two members with respect to their local frames. In doing so, members of different geometries (defined as free-fixed microhinges) can be utilized in a bent beam design. A stiffness matrix can be defined by inverting the compliance matrix of Eq. (3.19): In the case the two compliant segments of the bent beam are identical, the particularly-important in-plane stiffnesses are: 3. Microsuspensions 141 When the axial deformations are negligible compared to the bending deformations, Eqs. (3.27) and (3.28) can still be used by considering that the axia l compliances of the two segments are zero (axially rigid members). The mention was made in Chapter 2 that the stiffnesses defined by inverting the compliance matrix are different from the stiffnesses that are calculated as: and that the stiffnesses of Eqs. (3.27) and (3.28) should be used when forces need to be calculated based on known displacements. However, Eqs. (3.29) are used as definition relationships and their values can be obtained by using the transformation Eqs. (2.25) of Chapter 2 from the stiffnesses of Eqs. (3.26). The out-of-the-plane definition stiffness can be determined by applying a force at point 1 of Fig. 3.9 about a direction perpendicular to the bent beam’s plane and by calculating the corresponding displacement. By taking bending and torsion into account, the z-direction stiffness is: Example 3.4 Calculate the mam stiffnesses of a bent beam microsuspension with identical legs and of constant rectangular cross-section. Evaluate the errors in calculating by its definition – Eqs. (3.29) – as opposed to the compliance derived stiffness of Eq. (3.27). Solution: For this particular case, the linear in-plane stiffnesses are equal, namely: and the z-axis stiffness is: It has been considered that w << t (very thin cross-section) and therefore: 142 Chapter 3 The ratio of the x-axis stiffnesses becomes: and for and this ratio is almost constant with a 2.5 approximate value. 2.3 U-Springs Microsprings that have the approximate shape of the letter U (called here U-springs ) are mainly used in applications involving translatory motion of rigid bodies. Due to symmetry about the axial (motion) direction, the proof mass can translate about that axis, as suggested in Fig. 3.10. Figure 3.10 Proof mass in translatory motion with four U-springs attached frontally Other (parasitic) motions, either planar or out-of-the-plane (especially due to the self-weight of the proof mass) are also possible, hence quantifying the stiffnesses about the direction perpendicular to the motion direction and the direction perpendicular to the plane of the microdevice of Fig. 3.10 will also be done, in addition to formulating the main stiffness about the motion direction. Figure 3.11 pictures a U-spring with the reference frame that is used to define the linear stiffnesses of interest, and which are (the stiffness related to the main translatory motion of the proof mass shown in Fig. 3.10), (the stiffness defining the elastic properties of the U-spring when the body translates about a direction perpendicular to the main one and is contained in the plane of the microdevice) and (the stiffness which describes the spring behavior for the case of an out-of-the-plane motion about the z-direction, as indicated in Fig 3.11). 3. Microsuspensions 143 Figure 3.11 U-spring model with boundary conditions, main degrees of freedom, and corresponding forces As a result of these three translatory motions of the shuttle, the true boundary condition at point 1 in Fig. 3.11 is a forced translation about the x- axis. However, as a simplification to the real situation, it may be considered that point 1 is free to move, as also assumed previously with the bent beam microsuspension. Because the force acting at that point is basically directed about the same direction, the errors of considering point 1 as free are expected to be small. Three different configurations will be analyzed in the following: one with sharp corners, a second one where the short straight link of the model is substituted by half a circle, and a third variant with filleted corners. 2.3.1 U-spring with Sharp Corners (Configuration # 1) Configuration # 1 is formed of three elastic segments, as shown in Fig. 3.12. In order to keep the formulation valid for a generic case, they can have different but constant cross-sections. It will also be considered that only bending of each of the three segments contribute to the total strain energy of the spring. The in-plane compliances are calculated by applying the loads and as shown in Fig. 3.12, and by calculating the corresponding displacements and Castigliano’s displacement theorem is applied again in order to calculate these displacements. A compliance matrix of the type shown in Eq. (3.19) can be formulated, whose terms are: 144 Chapter 3 Figure 3.12 U-spring design with sharp corners The stiffness is the element of the stiffness matrix (which is the inverse of the compliance matrix consisting of the elements defined in Eqs. (3.35) through (3.40)) located on the first row and first column. Similarly, the stiffness is the element placed on the second row – second column position o f the same stiffness matrix. The definition stiffnesses are simply the inverses of the corresponding compliances, and The stiffness about the z-direction is calculated in the definition sense that has been introduced in the previous chapters and was also mentioned previously in Eq. (3.30) for instance. Its expression is found by taking the ratio of a force which is applied at point 1 in Fig. 3.12 about a direction perpendicular to the plane of the microdevice to the corresponding displacement namely: 3. Microsuspensions 145 Example 3.5 Analyze the change in the main compliance of a U-spring when the axial deformations are also taken into account. Consider that the three legs have identical constant rectangular cross-sections. Solution: In the case where the axial deformations are considered, the strain energy will include terms induced by the axial effects, in addition to bending- produced ones, but the procedure of calculating the main compliance, remains the same. For a constant rectangular cross-section, the ratio of the bending-related compliance to the compliance that considers both bending and axial effects becomes: Figure 3.13 Compliance ratio in terms of and Figure 3.14 Compliance ratio in terms of w and 146 Chapter 3 Equation (3.42) indicates that the model including bending effects generates a compliance about the main direction of action which is very slightly larger than the compliance yielded by the model that adds axial effects to bending. Figure 3.13 is the plot of the compliance ratio of Eq. (3.42) as a function of and when and Similarly, Fig. 3.14 is the plot of the same compliance ratio in terms of w and when As both figures indicate, the compliance ratio is in the very close vicinity of 1 when the design variables of Eq. (3.42) span relatively wide ranges, which indicates that neglecting the axial effects has little influence on the main compliance. Example 3.6 Find the definition stiffness of a U-spring about the y-direction in the case where the middle leg has a small length which implies considering the additional shearing effects and associated deformations. Compare the resulting stiffness with the regular one determined by means of the compliance of Eq. (3.38) in the case where and Solution: When the length is only about 3-5 times greater than the largest cross- sectional dimension, the deformation produced by the shearing force has to be accounted for in addition to bending. The displacement at point 1 about the y-direction in Fig. 3.12 is calculated by means of Castigliano’s displacement theorem as: where the subscripts in bending moments M and shearing force S indicate the specific segment out of the three ones making up together the U-spring. The linear stiffness about the y-direction can be expressed according to its definition as: whereas the same stiffness which only considers bending is: By constructing the ratio of the y-axis stiffness in Eq. (3.44) to the stiffness of Eq. (3.45), the plot of Fig. 3.15 can be drawn in terms of the lengths and 3. Microsuspensions 147 for It can be seen that the stiffness ratio is almost constant and equal to 1, which indicates that the shearing effects are not particularly large. Figure 3.15 Stiffness ratio in terms of and 2.3.2 U-spring with Circular Short Link (Configuration # 2) As Fig. 3.16 shows it, configuration # 2 incorporates a semi-circular portion instead of the straight segment 2-3 of the previous design. Figure 3.16 U-spring design with circular short link There are two possibilities, connected to the form factor of the semi- circular section. It is known (see Young and Budynas [1], for instance) that for thin curved beams, when the ratio of the radius R to the cross-sectional widt h w is greater than 10, the deformations of the curved beam can safely be treated by using the tools applicable to straight beams. By applying the same procedure that has been used for the U-spring configuration # 1, and by only taking bending of the three segments into account, the in-plane compliances o f the constant cross-section design are: [...]... sides Figure 3. 36 Two folded-beam springs attached to the sides of a moving proof mass The aim is again to determine the spring stiffnesses about the three possible translatory motions of the center mass, namely the x-direction (the motion direction indicated in Fig 3. 36) , the y-direction (the other in-plane direction, which is perpendicular on the x-direction) and the out -of- the-plane z-direction In... by means of Eq (1.122) By applying again Castigliano’s displacement theorem for the configuration of Fig 3. 16 in the presence of the tip loads and (not shown in Fig 3. 16) , the resulting displacements and can be found by means of the following compliances: 3 Microsuspensions 149 The out -of- the-plane definition stiffness is found by applying a force at the free point of the spring in Fig 3. 16, perpendicularly... segments have variable cross-sections, Eq (3.107) can be written as: 3 Microsuspensions 167 where the superscripts 1 and 2 denote the 2-3 and 4-5 members, and the subscript Fy-uy indicates the direct linear bending stiffness of the compliant members with respect to their local frames Figure 3.37 Half-model of a folded-beam spring Because one entire folded-beam spring is composed of two identical halves... the motion direction of the half model is: By taking in Eq (3. 96) results in: which is the stiffness for a U-spring formed of two parallel fixed-free beams of length When the cross-section of the two identical compliant members of Fig 3.34 is variable, Equation (3. 96) can be generalized to the form: Chapter 3 164 where is the direct linear bending stiffness of the compliant member 2-3 , calculated at point... that is being utilized to spring-couple to the translatory motion of a proof mass is shown in Fig 3. 36 where two Chapter 3 166 pairs of so-called folded beams are placed on the sides of the moving mass Unlike the other configurations where the springs have been coupled in a serial fashion at both ends of the mass and aligned with the motion direction, each of the folded-beam springs are placed in parallel... 10 As a conclusion, configuration # 1 is the most compliant about the x-direction, followed by configuration # 3 and configuration # 2, which is the stiffest Example 3.8 Compare the performance of the U-spring configurations # 1 and # 3 in terms of the out -of- the-plane stiffness about the z-direction in the case where w = 10 t and where c is a parameter Consider the stiffness equations according to... same Eqs (3 .60 ) through (3 .65 ) change into Eqs (3. 46) through (3.51), respectively, which define configuration # 2 All these calculations confirm the correctness of the equations derived here The out -of- the-plane stiffness for this configuration # 3 is: Example 3.7 Decide which of the three U-spring configurations is the most compliant about the main direction of motion when compliant members of all design... half of one folded-beam spring, as pictured in Fig 3.37 When only the links denoted by 2-3 and 4-5 in Fig 3.37 are compliant, and each of the segments has a constant cross-section, the simplest expression of the x-axis stiffness is: which simply considers that the two compliant segments behave as two beams in parallel with respect to the x-motion As a consequence, the resulting stiffness is the sum of. .. variable cross-section, and which can be of variable cross-section The stiffness is calculated at point 4 in Fig 3.35 with respect to a local frame The total stiffness of a sagittal microspring is twice the stiffness of one half because the two half components are connected in parallel, and therefore: The out -of- the-plane stiffness about the z-direction can be found by applying a force and by determining... it, very much similar to the action of a bow-arrow system The three definition stiffnesses, (both are in-plane stiffnesses) and (the out -of- the-plane stiffness), will be expressed for a sagittal spring configuration In order to do determine the stiffness, which defines the spring action about the direction of motion of the shuttle mass, as shown in Fig 3.33, half of the entire microsuspension will be . serpentine springs are formed by series connection of several base units. Figure 3.27 is a three-dimensional model of a serpentine unit with the local reference frame. The in-plane compliances. dimension, linearly for instance, a design such as the one sketched in Fig. 3. 26 can be conceived, but the scaling law can be, in general, different than the linear one exemplified here. Figure. drawing of a pair of serpentine springs connected to a proof mass that can move and alternatively extend and compress each spring. The in-plane motion about a direction perpendicular to the one indicated