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

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 thatare perpendicular can be util

stiffnesses by utilizing the matrix transformation of Eq (3.4), and thestiffness 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 aregiven 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 beseen that the stiffness of the straight beam increases noticeably, compared tothe stiffness of the inclined beam When the inclination angle and theparameter c increase towards their upper limits in the selected ranges, thestiffness 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 thatare 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 centralmass symmetrically While the body translates about one of the directionsindicated in the figure, the spring leg that is directed perpendicularly to themotion direction will bend, whereas the other leg will be subject to axialextension/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 befixed-free, as also indicated in the same figure The main deformations of abent beam spring are planar and they result from the two-dimensional motion

of the central mass However, the bent beam is also sensitive to z-axisparasitic loading, generated by the weight of the central mass As aconsequence, in-plane stiffnesses about the x- and y-directions, as well as theout-of-the-plane stiffness about the z-direction will be derived for thismicrosuspension.

Figure 3.8 Rigid body and four bent beam springs for planar motion

Figure 3.9 Geometry of a bent beam microsuspensionThe in-plane deformations of the bent beam can be studied by applyingthe loads and and by calculating the corresponding tipdisplacements and through Castigliano’s displacement theorem.The strain energy collects contributions from bending and axial loading onthe two segments, 1-2 and 2-3 The tip deformations can be related to the tiploads by means of a compliance matrix in the form:

The terms of the compliance matrix are:

The superscripts (1) and (2) refer to the first segment 1-2 and to the secondone 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 totheir local frames In doing so, members of different geometries (defined asfree-fixed microhinges) can be utilized in a bent beam design A stiffnessmatrix 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, theparticularly-important in-plane stiffnesses are:

When the axial deformations are negligible compared to the bendingdeformations, Eqs (3.27) and (3.28) can still be used by considering that theaxial compliances of the two segments are zero (axially rigid members).The mention was made in Chapter 2 that the stiffnesses defined byinverting the compliance matrix are different from the stiffnesses that arecalculated as:

and that the stiffnesses of Eqs (3.27) and (3.28) should be used when forcesneed to be calculated based on known displacements However, Eqs (3.29)are used as definition relationships and their values can be obtained by usingthe 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 aforce at point 1 of Fig 3.9 about a direction perpendicular to the bentbeam’s plane and by calculating the corresponding displacement By takingbending and torsion into account, the z-direction stiffness is:

Example 3.4

Calculate the mam stiffnesses of a bent beam microsuspension withidentical legs and of constant rectangular cross-section Evaluate the errors incalculating by its definition – Eqs (3.29) – as opposed to the compliancederived 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:

The ratio of the x-axis stiffnesses becomes:

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 ofrigid bodies Due to symmetry about the axial (motion) direction, the proofmass 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 tothe self-weight of the proof mass) are also possible, hence quantifying thestiffnesses about the direction perpendicular to the motion direction and thedirection perpendicular to the plane of the microdevice of Fig 3.10 will also

be done, in addition to formulating the main stiffness about the motiondirection

Figure 3.11 pictures a U-spring with the reference frame that is used todefine 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), (thestiffness defining the elastic properties of the U-spring when the bodytranslates about a direction perpendicular to the main one and is contained inthe plane of the microdevice) and (the stiffness which describes the springbehavior for the case of an out-of-the-plane motion about the z-direction, asindicated in Fig 3.11)

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 trueboundary 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 consideredthat point 1 is free to move, as also assumed previously with the bent beammicrosuspension Because the force acting at that point is basically directedabout 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 themodel 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 havedifferent but constant cross-sections It will also be considered that onlybending of each of the three segments contribute to the total strain energy ofthe spring The in-plane compliances are calculated by applying the loadsand as shown in Fig 3.12, and by calculating the correspondingdisplacements and Castigliano’s displacement theorem is appliedagain in order to calculate these displacements A compliance matrix of thetype shown in Eq (3.19) can be formulated, whose terms are:

Figure 3.12 U-spring design with sharp cornersThe stiffness is the element of the stiffness matrix (which is the inverse ofthe compliance matrix consisting of the elements defined in Eqs (3.35)through (3.40)) located on the first row and first column Similarly, thestiffness is the element placed on the second row – second column position

of 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 thathas been introduced in the previous chapters and was also mentionedpreviously in Eq (3.30) for instance Its expression is found by taking theratio of a force which is applied at point 1 in Fig 3.12 about a directionperpendicular to the plane of the microdevice to the correspondingdisplacement namely:

Example 3.5

Analyze the change in the main compliance of a U-spring when the axialdeformations are also taken into account Consider that the three legs haveidentical constant rectangular cross-sections

Solution:

In the case where the axial deformations are considered, the strain energywill 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 thebending-related compliance to the compliance that considers both bendingand axial effects becomes:

Figure 3.13 Compliance ratio in terms of and

Figure 3.14 Compliance ratio in terms of w and

Equation (3.42) indicates that the model including bending effects generates acompliance about the main direction of action which is very slightly largerthan 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 ofand when and Similarly, Fig 3.14 is the plot of thesame compliance ratio in terms of w and when As bothfigures indicate, the compliance ratio is in the very close vicinity of 1 whenthe design variables of Eq (3.42) span relatively wide ranges, which indicatesthat 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 casewhere the middle leg has a small length which implies considering theadditional shearing effects and associated deformations Compare theresulting stiffness with the regular one determined by means of thecompliance of Eq (3.38) in the case where and

where the subscripts in bending moments M and shearing force S indicate thespecific segment out of the three ones making up together the U-spring Thelinear stiffness about the y-direction can be expressed according to itsdefinition 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

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-circularportion instead of the straight segment 2-3 of the previous design

Figure 3.16 U-spring design with circular short linkThere are two possibilities, connected to the form factor of the semi-circular section It is known (see Young and Budynas [1], for instance) thatfor thin curved beams, when the ratio of the radius R to the cross-sectionalwidth w is greater than 10, the deformations of the curved beam can safely betreated by using the tools applicable to straight beams By applying the sameprocedure that has been used for the U-spring configuration # 1, and by onlytaking bending of the three segments into account, the in-plane compliances

of the constant cross-section design are:

By arranging these compliances into a 3 by 3 symmetric compliance matrix,the corresponding stiffness matrix can be obtained through inversion of thiscompliance matrix When the definition stiffnesses are needed, then simpleinversion of the individual compliances of Eqs (3.46) through (3.51) willproduce these stiffnesses.

For designs where the circular segment is relatively short (R < 10 w),Young and Budynas [1] recommend using the following bending energy:

where e is the eccentricity, which, for a rectangular cross-section, can be

calculated by means of Eq (1.122) By applying again Castigliano’sdisplacement theorem for the configuration of Fig 3.16 in the presence of thetip loads and (not shown in Fig 3.16), the resulting displacementsand can be found by means of the following compliances:

The out-of-the-plane definition stiffness is found by applying a force at thefree point of the spring in Fig 3.16, perpendicularly to the plane of the figure.This stiffness is the ratio of the applied force and the resulting deflection to itsequation is:

2.3.3 U-spring with Filleted Corners (Configuration # 3)

Configuration # 3 is sketched in Fig 3.17 where the middle segment iscomposed of two quarter-circles encompassing a straight line such that overall,this spring configuration is made up of five segments

Figure 3.17 U-spring design with circularly-filleted corners

For a thin configuration where the radius-to-width ratio R/w is larger than

10, and in the case where all segments have the same constant cross-section,the in-plane compliances can be determined by following the procedure thathas been used for the other U-spring configurations Their expressions are:

For a relatively-thick design (R / w < 10), the in-plane compliances are:

As Fig 3.17 indicates, when the radius of the two circular portions is zero, thecurrent design transforms into the design configuration # 1, whereas whenconfiguration # 3 changes into configuration # 2 Checks have beenperformed in order to verify whether the corresponding stiffnesses of eitherconfigurations #1 or # 2 are retrieved by using the particular geometricparameters limits mentioned above When R 0, Eqs (3.60) through (3.65)transform indeed into Eqs (3.35) through (3.40), respectively, which define

configuration # 1 Similarly, when the same Eqs (3.60) through (3.65)change into Eqs (3.46) through (3.51), respectively, which defineconfiguration # 2 All these calculations confirm the correctness of theequations 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 compliantabout the main direction of motion when compliant members of all designvariants have the same rectangular cross-section and can be inscribed each inthe same rectangle of sides equal to and Also known are

andSolution:

The compliances of the three U-spring configurations will becompared by analyzing compliance ratios It can be seen that while the firsttwo configurations have their compliances determined by the parametersgiven in this example, the third configuration can have various compliancesbecause the radius R can take any value from 0 to

The following compliance ratios are discussed:

where the superscripts 1, 2 and 3 denote the first, second and thirdconfiguration, respectively

Figure 3.18 is the plot of the first ratio defined in Eqs (3.73) as afunction of the radius R, in the case R / w > 10 for configuration # 3.Similarly, Fig 3.19 pictures the second ratio of Eq (3.73) As Fig 3.18indicates it, the design configuration # 1 is more compliant than the designconfiguration # 3, but they tend to be equal for small radii Configuration # 2

is less compliant than configuration # 3, but for large radii, the two designshave almost identical compliances Similar plots are drawn in Figs 3 20 and3.21 when R / w < 10 for configurations # 2 and # 3

Figure 3.18 Compliance comparison: configuration # 1 versus configuration # 3 (R / w > 10

for configuration # 3)

Figure 3.19 Compliance comparison: configuration # 2 versus configuration # 3 (R / w > 10

for both designs)

Figure 3.20 Compliance comparison: configuration # 1 versus configuration # 3 (R / w < 10

for configuration # 3)