Analytical modeling of an inclined folded-beam spring used in micromechanical resonator devices

Accurate estimation of the mechanical behavior of springs is crucial for the proper design of Microelectormechanical systems (MEMS). The main objective of this study is to derive a closed-form equation for the calculation of the stiffness of an inclined spring in the form of folded-beamss. The energy-based method was used to calculate the displacements of a folded-beams that was fixed at one end and giuded at the other end. The analytical model was then compared with the finite element method using ANSYS for different inclination angles of the folded-beams spring, showing good agreement. The angle on inclination has changed from zero to 180 degress, and stiffness of folded-beams is detemined. The derived expressions of the compliances were checked for the case of serpentine springs with inclination angle of zero, and different length ratios against the literature. It is found that neglecting small lengths for calculating the stiffness of the folded-beams spring is not justified. The influential geometrical parameters, including different lengths of the spring and inclination angle of the spring, on the stiffness are studied. It is found that the angle of inclination of the principal axes of spring constants depends on the geometrical parameters, and the angle of inclination has more effect on the stiffness of a folded-beams spring than the number of folds.

, Displacement of the spring in , a and b directions C ij Compliance coefficient in i direction due to unit load in j direction Any further distribution of this work must maintain attribution to the author-(s) and the title of the work, journal citation and DOI.
systems, piezoelectric vibration and pressure sensors, MEMS switches, timing devices, resonators, strain gauges, and fiber optic sensors. Potential applications of springs are diverse and include MEMS actuators, MEMS XY stage, RF switches, and energy harvesting devices [2]. The main functions of the springs in these devices are reduction of vibration, keeping elements in position, connection between elements, and application of repulsive force [2]. Among various MEMS inertial devices, an accelerometer is a device that senses the inertial reaction of a proof mass for the purpose of measuring linear or angular acceleration [3] and is used in automotive, biomedical, robotics, consumer, and military industries [4]. It is composed of a proof mass in the form of a rectangular plate suspended via beams (springs) to a fixed frame, whose displacement is proportional to the acceleration it receives [5]. The proof mass displaces with respect to the external frame due to the applied acceleration, which in turn causes contraction/extension of the supporting springs. Both displacement of the proof mass and induced force in the springs are used to measure the applied acceleration [6]. Generally, all micromachined accelerometers are modeled as a mass-spring-damper system [4]. One performance parameter of the accelerometer is the Q-factor (quality factor), which is related to the ratio of stored energy to dissipated energy. The quality factor of the accelerometer is defined as: where M is the mass of the proof mass, k is the spring constant of the suspension beams, and D is the damping coefficient. To manipulate the quality factor, where the mass is fixed, the spring stiffness and damping should be tuned to achieve the desired Q-factor. Another important parameter for accelerometers is their sensitivity. The ratio of the proof mass displacement to the applied acceleration is defined as the static sensitivity of an accelerometer. To increase the sensitivity of an accelerometer, the ratio of the proof mass to the supporting beam stiffness should be increased [7]. The total noise equivalent acceleration (TNEA) is calculated as [8,9]: where K B is the Boltzmann constant ( J K 1.38 10 23 / -), T is the absolute temperature in kelvin, D is the viscous damping coefficient, and M is the proof mass of an accelerometer. To reduce the noise, the proof mass should be increased and have the damping reduced. Thus, in general, the damping of the accelerometers should be minimal, and the proof mass should be maximal. However, the stiffness should be a trade-off between the quality and static sensitivity. Accurate estimation of the spring constant is therefore crucial and one of the most complicated aspects of vibration analysis.
The accelerometer can be a single-axis (single-degree of freedom) accelerometer, in which the proof mass is free to displace in one direction. In this case, the suspended beam can be one beam in the form of a cantilever beam [10], two suspended beams in the form of a double cantilever beam [11], two beams at the middle of the two parallel edges of the proof mass called a simple bridge or four beams at the corner of two parallel edges called a double bridge (also known as clamped-clamped flexure [12]). To increase the flexibility of the accelerometers, the straight beams are folded, and to have symmetry, they are used at four corners of the proof mass called folded-beam and crab-leg beams [12]. The suspension beam has also been used in the form of an in-plane frame around the proof mass and parallel to the fixed frame [13].
Another type of MEMS inertial sensor that uses a vibratory proof mass supported by springs is a gyroscope. A MEMS gyroscope is a vibratory gyroscope [14] in which a vibratory element in place of a traditional spinning element [15] is used to measure angular rotation or angular velocity with Coriolis effects [1]. The vibratory element or proof mass can be in the form of a beam, plate or shell suspended by flexible beams above a substrate [16]. The simplified equation of motion of MEMS vibratory gyroscopes is based on a spring-mass-damper system [17], which can be based on linear or torsional motion [16]. In linear MEMS gyroscopes, the proof mass should be free to vibrate in two orthogonal linear directions, in which the supporting beams are in the form of crab-leg, serpentine, U-shape, and frame [18][19][20]. The gyroscopes have a pair of principal elastic axes that define stable directions of a straight-line motion of the gyroscope [21]. If the principal spring constant is defined as k 1 and k , 2 the stiffness of the spring in the general direction is defined as [22]: where q is the angle between the principal axis and a axis, and k ab is the cross-axis stiffness of the spring. Mounting the spring on its principal axes eliminates any cross-axis motion. Appropriately mounting springs in the principal axes can affect the performance of MEMS by reducing parastic motion, increasing the quality factor, and eliminating cross-axis motion.
Reducing the actuation voltage of MEMS switches broadens the range of their application and enhances their application [23]. Among the different methods, lowering the switch spring constant is the best approach to reduce the actuation voltage [23,24]. Selection of the spring in the MEMS depends on the fabrication process, size limitation and, most importantly, stiffness. The folded-beamss are chosen due to their low value of stiffness in a restricted area [25]. Saffari et al [26] introduced a rotated serpentine spring to reduce the stiffness and actuation voltage of RF MEMS. Folded-beam springs are also used in MEMS optical switches due to their high reliability [27].
Fedder [12] analytically calculated the spring constants of a serpentine spring using the energy method. Iyer et al [28] used the energy method to derive the spring constant of U-spring, crab-leg and serpentine spring. Barrilaro et al [29] calculated the torsional stiffness of a serpentine spring using the energy method. Lu et al [30] proposed an expression for the calculation of the stiffness of classical and rotated serpentine springs. Wai-Chi et al [31] and Witter and Howell [32] determined the stiffness of a U-shape spring considering shear deformation. Liu and Wen [33] gave an equation for the calculation of a folded-beam spring. Urey et al [34] calculated the stiffness constant of flexural beams for five different modes of vibration used in micromechanical scanners. Kovacs and Vizvary [35] compared the sensitivity of an accelerometer with cantilever and bridge-type springs. Grandos-Rojas et al [36] optimized the number of folded-beams to have the required stiffness of the spring within the available space. Jiang et al [37] compared two spring configurations of folded-beams and coild springs from the point of view of having the same stiffness in both directions while occupying less area. Xia et al [38] used different types of folded-beam springs in a tri-axis linear vibratory gyroscope and adjusting the springs' dimensions minimized the frequency mismatches between different modes of vibration. Zhou and Dowd [39] proposed a tilted folded-beam spring for comb-drive actuators. Rouabah et al [40] numerically compared the stiffnesses of square and curved serpentine springs for out-of-plane motion in an electrostatic actuator MEMS. Kaya et al [41] and Petsch and Kaya [42] numerically studied the dynamic behavior of serpentine springs with variable lengths mounted at 45 degrees in an accelerometer. Yamane et al [43] analyzed the spring constant of a serpentine beam in the z direction to suspend the high-density proof mass of an accelerometer. Guo et al [44] determined the stiffness of a curve serpentine spring in the form of an S-shape in a three-axis gyroscope.
The quality factor, stability and sensitivity of MEMS resonators depend on the stiffness of the springs. However, depending on the type of MEMS, the stiffness should be maximum or minimum, and in most cases, it is desirable to have minimum stiffness in one direction and maximum stiffness in the orthogonal direction [45]. Stiffness calculations of folded-beam springs have been used for years in MEMS resonators; however, stiffness of folded-beam springs in a general direction never received detailed attention in the literature. Advance manufacturing methods make it possible to use incline springs for better performance and optimize the use of resticted space. In addition, due to fabrication errors, the stiffness matrix would not be diagonal, and nondiagonal terms cause coupling of motion in two orthogonal directions, leading to quadrature error for the drive mode in gyroscopes [46,47]. Thus, it is necessary to have the spring constant of folded-beam springs in a general direction of motion to determine the principal axes of stiffness.
Research on vibration and spring performance is interdisciplinary and involves fields such as electrical, mechanical, and material science. The main aim of the present study is to use the energy method to derive the spring constant of a folded-beam when it is free to dispalce in any direction. A literature-based equation and finite element method (FEM) are used to validate the derived expressionBy mounting a spring in an inclination angle to achieve benefits that go beyond what is possible with conventional arrangements. In the next section, the energy method is used to calculate the stiffness of a general folded-beam spring, followed by the numerical method. In section four, the stiffness of different types of folded-beam springs, including serpentine spring, U-shape spring, and crab-length beams, are determined and discussed for different inclination angles, and in section five, the conclusions are given.

Stiffness analysis of a folded-beam free to displace in a general direction
The serpentine and crab-leg springs consist of folded-beams with rectangular cross sections (figure 1).
In micromechanical chips, one end of the springs is fixed to the surrounding frame, and the other end is connected to the proof mass, which can displace with the proof mass in any direction. For an inclined spring and/or when the proof mass moves in any direction, it is necessary to have the spring constant of this spring in a general direction. Energy-based Castigliano's method used to derive the displacement of a folded-beam spring. The bending moments, M x ( ) at all parts are calculated (appendix), and the displacements of the free end of the folded-beam are calculated as follows: where d a and d b are displacements of the spring at the free end in the a and b directions, respectively z q is the rotation of the free end around the z axis (figure 1), F a and F b are the applied forces at the free end in the a and b directions, respectively, M q is the applied bending moment around the z-axis at the free end, and EI is flexural rigidity of folded-beam.
Substituting the bending moment from the appendix in equations (4)-(6) yields the following relations for calculating the displacements: Equations (7)-(9) can be expressed as follows: where C ij are the compliance coefficients of the folded-beam spring and are given in the appendix. Thus, the stiffness of the spring in the a and b directions is defined as:

Numerical method
To validate the results of equations (11)-(13), the finite element simulation software, ANSYS v.2022 R [48] was used using solid elements, solid185, which has eight nodes with three degrees of freedom at each node, and translations in the nodal x, y, and z directions (figure 2). A fixed boundary condition at one end is imposed by restraining all degrees of freedom of all nodes. The other end is free to move only in the direction of ; a thus, degrees of freedom in the direction of b and rotation around z-axis are restrained, and in the direction of , a all nodes are coupled. Young's modulus is selected as 1.60 10 11 Pa, and Poisson's ratio is selected as 0.22. A unit load is applied in the direction of , a and the displacement in this direction is determined.

Results and discussion
The equations of the compliances (appendix) are used to investigate the effect of different geometrical parameters, including the number of folds, the ratio of the length of connector beams to span beams, and the ratio of end beams to span beams, on the stiffness of the inclined folded-beams. To demonstrate the applicability of the method, the stiffness of serpentine springs with two different lengths of first and last span beams are determined and compared with available equations in the literature, followed by U-shape springs and crablength springs. sin As seen, in equation (14), there are many terms with sentences such as l l ,   Substituting equation (14) in equation (11) for the case of 0 a = and neglecting the length ratio l l ,   Barillaro et al [29] have given the following expression for the calculation of k :  (11), in which all terms are considered (developed code in MATLAB), and FEM using ANSYS.
As seen, the results of equation (15) are very close to the Barillaro et al [29] results for the case of the parameter m larger than one; however, for the value of m = 0, the difference can reach up to 16%. Additionally, it can be concluded that the results of the current study when all terms are considered, equation (11), are very close to the FEM using computer code, ANSYS, regardless of the value of the parameter m and the ratio of l l . However, the results of equation (15), in which small values of lengths, l , p are ignored, have deviations from the FEM and equation (11) for small values of the parameter m, and for the case of m = 0 ( figure 3(a)), the deviation is as high as 37%.
For the case of 0, a = using equation (12), the stiffness in the y direction can be calculated. Barillaro et al [29] have given the following expression for the calculation of k :  (17) for the case of m = 0. Figure 6 depicts the results of equation (12) for calculation of the stiffness of a horizontal serpentine spring in y direction, which are compared with equation (17) and FEM using computer code ANSYS. For a small value of the parameter m, there are significant differences between the results of the current study and Barillaro et al [29]; however, for the parameter m larger than four, the difference is negligible, and the stiffness approaches zero. Additionally, the results of the current study are the same as those of the FEM regardless of the value of the parameter m.
In figure 7, the results of the current study for calculation of the stiffness of an inclined serpentine spring in x direction for the case of l l l 0.5 ,  (figure 1(b)). As seen, there are very good agreements between the results of the present study and FEM.
Comparing table 1 with figure 7, one can conclude that the angle of inclination, a ( figure 1(b)) has more effect on the stiffness of a folded-beam spring than the number of folds. In addition, in a restricted area to position a spring, inclining occupied less area than increasing the number of folds. Figure 8 shows the stiffness of a horizontal serpentine spring with ratios of l l l 0.5 ,    As seen, the results of the current study are the same as Fedder [12] and have very good agreements with FEM and regardless of the value of the parameter m. Figure 10 shows the displacement of the folded-beam determined using computer code ANSYS for the case ofl l l ,  Figure 11 shows the stiffness of a serpentine spring for the case of l l l 1 2 0 = = when the parameter m is equal to one and ten.  The principal axes of the spring constant in the case of m = 1 have an angle of 90 degrees with respect to the x-or y-axes, and for the case of m = 10, the principal axis has an angle of zero degrees with respect to the x-and y-axes.
In figure 12, the variation in the angle of the principal axis with respect to the x-axis for different values of the parameter m and two different length ratios of l l l l 1 0 2 0 = is depicted. When the parameter m is between zero and three, for the case of l l l 1 2 0 = = (figure 9), the angle between the principal axis and x-axis is approximately 90 degrees; therefore, the x-and y-axes are principal axes; however, the y-axis corresponds to maximum value of stiffness and x-axis correspond to minimum value of stiffness. For the parameter m larger than five, the angle of the principal axis with respect to the x-axis is zero, which means that the x-axis is the principal axis that has maximum value of stiffness, and the y-axis has minimum value of stiffness. For the case of l l l 0.5 1 2 0 = = ( figure  3), when the parameter m is less than seven, the x and y-axes are not principal axes, and due to the applied force in the x direction, the spring would displace in both directions of x, and y. However, when the parameter m is larger than seven, the angle of principal axis with respect to x-axis is 180 degrees. This means that, x is principal axis correspond to maximum value of stiffness, and y-axis correspond to minimum value of stiffness.

U-shape spring
In a U-shape spring (figure 13), the equations given in the appendix for the calculation of compliance are simplified as follows:    = at 0, , a = k ab is zero, which means that x and y are principal axes of a U-shape spring with equal length, and since k b is higher than k , a thus y-axis correspond to maximum value of stiffness and x-axis correspond to minimum value of stiffness. For the case of l l 0.5 , ab is zero; thus, the principal axes have rotated by three degrees with respect to the x-and y-axes. Iyer et al [28] determined the crossaxis stiffness of a U-shape spring when l 0 is close to l 1 as follows: As mentioned earlier, for U-spring with equal length of long beams, the x-and y-axes are principal axes of stiffness; thus, the cross-axis stiffness, k , xy is zero.
Wai-Chi et al [31] proposed following equation for calculating the stiffness of a folded-beam in the form of a U-shape spring with equal length in the x direction considering shear deformation:  As seen, the results of equation (23) and FEM are the same regardless of the ratio of l l , p 0 and the results of equation (26) are higher than equation (23). However, as the length of the short beam and/or its height increase, the results of equation (26) approach to the results of equation (23). The results of equation (25) are excluded because it yields a negative value for a large value of the ratio of l l p 0 . It worth to mention that figure 16 also valid for the case of l 0.1 p = l 0 and the ratio of I p /I 0 varies from 1.0 to 10. Liu and Wu [49] proposed following equation for calculating stiffness of a U-shape beam called single-stage folded-beam, by neglecting the short beam and difference between length of two long beams:   Table 3 compare the stiffness of a U-shape beam spring calculated by equation (27) and current study.
As seen, when the ratio l l 1 0 approaches to one, and l l p 0 approaches to zero, the results of current study and Liu and Wen [49] are very close, however for the case of l l 1.10 , = the deviation is about 25%.

Crab length spring
Crab length springs consist of two perpendicular beams ( figure 17). Figure 18 depicts the stiffness of a crab length spring as a function of inclination angle, . a When the crab length spring is mounted parallel to the y-axis, its cross-axis stiffness and stiffness in the x-direction are almost zero, and the stiffness in the y-direction is maximum. For the case of 0, a = by substituting equation (21) for the case of l 0 1 = in equations (11) to (13), the stiffness in x, y and cross-axis are determined as follows:  [28] gave the following equation for the calculation of the cross-axis stiffness of a crab-length spring, which is the same as equation (30): Table 3. Spring constant of U-shape beam spring in the x direction, k .

( ) =
where n is number of long beams, which is equal to 2m + 2 in figure 1, t and W are thickness and width of the cross section of the beam. Table 4 depicts the results of equation (31) and current study using the developed code in MATLAB for the case of I I ,   = and m = 0, the deviation reaches up to 55%. Figure 22 depicts variation of stiffness of this type of serpentine spring as a function of spring inclination angle, a for the case m = 0.
As seen, for this serpentine spring prinscipal axes have 2 degrees in respect to x-and y-axes.

Conclusion
In this study, a closed-form equation for calculating the stiffness of an inclined folded-beam spring is derived using the energy method. The inclination angle has changed from zero to 180 degrees, and number of folds from zero to ten. First the influence of small lengths in the stiffness of serpentine spring are examined. The results show that for an inclined serpentine spring with an angle between 15 and 165 degrees, neglecting the small lengths of the folded-beam yields inaccurate results. When inclination angle of spring is zero, for calculation of stiffness in the x direction, this neglect is justified only when the number of folded-beams are large. The derived equation is examined against the literature and numerical method using the computer code ANSYS for different types of folded-beam springs including serpentine springs with different length ratio of long and short beams, U-shape springs, and crab-length spring. Mounting the spring in principal axes of spring is crucial in reduction of unwanted parasitic motion, reduction of noise and increasing quality factor. The influential geometrical parameters, including different lengths of short and long beams and inclination angle of the spring on the stiffness are studied, and principal axes of the spring constant are determined. It is found that, the angle of the principal axes of the spring constant depends on the geometrical parameters of the folded-beams. For U-shaped springs, the principal axes are parallel to the beams when the lengths of both long beams are equal. For serpentine springs, the position of the principal axes depends on the number of folded-beams and the ratio of the length short beams to long beams.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).

Appendix
The bending moment at any cross section of the folded-beam spring (figure 23) is defined as: