## 1 Introduction  
Electrostatically actuated MEMS cantilevers are one of the most typical structures in the field of micro-electromechanical systems (MEMS) and are widely used in many kinds of micro-sensors [1, 2], micro-switches [3, 4] micro-resonators [5] and many other kinds of micro-actuators [6, 7] such as air-bag sensors, particle sensors, mass sensors, optical switches, display subdevices, micro-mirrors, and RF switches because of their simple structure, large actuation force, and for Si-based MEMS, the IC compatibility. Pull-in voltage and actuation speed are two of the most important properties of electrostatic MEMS cantilevers [8]. For many applications in sensors and switches, lower pull-in voltage reduces the energy consumption, extends the applicable scope, and enhances reliability [9, 10]. In MEMS resonators, e.g., the driving voltages need be controlled accurately, and the pull-in phenomenon must be avoided when the devices are working [11]. Higher responsiveness saves operating time and broadens applications, and it is always a huge advantage for all applications. The manufacturing of compact MEMS cantilevers with fast response has been demonstrated via batch manufacturing processes in large quantities at low cost [12]. It has been proven that the dynamic and static pull-in features of cantilevers are affected significantly by small-scale structural and geometric parameters as well as non-linearities [13]. Moreover, higher resonant frequencies and thus faster actuation speed can be achieved as the microstructures become smaller [14]. All these points define research gaps and a strong motivation to identify contributions to design rules leading to improvements in MEMS dynamics regarding pull-in voltage and actuation speed. Many studies and simulations on MEMS cantilevers have been reported. [PERSON] et al. presented a U-shaped and rectangular triple-coupled micro-cantilever and simulated corresponding eigenfrequencies, surface stresses and displacements. The results showed that the U-shaped structure reveals higher sensitivity than the rectangular one [15]. [PERSON] and [PERSON] calculated the transfer function relating the displacement to the acting force and simulated the time response following an acting force [16]. [PERSON] et al. designed a mass sensor based on a silicon MEMS cantilever which is horizontally actuable and simulated eigenfrequency and displacement under voltage of the sensor with and without added mass [17]. [PERSON] et al. investigated and optimized MEMS cantilevers based on the simulation results of vibration modes, effect of slots and effect of edge serration [18]. [PERSON] et al. designed and fabricated an electrostatically actuable cantilever arrays as an audible frequency generator and simulated the vibration response under pulsed signal input [19]. In the study of uniaxial capacitive accelerometers, [PERSON] et al. simulated the frequencies of combs of cantilevers. The results have good agreements when a reduced thickness is considered [20]. [PERSON] et.al proposed an analytical model for static and dynamic pull-in of curled microcantilevers and found an excellent agreement of the present results [21]. Silicon is the by far dominating MEMS material. However, there are also metallic materials in use for specific MEMS devices, such as optical MEMS shutter arrays for smart glass applications [22; 23; 24; 25; 26; 27; 28; 29; 30; 31] to modulate daylight in windows, for space instrumentation [28], for camera shutters [32], and for artificial eyelids for sensor protection [33] and displays [22; 23; 24; 30; 34; 35]. In the anchor region, these MEMS shutters are fixed to the transparent conductive oxide layer located on the glass substrate. Due to intrinsic stress, these shutter blades are curled in the open state and are uncurled by electrostatic actuation to close them toward the glass substrate. For active daylight steering applications, our flat metallic micromirrors [25; 26] have been developed, which are flat in the mirror region by stress compensation.