## 1 Introduction  
Cantilever devices have been used as the key sensing elements in a number of instruments, including the atomic force microscope (AFM), and mass, fluid, and viscosity sensors. To produce quantitative data from cantilever-based instruments, the spring constant of the cantilever must be accurately determined. For example, accurate spring constant calibration of an AFM cantilever is critical to precisely measure adhesive forces between the tip and the surface of a sample [1; 2; 3; 4; 5], measure the mechanical properties of sample surfaces [6; 7; 8; 9], and determine the appropriate force to accurately determine the thickness of 2D materials, such as graphene [10]. If a cantilever is used as a mass sensor to detect biological or chemical species, an accurate spring constant value is required to determine the additional mass [11; 12]. Therefore, since the development of the AFM, a number of techniques have been developed to determine the spring constant of cantilevers. These techniques can be divided into different categories and include methods based on Euler beam equations [13; 14; 15], the resonant frequency response of the cantilever [16; 17; 18; 19; 20; 21], thermal noise-based methods [22; 23; 24; 25; 26; 27], loading the cantilever with a known force [28; 29; 30; 31; 32; 33], or a combination of these [34]. Some of the most recent research involving cantilever calibration and dynamics include determining the spring constant of higher modes, which has applications to AFM force spectroscopy and chemical and biological sensing [35]. The calibration of spring constants for AFM cantilevers of arbitrary shape has also been reported using resonance frequency measurements acquired in air and fluid [36]. The calibration of the torsional spring constant of AFM cantilevers has also been explored and is important to the study of friction and tribology on the nanoscale. Recent work by [PERSON] et al. involved a method to determine the torsional spring constant using the properties of viscous environments [37], while [PERSON] et al. developed a technique for calibrating the static and dynamic torsional spring constant of AFM cantilevers by directly measuring the thermally driven motion of the cantilever with an interferometer [38]. [PERSON] et al. also demonstrated the calibration of the flexural and torsional spring constants and the deflection sensitivities for AFM cantilevers using a noninvasive thermal noise method [39]. Depending on the method employed, there are a number of factors that can complicate the calibration of the cantilever spring constant. One of these factors is the plan-view (or top-down view) shape of the cantilever. For AFM cantilevers, there are a number of commercially available geometries but, in general, cantilevers are divided into two shapes--beam shaped and V-shaped where the two different shapes evolved for different reasons. Beam shapes are an obvious choice based on the simplicity of the design while V-shaped cantilevers were developed since they were assumed to provide greater lateral stability when scanning and significantly reduce torsional twisting of the cantilever. However, it is important to note that the research by [PERSON] et al. [40; 41] disputed this assumption with strong theoretical and experimental evidence. A relatively recent addition to the design of cantilevers is the trapezoidal plan-view shape. Bruker, in particular, has developed a line of cantilevers based on this shape for their fast-scanning AFM system. This geometry was selected by [PERSON] to provide high-performance fast-scanning probes since it is known to reduce the cantilever settling time and facilitates faster scanning [42; 43]. Recent work by [PERSON] and [PERSON] [44] also showed that for biological and chemical sensing applications, the trapezoidal cantilever shape offered superior mass sensitivity in fluid environments compared to a range of other geometries. [PERSON] et al. conducted research into using and modifying trapezoidal-shaped cantilevers for mass sensing [45]. [PERSON] et al.