## 1 Introduction
Based on the study by [PERSON] et al., it was found that the dissipated power \(\mathrm{P_{dis}}\) of the cantilever driven at the fundamental resonant frequency and the phase lag angle can be related as shown in Equation (1) [8]:  
\[\mathrm{P_{dis}}=\frac{\pi\mathrm{k}\mathrm{f}_{0}\mathrm{A}\mathrm{A}_{0}}{ \mathrm{Q}}\bigg{(}\sin\phi-\frac{\mathrm{A}}{\mathrm{A}_{0}}\bigg{)} \tag{1}\]  
where \(\mathrm{A}_{0}\) is the free oscillation amplitude, \(\mathrm{f}_{0}\) is the resonant frequency (i.e., first eigenmode in this case), \(\mathrm{k}\) is the spring constant of the cantilever, and \(\mathrm{Q}\) is the quality factor. The total energy dissipated per oscillatory cycle \(\mathrm{E_{dis}}\) of the cantilever can be written as Equation (2). \[\mathrm{E_{dis}}=\mathrm{P_{dis}}\times\mathrm{T}=\frac{\pi\mathrm{k}\mathrm{ A}\mathrm{A}_{0}}{\mathrm{Q}}\bigg{(}\sin\phi-\frac{\mathrm{A}}{\mathrm{A}_{0}} \bigg{)} \tag{2}\]  
where \(\mathrm{T}\) is the period of an oscillation cycle. Although there can be many sources for dissipation of energy within an oscillation (such as capillary forces, electrostatic charges, or magnetic forces), in a controlled setting, it is assumed to be due to the viscoelastic interaction and the hysteresis in the surface adhesion interaction between the tip and surface. Therefore, the dissipated power can be also written as \(\mathrm{P_{diss}}\equiv\mathrm{F_{fs}}.\mathrm{z}\) and the stored power of the oscillating cantilever can be written as \(\mathrm{F_{fs}}.\mathrm{z}\). Because of this relationship, [PERSON] and [PERSON] showed that the \(\tan\delta\) can be directly found by amplitude and phase signals, so it can be written as [11]:  
\[\tan\delta=\frac{\sin\phi-\mathrm{A}/\mathrm{A}_{0}}{\cos\phi} \tag{3}\]  
It should be noted that the ratio of instantaneous oscillation amplitude (A) to free oscillation amplitude (A\({}_{0}\)) make this relationship independent of actual amplitude values in length units. Therefore, Equation (3) can be re-written as the following, where V and V\({}_{0}\) represent the voltage values of instantaneous oscillation amplitude and free oscillation amplitude, respectively. \[\tan\delta=\frac{\sin\phi-\text{V}/\text{V}_{0}}{\cos\phi} \tag{4}\]  
It should be noted that based on the Equation shown in (4), the \(\tan\delta\) can be found without the need for accurate calibration of actual amplitude values. Although there are many studies that have done different types of analysis on this topic, there is still a gap in the field to understand the true effect of tip-velocity (i.e., excitation frequency) on the observable aspects in loss tangent AM-AFM measurements. [PERSON] et al. have examined the effect of oscillation amplitudes [11]. [PERSON] et al. showed that the AM-AFM loss tangent measurements overestimated the values compared to bulk measurements [13]. In a recent study, [PERSON] et al. have also investigated the effect of tip radius on the measured values [14]. It was found that ultra-sharp tips are advantageous to eliminating the unwanted dissipative processes. In this study, we focus on the effect of excitation frequency and the penetration depth caused by the tip trajectory on different types of materials and compare them with published bulk material properties. [PERSON] et al. have suggested using cantilevers with frequencies over 300 kHz [12; 13; 15]. A practical method to do so is using the higher eigenmode of a typical AM-AFM cantilever to achieve this guideline. However, it is still unknown how the penetration of the tip into soft matter can play a role in the observable aspects of loss tangent measurements.