Theoretical and experimental study of the dynamic response of absorber-based, micro-scale, oscillatory probes for contact sensing applications Free

Tuning fork and other quartz-based oscillators are commonly used as timers in clocks, AFM probes,^1–4 touch sensitive sensors, micro-robotics fingers,^5–7 and micro balances.⁸ Micro-scale oscillatory probes made by attaching carbon, glass, or tungsten fibers to a quartz tuning fork tine provide a capability to both oscillate the fiber and monitor changes in its response when the fiber comes into proximity with external objects or is immersed in fluids. These probes have been used for a broad range of applications including for surface modification studies in vortex machining process,^9–11 touch probes in coordinate measuring machines (CMMs),⁶ touch-sensitive micro-robotic fingers,⁵ and in high speed fluid-flow in microfluidics studies.¹² Another study plans to use these tuning fork-based micro probes for investigation on micro-particles dynamics around dynamically vibrating objects for non-contact manipulation of particles immersed in fluids, which can be used for assembly and sorting/shepherding particles.^13–16 

To better control these processes, it is necessary to interpret the measured responses of the probes in terms of the fiber and tuning fork dynamics. Hence a mathematical model of oscillating fibers attached to a tuning fork tine is provided in this article. This will be represented in two main sections, the theory behind the two serially connected beams model and the lumped absorber model, results from which are compared with experimental data and finite element analysis (FEA).

In most of the micro-probe applications, a tuning fork is used as an oscillation mechanism to drive a fiber attached to it. This section presents two vibration models of the tuning fork-based probes one is two Euler-beams serially connected and the other is a simplified lumped model approximation. A 3D solid model of a 75 μm diameter tungsten fiber attached to a tuning fork tine with the important geometric parameters is shown in Figure 1.

The blue and red regions in the model represent the electrodes to which an oscillating potential difference is applied.

Using the Euler beam equation,¹⁷ mode shapes for these two beams are given by

The parameters y₁ and y₂ are the lateral deflections of the tuning fork tine and fiber as a function of axial distances x₁ and x₂ measured from the fixed end, respectively. Additionally, parameters m, E, L, and I represent the mass, elastic modulus, length, and second moment of area about the neutral axis of bending with subscripts 1 and 2 representing the tuning fork tine and the fiber, respectively. To determine mode shapes and natural frequencies, it is necessary that the above equations satisfy the eight boundary conditions (a)–(f)

From the first of these conditions, Equations (1) and (2) can be rearranged in a simplified form

To solve for the natural frequency of this combined system it is necessary to express the α coefficients in terms of a common eigenvalue form

The description of these parameters and their values in the model are given in Tables II and III. In Equation (6), the subscript s represents the modal frequency. It is important to note that the constant β is a function only of the materials and geometry of the fiber and tine.  Appendix details the derivation of the equations of this section.

Equations (A3)–(A7) (provided in the  Appendix) can be written in matrix form

Substituting ψ_s = α_s2L₂  and   s = 1, 2, …, ∞ Equation (10) can be expressed as

A Matlab™ program is used to find the eigenvalues of this matrix for given κ, β, and γ. The length of the fiber is varying by etch increment in the program, and the obtained eigenvalues (ψ_s2) of matrix in Equation (11) are the natural frequencies of the system, representing different modal frequencies of the probe at each fiber length.

Generally, the tuning fork tine is excited with an applied distortion characteristic of the first mode shape for lateral vibration in the plane of the two tines. Theoretically, an excitation force distributed along the tine corresponding to this mode shape will excite only modes with single node lateral deflection. Because the system to be modeled comprises the tuning fork tine with a relatively small fiber attached (in practice, with a small amount of epoxy adhesive) at the free end, to a first approximation, the tine can be modeled as a single degree of freedom system. Assuming that the attached fiber has a little influence on the tine, it is reasonable to expect that the fiber will only have major significance on the tine response if the modes of the fiber are correspondingly close to that of the tine. Indeed, if the fiber attached to the tine has a coincident eigenvalue, it can act as an absorber. To understand the expected behavior of a fiber probe as a function of fiber dimensions (later evaluated using electrochemical etching), a lumped model of the complete probe system is shown in Figure 2.

The various parameters of this model are given in Table I.

Generally, values for the equivalent stiffness and mass of the fiber elements are chosen to result in similar modal frequencies. Damping coefficients are more difficult to assess. For tuning fork tines in air, experience shows Q values ranging from a few hundred up to a few thousand. The fibers are near to ideal being essentially bonded to the ends of the tines and extending freely. Hence a low value for the intrinsic modal damping has been assumed. For simplification of computing, each mode is assumed to have a similar damping ratio for which free vibration exponential decay will be proportional to the modal frequency (i.e., damping proportionate to stiffness in modal analysis models).¹⁸ 

For the tuning fork tine undergoing a first mode oscillation, Rayleigh’s method produces a reasonable estimate of the natural frequency. For this method the Rayleigh’s quotient can be derived by assuming a deflection shape corresponding to the deflection of the beam due to a load at the free end. In this case, the stiffness at this free end, k_f, can be derived from the equation

All symbols and their values used for theoretical calculation are defined in Table II. Additionally, it is known that the tuning fork in its free state (i.e., with no fiber attached) has a defined first mode frequency typically around 32 kHz (or 40 kHz in some cases). Hence, its representation as a single degree of freedom system can be represented by the lumped equation from which its free-state, undamped natural frequency, ω_f, can be expressed as

For a given free state natural frequency the above equation can be rearranged to give an expression for the equivalent lumped mass. Based on these values, the damping coefficient of the system is readily given by

In all cases x represents the modal coordinate for displacement at the free end. For the fiber, each mode is represented by a single degree of freedom system with energy storage and dissipation elements. Equating the elements of the individual modes with a discrete lumped mass it is reasonable to assume that a system in which at the extreme stiffness values will converge to a lumped system of comprising the tine plus total fiber mass. From a modal equation for the fiber as a cantilever independent of the tine (this will occur when the fiber is acting as a perfect absorber), it is found that the modal mass is constant for each modal coordinate while the stiffness will increase in proportion to the eigenvalue. Based on these assumptions, the lumped parameters can be obtained from the eigenvalues given by

Therefore,

Two different fibers, i.e., carbon and tungsten are used in the experiments for comparison with the theoretical results (represented in Section IV). The description of two fiber parameters and their values used in theoretical calculation are given in Table III.

The root $α i L$ are obtained from the boundary conditions of the beam. For a free cantilever, these roots for the first ten modes are given in Table IV. Due to the slow convergence of modal series and sensitivity to small variations in constants of subsequent simulations, the number of significant digits provided in this table is often necessary for modal analysis.

Equations (12)–(16) provide the necessary relationships between the probe assembly and lumped model. The dissipation function, D, potential, U, and kinetic energy, T, for the lumped model in Figure 2 are given by

Lagrange equation provided below is used to obtain the equations governing motion for the model. Therefore equations in (17) are substituted in the Lagrange equation,^17–19 

Correspondingly, the equations governing motion of this system are given by

Figure 3 is a linear system diagram relating the output displacement of each coordinate for which y is the displacement generated by the input electrical voltage that causes a static deflection closely approximating the shape of the first mode of the tuning fork tine. x_o represents the displacement of the tuning fork tine, and x_i represents the displacement of the fiber in its i-th mode. The steady stated frequency responses of the individual coordinates of this system can be obtained using the assumed solutions

Substituting (20) into (19) and rearranging yield an expression for the steady state frequency response at coordinates x given by

A Matlab code is developed to compute these frequency response functions as the fiber length is shortened. In practice for a tuning fork based probe, it will be the tine displacement that is sensed and therefore will be measured as the frequency response of the probe, see Section IV C.

Fiber probes are fabricated by manually aligning the fiber and tuning fork tine under an optical microscope using a multi-axis micrometer driven stage. Once aligned the tungsten and carbon fibers are attached by high strength epoxy (Norland optical adhesive #61). Figure 4 shows a 75 μm diameter tungsten fiber glued to the upper tine of a quartz tuning fork.

An electrochemical etching apparatus is used for precise material removal for the purpose of controlled shortening of the length or diameter reduction of the fiber. The etching solution consisted of deionized water (15.3 MΩ cm resistivity) with a 1 molar solution of potassium hydroxide (KOH). Mass was removed from the probes in small increments (5–10 μg) and measured from the electrode charge transfer during etching (using chronocoulometry). Figure 5 shows the etching experimental setup. Voltage is applied across the carbon rods and the probe fiber. Current through this cell (i) is in series with a resistor (R_ref) that provides a voltage (V_R) that is amplified (by factor of G which generates V_O). This output voltage is then integrated to measure the amount of charge (q) from which mass removal is calculated. For the conditions of these experiments, 1 tungsten atom is removed for every 6 electrons through the cell. More details about the development of the etching instrument is presented in Ref. 20.

After shortening the length (or diameter) of the fibers, a lock-in amplifier is used to generate a frequency sweep to obtain the new frequency response of the probe.

Results presented in this section illustrate the correlation between the two theoretical approaches with experimental data and FEA results. First, the result of two beams serially connected method, predicting the behavior of probes versus the fiber varying length and diameter compared with experimental data is represented. Next, the amplitude and phase frequency response of the probe obtained from lumped absorption model is illustrated. Finally, the natural frequencies of probes and their modes shapes obtained from FEA simulation and the comparison with experimental data are provided.

Figure 6 illustrates the first ten modes of the carbon probe obtained from the theory for lateral vibration of two beams serially connected, Section II A. In this figure the asymptotic loci of markers represented with blue dots show the modes of the probe which demonstrates the natural frequencies relationship with length of the fiber. The asymptotic lines represented with pink circles correspond closely to the natural frequencies for the carbon fiber modeled as a single cantilever beam. The consistency between these two lines indicates that the carbon probe is behaving similar to a cantilever beam.

Oscillation of carbon fiber with 7 μm diameter attached to tuning fork shows a clear illustration of modal frequencies. The photograph of these observations (obtained experimentally) for third, fourth, and fifth modes is shown in Figure 7 as an illustration.²¹ 

Figure 8 shows the frequency-fiber length plot for the tungsten probe. In this figure, the difference between lines with blue dots and pink circles is more significant than the one depicted in Figure 6. It is speculated that since the tungsten fiber has larger diameter and is heavier than the carbon fiber, it therefore results in the probe deviating further from simple cantilever type behavior. Generally, for “touch” sensitivity, the probe length is optimal when the fiber oscillates at maximum (or near to maximum) amplitude. This corresponds closely to the zeros in the frequency response plot. Not coincidentally, these are also located in a region where the probe, and closely related independent fiber, frequencies coincide closely to that of the independent tine first mode resonance. It might be expected that the fiber will have a node at the attachment point at the tip of the tine at which point it will act as an absorber. Consequently, under these combined favorable conditions, the energy supplied to the tine will be maximally transferred to the fiber that is typically excited in a much higher mode. For a simple cantilever, the maximum total potential, V, and kinetic energy, T, associated with oscillation of the fiber are

where q_s represents the modal (or normal) coordinate associated with the sth mode. In fact, under these optimal conditions, the fiber may be considered to be an optimal absorber for the tine.

Also shown in Figures 6 and 8 are eigenvalues of the matrix that, while exhibiting asymptotic behavior over short changes in length, these then jump to form a series of short segments that correspond to a nearly constant frequency. These lines correspond to the natural frequencies of the tuning fork tine in different modes, for which four of them are highlighted in Figure 6. For the longer fiber lengths, values for the first five frequencies have been extracted from these theoretical plots, and the ratio between them is compared with different cantilever beam types, see Table V. The ratio of the modal frequencies shows that the tuning fork tine is behaving as beam with frequency being between clamped-pinned and clamped-free conditions. One side of the tine is assumed to be clamped as it is stated in the boundary condition (Equation (3)). The other end of the tine will experience varying dynamic forces that will depend on the free natural frequencies of both beams. For high frequencies, forces on the tine due to fiber mass and the second moment of mass will be relatively large, effectively acting to a pin the “free” end of the tine. For lower frequencies however, the effect of the fiber mass is less significant thus, the fiber forces at the end of the tine become less significant and it responds more closely to being free at the end.

Figure 9 shows the experimentally measured frequency response in terms of magnitude and frequency as a function of the length of the tungsten fiber. The initial length of the fiber used was 14.11 mm after which it was etched down to 0.83 mm in non-equal steps (due to difficulties associated with controlling the immersed length of the fiber tip in the electrolyte solution). It can be noticed that the natural frequencies of the probe at different modes are changing asymptotically with respect to the length of the fiber similar to the observations obtained from theoretical graphs in Figures 6 and 8. This will be further discussed in Section IV C.

Another study has been carried out to examine the effect on probe’s natural frequency of changing the fiber diameter. Figure 10 plots the theoretical eigenvalues for a tungsten fiber of length 8.16 mm attached to a tuning fork tine. Different colors in the graph show different modes of the probe and demonstrate a linear relationship with varying fiber diameter.

This can be predicted from the vibrating cantilever beam equations. The natural frequencies of a cantilever beam is obtained from

Substituting I and m for a cylindrical rod yields

Therefore, the natural frequency is expected to vary linearly with the diameter of the rod and hyperbolically with length. The result of the experimental test where a tungsten probe (8.16 mm long fiber attached to tuning fork) is etched in step of 5 μg (decreasing diameter) is shown in Figure 11.

Figure 12 represents the amplitude frequency response plot showing the magnitude of H_oy (Equation (21)) as a function of length of a tungsten fiber of radius 37.5 μm.

The phase response of H_oy as a function of frequency and diameter is shown in Figure 13.

Similarly, the amplitude frequency response of individual lumped modes H_iy (Equation (22)) is plotted relative to the length of the fiber. These plots for the first four modes of a tungsten probe are shown in Figure 14. The same asymptotic lines represented in Figures 6 and 8 are noticeable in these images. Most significantly, and as expected from this lumped model approximation, the response for each fiber dominates while, because all coordinates of the system response share common roots, the influence of the surrounding system remains. However, it is apparent that the resonant peak for each fiber mode passes through the location of the zero in the tuning fork response.

A frequency study using FEA software available in SolidWorks™ has been carried out on a 3D solid model of the probes. Figure 15 shows the fifth mode shape of the oscillating carbon probe acquired from FEA, and its consistency with the experimental image.

The third and fourth mode shapes of tungsten probe with 5.5 mm fiber length obtained from an FEA study are shown in Figure 16.

The natural frequencies of first five modes for tungsten probe (fiber length = 5.5 mm) acquired from theoretical calculation (two beams serially connected method) and FEA is represented in Table VI, which represent a good correlation between two approaches.

It is noted that the theoretical model predicts frequencies that are higher than of those obtained from FEA. Two possible sources of this difference are mesh size used and the fact that the fiber bonds a short distance on to the end of the tine. Consequently, the additional mass due to the short length of fiber on the end of the tine is not included in the theoretical model. Such a reduction in mass would be expected to result in higher natural frequencies.

Micro-scale oscillatory probes are widely used in metrology, manufacturing, and assembly for precision applications. Two beams serially connected and lumped model approximate vibration models are developed in this paper for detailed understanding of the dynamics of these probes.

Experiments have been carried out with tungsten and carbon probes where the fiber length or diameter was electrolytically etched and their frequency response was acquired. The results from theoretical models and the experimental data displayed natural frequencies of probes to retain an asymptotic relationship with the fiber length and a linear relationship with the fiber diameter. Additionally, the tuning forks in these probes can be modelled with eigenvalues varying between clamped-pinned and clamped-free beams that depend on how closely the fiber resonance (a function of the fiber length or diameter) coincides with that of the tine. Moreover, a frequency study in SolidWorks predicted the probe mode shapes and its natural frequencies which were consistent with the data obtained from experiments, hence it rules out the necessity of complex FEA studies.

A key functional characteristic of these probes is the matching of the fiber modes to that of the first mode of the tine. Generally, the electrode pattern on the tuning fork tines is designed to excite the fundamental mode. Consequently, this excitation shape is not suited for exciting other probe modes. When the frequencies of fiber modes that are not closely matched to that of the tine, even though they may independently have a relatively high Q value, the tine excitation, and therefore the fiber, will have a low amplitude. The approaching and receding fiber modes are visible in Figure 10 for tungsten fiber lengths at 3 mm and 13 mm. When the fiber length contains a mode close to the tine natural frequency, the energy of excitation is transferred into the fiber that then acts as an absorber for the tine mode, resulting in a zero for the frequency response of the tine (i.e., the frequency response that is measured by these types of probe).

While the focus of this study is the analysis of probes undergoing planar oscillations, in practice there are additional modes resulting in out-of-plane motion. Planar analysis is justified by the fact that the excitation force is applied only in this plane and that the results from both theory and experiment are consistent in terms of pole and zero locations in the probe response. Non-linear effects have been observed, more commonly occurring with high Q systems or those having measurable assembly errors such as misalignment between the axis of the fiber and that of the tine. Additionally, when these probes are used as contact sensors, the dynamics of the interaction is typically chaotic. Both of these topics, while interesting, are outside of the scope of this paper. However, there are applications for which multi-directional oscillators for contact sensing applications are of interest, particularly in the field of micrometer-scale touch-sensors for coordinate measuring machines.^22–25 It is expected that the technique for optimizing performance for the planar probe could be similarly determined for any resonant probe designs comprising slender elements attached to an oscillator operating at fundamental or higher modes.

While the probes of this study might oscillate in a single plane, the frequency response does change when the probe is brought into proximity with a surface having any orientation relative to the direction of oscillation. Studies to quantify this orientation dependence have yet to be undertaken. Additional studies are also necessary to determine the optimal parameters for probe designs in terms of subsequent signal sensitivity as a function of proximity to a specimen surface. Sensitivity is also complicated by the choice of excitation parameters (amplitude, frequency, and phase) in the region of the resonant peak. Studies to quantify these parameters are planned, ultimately aimed at achieving optimal performance in terms of bandwidth and signal to noise ratio of micro-resonator probes.

This material is based upon work supported by the National Science Foundation (NSF) under Award Nos. DMI 1000656 and DMI 1538814. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Furthermore, the authors would like to thank Dr. Bartosz Nowakowski for his help in the development of the electrolyte etching instrument and the Center for Precision Metrology (CPM) for providing facilities in fulfilling this project.

Dropping the s subscript, Equations (4) and (5) can be expressed in the form

Satisfying the rest of the boundary conditions is not as straightforward. Consider each of the conditions in Equation (3) in turn

or

or

or