JP2010071798A - Contact type displacement sensor - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a contact type displacement sensor which reduces the effect of friction or stick slip occurring between a stylus and a measuring object and enables measurement of a shape by a scanning method. <P>SOLUTION: The contact type displacement sensor 1 which senses displacement of the measuring object 100 includes a movable shaft 3 which is so fitted as to be movable back and forth in relation to the measuring object 100, the stylus 4 which is fitted to the tip of the movable shaft 3 and made to scan on the surface of the measuring object 100, and a vibrator 5 which is fitted between the stylus 4 and the movable shaft 3. The vibrator 5 is made to vibrate at a frequency in a suitable frequency band which is determined on the basis of the spring constant of the vibrator 5, the spring constant and a viscosity coefficient in the fitting part of the vibrator 5 and the stylus 4, the spring constant and the viscosity coefficient in the fitting part of the vibrator 5 and the movable shaft 3, the equivalent mass of the vibrator 5, the mass of the movable shaft 3 and the mass of the stylus 4, while the stylus 4 is vibrated in the axial direction at a prescribed amplitude, and the movable shaft 3 hardly vibrates. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to a contact-type displacement sensor used for shape measurement or the like, and more particularly to a contact-type displacement sensor that is attached with a stylus having a vibrator and performs shape measurement by a scanning method.

  The measurement by the contact-type displacement sensor uses a thin stylus, contacts its tip with the measurement surface to be measured, and measures the shape by following the measurement surface. Shape measurement using a contact-type displacement sensor is an important means in the industrial field because it is resistant to disturbances, does not require special conditions such as vacuum, and can measure with high accuracy even if oil is attached to the measurement target. It has become. In addition, the contact displacement sensor has a feature that it can measure a shape having a steep angle that cannot be measured by light scattering in a microscope or the like using light.

Measurement methods using a contact-type displacement sensor include a point measurement method and a scanning method. The point measurement method is a method in which a contact-type displacement sensor is brought into contact with a measurement object at a certain measurement point, the position thereof is measured, and then once separated from the measurement object and moved to the next measurement point. As an apparatus for performing shape measurement by the point measurement method, for example, an apparatus described in Patent Document 1 is known.
JP 2007-309684 A

However, the measurement using the point measurement method apparatus described in Patent Document 1 has a problem that the measurement time is long.
In contrast, the scanning method is a method of measuring the shape of the contact-type displacement sensor and the measurement object in contact with each other. Therefore, although the measurement time is shorter than that of the point measurement method, the scanning method is performed between the stylus and the measurement object. There is a problem that the stick-slip phenomenon occurs.

In particular, when measuring a micro shape having a size of 100 micrometers (μm) to 200 μm such as a micro part or a micro lens array constituting a micro machine, a thin and long stylus shaft having a tip of several tens of μm is required, and further, friction or The effect of stick-slip is increased.
In order to solve this problem, a method of measuring by increasing the contact pressure of the contact type displacement sensor is conceivable, but there are problems such as damaging the measurement object and bending the tip of the stylus.

  The present invention has been made in view of the above circumstances, and provides a contact-type displacement sensor capable of measuring the shape by a scanning method while reducing the influence of friction and stick-slip generated between the stylus and a measurement object. The purpose is to do.

  The present invention is a contact-type displacement sensor that senses the displacement of an object to be measured, and includes a movable shaft that is attached to the object to be measured so as to be able to advance and retreat, a tip of the movable shaft, A stylus whose surface is scanned, and a vibrator attached between the stylus and the movable shaft, wherein the vibrator has a spring constant of the vibrator, and a spring constant at an attachment portion of the vibrator and the stylus. And a viscosity coefficient, a spring constant and a viscosity coefficient at the attachment portion of the vibrator and the movable shaft, an equivalent mass of the vibrator, a mass of the movable shaft, and a mass of the stylus within a suitable frequency band. Vibrating at a frequency, the stylus vibrates in an axial direction with a predetermined amplitude, and the movable shaft hardly vibrates.

  In the present invention, “almost no vibration” means both a state where there is no vibration at all and a state where the vibration is small enough to be ignored in view of the detection limit of the contact displacement sensor. including.

  According to the contact type displacement sensor of the present invention, when the vibrator is vibrated at a frequency within the preferred frequency band, the stylus is vibrated in the axial direction with a predetermined amplitude, and is intermittently separated from the surface of the object to be measured. The On the other hand, since the movable shaft hardly vibrates, the detection value of the contact displacement sensor is not affected.

The preferred frequency band includes the vibration frequency of the vibrator ω, the spring constant k _i , the sum of 1/2 of the equivalent mass of the vibrator and the mass of the movable shaft, m _h , and the equivalent of the vibrator. The sum of 1/2 of the mass and the mass of the stylus is _mg, the spring constant and the viscosity coefficient of the vibrator and the stylus attachment part are k _g and c _g , respectively, and the attachment part of the vibrator and the movable shaft Based on the following formulas 1 and 2, the curves relating to the transitions of ξ _g and ξ _h for each vibration frequency of the vibrator are obtained by setting the spring constant and the viscosity coefficient at k _h and c _h , respectively. It may be determined. In this case, since the transition for each vibration frequency of the vibrator can be easily grasped by the curve, a suitable frequency band can be determined easily and accurately.

  The vibrator may be a piezo element. In this case, the vibration frequency of the vibrator can be suitably controlled by arbitrarily modulating the applied voltage.

  According to the contact-type displacement sensor of the present invention, it is possible to reduce the influence of friction and stick-slip generated between the stylus and the measurement object and perform shape measurement by the scanning method.

  An embodiment of the present invention will be described with reference to FIGS. FIG. 1 is a diagram showing a contact-type displacement sensor (hereinafter simply referred to as “displacement sensor”) 1 of the present embodiment. As shown in FIG. 1, the displacement sensor 1 is a displacement sensor for measuring the surface shape of the measurement object 100, and includes a main body 2 and a movable shaft 3 slidably attached to the main body 2. The stylus 4 is attached to the tip of the movable shaft 3, and the vibrator 5 is attached between the stylus 4 and the movable shaft 3.

The main body 2 has a known configuration in which the movement of the movable shaft 3 is read as a displacement amount with a built-in linear encoder (not shown).
The movable shaft 3 is attached to the main body 2 through an air bearing (not shown) so as to be able to advance and retract, and moves forward and backward according to the surface shape of the object 100 when the object 100 is scanned with the stylus 4. The measurement pressure of the movable shaft 3 is as low as 1 millinewton (mN) by the air bearing, but it may be attached to the main body 2 by another mechanism as long as it can slide smoothly.

  FIG. 2 is an enlarged view showing the stylus 4 and the vibrator 5. The stylus 4 is formed with a pointed tip, and the base end is fixed to the vibrator 5. The vibrator 5 can vibrate at a specific frequency in the axial direction of the stylus 4 and the movable shaft 3, and is a piezo element, a magnetostrictive element, a crystal vibrator, a micro-electro-mechanical system (MEMS). Etc.) can be suitably employed. A holder 6 for attaching the stylus 4 and the vibrator 5 to the movable shaft 3 is attached to the base end of the vibrator 5. The holder 6 has a screw portion 6A, and the screw portion 6A is screwed into a screw hole (not shown) of the movable shaft 3, so that the stylus 4, the vibrator 5, and the movable shaft 3 are integrally fixed substantially coaxially. For fixing the stylus 4 and the vibrator 5 and between the vibrator 5 and the holder 6, an adhesive made of a thermosetting resin or the like can be suitably used.

  When measuring the surface shape of the device under test 100 using the displacement sensor 1 configured as described above, the device under test 100 is attached to the spindle 101 as shown in FIG. Then, in a state in which the object to be measured 100 is rotated by rotating the spindle 101, the surface of the object to be measured 100 is scanned with the tip of the stylus 4, and the displacement amount of the movable shaft 3 at each rotation angle of the object to be measured 100. Is recorded by the linear encoder of the main body 2, and the surface shape of the DUT 100 is obtained by reconstructing the recorded values of the linear encoder.

  At this time, if the tip of the stylus 4 and the surface of the object to be measured 100 are always in close contact with each other and scanned, stick slip or the like occurs between the stylus 4 and the object to be measured 100, and the accuracy of the shape measurement is reduced. There is. In particular, as shown in FIG. 3, for example, the measurement surface of the DUT 100 has a high aspect ratio in which the ratio of the amplitude d to the wavelength λ is about 2/3 to 1/2 and the maximum inclination θ1 is 70 degrees to 30 degrees. In the case of such a shape, stick slip or the like is more likely to occur.

  Therefore, in order to reduce the occurrence of stick-slip or the like, as shown in FIG. 4, the vibrator 5 is vibrated in the axial direction of the stylus 4, and scanning is performed while the stylus 4 is intermittently separated from the surface of the object 100 to be measured. Let By doing so, the influence of friction and stick-slip can be reduced even when measuring a micro shape such as a micro part or a micro lens array constituting the micro machine.

  However, if the vibration of the vibrator 5 is transmitted to the movable shaft 3 at this time, the displacement amount of the movable shaft 3 is changed by the vibration as shown in FIG. Noise is added. In order to avoid this, the vibration of the vibrator 5 is transmitted only to the stylus 4 (that is, the state shown in FIG. 4 is obtained), or the vibration is transmitted to the movable shaft 3, but the movable shaft at that time 3 vibrates the vibrator 5 at a frequency belonging to a predetermined frequency band (hereinafter referred to as “preferred frequency band”) such that the amplitude of 3 becomes a value that can be ignored with respect to the detection limit of the displacement sensor 1. It is necessary to let

  A study for setting a suitable frequency band will be described. First, in order to set a suitable frequency band, the movable shaft 3, the stylus 4, and the vibrator 5 are considered as a model in which two carriages 3A and 4A are connected by a spring 5A as shown in FIG. Here, it is assumed that the carriage 3A and the spring 5A, and the carriage 4A and the spring 5A are attached with predetermined spring constants and viscosity coefficients, respectively. The carriages 3A and 4A correspond to the movable shaft 3 and the stylus 4, respectively, and the spring 5A corresponds to the vibrator 5. Therefore, in the following description, the carriages 3A and 4A will be described by replacing the movable shaft 3 and the stylus 4, and the spring 5A with the vibrator 5, respectively.

m _g is the sum of 1/2 of the equivalent mass of the vibrator 5 and the mass of the stylus 4, m _h is the sum of 1/2 of the equivalent mass of the vibrator 5 and the mass of the movable shaft 3, and Let the spring constants before and after the center point be k _i , respectively. _Let ξ _g and ξ _{h be} the movement amounts of the stylus 4 and the movable shaft 3, respectively, and consider the case where the central portion of the vibrator 5 is vibrated by a · sin ωt. At this time, the portion on the stylus 4 side from the center point of the vibrator 5 is represented by the equivalent circuit of FIG.

When this is solved, the movement amount ξ _g of the stylus 4 is expressed by the following equation (3).

Similarly, the movement amount ξ _h of the movable shaft 3 is expressed by the following equation (4).

An example of a simulation of a suitable frequency band using the above equations 1 and 2 is shown in FIGS. In this simulation, 1 milligram mass of the stylus 4 (mg), the mass 37 grams of the movable shaft 3 (g), the spring constant _{k i} of 4.0 × 10 ⁷ N / m of the oscillator 5, the oscillator 5 the equivalent mass 8 mg, the spring constant _{k g} and damping ratio zeta _g each 3.0 × 10 ⁵ N / m in the mounting portion of the vibrator 5 and the stylus 4, 0.3, vibrator 5 and the mounting portion of the movable shaft 3 The calculation was performed with the spring constant k _h and the damping ratio ζ _h at 1 × 10 ⁴ N / m and 0.3, respectively. Note that the damping ratio ζ _g is obtained by the formula 1 from the mass _mg , the spring constant k _g , and the viscosity coefficient c _g (the damping ratio ζ _h is similarly obtained by the formula 2).

  FIG. 8 is a graph showing the relationship between the vibration frequency of the vibrator 5 and the amplitude of the stylus 4, and FIG. 9 is a graph showing the relation between the vibration frequency of the vibrator 5 and the amplitude of the movable shaft 3. The amplitude of the stylus 4 and the movable shaft 3 is shown as a ratio to the amplitude of the vibrator 5. That is, as the amplitude is closer to 0, the vibration of the vibrator 5 is not transmitted.

  When the graph of FIG. 8 is seen, the amplitude of the stylus 4 rapidly increases when the vibration frequency of the vibrator 5 exceeds 20 kilohertz (kHz), and reaches a peak when the vibration frequency reaches about 36 kHz. About 100 kHz, it is about 0.3, which is 1/3 of the amplitude of the vibrator 5. In the region R1 of the frequency band 20 to 50 kHz, the amplitude of the stylus 4 may change abruptly even if the vibration frequency is slightly changed. Therefore, there is a possibility that the vibrator 5 is damaged. It is not appropriate as a vibration frequency. Therefore, it is preferable that the band like the region R1 is excluded from the preferred frequency band.

  On the other hand, when viewing the graph of FIG. 9, the amplitude of the movable shaft 3 converges to 0 after reaching a peak when the vibration frequency of the vibrator 5 is close to 80 hertz (Hz). The amplitude is about 0.05 and 1/20 of the amplitude of the vibrator 5, and the vibration of the vibrator 5 is hardly transmitted, or even if it is transmitted, it is small enough to be ignored with respect to the detection limit of the displacement sensor 1. I understand that.

Returning to FIG. 8 again, it can be seen that the amplitude of the stylus 4 when the vibration frequency of the vibrator 5 is 1 kHz is about 1.000 and vibrates with substantially the same amplitude as that of the vibrator 5.
From the above, the approximate suitable frequency band in this simulation can be set as a range of 1 to 15 kHz and 50 to 100 kHz, for example.

In order to verify the simulation results described above, an experiment was performed using the experimental system shown in FIG. A displacement sensor 1A in which the mass of the stylus 4, the mass of the movable shaft 3, the spring constant k _i and the equivalent mass of the vibrator 5 are set to be the same as those in the above simulation is prepared, and the tip of the stylus 4 is brought into contact with the rigid body 102 . Then, the movement of the movable shaft 3 was measured from the front with the fiber sensor 103 having a maximum sampling frequency of 160 kHz.

  FIG. 11 is a graph showing the output of the fiber sensor 103 and the input voltage waveform to the vibrator 5 when the vibrator 5 is vibrated at 100 Hz in the experimental system of FIG. A curve C1 shows an input voltage waveform, and a voltage of about 10 V is applied at 100 Hz. On the other hand, the output of the fiber sensor 103 has an amplitude of about 150 nanometers (nm) as shown by the curve C2. This indicates that the vibration of the vibrator 5 is transmitted to the movable shaft 3 and the movable shaft 3 is vibrating.

  FIG. 12 is a graph showing the output of the fiber sensor 103 and the input voltage waveform to the vibrator 5 when the vibrator 5 is vibrated at 1 kHz in the experimental system of FIG. A curve C3 shows an input voltage waveform, and a voltage of about 10 V is applied at 1 kHz. On the other hand, the output of the fiber sensor 103 shown by the curve C4 has an amplitude of about several nanometers that can be ignored with respect to the detection limit of the displacement sensor 1A, and the vibration of the vibrator 5 is hardly transmitted to the movable shaft 3. Was confirmed.

  As described above, when the vibrator 5 is vibrated at a frequency of 1 kHz, the movable shaft 3 hardly vibrates. Therefore, no noise is generated in the detection value of the displacement sensor 1A detected by the linear encoder of the main body 2, and the stylus is used. Only 4 is vibrating. That is, it was confirmed that 1 kHz is a frequency belonging to the preferred frequency band under these conditions.

Next, FIGS. 13 and 14 show the results of analyzing the input voltage and the output of the fiber sensor 103 using a fast Fourier transform (FFT) analyzer. A curve C5 in FIG. 13 indicates a gain, and a curve C6 in FIG. 14 indicates a phase.
13 and 14, similarly to the simulation results, it is understood that when the vibration frequency of the vibrator 5 is 1 kHz, the gain becomes −40 decibels (dB) or less and is hardly transmitted to the movable shaft 3. . Moreover, it turns out that the tendency becomes so strong that the frequency is increased.

Further, an experiment was conducted in order to investigate the relationship between the vibration of the vibrator 5 and the effect on the friction and stick slip generated in the stylus 4. FIG. 15 is a diagram showing an experimental system for performing the experiment.
The force sensor 104 is mounted on the uniaxial automatic stage 105, and the stylus 4 and the force sensor 104 are brought into contact with each other. At this time, the angle θ ₂ formed by the sensitivity direction of the force sensor 104 and the sensitivity direction of the displacement sensor 1A shown in FIG. 15 is set to 110 degrees.

  In this state, the uniaxial automatic stage 105 is moved in the direction indicated by the arrow in FIG. 15, and the outputs of the force sensor 104 and the displacement sensor 1A are measured to examine changes such as friction and stick-slip. If stick-slip does not occur and the friction is constant, the outputs of the force sensor 104 and the displacement sensor 1A are constant even if the single-axis automatic stage 105 is moved. On the contrary, if there is a change in stick-slip or friction, the output of the force sensor 104 and the displacement sensor 1A will change.

16 and 17 are graphs showing the experimental results of the experimental system of FIG. 15, and show the outputs of the force sensor 104 and the displacement sensor 1A when the single-axis automatic stage 105 is reciprocated once in the direction of the arrow.
FIG. 16 is a graph showing a result of an experiment performed without vibrating the vibrator 5. The horizontal axis is the sampling time, the left vertical axis is the output of the displacement sensor 1A, and the right vertical axis is the output of the force sensor 104. A curve C7 shows the output of the displacement sensor 1A, and a curve C8 shows the output of the force sensor 104, but there is a difference in output between the forward path (leftward movement in FIG. 15) and the backward path (rightward movement in FIG. 15). . Both outputs of the displacement sensor 1A and the force sensor 104 vibrate greatly on the return path, which is considered to be caused by stick slip.

FIG. 17 is a graph showing experimental results when the vibrator 5 is vibrated with the vibration frequency set to 80 kHz. As shown in FIG. 17, it can be seen that the difference in the shapes of the curves C7 and C8 between the forward path and the return path is smaller than the result of FIG.
From the above results, it was confirmed that the occurrence of stick-slip and friction can be reduced by vibrating the vibrator 5 at a frequency belonging to the preferred frequency band.

Finally, using the displacement sensor 1A, the measurement object having the shape of the measurement surface as shown in FIG. 3 is attached to the spindle 101 as shown in FIG. Show.
FIG. 18 is a graph showing the results of measurement without vibrating the vibrator 5. The horizontal axis shows the rotation angle of the spindle 101, the left side of the vertical axis shows the output of the displacement sensor 1A, and the right side of the vertical axis shows the repeat error when the measurement is performed twice. A curve C9 shows the surface shape of the measurement object obtained by the displacement sensor 1A, and a curve C10 shows a repetition error. The value of the repeated error is large at a place where the slope of the curve C9 is steep, and it can be estimated that an error has occurred due to the effect of stick-slip.

  FIG. 19 is a graph showing the result of measuring the surface shape of the object to be measured by vibrating the vibrator 5 with the vibration frequency set to 10 kHz. The repetition error is smaller than the experimental result of FIG. 18, and it is shown that the effect of stick-slip can be reduced by vibrating the vibrator 5 and vibrating the stylus 4 even at a steep place. It was.

  As described above, according to the displacement sensors 1 and 1A of the present invention, the vibrator 5 is vibrated at a frequency belonging to the preferred frequency band, so that only the stylus 4 is vibrated without substantially vibrating the movable shaft 3. Can do. As a result, the output value detected by the main body 2 does not include noise, and the tip of the stylus 4 is intermittently pulled away from the surface of the object to be measured, and the occurrence of stick-slip or the like is preferably reduced. Therefore, more accurate measurement can be performed in a short time.

The preferred frequency band naturally changes in conjunction with changes in the mass of the stylus 4, the mass of the movable shaft 3, the spring constant k _i and the equivalent mass of the vibrator 5, etc. By acquiring the graphs as shown in FIG. 8 and FIG. 9, it is possible to determine a suitable frequency band in the same manner even in a displacement sensor having an arbitrary design parameter.
In addition, the control unit including a program or the like automatically performs these calculations so that the suitable frequency band is automatically determined by the control unit and the vibrator is automatically vibrated at the vibration frequency belonging to the suitable frequency band. A displacement sensor may be configured.

Although one embodiment of the present invention has been described above, the technical scope of the present invention is not limited to the above-described embodiments, and various modifications can be made without departing from the spirit of the present invention. is there.
For example, in the above-described embodiment, an example in which the tip of the stylus is sharp has been described, but instead, a stylus 14 in which the tip 14A is formed in a spherical shape is used as in the modification shown in FIG. May be. Alternatively, it may be a hemispherical tip. The tip 14A having such a shape can be suitably formed using a ruby sphere, a glass sphere, or the like.

It is a figure which shows the contact-type displacement sensor and to-be-measured object of one Embodiment of this invention. It is an enlarged view which shows the stylus and vibrator | oscillator of the contact-type displacement sensor. It is a figure which shows an example of the surface shape of a to-be-measured object. It is a figure which shows the movement of the stylus and movable axis | shaft of the contact-type displacement sensor when the vibrator is vibrated with the frequency which belongs to a suitable frequency band. It is a figure which shows the motion of the stylus and the movable shaft when the vibrator is vibrated at a frequency not belonging to the preferred frequency band. FIG. 2 is a model diagram of a stylus and a movable shaft shown in FIG. 1. It is a figure which shows the equivalent circuit of the part by the side of a stylus from the center point of the model figure shown in FIG. It is a graph which shows the relationship between the vibration frequency of the vibrator | oscillator in the example of simulation, and the amplitude of the stylus. It is a graph which shows the relationship between the vibration frequency of the vibrator | oscillator in the example of simulation, and the amplitude of the same stylus in the same simulation and the vibration frequency of the vibrator | oscillator in the same simulation, and the amplitude of the movable axis. It is a figure which shows the experimental system for verifying the result of the simulation. In the same experimental system, it is a graph which shows the relationship between the input voltage and the output of a fiber sensor when the vibrator is vibrated with the frequency which does not belong to a suitable frequency band. In the same experimental system, it is a graph which shows the relationship between the input voltage and the output of a fiber sensor when the vibrator is vibrated with the frequency which belongs to a suitable frequency band. It is a graph which shows the result of having analyzed input voltage and the output of a fiber sensor by fast Fourier transform. It is a graph which shows the result of having analyzed input voltage and the output of a fiber sensor by fast Fourier transform. It is a figure which shows the experimental system for examining the friction and stick slip between the vibrator | oscillator and the stylus. It is a graph which shows the experimental result conducted without vibrating the same vibrator in the same experimental system. It is a graph which shows the experimental result conducted by vibrating the vibrator with the frequency which belongs to a suitable frequency band in the same experimental system. It is a graph which shows the result of having measured the surface shape of the to-be-measured object, using the contact displacement sensor, without vibrating the vibrator. It is a graph which shows the result of having measured the surface shape of the to-be-measured object by vibrating the vibrator with the frequency which belongs to a suitable frequency band using the contact type displacement sensor. It is a figure which shows the stylus in the modification of the contact-type displacement sensor of one Embodiment of this invention about the relationship between the vibration frequency of the vibrator | oscillator in the example of simulation, and the amplitude of the stylus.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1, 1A Contact-type displacement sensor 3 Movable shaft 4, 14 Stylus 5 Vibrator 100 Measured object

Claims (3)

A contact-type displacement sensor for measuring the surface shape of an object to be measured,
A movable shaft attached to the object to be measured so as to be able to advance and retreat;
A stylus attached to the tip of the movable shaft and scanned over the surface of the object to be measured;
A vibrator mounted between the stylus and the movable shaft;
With
The vibrator includes a spring constant of the vibrator, a spring constant and a viscosity coefficient at a mounting portion of the vibrator and the stylus, a spring constant and a viscosity coefficient at a mounting portion of the vibrator and the movable shaft, and an equivalent of the vibrator. Oscillated at a frequency within a preferred frequency band determined based on the mass, the mass of the movable shaft, and the mass of the stylus,
The contact-type displacement sensor, wherein the stylus is vibrated in an axial direction with a predetermined amplitude, and the movable shaft hardly vibrates. The preferred frequency band includes the vibration frequency of the vibrator ω, the spring constant k _i , the sum of 1/2 of the equivalent mass of the vibrator and the mass of the movable shaft, m _h , and the equivalent of the vibrator. sum m _g of 1/2 and the stylus of the mass of the _mass, the vibrator respectively k _g the spring constant and viscosity coefficient of the mounting portion of the _stylus, c _g, the attachment portion of the movable shaft and the transducer Let the spring constant and viscosity coefficient at k be k _h and c _h , respectively.

as well as

2. The contact-type displacement sensor according to claim 1, wherein a curve relating to a transition of each of the vibration frequencies of the vibrator of the ξ _g and the ξ _h is obtained based on the mathematical formula.   The contact displacement sensor according to claim 1, wherein the vibrator is a piezo element.

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