1 Introduction

Vibration measurement plays a quite important role in micro-mechanical system and fine processing technology [1,2,3,4,5,6]. Currently, vibration measurement methods can be classified to electrical measuring and optical measuring based methods according to the conversion principles of vibrational signal. The electrical measuring based methods convert the vibration signals to electronic signals for processing to obtain vibration parameters. These methods usually use magnetoelectric [7], piezoelectric and strain gauge sensors [8, 9] which are light-weighted, small-sized and low-cost, but in the meantime susceptible to on-site electromagnetic noises, which will lead to reduction of reliability and resolution. The optical measurement methods [10,11,12] typically include holographic optical technique [13] and laser Doppler interferometry (LDI) technique [14,15,16], which have the advantage of non-contact, little influence from vibration environment and high resolution. But the measuring systems are usually large and expensive, not suitable to be installed on the vibrating object for long real-time vibration monitoring. Hence, it is of huge practical significance to develop a low-cost vibration sensor for real-time vibration monitoring with good anti-disturbance ability.

In this paper, a vibration sensor with high resolution, wide measurement range and strong anti-disturbance ability is developed. The sensor is based on Phase Grating Interference (PGI) and spatial polarization phase-shifting techniques to generate interference signals corresponding to vibration. Differential operations are conducted by four photodetectors to make the signals more reliable, which are then used for fringe counting and A/D subdivision to gain nanometer resolution of vibration measurement. To test functions and characteristics of the sensor, a vibration actuator is designed based on piezoelectric transducer to carry out a series of comparison experiments between the designed sensor and LDI. The experiment results verify the practicality and effectiveness of the novel vibration sensor.

2 Vibration measurement system

2.1 Vibration sensor based on phase grating interferometry

The principles of phase grating measurements are demonstrated in Fig. 1, which is applied to the proposed vibration sensor. When a laser beam incident onto the phase grating, it is diffracted into +1 and −1 order of diffraction lights, which are combined to interfere through series of optical components to obtain four interference signals with phase difference of 90° by spatial polarization phase shifting. The optical signals are captured by four photodetectors for analysis, so that phase grating displacement is measured.

Fig. 1
figure 1

Principle of phase grating interferometry

Full size image

According to optical Doppler effects [17], frequency shift Δf will occur when the grating moves perpendicularly to its reticule. Δf equals to product of quantity of the passed grating reticule per second and the order of diffraction fringe. Frequency shift is independent of incident angle and wave length of light. Δf of ±1 order fringe is expressed as:

$$\left\{ \begin{aligned} &\Delta f_{( + 1)} = \nu /d \hfill \\ &\Delta f_{( - 1)} = - \nu /d \hfill \\ \end{aligned} \right.$$
(1)

where v is the kinematic velocity of grating along normal direction, d is the grating pitch.

The frequency of interference fringe on the photodetector is

$$\Delta f = \Delta f_{( + 1)} - \Delta f_{( - 1)} = \frac{2v}{d}$$
(2)

Grating displacement can be calculated by

$$S = \int_{0}^{t} {\nu dt} = \int_{0}^{t} {\frac{\Delta f \cdot d}{2}dt} = \frac{d}{2}\int_{0}^{t} {\Delta fdt} ,$$
(3)

where \(\int_{0}^{t} {\Delta fdt}\) is the quantity of fringes passing through a photodetector within time t and denoted as n, yielding

$$S = dn/2$$
(4)

Grating moves a distance of d/2 within each dimming and brightening change cycle of interferometry signal. Signal impurity will hamper realization of electronic subdivision with high quality. To achieve qualified signal, light polarized transformation elements such as PBS and NPBS are used to produce spatial phase shift [18, 19] and generate four paths of signals with phase difference of 90°.

A laser beam incidents onto phase grating to produce diffraction lights of ±1 order. Then the two lights are reflected by Mirror1 and Mirror2 to enter Polarizer1 and Polarizer2, respectively. By adjusting the angles of the two polarizers, the two lights can make a polarization shift of 90°. After passing through two polarizers, wave equations of the two lights are

$$E_{( + 1)} = A_{0} \sin \left( {\omega t - 2\pi \frac{v}{d}t + \delta_{ + 1} } \right),$$
(5)
$$E_{( - 1)} = A_{0} \sin \left( {\omega t + 2\pi \frac{v}{d}t + \delta_{ - 1} } \right),$$
(6)

where δ +1 and δ −1 are phase changes caused by optical path variation and set as 0. Then the wave equations can be

$$E_{( - 1)} = A_{0} \sin \left( {\omega t + 2\pi \frac{v}{d}t} \right) = A_{0} \sin (\phi_{ - 1} )$$
(7)
$$E_{( + 1)} = A_{0} \sin \left( {\omega t - 2\pi \frac{v}{d}t} \right) = A_{0} \sin (\phi_{ + 1} )$$
(8)

The polarized light passing through NPBS are divided into two paths, of which one enters wave plate1 and PBS1 and the other enters wave plate2 and PBS2. For simplification, the transmission index and reflection index are set as 50% by ignoring light intensity loss in NPBS. The vibration equations can be obtained as:

$$\left\{ {\begin{array}{*{20}c} {E_{( - 1,R)} = \frac{{A_{0} }}{\sqrt 2 }\sin (\phi_{ - 1} ) = E_{( - 1,T)} } \\ {E_{( + 1,T)} = \frac{{A_{0} }}{\sqrt 2 }\sin (\phi_{ + 1} ) = E_{( + 1,R)} } \\ \end{array} } \right.,$$
(9)

where E (+1,T) and E (−1,T) are wave equations of ±1 order diffraction lights after transmission of NPBS. E (+1,R) and E (−1,R) shown are wave equations of ±1 order diffraction lights after reflection of NPBS.

Specially, the Polarizer1 is placed with polarization direction parallel to the paper surface and denoted as x direction in Fig. 2 and the Polarizer2 is placed with polarization direction vertical to the paper surface denoted as y direction in Fig. 2. The wave plate1 is placed so that the coordinate system consisting of its fast axis denoted as yʹ axis and slow axis denoted as xʹ axis has an angle of 45° relative to xy coordinate system as shown in Fig. 2a. After passing through wave plate1, the +1 order polarized light is transformed to circularly polarized light of which the wave equation can be expressed by components in xʹ and yʹ axis, i.e. E (+1,R,x′) and E (+1,R,y′) in Fig. 2a. The circularly polarized light formed by −1 order light after passing through wave plate1 can be expressed by E (−1,T,x′) and E (−1,T,y′) in Fig. 2a. They will enter PBS1. These four components can be expressed by

$$\left\{ \begin{aligned} &E_{( + 1,R,y')} = \frac{{A_{0} }}{2}\cos \left( {\phi_{ + 1} + \frac{\pi }{2}} \right) \hfill \\ &E_{( + 1,R,x')} = \frac{{A_{0} }}{2}\cos (\phi_{{{ + }1}} ) \hfill \\ \end{aligned} \right.{\kern 1pt} {\kern 1pt}$$
(10)
$$\left\{ \begin{aligned} &E_{( - 1,T,y')} = \frac{{A_{0} }}{2}\cos \left( {\phi_{{{ - }1}} + \frac{\pi }{2}} \right) \hfill \\ &E_{( - 1,T,x')} = \frac{{A_{0} }}{2}\cos (\phi_{ - 1} ) \hfill \\ \end{aligned} \right.$$
(11)
Fig. 2
figure 2

Schematic diagram of polarization conversion for wave plate1 (a) and for wave plate2 (b)

Full size image

Similarly, the circularly polarized lights formed by ±1 order lights after passing through wave plate2 can be expressed as the components in xʹ and yʹ axis in Fig. 2b, and they will enter PBS2. These components are

$$\left\{ \begin{aligned} &E_{( + 1,T,y')} = \frac{{A_{0} }}{2}\cos \left( {\phi_{ + 1} + \frac{\pi }{2}} \right) \hfill \\ &E_{( + 1,T,x')} = \frac{{A_{0} }}{2}\cos (\phi_{{{ + }1}} ) \hfill \\ \end{aligned} \right.$$
(12)
$$\left\{ \begin{aligned} &E_{( - 1,R,y')} = \frac{{A_{0} }}{2}\cos \left( {\phi_{{{ - }1}} + \frac{\pi }{2}} \right) \hfill \\ &E_{( - 1,R,x')} = \frac{{A_{0} }}{2}\cos (\phi_{ - 1} ) \hfill \\ \end{aligned} \right.$$
(13)

After entering PBS1 and PBS2, respectively, the circularly polarized lights will be polarized into two linearly polarized lights with vibration directions perpendicular to each other, and they are transmitted or reflected corresponding to the transmission axis. Since PBS1 is mounted with its mounting surface having an angle of 45° respective to the paper surface, the reflection light from PBS1 has an angle of 45° relative to the x axis. According to the characteristics of PBS [20], the transmission direction of emergent lights from PBS1 in point “a” is along the xʹ axis and that in point “b” is along the yʹ axis as shown in Fig. 3a. The wave equation E a for point “a” is synthesized by the components along xʹ axis i.e. E (+1,R,xʹ) and E (−1,T,xʹ) as:

$$E_{\text{a}} = E_{{( + 1,R,x^{\prime})}} + E_{{( - 1,T,x^{\prime})}} = \frac{{A_{0} }}{2}\left[ {\cos (\phi_{{{ + }1}} ) + \cos (\phi_{ - 1} )} \right]$$
(14)

Similarly, the wave equation E b for point “b” can be expressed as:

$$E_{b} = - E_{{( + 1,R,y^{\prime})}} + E_{{( - 1,T,y^{\prime})}} = \frac{{A_{0} }}{2}\left[ {\cos \left( {\phi_{{{ + }1}} + \frac{3\pi }{2}} \right) + \cos \left( {\phi_{ - 1} + \frac{\pi }{2}} \right)} \right]$$
(15)
Fig. 3
figure 3

Schematic diagram of polarization conversion for PBS1 (a) and for PBS2 (b)

Full size image

PBS2 is mounted with its mounting surface parallel to the paper surface, so the transmission axis of PBS2 for emergent light in point “c” is along the direction of y axis and in point “d” is along the direction of x axis as shown in Fig. 3b. The components of E (+1,T,xʹ) and E (+1,T,yʹ) in the y axis are denoted as purple arrows, and the components of E (−1,R,xʹ) and E (−1,R,yʹ) in the y axis are denoted as red arrows, and these components constitute the wave equation E c for point “c” as

$$\begin{aligned} E_{c} = \frac{{E_{( + 1,T,y')} }}{\sqrt 2 } + \frac{{E_{( + 1,T,x')} }}{\sqrt 2 } - \frac{{E_{( - 1,R,y')} }}{\sqrt 2 } + \frac{{E_{( - 1,R,x')} }}{\sqrt 2 } \\ = \frac{{A_{0} }}{2\sqrt 2 }\left[ {\cos \left( {\phi_{ + 1} + \frac{\pi }{2}} \right) + \cos (\phi_{ + 1} ) + \cos \left( {\phi_{ - 1} + \frac{3\pi }{2}} \right) + \cos (\phi_{ - 1} )} \right] \\ = \frac{{A_{0} }}{2}\left[ {\cos \left( {\phi_{ + 1} + \frac{\pi }{4}} \right) + \cos \left( {\phi_{{{ - }1}} + \frac{3\pi }{4}} \right)} \right] \\ \end{aligned}$$
(16)

Similarly, the wave equation E d for point “d” should be as

$$\begin{aligned} E_{d} = - \frac{{E_{( + 1,T,y')} }}{\sqrt 2 } + \frac{{E_{( + 1,T,x')} }}{\sqrt 2 } + \frac{{E_{( - 1,R,y')} }}{\sqrt 2 } + \frac{{E_{( - 1,R,x')} }}{\sqrt 2 } \\ = \frac{{A_{0} }}{2\sqrt 2 }\left[ {\cos \left( {\phi_{ + 1} + \frac{3\pi }{2}} \right) + \cos (\phi_{ + 1} ) + \cos \left( {\phi_{ - 1} + \frac{\pi }{2}} \right) + \cos (\phi_{ - 1} )} \right] \\ = \frac{{A_{0} }}{2}\left[ {\cos \left( {\phi_{ + 1} - \frac{\pi }{4}} \right) + \cos \left( {\phi_{{{ - }1}} + \frac{\pi }{4}} \right)} \right] \\ \end{aligned}$$
(17)

From (14) to (17), phase differences of the four lights at points “a”, “b”, “c” and “d” are 0, π, 3π/2 and π/2, respectively. Setting θ = ϕ +1 − ϕ −1, light intensities are derived from

$$\left\{ \begin{aligned} &I_{a} = E_{a}^{2} = A_{0}^{2} (1 + \cos \theta )/2 \hfill \\ &I_{b} = E_{b}^{2} = A_{0}^{2} [1 + \cos (\theta - \pi )]/2 \hfill \\ &I_{c} = E_{c}^{2} = A_{0}^{2} [1 + \cos (\theta - 3\pi /2)]/2 \hfill \\ &I_{d} = E_{d}^{2} = A_{0}^{2} [1 + \cos (\theta - \pi /2)]/2 \hfill \\ \end{aligned} \right.{\kern 1pt} {\kern 1pt}$$
(18)

The optical signals are captured by four photodetectors and then differentially amplified to eliminate disturbance of DC component and noises. On one hand, these signals will be modulated to square wave first and then applied for quantity counting of the interference fringes to realize direction identification when grating starts vibrating. On the other hand, after A/D subdivision, they are used to gain phase θ by arctangent operation. As a result, a higher displacement resolution can be achieved. Based on the above theory, the vibration amplitude of the sensor can be calculated by

$$L = \frac{d}{2}n + \arctan \left( {\frac{{I_{c} - I_{d} }}{{I_{a} - I_{b} }}} \right) \times \frac{d}{4\pi }$$
(19)

Grating #43-209, 1200 line/mm from Edmund Optics and A/D7656-1 from AD are applied in this study. The A/D7656 is a 16-bits chip with a response frequency of 1.5 MHz. Photodetector has a much higher frequency than A/D. Therefore, the stable response frequency of the sensor is approximately 0–15 kHz. Theoretically, a cycle change of signal is detected by photoelectric detector after a grating constant movement. The signal is divided into 1/4 periodic signal by phase-shift spatial polarization, and subsequently subdivided 16 times by the A/D chip. The theoretical distinguishability of vibration displacement measurement is

$$s_{0} = \frac{1}{2} \times \frac{1}{1200} \times \frac{1}{4} \times \frac{1}{{2^{16} }}$$
(20)

Considering the influences of optical signal noises, circuit noises and sampling synchronization etc., the PGI sensor here can achieve a measurement resolution of 1 nm.

2.2 Vibration generator

The vibration generator consists of two key parts: a piezoelectric transducer (PZT) and a translational flexible hinge. The response of PZT is fast and the displacement output is stable with a large range [21]. When driven by an alternating voltage, PZT vibrates as the alternating frequency. A PZT with a voltage range of 0–120 V and response time lower than 300 ns is selected as a vibration source. However, due to small differences of deformation in each layer of the piezoelectric ceramic and errors in bonding operation, a good translational vibration cannot be guaranteed. The measurement accuracy of PGI vibration sensor will be affected significantly. Therefore, contrast experiments are carried out between PGI vibration sensor and laser Doppler interferometer (LDI). The precondition of contrast experiments is that the mirror of LDI as a reflective target should be mounted on a place where translational vibration of PGI vibration sensor is exported. A flexible hinge is designed as a vibration transmitting component as well as a device holding up the mirror.

The translational flexible hinge is a parallel spring with a linear range from tens to hundreds of microns. The resolution of the flexible hinge can reach submicron order [21]. Lubrication is non-essential for no mechanical friction. Meanwhile, the parallel spring flexible hinge does not produce heat or noises, but shows high motion sensitivity and can guarantee translational movements [22]. The material chosen for the part is 4Cr13. Optimization is carried out according to material parameters, instalment dimension and actual machining. The obtained optimum radius is 5 mm and the minimum thickness of thin-wall is 0.5 mm. The vibration frequency of PZT could reach above 10 kHz. As a key part in the transmission of vibration from PZT, the response performance and following characteristics of the flexible hinge directly determine the frequency range provided by vibration generator. Thus, it is necessary to do harmonic response analysis under dynamic cyclic load. In the analysis, the harmonic response frequency is set between 0 and 250 Hz. Within this range, the structure response under cyclic load is calculated to gain the relationship of response amount varying with frequency. Succulently, the peak frequency and structure stress under such value can also be obtained.

Figure 4a shows that the amplitude peak does not appear in 0–250 Hz. This indicates that resonance of translational flexible hinge will not occur under frequency in this range. Figure 4b shows the amount of phase fluctuation is 0.6° during 0–250 Hz. This illustrates that the translational displacement of flexible hinges within this frequency range can produce a good response. Figure 4c shows that fatigue damage will not occur under 250 Hz because the maximum of equivalent stress is 24.856 Mpa, which is less than the tensile strength of the material. The flexible hinge utilized as a vibration transmission component in vibration generator meets the requirement above.

Fig. 4
figure 4

Harmonic response analysis for the flexible hinge. a Frequency-amplitude curve, b Frequency-phase angle curve, c Equivalent stress distribution

Full size image

2.3 Measurement system setup

As shown in Fig. 5, the vibration measurement system is composed of a PGI vibration sensor, a vibration generator and a displacement regulating mechanism. The PGI vibration sensor and the vibration generator are installed as shown in Fig. 6. Phase grating is mounted on the top of a flexible hinge. When connected to an alternating voltage, PZT guides the flexible hinge to vibrate. The vibration is delivered to the grating and induces its translational motion. By monitoring the movement of the grating, PGI vibration sensor records vibration information of the generator. The position adjusting mechanism is designed for distance adjustment between the vibration actuator and the sensor. When the motor drives the screw rotating, inclined block mounted on the screw nut pushes cylindrical pin connected to the bottom of the vibration generator, which makes the vibration generator move along the vertical direction. In this way, the related position of the vibration actuator and the sensor is altered.

Fig. 5
figure 5

Vibration measurement system

Full size image
Fig. 6
figure 6

PGI vibration sensor and vibration generator

Full size image

1, 2, 3, 4, 11, 12, 13, 15, 17, 19—Components for optical adjustment; 5—Laser; 6, 7, 9—Components for angle adjustment; 8—Flexible hinge; 10—Phase gating; 14, 27—Polarizer; 18,23—PBS; 20, 21, 22, 24—Photodetectors; 16, 25—Wave plate; 26—NPBS; 28—PZT.

3 Experiments

Contrast experiments are carried out to verify the vibration measuring function of the PGI sensor by using MCV-500 LDI (Optodyne,Inc.,US) with a minimum resolution of 0.02 μm and response frequency of 0–350 MHz, as shown in Fig. 7. The mirror is installed on the top of phase grating as a reflective target for LDI. The excitation alternating voltage is input with an amplitude varying from 25 to 150 V every 25 V and a frequency variation range of 50–250 Hz in a 50 Hz interval. Each experiment is repeated 5 times. The results are shown in Table 1, in which “√” means periodic vibration and “O” means aperiodic vibration. Experiments show that following characteristics of the flexible hinges with PZT are limited and aperiodic vibration is induced when driving frequency and voltage are high.

Fig. 7
figure 7

Calibration and verification system using laser Doppler interferometer. 1 Mechanical structure; 2 MCV-500 laser head, 3 laser power; 4 system control box; 5 MCV-500 control box; 6 PC; 7 the mirror as a reflective target

Full size image
Table 1 Contrast experiments
Full size table

Since the experimental data in this study is very huge, only two typical experimental results are displayed. The flexible hinge produces a periodical vibration and agrees with the results of LDI very well as shown in Figs. 8 and 9, which demonstrates the good following characteristics of the flexible hinge. Figure 10 reveals an approximately linear dependence of vibration amplitude on an excitation voltage at 50 Hz. This is coincident with the fact that piezoelectric extension changes proportionally to voltage variation and proves that sensor vibration measurement accords with driving source changes. In order to make sure that the “O” results in Table 1 are caused by failure of following characteristics of flexible hinges, but not by measuring errors of PGI sensor, a set of experiment data is extracting and exhibited in Fig. 11. It can be seen that though the vibration amplitude varies randomly, there exists a great agreement in the results between PGI and LDI. This means that the “O” results of PGI in Table 1 are correct and vibration measurement with PGI sensor is effective. Partial results of periodical vibration in Table 2 extracting from a large number of duplicate measurements reveal a deviation of 0–30 nm compared with LDI. This illustrates that PGI vibration sensor possesses a good and stable measuring resolution.

Fig. 8
figure 8

LDI and PGI vibration measurement under 50 Hz, 150 V. a Amplitude of sampling, b Spectrogram of sampling

Full size image
Fig. 9
figure 9

LDI and PGI vibration measurement under 50 Hz, 200 V. a Amplitude of sampling, b Spectrogram of sampling

Full size image
Fig. 10
figure 10

Amplitude variation with excitation voltage at 50 Hz

Full size image
Fig. 11
figure 11

Amplitude under excitation voltage of 150 Hz, 100 V when flexible hinges suffer from failure of following characteristics

Full size image
Table 2 Partial results of periodical vibration in the contrast tests, each data representing an average of five measurements
Full size table

4 Conclusions

In this paper, making use of the ability of phase grating for dynamical displacement measurement, a novel vibration sensor based on phase grating interferometry is developed. To achieve high resolution for vibration measurement, spatial polarization phase-shift interference technology is adopted for interference signal counting and subdivision. To test the characteristics of the sensor, a vibration generator is also developed. Through comparative experiments with laser Doppler interferometer, the reliability and accuracy of the sensor are verified. Due to response limitation of the flexible hinges, the vibration generator in these experiments cannot provide periodic vibration with higher frequency to test the proposed PGI sensor. If a A/D chip with higher frequency is applied, the measuring frequency of the PGI sensor could be higher.