The calibration of sinusoidal excitation force of piezoelectric ceramic exciter

2.1. PCE excitation feedback system

PCE excitation feedback system includes four main components, their detailed parameter and functions are:

1) PCE

A schematic of self-designed PCE excitation feedback system is shown in Fig. 1. The selected PCE’s item number is P-885.10 produced by PI in Germany, its dimensions is 5 mm×5 mm×9 mm, its capacitance is 0.6 μF with unipolar characteristics, weighs only 2 g. When it is being used, 2.5 V to 100 V bias voltage is needed to driver it to work in the positive voltage range securely, and it should be glued firmly on the measured specimen to make sure that its excitation force do not change gradually due to looseness of its contact area.

2) HPEA

High-voltage piezoelectric amplifier (Rhvd3c110v, made by RongZhiNaXin Corporation) is connected to data acquisition instrument (DAI) and is used to amplify low-voltage excitation signal from DAI to the higher level with a fixed gain (24 times magnification), and to provide stable bias voltage for PCE from 25 V to 110 V. Its voltage output resolution is 5 mV, the maximum output power is 100 W, the communication rate is 9600 Bps and the amount of output voltage can be displayed in real time by LCD screen.

3) Feedback attenuator

Feedback attenuator (jointly developed by the author’s research group and NanJingHongBin Corporation) is used to adjust bias voltage to 0 V and convert high-voltage excitation signal (10 V-200 V) applied on the PCE by HPEA into the low-voltage feedback signal (0 V-10 V) with a fixed attenuating ratio (correspond with HPEA’s magnification). It has two BNC ports, one is contacted to PEC’s two pins to receive high-voltage signal, and the other one is contacted to DAI to provide feedback signal which can be monitored and recorded by DAI simultaneously.

4) DAI and computer

LMS SCADAS Mobile Front-End is used to record excitation voltage signal of PCE through feedback attenuator and other response signals acquired by vibration sensors with 102.4 kHz sample rate within 1 mV-10 V input range (with16-channel data acquisition module) as well as to generate sinusoidal excitation signals (with 2-channel source module). It can also output sweep, random, chirp and other excitation signals with 24-bit resolution. LMS Test.Lab software, version 10 B, is used to control the data acquisition and generation. Besides, Dell notebook computer (with Intel Core i7 2.93 GHz processor and 4 G RAM) is used to operate LMS Test.Lab 10 B software and store measured data.

2.2. Influences on linear excitation capability of PCE

Some major factors which influence linear excitation capability of PCE are listed, such as excitation frequency, excitation voltage and bias voltage. Still, other factors like PCE’s size, material, capacitance, output power and magnification of HPEA and other hardware parameters undoubtedly affect PCE’s excitation performance. However, they appear to be particularly reliant on their relevant hardware performance that have been determined by manufacturers, rather than users or researchers, so we limit our discussion about linear excitation capability of PCE to the cases of practical application in this research.

2.2.3. Influence of excitation frequency

Fixed bias voltage at 75 V, set output signal voltage as 1 V and changed excitation frequency from 300 Hz to 6500 Hz in the software, experimental amplitudes and corresponding linearity criterion were obtained and listed in Table 2. Excitation voltage signal of PCE in time and frequency domain under 300 Hz and 4000 Hz were shown in Fig. 2 and Fig. 3. From which we can find out that excitation frequency has strong effect on linear excitation capability of PCE. When it is below 2800 Hz, base-frequency amplitude is dominant in the spectrum, where linear criterion $S_2\le$0.01 and $S_3\le$0.01, and thus excitation voltage signal is or close to linear. However, its harmonic amplitudes gradually raise with the rise of excitation frequency, i.e., 2-harmonic and 3-harmonic amplitudes are getting increasingly higher when excitation frequency is above 2800 Hz.

Fig. 2Excitation voltage signal in time and frequency domain at 300 Hz with 1 V output signal voltage

a) Excitation voltage signal in time domain

b) Spectrum of excitation voltage signal

Fig. 3Excitation voltage signal in time and frequency domain at 4000 Hz with 1 V output signal voltage

a) Excitation voltage signal in time domain

b) Spectrum of excitation voltage signal

Table 2Excitation voltage amplitude and linearity criteria at different excitation frequencies

Excitation frequency (Hz)	Output signal voltage (V)	Excitation voltage amplitude (V)			Linear criterion $S_2$	Linear criterion $S_3$
		$A_0$	$A_1$	$A_2$
300	1	0.98	0	0	0	0
600	1	0.98	0	0	0	0
900	1	0.97	0	0	0	0
1200	1	0.96	0	0	0	0
1500	1	0.95	0.003	0	0.003	0
1800	1	0.94	0.005	0	0.005	0
2100	1	0.93	0.006	0	0.006	0
2400	1	0.91	0.007	0	0.008	0
2700	1	0.9	0.008	0.003	0.009	0
2800	1	0.87	0.01	0.006	0.01	0.007
4300	1	0.86	0.15	0.06	0.17	0.070
5800	1	0.76	0.17	0.07	0.22	0.092
6500	1	0.71	0.18	0.08	0.25	0.113

2.3. Linear excitation equation

From the conclusion in Section 2.2, for objectively evaluating linear excitation capability of PCE, it should take into account the influences of excitation frequency and output signal voltage simultaneously (we will discuss how to convert output signal voltage into sinusoidal excitation force by some calibration method in Section 3). Since 2-harmonic frequency has higher effects on linear excitation than 3-harmonic frequency ($S_2>S_3$), test data corresponding to $S_2\approx$1 was chosen, as shown in Table 6, to establish linear excitation equation which could help us to better regulate PCE to produce linear sinusoidal excitation signal.

If linear evaluation index $I\text{,}$ of PCE is introduced, the linear excitation equation may be characterized by Eq. (2):

where $\mathbf{f}=[f_1,f_2,\dots ,f_i]$ is a vector of critical frequencies of linear excitation, $\mathbf{V}=[V_1,V_2,\dots ,V_i]$ is a vector of output signal voltages. Additionally, $b$ is the fitting coefficient from experimental data.

It can be observed from Eq. (2) that when $V_2=$1 V, the value of $I$ remains a fixed value no matter what the value of $b$ is. From this fact, we can deduce that:

Based on experimental data from the Table 6, it can be known that $b$ should be larger than 1 to ensure the value of $I$ being a constant. Using least squares method we can determine the value of $b$, which can be expressed as:

where, $\mathrm{\Delta }I$ refers to the root mean square deviation.

Substitute Eq. (3) into the Eq. (4) and solving Eq. (4) by different fitting coefficient $b$, until we finally get $b=$1.072 at which $\mathrm{\Delta }I$ is reach to its minimum, and the fitting coefficient/root mean square deviation relationship was showed in Fig. 4, whose horizontal and vertical axis is $b$ and $\mathrm{\Delta }I$, respectively.

Fig. 4The fitting coefficient/root mean square deviation relationship

2.4. Applications of the linear excitation equation

According to Eq. (2), PCE can be controlled effectively to produce sinusoidal excitation signal without nonlinear harmonic amplitudes. For example, if we want to use it to excite a structure or specimen at 1500 Hz, then we estimate the maximum output signal voltage should be below 2.06 V. Conversely, the critical frequency of linear excitation can also be acquired under the given output signal voltage. Moreover, we can further get its linear frequency-confidence interval, as seen in Fig. 5, by associating different output signal voltage with different colors, to evaluate its linear excitation capability.

Fig. 5Linear frequency-confidence interval of PCE

3.1. Dynamics model of the cantilever beam subjected to sinusoidal excitation force of PCE

Experimental procedure of using PCE to excite the cantilever beam can be simulated by the following model, as shown in Fig. 6, here a classic Euler-Bernoulli beam with uniform cross-section is assumed and its bending vibration is of the major concern. $x$-axis is presumed to locate in the centerline of beam and $d$ is the distance between clamp end and the geometric center of PCE, the length of the beam and PCE along the $x$-axis is $l$ and $2\mathrm{\Delta }$ respectively. Adopt distributed transverse loading, $q(x,t)$, acting on the differential segment of beam to simulate sinusoidal excitation force, its expression is shown as:

where $F$ refers to amplitude of sinusoidal excitation force in N, $\omega$ refers to angular excitation frequency in rad/s.

Fig. 6Dynamics model of the cantilever beam excited by PCE

Considering viscous damping, the bending vibrations of a uniform beam subjected to dynamic loading $q(x,t)$ are described by the following equation:

where $E$, $I$, $m$, $y(x,t)$ are the Young’s modulus, second moment of area, mass per unit length, and transverse­displacement response. And $c$ and $c_r$ is damping coefficient of displacement velocity and deformation velocity respectively.

Introducing normal coordinates and solve the Eq. (6) by mode superposition method, we get displacement response of the cantilever beam as:

where $y_i\left(x\right)$ represents vibration amplitude of $i$-modal shape, ${\eta }_i\left(t\right)$ represents the value of $i$-modal coordinate.

In text books [26], that modal shape function, $y_i\left(x\right)$, of a cantilever beam is usually given by the following:

where $D_i$ represents excitation amplitude of vibration system related to amount of input energy, and $k_i$ represents a parameter associated with $i$-natural frequency.

Introducing modal damping ratio ${\zeta }_i$ and natural frequency ${\omega }_i=A_i^2\sqrt{EI/m{l}^4}$ ($A_i=$1.875, 4.694, 7.855, …, which is associated with mode shape) for the $i$th mode and two constant coefficients, $\alpha$, $\beta$, make the following assumptions:

Substitute Eq. (7), Eq. (9)-(11) into the Eq. (6), and according to the orthogonality conditions of any arbitrary shape, normal­coordinate response equation can be normalized as:

where $F_i\left(t\right)$ is the generalized loading associated with mode shape $y_i\left(x\right)$, its expression is:

The Duhamel solution of Eq. (12) under zero initial conditions is:

where ${\omega }_i^{\text{'}}={\omega }_i\sqrt{1-{\zeta }_i^2}$, is the natural frequency for the $i$th mode with damping.

Dynamic loading $q(x,t)$ of PCE can be equivalent to the distributed loading on the differential segment of beam, bring it into the Eq. (13) yields:

Substitute Eq. (15) into the Eq. (14) and add Eq. (8), by using Eq. (7) we can get dynamic response at any point in cantilever beam subjected to sinusoidal excitation force of PCE.

3.2. Calibration principle based on the dynamics model of the cantilever beam

Schematic of calibration principle, as seen in Fig. 7, for determining the level of sinusoidal excitation force based on the dynamics model of the cantilever beam mainly involves three mapping: (I) Experimental process vs. simulation process; (II) Output signal voltages applied on the PCE by high-voltage piezoelectric amplifier vs. theoretical excitation force applied on dynamics model; (III) Response results of experiment vs. calculated results of dynamics model. On one hand, we carry out experimental test to get output signal voltages on the basis of PCE excitation feedback system and displacement response of cantilever beam specimen by Laser Doppler Vibrometer (its velocity response signal needs to be processed through the integral transform). On the other hand, at the same position of excitation and response signal, we use dynamics model of the cantilever beam proposed in Section 3.1 to calculate dynamic response subjected to some level of theoretical excitation force. When the calculated result of dynamics model have a good agreement with test result, i.e., its error is below 3 %, we regard this level of theoretical excitation force is equal to actual sinusoidal excitation force produced by PCE, and thus we successfully solve concerned calibration problem.

Fig. 7Calibration principle of sinusoidal excitation force of PCE based on the dynamics model of the cantilever beam

3.3. Calibration process

To obtain sinusoidal excitation force of PCE, nature frequencies, modal damping ratios and displacement responses of cantilever beam specimen are needed to be measured precisely as to update the dynamics model. The calibration process can be summed up as following:

1) Set up calibration system of sinusoidal excitation force of PCE.

There are three basic steps to set up calibration system of sinusoidal excitation force of PCE based on beam theory. The first is that using a torque wrench to hold beam specimen tightly in the clamping fixture to ensure the beam is in clamped-free boundary conditions. The second is choosing Laser Doppler Vibrometer to reveal displacement response at the top position of beam without contact. The third is setting up PCE excitation feedback system to record excitation voltage signal of PCE through feedback attenuator and response signal from Laser Doppler Vibrometer. To reduce the influence resulting from added mass and stiffness of PCE, it needs to be placed closely to the clamped end of cantilever beam and glued firmly by super glue 502.

2) Measure natural frequencies of cantilever beam specimen.

Experimental Modal Test was used to obtain frequency response function (FRF) of cantilever beam specimen. Firstly, random vibration signal amplified by HPEA is employed to driven PCE, and then excite cantilever beam specimen to vibrate. Next, the digital filtering technology is adapted to drop down the impact of the test noise to obtained response signal. At last, estimate FRF by the recorded excitation and response signals, from which natural frequency can be determined through each resonant peak, besides, modal damping ratio can also be identified by the half-power bandwidth method which is calculated by measuring the bandwidth of the frequency curve (or approximately 3 dB) down from the resonant peak. In order to obtain the accurate results, several different test points can be chosen and the average is used as the final experimental results, for providing inherent as well as damping parameters of the real specimen for dynamics model.

3) Obtain displacement amplitude under different output signal voltages.

Select frequency range in which PCE often works and divide this range into a number of tested frequency point, and sine excitation experiments are performed on each of these frequencies. On one hand, get the spectrum of velocity response signal in the laser point and convert it to displacements through integration process, consequently identify displacement amplitude. On the other hand, mark and record output signal voltages corresponding to this displacement amplitude.

4) Establish dynamics model and update this model through measured data.

Testing the geometry and physical parameters of specimen is the basis for establishment of dynamics model. As important input parameters for dynamics model, length, width, thickness of the cantilever beam and positions of measuring points are acquired by vernier caliper or micrometer yet with test error; physical parameters is somewhat unclear due to thermal deformation of the cutting process, hence they all need to be modified slightly before being as final input data for dynamics model. According to the description in Section 3.1, establish dynamics model of the cantilever beam subjected to sinusoidal excitation force of PCE. Firstly, roughly calculate natural frequencies based on this model. Then, update dynamics model by modifying material and geometric parameters until errors between theoretical and experimental result for each order natural frequency is less than 5 %.

5) Calculate dynamic response amplitude and calibrate sinusoidal excitation force.

Calculating dynamic response and calibrate sinusoidal excitation force of PCE based on the updated dynamics model can be divided into two steps. First, put measured damping parameter from step (2) into this model so as to get more reliable response result, since vibration response of the structure is highly dependent on damping performance of each mode. Then, at the same position of excitation and response signal, using this model to calculate dynamic response amplitude subjected to certain level of theoretical excitation force. Adjusting this force to levels where calculated result is close to the test result (its error is below 3 %). Thus, this level of theoretical excitation force can be regarded as actual sinusoidal excitation force produced by PCE.

6) Draw calibration curve of PCE at certain excitation frequency.

Repeat steps (3) to (5), change output signal voltage to get a series of value of displacement amplitude, and map each of them with calculated result, in this way we can clearly establish the relationship between output signal voltage and sinusoidal excitation force from updated dynamics model. Besides, calibration curve as well as its mathematical expression can be obtained by least square fitting technique, and the more we get value of vibration response at certain excitation frequency, the more precise calibration curve we can obtain.

4.1. Calibration system setup

According to the calibration process in Section 3.3, calibration system of sinusoidal excitation force of PCE was set up by adding cantilever beam specimen and Laser Doppler Vibrometer to the PCE excitation feedback system in Fig. 1. A Ti-6Al-4V beam for this study is 280 mm×20 mm×1.5 mm (Young’s modulus is 110.32 GPa, Poisson’s ratio is 0.31, and the density is 4420 kg/m³), the effective test area was 250 mm×20 mm×1.5 mm with 30 mm in clamped region. A two piece fixture was designed to sandwich the beam specimen and four bolts, two beyond both ends of the beam and two near its centerline, were adjusted to 40 Nm to provide enough clamping force. The selected PCE, of type P-885.10 produced by PI in Germany with dimensions 5 mm×5 mm×9 mm, was glued firmly on the centerline of beam with 7 mm above the clamped end. Laser Doppler Vibrometer (Polytec PDV-100) was used to reveal the frequency and vibration displacement of the beam. Because it was not configured to take displacement measurements, its velocity measurements were converted to displacements by dividing $2\pi f\text{,}$ where $f$ represents excitation frequency, and the distance from laser point to beam’s free end is 5 mm. Other instruments, such as HPEA, feedback attenuator, DAI and computer, were already described in Section 2.1. It is necessary to note that each distance parameter was measured by vernier caliper for three times and the average was used as test result, tests of nature frequencies, modal damping ratios and displacement responses of cantilever beam specimen were run at room temperature.

4.2. Test procedures and calibration results

4.2.3. Draw calibration curve

According to theoretical and experimental data in Table 8, we can draw force/output signal voltage calibration curve at 300 Hz and thus we successfully determined the level of sinusoidal excitation force of PCE under this excitation frequency, as showed in Fig. 8, whose horizontal and vertical axis was output signal voltage and sinusoidal excitation force respectively. As it is clear from the plots, the excitation force raises when the output signal voltage increases. Besides, using least square fitting technique, we can further obtain its mathematical expression, $y=\text{0.054}+\text{0.086}x$, where $x$ represents output signal voltage and $y$ represents target force, it is obvious these two variables have a linear relationship. It should be noted that this calibration curve is based the calibration system as well as related hardware devices designed and configurated by the author’s research group, and different hardware devices are likely to cause different calibration results of sinusoidal excitation force of PCE at different frequencies. However, this calibration method can still be an important reference for solving similar calibration problems.

Fig. 8The calibration curve of sinusoidal excitation force of PCE at 300 Hz

4.3. The application of calibration curve of PCE

Repeated the above steps, calibration curves of PCE at other interested frequencies can be obtained. Thus, excitation force level of PCE can be clearly determined, which is one of most important parameters in accurate test of dynamic response. Therefore, PCE, with lighter quality and higher excitation frequency compared to the electromagnetic exciter, not only can be applied in the test area of inherent and damping characteristics, but also in the test area of vibration response of thin cylindrical shell by Laser Doppler Vibrometer, as seen in Fig. 9. FRF measurement (velocity over drive voltage) had been measured between the laser response point and one single excitation point with PCE pasted there, chirp excitation was applied on the tested shell by PCE with the following parameters: 0-3000 Hz frequency range; 70 % chirp length; 0.15 Hz frequency resolution; 10 averages; uniform windows. By identifying FRF of thin cylindrical shell, as given in Fig. 10, the first order natural frequency was 300 Hz.

Fig. 9Photograph of the tested thin cylindrical shell being excited by PCE

Next, sine excitation experiment at 300 Hz was performed which was hope to excite cylindrical shell under first order resonance, excitation voltage signal of PCE at 300 Hz with 3 V output signal voltage was shown in Fig. 11 and response signal of thin cylindrical shell was shown in Fig. 12. Using the calibration curve in Fig. 8, we had known that the sinusoidal excitation force corresponding to 3 V voltage was about 0.32 N. Beside, we can acquire response level, about 0.0009 m/s (equal to 0.48 um), by identifying the peak of base frequency in the spectrum of response signal. It should be noted that some harmonic amplitudes was also observed in this case, the main reasons for this were: (1) For this PCE, of type P-885.10, its excitation force was a bit small, so it did not adequately excite the tested shell under resonance. Thus, response measurement was interfered by environmental noise. (2) Influences of experimental boundary condition or nonuniform thickness of the tested shell may lead to the rise of harmonic frequencies. Therefore, for obtaining desired response data, we needed to use larger PCE or adopt multi-point excitation technique with more PCEs to provide sufficient excitation energy for the tested shell; in the meantime, we should reduce the nonlinear effects of above influences.

Fig. 10FRF of the tested thin cylindrical shell

Fig. 11Excitation voltage signal in time and frequency domain at 300 Hz with 3 V output signal voltage

a) Excitation voltage signal in time domain

b) Spectrum of excitation voltage signal

Fig. 12Response signal of thin cylindrical shell in time and frequency domain

a) Response signal in time domain

b) Spectrum of response signal