A Full-Field Wideband Modal Testing Method for Turbine Blades

Abstract

In order to accurately measure the low- and high-order modes of turbine blades, this study proposes a method that integrates piezoelectric ceramic excitation with time-averaged electronic speckle pattern interferometry (TA-ESPI). The piezoelectric exciter provides wideband and high-output excitation, effectively stimulating both low- and high-order blade modes. The TA-ESPI technique captures the vibration signals with high displacement sensitivity, enabling full-field mode shape measurement without the need for high-speed cameras. Additionally, an equivalent strain principle based on deflection curvature is introduced to extract strain modes from displacement modes for identifying potential fatigue failure areas (PFFAs). Experimental validation on an aero-engine turbine blade successfully identified the first eight natural frequencies (up to 7753 Hz), a significant advancement over the conventional method which identified only two. The measured first-order frequency showed a high agreement (deviation < 2%) with the impact hammer test, confirming accuracy. The extracted strain modes for the first six orders clearly revealed the PFFAs, which aligned well with actual failure regions. The proposed method proves to be an effective, practical, and efficient modal testing technique for turbine blades, offering a substantial improvement in bandwidth over established methods.

1. Introduction

Turbine blades mounted on aero-engines are often subjected to complex cyclic loads from the environment, causing them to undergo repeated bending and twisting vibrations [1]. When the frequency of the external loads approaches the blade’s natural frequency, resonance occurs and the blade’s vibration amplitude increases dramatically, which may induce fatigue fractures in the high-stress regions of the blade [2]. In engineering practice, blades commonly experience both low-cycle fatigue (LCF) and high-cycle fatigue (HCF). LCF is typically caused by low-frequency resonance, with potential fatigue failure areas (PFFAs) concentrated near the blade root, while HCF usually results from high-frequency resonance, where PFFAs predominantly occur near the blade tip [3]. Therefore, conducting accurate and efficient modal testing across a wide frequency range is essential for optimizing blade design, predicting fatigue life, and ensuring operational safety [4].

Currently, methodologies for acquiring the modal parameters of turbine blades can be broadly classified into three categories: numerical simulation, contact-based experimental testing, and non-contact optical measurement. Numerical methods, primarily the finite element method (FEM), construct a blade model using software to solve motion equations and extract modal parameters [5,6,7,8,9]. Although theoretically capable of solving for higher-order modes, FEM accuracy is critically dependent on model fidelity, making its application challenging for blades with complex geometries and uncertain material properties [10]. Contact-based experimental techniques, such as those employing dynamic strain gauges or accelerometers [11,12], offer high sensitivity. However, each sensor provides only single-point data, and deploying multiple sensors introduces parasitic mass and stiffness, which can significantly alter the dynamic characteristics of lightweight blades [13].

With advancing optical technologies, non-contact methods have been progressively applied [14]. Laser Doppler vibrometer (LDV) utilizes the Doppler effect to acquire vibration data with high precision and a broad frequency response [15,16]. Nevertheless, LDV measures vibration point-by-point, failing to provide instantaneous, synchronized full-field mode shapes [17]. Digital image correlation (DIC) serves as a non-contact, full-field deformation measurement technique [18], but it requires high-frame-rate cameras and substantial computational resources [19]. Moreover, its displacement sensitivity is constrained to the micron-level, which is often insufficient for capturing the nano-scale vibration amplitudes of high-stiffness turbine blades [20]. In contrast, electronic speckle pattern interferometry (ESPI) traces deformation to the wavelength of light, enabling nano-scale displacement detection [21]. Crucially, ESPI can capture steady-state vibration patterns through the time-averaged (TA) effect within a single camera exposure, eliminating the need for high-speed cameras [22], making it inherently suitable for full-field modal measurement.

In addition, during modal testing, it is necessary to excite the turbine blade to induce the vibrational response. Common excitation methods include pulse excitation and sweep frequency excitation [23]. Pulse excitation typically uses a force hammer to strike the structure’s surface, applying a short impulse force to induce multi-frequency vibrations. Achieving an effective impulse excitation often necessitates repeated strikes with a force hammer. This repetitive action may inadvertently damage the blade. Alternatively, sweep frequency excitation employs an exciter to induce steady-state vibrations in the turbine blade. The modal characteristics of the structure can then be identified based on resonance amplification effects [24]. Conventional exciters, such as electromagnetic shakers, are commonly modeled as single-degree-of-freedom systems. While they can generate substantial excitation forces at low frequencies, their output attenuates markedly at higher frequencies. Given the characteristically high stiffness and high natural frequencies of turbine blades, these traditional exciters struggle to excite the blades’ modes effectively.

In summary, while TA-ESPI itself offers excellent sensitivity, its application in wideband modal testing is constrained by the lack of suitable exciters capable of effectively stimulating both low- and high-order modes of high-stiffness blades. Furthermore, the critical task of directly extracting strain modes—indispensable for PFFA identification—from full-field optical displacement data remains an underexplored challenge.

To address these interconnected limitations, this study proposes a novel, integrated testing methodology. The core innovation lies in combining wideband piezoelectric ceramic excitation with TA-ESPI full-field measurement, augmented by an equivalent strain principle based on deflection curvature. Specifically, (1) the piezoelectric exciter provides high-output, broadband excitation to effectively stimulate both low- and high-order blade modes. (2) The TA-ESPI technique synchronously captures full-field mode shapes with nano-scale displacement sensitivity, without requiring high-speed cameras. (3) The equivalent strain principle enables the direct extraction of strain modes from the measured displacement fields, offering a non-contact solution for identifying PFFAs. This integrated approach simultaneously overcomes key bottlenecks in bandwidth, full-field capability, measurement sensitivity, and strain modal analysis, presenting a comprehensive and practical solution for advanced turbine blade modal testing.

The remainder of this paper is organized as follows. Section 2 elaborates on the fundamental principles of the proposed method, including the TA-ESPI technique, broadband piezoelectric excitation, and the equivalent strain mode theory. Section 3 describes the experimental setup and presents the direct results of the modal testing. Section 4 provides a detailed discussion on the derived strain modes and the identification of potential fatigue failure areas. Finally, Section 5 concludes the paper by summarizing the main findings and contributions.

2. Principle and Methodology

2.1. The Principle of TA-ESPI

The ESPI technique uses laser speckle patterns as the information carrier for the surface deformation of the measured object. By analyzing the variations in the speckle pattern captured by the camera, the deformation of the object’s surface can be inferred. Figure 1 shows a schematic diagram of the ESPI measurement optical path. The laser is first expanded by a beam expander and then split into two beams by a beam splitter. These two beams shining on the surfaces of the turbine blade to be measured and the reference plate. The beam diffusely reflected from the blade is called the object beam, and the beam diffusely reflected from the reference plate is called the reference beam. The object and reference beams are combined by the beam splitter, and finally, they interfere on the target plane of the camera, forming a speckle pattern.

When using an exciter to perform slow sweep frequency excitation on the blade, at time $t$, the blade approximately undergoes steady-state vibration, and the speckle pattern on the target plane can be expressed as

(1)$I=I_O+I_R+2\sqrt{I_O{I}_R}\mathit{cos}[\phi +\mathrm{\Omega }\mathit{sin}(\omega t\left)\right]$

where $I_o$ and $I_R$ represent the intensity of the object beam and the reference beam, respectively. $\phi$ is the initial phase, $\mathrm{\Omega }=4A\pi /\lambda$ is the phase generated by the amplitude $A$ of the measured blade, and $\omega$ is the angular frequency of the blade vibration.

Since the blade undergoes steady-state vibration, the intensity of the images captured by the camera is the integral of the varying speckle pattern intensity during the camera’s exposure time, meaning it has a time-averaging effect. When the vibration frequency of the blade is higher than the camera’s frame rate, the speckle pattern captured by the camera can be expressed as

(2)$I=I_O+I_R+2\sqrt{I_O{I}_R}\mathit{cos}\phi J_0\left(\mathrm{\Omega }\right)$

As seen above, the speckle interference intensity under the time-averaging effect is modulated by the zeroth-order Bessel function $J_0$. Due to the presence of background light intensities $I_o$ and $I_R$, the fringes representing the mode shape in the speckle pattern are difficult to identify. In this study, real-time subtraction is used on the images, and an additional phase is introduced to remove the background light intensities, allowing the blade’s mode shape fringes to be revealed. When the object undergoes steady-state vibration, introducing the additional phase $\Delta \phi$, the speckle pattern captured by the camera can be expressed as

(3)$I_1=I_O+I_R+2\sqrt{I_O+I_R}\mathit{cos}(\phi +\Delta \phi \left)J_0\right(\mathrm{\Omega })$

Perform subtraction between the two patterns before and after the introduction of the additional phase, and the absolute value is taken, which is displayed in real-time on the screen. This allows for obtaining the fringe pattern that represents the blade’s mode shape, which can be expressed as

(4)${I\left|I_1-I\right|\sqrt{I_O{I}_R}\left|\mathit{sin}(\phi +\Delta \phi /2\left)\mathit{sin}\left(\Delta \phi /2\right)J_0\right(\mathrm{\Omega })\right|}_m$

where $\left|\mathit{sin}(\phi +\Delta \phi /2)\right|$ is the random term related to the speckle pattern, $\left|J_0\left(\mathrm{\Omega }\right)\right|$ represents the mode shape modulation term, and $\left|\mathit{sin}\left(\Delta \phi /2\right)\right|$ represents the overall grayscale modulation. Clearly, when $\Delta \phi =\pi$, the contrast of the mode shape fringes is at its highest. The function $\left|J_0\left(\mathrm{\Omega }\right)\right|$ and its modulated fringes are shown in Figure 2. It can be observed that the function $\left|J_0\left(\mathrm{\Omega }\right)\right|$ is quasi-periodic. This function reaches its maximum value at the region where the blade amplitude is zero, i.e., at the nodal line. Several maxima of this function gradually decrease as the blade amplitude increases. These characteristics determine that the structure’s vibration mode consists of black and white fringes, with the brightest fringes representing the nodal lines of the blade.

2.2. Broadband Piezoelectric Excitation

The electromagnetic shaker is a commonly used frequency sweeping exciter in modal testing, as shown in Figure 3a. The mechanical structure of this shaker can be simplified into a single-degree-of-freedom vibration model, as shown in Figure 3b, with its amplitude–frequency curve depicted in Figure 3c. From the amplitude–frequency curve, it can be seen that when the excitation frequency of the shaker exceeds the system’s natural frequency, the excitation force output by the shaker decreases rapidly. When the excitation frequency exceeds three times the natural frequency, the excitation force output by the shaker nearly decays to zero [25]. Therefore, the working frequency range of this kind of exciter is relatively narrow, generally less than 500 Hz. Since the blades are normally small in size and high in stiffness, their fundamental frequency is generally above 500 Hz, making it difficult for traditional exciters to excite the modes of turbine blades.

To meet the requirements for exciting the modes of blades, it is necessary to select an exciter with a wide working frequency range and a large output force. Therefore, this study chooses piezoelectric ceramics as the exciter for the blades. Piezoelectric ceramics exhibit both direct and inverse piezoelectric effects, as shown in Figure 4. When piezoelectric ceramics are used as an exciter, their inverse piezoelectric effect is utilized. The inverse piezoelectric effect refers to the phenomenon where, when an electric field is applied to a piezoelectric material, mechanical deformation occurs in certain directions of the material. When the polarity of the electric field changes, the direction of deformation also changes. When the electric field disappears, the deformation of the piezoelectric material disappears as well, and the amount of deformation is proportional to the strength of the external electric field. Compared to traditional exciters such as force hammers, shakers, or vibration tables, piezoelectric ceramics offer advantages such as their light weight and fast response.

2.3. Equivalent Strain Modes

The mode shape fringes obtained through the TA-ESPI method are modulated by the zeroth-order Bessel function $\left|J_0\left(\mathrm{\Omega }\right)\right|$. The dark fringes correspond to the zero points of the zeroth-order Bessel function, and the amplitudes of the blade at the zero points are known. Therefore, by extracting the skeleton lines of the vibration mode fringes and assigning amplitude values to them, followed by surface fitting, the full-field amplitude distribution of the mode shape can be obtained. However, when assigning amplitude values to the skeleton lines, it is necessary to consider the positive and negative signs of the amplitude, as the vibrations of the entire field are not in the same direction. Since the blade vibration involves energy transfer, especially at high-order modes, energy is transmitted between multiple nodes and anti-nodes [26]. At the nodal lines, the energy is at its lowest, while at the anti-nodal lines, the energy is at its maximum. During the vibration process, the energy distribution satisfies the following formula:

(5)$E={\displaystyle \frac{1}{2}}M(x,y)\theta (x,y)$

where $E$ represents energy, $M(x,y)$ is the bending moment, and $\theta (x,y)$ denotes the torsion angle of the blade. In the vibrating system, energy is converted between potential and kinetic energy. To ensure the symmetrical distribution of energy and the stability of the system, the bending moment and vibration direction will reverse. The vibration directions on either side of the nodal line must be opposite. Therefore, the signs of the amplitude values assigned to the skeleton lines on both sides of the nodal line should be opposite, while the signs of the amplitude values assigned to the skeleton lines between two nodal lines should be the same.

By computing the second-order partial derivatives of the obtained blade surface mode shape $u(x,y)$ along the x-direction and y-direction, the local curvature components of the blade’s mode shape at each point can be determined along the x-direction and y-direction:

(6)$\left\{\begin{array}{c}{\kappa }_x={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\\ {\kappa }_y={\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\end{array}\right.,$

In the theory of bending of plates, the relationship between the local strain and the local curvature is as follows [27]:

(7)$\epsilon =-z\kappa$

where $\epsilon$ is the local strain at a point on the blade surface, $z$ is the perpendicular distance from the point to the neutral axis of the blade, and $\kappa$ is the local curvature at the point. Then, from Formulas (6) and (7), the relationship between the components of local strain along the x-direction and y-direction and the mode shape is given by

(8)$\left\{\begin{array}{c}{\epsilon }_x=-z{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\\ {\epsilon }_y=-z{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}\end{array}\right.,$

It can be seen that the relationship between the local strain and the second-order partial derivatives of the mode shape is linear. Therefore, the strain modes in the x-direction and y-direction at each point of the blade can be obtained by taking the second-order partial derivatives of the displacement mode along the x-direction and y-direction.

3. Turbine Blade Modal Experiment

3.1. Modal Testing System and Blade Sample

Figure 5 shows the schematic diagram of modal testing system for aero-engine turbine blades based on piezoelectric ceramic excitation and TA-ESPI, and the actual experimental setup are photographed as in Figure 6. The system consists of an interference module and an excitation module. The interference module includes a laser, a beam expander, a beam splitter, two mirrors, and a CCD camera, while the excitation module is made up of a signal generator and a piezoelectric ceramic exciter. The laser has a power of 20 mW and generates coherent green light with a wavelength of 532 nm. The programmable CCD camera has a resolution of 2048 × 1536 pixels. The signal generator can produce sine wave signals in the range of 0–200 kHz.

The test object is a specific type of turbine blade from a CFM56 aero-engine (CFM International, Cincinnati, OH, USA). The blade is made from a high-temperature resistant titanium alloy, and the reference material parameters are shown in Table 1. The blade structure consists of the tip, body, root, and tenon, with a thermal barrier coating sprayed on the surface. Figure 6b illustrates the front surface of the blade, which has a matte paint coating to scatter the light. Figure 6c shows the back surface of the blade, with a piezoelectric ceramic exciter adhered to it.

During the experiment, the root of the blade is clamped and fixed using a vice, and a piezoelectric ceramic exciter is adhered to the back of the blade. The exciter is a piezoelectric ceramic disk with an outer diameter of 30 mm and a center thickness of 0.4 mm. Since the nodal lines of the blade are typically located at the transverse center and root, to ensure the excitation effect of the piezoelectric ceramic exciter is significant, the copper side of the piezoelectric ceramic sheet is adhered to the longitudinal center left side of the back of the blade. The piezoelectric ceramic exciter is controlled by the signal generator to output the sinusoidal excitation force.

A computer controls the CCD camera, which displays and processes the speckle patterns captured by the camera in real-time. By using frequency sweeping excitation and the real-time subtraction mode of patterns, the natural frequencies of the blade can be determined. Once the natural frequencies are identified, steady-state excitation is applied to the blade, and an additional phase of π is introduced to obtain the mode shape fringe patterns. The fringe pattern is then processed through skeleton line extraction, value assignment, and surface fitting to obtain the amplitude distribution of the mode shape. Solving the curvature of the amplitude can further yield the strain mode information of the blade.

3.2. Modal Testing Results

The natural frequencies and mode shapes of the turbine blades were measured using the principles outlined in Section 2.1. A total of three blades of the same kind of type were tested, and the first eight orders of natural frequencies measured are shown in Table 2, the natural frequency test results for the three blades of the same model are consistent, indicating the uniformity of the blade’s manufacturing process.

The first-order frequency of this model is higher than 400 Hz, and the second-order frequency exceeds 1500 Hz, indicating that the alloy material of the blade has relatively high elastic modulus and shear modulus. Using the method proposed in this paper, it is easy and accurate to measure the blade’s natural frequencies above 7 kHz, which is nearly impossible to achieve with traditional modal testing techniques.

To provide a direct quantitative comparison, we have conducted additional validation using a commercial modal testing system (DH 5923, from Donghua Testing Technology Co., Ltd. Taizhou. China), which represents a standard impact hammer test, as shown in Figure 7. The blade was excited by an impact hammer, and the response was measured by an eddy current sensor with a sampling frequency of 10 kHz.

The results clearly demonstrate the superiority of our proposed method:

(1). Wideband capability of our method: The conventional hammer impact method, despite its high sampling rate, could only identify the 1st-order bending natural frequency (478 Hz) and 1st-order twist frequency (1992 Hz) through spectrum analysis due to the rapid attenuation of impact energy and limitations in sensor signal-to-noise ratio at high frequencies. In stark contrast, our proposed method successfully identified the first eight orders of natural frequencies, up to 7753 Hz.

(2). High precision: For the common 1st-order frequency, the measurement results from both methods showed a difference of less than 2%, confirming the high accuracy of our technique.

(3). Full-field made shape visualization: The sensors in the traditional hammer impact method are single-point and cannot provide the vibration patterns. The method proposed in this paper can directly obtain the full-field of vibration modes, as shown in Figure 8.

The first 8 order mode shape fringe patterns are depicted in Figure 8. The mode shape characteristics of the blade can be determined based on the positions of the nodal lines and the fringe distribution. The blade root is fixed by the vice, and the amplitude is always zero, representing the fixed nodal line for each mode shape, displayed as bright fringes. In Figure 8a, the mode shape fringes are primarily horizontal and nearly parallel to the nodal line at the fixed end, indicating that this is the first-order bending mode. Since the blade itself is not a flat plate, the fringes show a slight curvature. The mode shape fringes shown in Figure 8d exhibit two horizontal nodal lines, indicating a second-order bending mode. In the mode shape fringes shown in Figure 8c, a longitudinal bright fringe appears at the center of the blade, with the remaining fringes are distributed on either side of this central fringe. Since the blade root is the fixed end, the scattered fringes, except for the central bright fringe, curve toward the edges of the blade as they approach the fixed end. Therefore, the longitudinal bright fringe can be identified as the nodal line, representing the torsional axis of the blade. This mode shape can be classified as the first-order torsional mode.

In the mode shape fringes shown in Figure 8b, two nodal lines were observed. In addition to the nodal line at the blade root, another nodal line appeared at the mid-span of the blade. This line is neither horizontally nor longitudinally distributed, indicating that this mode is a combination of bending and torsional vibrations. By analogy, in Figure 8e–h, the fringe patterns of the blade show irregular shapes and distributions of the nodal lines, and the remaining scattered fringes exhibit poor symmetry, indicating complex coupled modes.

4. Discussion

The mode shapes fringe patterns of the blade obtained through TA-ESPI do not intuitively display the amplitude of the mode shape. To obtain the amplitude distribution of the full-field mode shape, operations such as extracting the skeleton line of the mode shape fringes, assigning values, and surface fitting are carried out. We use AutoCAD software (version: 2023) to extract the centerline of the blade speckle fringe patterns. Based on the positions of the blade’s nodal lines and the amplitude differences between the centerlines of the speckle fringes, we calculate the amplitude direction and magnitude at the remaining speckle fringe centerlines. Values are assigned to each centerline, and the three-dimensional coordinates of the points on each centerline are extracted. Using MATLAB software (version: 2024b), we perform quintic polynomial fitting on the extracted centerlines to obtain the vibration amplitude surface on the blade’s surface. By further calculating the second partial derivatives of the amplitude surface, the curvature distribution of the deflection surface is obtained, which in turn provides the strain mode information.

In this study, the speckle fringe patterns of the first six order mode shapes of the blade are processed to obtain the amplitude surfaces and strain distributions of the first six modes, as shown in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. Figure 9 shows the blade in the 1st mode, where the maximum surface amplitude occurs at the free end of the blade. The lateral strain (${\epsilon }_x$) is very small, while the longitudinal strain (${\epsilon }_y$) is relatively large, with the maximum value located on the left side of the blade root. As a result, the potential failure area of the blade in the 1st mode is located at the root of the blade. If environmental loads are continuously coupled with the blade’s first bending frequency, the blade is prone to fatigue, leading to complete fracture failure at the red position shown in the figure.

Figure 10 shows the blade in the 2nd mode, where the maximum surface displacement occurs on the tip at the free end. The lateral strain of the blade remains very small, while the longitudinal strain is relatively large, with the strain being more significant at three-quarters of the blade’s height. Thus indicating that when environmental loads are continuously coupled with the first bending vibration, fatigue is likely to occur on the middle section of the blade body, leading to crack formation and eventual tearing.

Figure 11 shows the blade in the 3rd mode, with the vibration mode being the first-order twist vibration mode. For the strain mode, similar to the first bending vibration mode, the component is negligible in the x-direction and obvious in the y-direction. The large strain in the y-direction is mainly concentrated at the right side of the blade root. Therefore, if environmental loads are continuously coupled with the second bending vibration, the blade is prone to fatigue failure at the right root. Figure 12 shows the blade in the 4th mode, with the vibration mode being a second-order bending mode and the strain modes being similar to that of the 2nd order mode.

Figure 13 shows the blade in the 5th mode, with the vibration mode being the combination of the second-order bending and the first-order twist. In the longitudinal strain mode (${\epsilon }_y$), significant strain is found at the right tip and root of the blade. Compared to the previous modes, the tip of the blade becomes a new PFFA. Therefore, when fatigue failure occurs in this strain mode, the blade is likely to fracture at the root and tip. Figure 14 shows the blade in the 6th mode. The 6th order strain mode is similar to the 5th order strain mode, except that the larger longitudinal strain occurs on the left side of the blade tip and the root. Based on the above analysis of the strain modes of the blades, the reason for the location of the PFFAs has been reasonably explained.

5. Conclusions

This study presented a modal testing method for turbine blades that combined piezoelectric ceramic excitation with TA-ESPI. By utilizing the advantages of piezoelectric ceramics, such as wideband frequency and strong output force, both low- and high-order modes of the turbine blades were effectively excited. Based on the time-averaging effect of the ESPI technique, high-frequency modes of the blades were measured with high displacement sensitivity, without the need for high-frame-rate cameras, which highlighted the advantages of low cost, simple operation, high precision, and a wideband capability for natural frequency measurements. This study also proposed an equivalent strain principle based on curvature to extract strain modes from the measured mode shapes, which served as a reliable reference for identifying the potential fatigue failure areas (PFFAs) on the blades.

Through modal testing on a specific type of aero-engine turbine blade, the reliability and effectiveness of the proposed method were validated. The method successfully identified the first eight natural frequencies, with the highest order reaching 7753 Hz, a significant advancement over conventional methods. The measured first-order natural frequency (469 Hz) showed excellent agreement (deviation < 2%) with the impact hammer test, confirming the accuracy of the method. By extracting the first six order equivalent strain modes, the PFFAs on the blade were clearly identified. The results showed that the lateral strain of the tested blade was negligible, while the longitudinal strain was predominant. Specifically, in low-order mode shapes (e.g., 1st and 3rd orders), the maximum strain occurred at the blade root. In high-order bending vibrations (e.g., 2nd and 4th orders), the maximum strain was located at the mid-span of the blade. For high-order bending–torsional combined mode shapes (e.g., 5th and 6th orders), the maximum strain occurred simultaneously at both the blade root and the blade tip. These findings were highly valuable for analyzing the causes of fatigue failure in blades during actual operation.

The method was validated on a specific blade type under simplified boundary conditions (a rigid clamp). It is recognized that this differs from the complex contact interface of a real blade root, which may affect the modal parameters. Therefore, the applicability to more complex geometries, composite materials, and real boundary conditions requires further investigation. Future research will focus on extending the technique to rotating blades under thermo-mechanical loads to enhance its practical utility in blade health monitoring.

Author Contributions

Methodology, Y.M.; investigation, L.Z. and K.L.; resources, Y.Z.; data curation, L.Z.; writing—original draft preparation, K.L. and L.Z.; writing—review and editing, Y.M.; visualization, K.L.; supervision, D.J.; project administration, Y.Z. and D.J.; funding acquisition, Y.M. and D.J. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data that support the findings of this study are available upon reasonable request from the authors.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

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Figures and Tables

Figure 1 TA-ESPI measurement optical path.

Figure 2 The zeroth-order Bessel function $\left|J_0\left(\mathrm{\Omega }\right) \right|$: (a) function curve, and (b) modulated fringes.

Figure 3 Traditional excitation device: (a) electromagnetic shaker, (b) simplified vibration model, and (c) frequency response function.

Figure 4 Schematic diagram of piezoelectric effect: (a) direct piezoelectric effect and (b) inverse piezoelectric effect.

Figure 5 Schematic diagram of the modal testing system.

Figure 6 (a) Actual detection system diagram and turbine blade for testing (b) the front surface and (c) the back surface.

Figure 7 A commercial modal testing system for method valuation and comparison.

Figure 8 Speckle fringe patterns of the mode shapes of the turbine blade: (a–h) correspond to the 1st to 8th order modes, respectively.

Figure 9 The 1st-order modal (a) displacement mode and (b) strain mode in the x-direction and (c) in the y-direction.

Figure 10 The 2nd order modal (a) displacement mode and (b) strain mode in the x-direction and (c) in the y-direction.

Figure 11 The 3rd order modal (a) displacement mode and (b) strain mode in the x-direction and (c) in the y-direction.

Figure 12 The 4th order modal (a) displacement mode and (b) strain mode in the x-direction and (c) in the y-direction.

Figure 13 The 5th order modal (a) displacement mode and (b) strain mode in the x-direction and (c) in the y-direction.

Figure 14 The 6th order modal (a) displacement mode and (b) strain mode in the x-direction and (c) in the y-direction.

Table 1

Titanium alloy material parameters.

Parameter	Young’s Modulus	Poisson’s Ratio	Bulk Modulus	Shear Modulus	Density
Symbol	E	μ	K	G	ρ
Value	96 GPa	0.36	114.29 GPa	35.294 GPa	4620 kg·m−3

Table 2

Measurement results of the first 12 natural frequencies of the turbine blade (Hz).

	Mode	1	2	3	4	5	6	7	8
Sample
Blade 1		464	1523	1864	2869	3530	6516	6946	7732
Blade 2		471	1510	1950	2931	3540	6511	6934	7750
Blade 3		474	1574	1954	2940	3536	6509	6929	7779
Average		469	1535	1922	2913	3535	6512	6936	7753

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