Analytical and Experimental Modal Analysis of a Bladed Disk

The conference paper discusses a study on the modal behavior of a bladed disk in rotating machinery system using analytical and experimental modal analysis techniques. The process involves measuring natural frequencies under different operating conditions. The modal properties from both approaches are compared, and the effects of rotational speed on the modal behavior of the bladed disk are presented and discussed. The analytical and experimental findings provide a comprehensive understanding of the bladed disk’s modal characteristics under varying rotational speeds. The integration of these different modal analysis techniques enhances the accuracy of the results and enables a more thorough investigation of the bladed disk’s behavior in rotating machinery systems.


Introduction
The Rotating blades play a pivotal role in various engineering applications, ranging from aircraft engines and gas turbines to power generation and industrial machinery.The reliable operation and performance of these systems depend on an in-depth understanding of blade behavior and vibration characteristics.Analytical and experimental modal analysis of bladed disks has emerged as a crucial area of research, offering valuable insights into the dynamic behavior of these rotating structures.In this context, the measurement of vibration in rotating blades holds paramount importance, As it enables engineers and researchers to identify potential issues, optimize designs, and ensure the overall safety and efficiency of the systems [1].
The measurement of vibration in rotating blades is critical for several reasons.First and foremost, vibration is an inherent characteristic of rotating systems and, if left unchecked, can lead to detrimental effects such as fatigue, resonant excitation, and even catastrophic failures.Understanding the modes of vibration and their associated frequencies is essential for predicting and mitigating these adverse effects.Moreover, accurate vibration measurements provide valuable data to validate analytical models and simulation tools, enhancing the accuracy of predictive analyses.Real-world experiments allow engineers to fine-tune their designs, validate assumptions, and optimize structural configurations for enhanced performance and durability [2].To measure the vibration of rotating blades, several advanced techniques have been developed, each catering to specific needs and scenarios.Blade Tip Timing (BTT) is a non-contact method that relies on measuring the time instant when the blade passes a sensor near its tip.This technique provides valuable information about blade tip displacement, blade-to-blade variations, and can be used to estimate blade mode shapes [3].The Strain Gauge (SG) method involves attaching strain gauges to the surface of the blades, which sense and record strain variations during operation.This technique is particularly useful for understanding the stress distribution and dynamic response of the blades under various operating conditions [4].Additionally, Laser Doppler Vibrometry (LDV) is a non-intrusive optical measurement technique that captures blade vibrations with high precision.LDV allows for remote measurement of blade displacements, velocities, and frequencies, making it suitable for delicate and sensitive structures [5].
The Campbell diagram is a graphical representation of the system's natural frequencies as a function of rotational speed.It visually presents the modes of vibration that can be excited during operation.Analyzing the Campbell diagram helps engineers identify critical speed regions where resonance may occur, allowing for the implementation of appropriate design modifications to avoid harmful vibrations [6].The link between the Campbell diagram and the blade tip timing method lies in their shared emphasis on rotational speed.Blade tip timing measurements provide valuable data points for constructing the Campbell diagram.By measuring the time instant when each blade passes a reference point, engineers can deduce the blade's frequency of vibration and its dependence on rotational speed.This data, when compiled for multiple blades, aids in constructing the Campbell diagram, which becomes a powerful tool for predicting and analyzing blade behavior at different operating speeds.

Blade tip timing principle
Blade Tip Timing (BTT) systems is a technique used to measure the vibration characteristics of rotating blades.It involves attaching sensors at the casing of the turbine to measure the blade's vibration as it rotates.The vibration signal of a blade can be modeled as a sinusoidal function with a frequency that is proportional to the engine's rotational speed.Specifically, we assume that the vibration signal can be expressed as follows: Where A is the amplitude of the signal, () is the frequency of the signal at time t, and φ is a phase shift.We further assume that() can be expressed as follows: Where n is an integer representing the number of revolutions and N is the total number of blades in the engine [7].Based on these assumptions, we can express () in terms of engine orders as follows: Where  =   ⁄ is the k-th engine order, and  = 1,2, . . ., .Note that  represents the frequency of vibration relative to the rotational speed of the engine.To detect blade vibration using a single sensors, we need to estimate  from ().We need to estimate  from ().This can be done by re-sampling () at a frequency equal to an integer multiple of  (the fundamental frequency of rotation).We define a new time variable τ as follows: Where n represents an integer number of revolutions made by the blade since some reference point.
Then we define a new signal () as follows: Note that () is periodic with a period of 1  ⁄ , and its Fourier series can be expressed as follows: Where d is the DC component of (),  and  are the Fourier coefficients of the sine and cosine terms, respectively, and  = 1,2, . . ., ∞.Note that the Fourier coefficients depend on  and A. To estimate  from (), we need to identify the frequency components.The identification process depends on prior inputs about the natural frequencies and mode shapes of the blades and the rotation speed which is obtained from experiments or the Finite Element Method (FEM).The role of the FEM is also to convert time vectors representing displacement to physical quantities like stress [8].A reliable computational model can reduce errors and provide a precise image of the system being studied.The FEM model in its turn should be validated using a high fidelity experimental tool like the laser Doppler Vibrometer, a lab instrument able to achieve a very high level of fidelity [9].

Geometry description and model analysis
Modal analysis is crucial for understanding the dynamic behaviour of structures and systems.It provides insights into natural frequencies, mode shapes, and damping ratios, which are essential parameters for Blade tip timing method as a non contact method for damage detection, structural health monitoring, performance evaluation, and troubleshooting.The bladed disk in Figure 1 is made up of 60 stainless steel blades with a density of 7800 Kg/m3, elasticity modulus 216 GPa, and Poisson's ratio of 0.3.The structural model of the blade for modal analysis is considered as a mass-spring system, where the blade is cantilevered and fixed at the roots while being free at the tip.This is done using ANSYS modal analysis toolbox [10].
The results presented in figure 2 shows the natural frequencies of the single blade model.The extraction of mode shape for the bladed disk assembly is demonstrated in figure 3 using ANSYS, the shapes are bending for the 1 st and 2 nd modes and torsional for the 3 rd mode.The mode shapes for single blade model is shown in Figure 4. the mode shape associated with the 1 st and 2 nd natural frequencies is bending while the 3 rd mode is torsional.

Experimental procedure and results
The study used the Laser Doppler Vibrometer to identify the natural frequencies of the blade.The test rig of the LDV laboratory is equipped with a motor that rotates the bladed disk while the LDV is equipped with a derotator that enables measurement under rotation regime (11)(12).The Rig maximum speed can reach up to 6000 RPM.The rig is also equipped with 16 electromagnets that can be used as auxiliary forced vibration generators.The entire model wheel, including sensors and electromagnets, is enclosed by a pressure-resistant glass front wall.Two vacuum pumps decrease the air pressure in this chamber down to 5 mbar, so that conditions near to vacuum can be created.For this experiment, the max speed was set at 3000 RPM that corresponds to 50 Hz.The first three natural frequencies will be excited under this condition.We avoided proceeding for higher speed as the objective was to check the prediction of the FEM and to avoid any possible unexpected damage on the rig.The scanning points were created; Measurement points are where the LDV will take measure and plot displacement behavior.The results of the FFT of the tip velocity due to a hummer punch test are presented in Figure 6.The relative error between the experimental and analytical results in Figure 6 is 2.15% for the 1st mode, 0.2% for the 2nd mode, and 2.9% for the 3rd mode for a speed range of 0 to 3000 rpm.

Conclusion
The paper discusses the modal behaviour of a bladed disk, which is important for various industrial applications.It presents both analytical and experimental modal analysis approaches, to predict and determine the modal properties of the structure.The results provide valuable insights into the behavior of bladed disks and can be used to improve their design and performance.The analytical modal analysis approach involved developing a finite element model of the bladed disk, which was used to predict the modal properties of the structure.The experimental modal analysis approach involved conducting vibration tests on the bladed disk and analyzing the resulting response data to determine the modal properties.Overall, the paper provides valuable insights into the modal behavior of a bladed disk and demonstrates the effectiveness of both analytical and experimental modal analysis techniques for studying rotating machinery systems.The results of this study can be used to improve the design and performance of bladed disks in various industrial applications.

Figure 1 :
Figure 1: a-Single blade model b-3D model of the bladed disk c-The bladed disk mounted on the rig.

Figure 2 :
Figure 2: Modal analysis results for single blade model

Figure 3 :
Figure 3: mode shapes of the blade disk

Figure 4 :
Figure 4: Modal analysis, a-bending mode b-bending mode c-torsional mode

Figure 6 :
Figure 6: Modal analysis results for single blade model via LDV

Table 1
summarizes the relative errors observed for the first, second, and third vibration modes of two opposite blades.The relative error values for each mode are presented as percentages, providing a quantifiable measure of the discrepancies between FEM predictions and LDV measurements.

Table 1 :
Comparative analysis of Modal Characteristics: Opposite blades -FEM vs. LDV measurement