An Increase in the Energy Efficiency of Axial Turbines by Ensuring Vibration Reliability of Blade Milling

Abstract

Ensuring the vibration reliability of power equipment is one of the fundamental problems in modern power machinery. This problem has become more critical due to a permanent increase in the machining performance of high-speed milling of axial turbine blades. This article aims to identify reliable vibration parameters for high-speed milling of turbine blades to increase the energy efficiency of gas and steam turbines. For this purpose, mathematical models of free and forced oscillations of turbine blades during machining were developed. As a result of considering experimental and finite element analysis data, critical frequencies and corresponding mode shapes of free oscillations were identified using regression procedures by the best fit of analytical and empirical approaches. Additionally, after considering the forced oscillations of blades during high-speed machining, the magnitude of the specific cutting force and external damping ratio in the system ‘axial turbine blade and milling head’ were evaluated. The resulting magnitude of forced oscillations during machining was calculated. Finally, the amplitude–frequency response was also assessed, considering the machining parameters. Overall, the proposed methodology increases energy efficiency due to a decrease in the obtained machining quality of turbine blades.

1. Introduction

The blade is one of the most precise and expensive parts in turbomachines’ rotor assemblies. Several parameters of the geometry of each section characterize the complex surface of turbomachines blades. The blade profile affects the dynamic gas characteristics of the flow part in general and its bearing capacity and fatigue resistance under cyclic loads [1]. In non-compliance with the technical requirements, the possibility of resonance stresses of separate blades is the cause of asymmetry and oscillations of the whole turbine wheel. Thus, in this article, the methodology of ensuring vibration reliability of blade milling is applied to gas and steam turbine blades.

A decrease in the roughness of the surface of turbine blades increases its corrosion resistance. The lower the roughness, the smaller the area of contact with the corrosive environment; consequently, the influence on the part’s surface is reduced. Additionally, it is known that grooves and deep and sharp scratches are stress concentrators that can lead to the destruction of the part during operation [2].

The high class of cleanliness of the surface of the working part of the blade reduces the consumption of gas and steam energy due to friction, thus affecting the efficiency of turbomachines. Boyle et al.’s experimental results [3] showed that the influence of blade roughness on turbomachines is highly related to the Reynolds number. Reynolds number represents the relative importance of inertial force to the viscous drag force.

At low Reynolds numbers, turbine blades with low surface roughness act as hydraulically smooth, but at high Reynolds numbers, even a low surface roughness could significantly affect the steam turbine efficiency. Hydraulic smoothness is characterized by the degree of surface microstructure penetration into the laminar portion of the flow. The surface is hydraulically smooth if the peaks of the surface microstructure do not penetrate the laminar layer [4]. Therefore, surface finishes as smooth as possible are crucial for design applications, implying high Reynolds numbers [5].

The surface roughness can debilitate the separation bubble feature at a low Reynolds numbers, thus reducing the aerodynamic loss. Still, at a high Reynolds number (Re), the surface roughness can activate transient progress, so the aerodynamic loss increases crucially. The loss is significantly large at a high Re with a rough surface. Compared to smooth turbine blade assembly, considered as a baseline, the total pressure loss coefficient of the assembly increases by up to 2.3 times for a 110 μm roughness at an Re = 3 × 10⁵ [6].

A numerical study by Zhihui et al. [7] represented how wall roughness affects the efficiency of an axial compressor. The roughness of the shroud hub and blade were 5, 20, and 40 µm, respectively. It was observed that the loss in the peak efficiency of the compressor was up to 95.3% due to the roughness of the blade, only 3.6% due to the roughness of the surface of the hub, and 1.1% due to the external casing surface roughness.

After considering general conditions of vibrational strength, the roughness of the surfaces of the working part should not be assigned rougher than Ra = 1.25–0.63 µm for turbine blades made from a material with a tensile strength of less than 736 MPa and Ra = 0.63–0.32 µm for turbine blades made of materials with a tensile strength of more than 736 MPa.

According to the above-mentioned results, a significant factor in the processing of blades is ensuring the vibration reliability of the process [8] and detuning from the resonance modes [9]. There are two main types of vibrations. Forced vibrations can be decreased by changing the acting force and frequency. Self-excitation vibrations require complicated external dumping techniques on the affected body. Forced and self-excited vibrations reduce the productivity of the machining and can lead to damage of the tool [10].

Research into forced vibrations in [11], dedicated to the continuous control of spindle speed and feed rate, details the spindle rate optimization strategy based on an overall energy evaluation of the part’s forced vibrations due to cyclic cutting forces. The proposed approach identifies sections of the blade with a critical increase in the vibration level at various spindle speed values. For each section level of the part, the mean vibration energy is determined by applying FEM analysis of the blade and its excitations by cutting forces.

For the profile contours, including turbine blade ones, milling shoulders, and grooves, down milling is recommended, as it provides better results for the Ra surface roughness and non-sticking chip. Moreover, for an annealed type of steel, the depth of cut increases vibration more than the speed and the feed rate. However, in the case of hardened steel, the most influential vibration factor is the cutting speed. Increasing the machined part’s hardness decreases the tool’s lifespan and increases vibration excitations during machining [12].

Considering the trend of machining coolant fluid rejection in the industry, studies related to the processing with a minimum quantity of lubrication resulted in an 88% improvement in Ra and a 91% improvement in Rz. In the experiments carried out with Minimum Quantity Lubrication (MQL), less tool wear was seen than with dry cutting. Machining with MQL prolonged the tool life of the chemical vapor deposition (CVD) tools by 3.7 times and the physical vapor deposition (PVD) tools by 3.0 times compared to dry machining [13]. Additionally, the MQL system could reduce the cutting zone temperature by nearly 100 °C in both tools (PVD and CVD) at all cutting speeds [14].

The establishment of mathematical models allows for predicting the mechanical system response. A conventional empirical strategy analyzes the behavior of specific features that are supposed to be necessary as a function of the factor. However, as with most parts of the machining processes, including milling, the numerous quality parameters measured are significantly correlated and have different optimization goals. Thus, individual analyses of each response can lead to an opposing optimum since the factor levels that enhance one response can, in other circumstances, degrade another.

For the most part, optimization strategies rely on a single information set, for example, a CNC program executed in the CAM system and sent to the machine through a postprocessor. However, optimization capabilities can be improved by processing different pieces of information in the product development process (e.g., design, technical preparation, time management, production, and quality assurance) [15], considering all stages as a complex approach to increase both the efficiency of the product manufacturing process and its efficiency as part of the entire assembly.

Response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. Optimizing process parameters for CNC milling (feed, cutting speed, and DOC) allows to influence the material removal rate, the surface roughness, the tool wear [16], and the dynamic stability [17]. Principal component analysis (PCA) is a statistical technique for reducing the dimensionality of a dataset [18] and has also been applied in error optimization studies. It can extract the measured vibration signal dominant harmonics [19]. RSM integrated with PCA is a functional approach to model and create equations for predicting and optimizing [20], particularly using the fewest experiments possible [21].

Globally, there is rising attention to the total energy consumption of manufacturing, because 83% accounts for machining in the USA and 85% in complete global manufacturing [22]. Thus, these volumes of production are relevant for optimization. Industry 4.0 has been a critical ambition in the area, directing industry toward a high level of automation and data integration dependent on Cyber-Physical Systems (CPS) [23]. CPSs give a feedback loop between the control of the physical process and software algorithms in the enterprise [24]. Ordinary manufacturing tasks, such as optimizing operations, can be reached through intelligent process integration [25].

The primary focus of this study is to find the combination of machining reliable vibration parameters in high-speed milling of turbine blades to increase the energy efficiency of axial turbines. For this purpose, mathematical models have been established considering empirical data and FEM simulations.

2. Materials and Methods

The mathematical model of the blade’s vibration state during milling in a complex plane is as follows (Figure 1):

$$\frac{d^2}{d{z}^2}\left[EI\left(z\right)\frac{d^{2}W\left(z,t\right)}{d{z}^2} \right]+\mu \left(z\right)\frac{d^{2}W\left(z,t\right)}{d{t}^2}+\beta \left(z\right)\frac{dW\left(z,t\right)}{dt}=f_0{e}^{i\omega t},$$

(1)

where z—longitudinal coordinate, m; t—time, s; E—Young’s modulus, MPa; I(z)—cross-sectional moment of inertia, m⁴; W(z, t)—complex deflection of the blade, m; μ(z)—specific linear mass, kg/m; β(z)—specific damping factor, N·s/m²; f₀—specific cutting force, N/m; i—imaginary unit; t—time, s; and ω—cutting speed, rad/s.

The first term of Equation (1) corresponds to the elastic bending force in a blade and the second and the third components correspond to inertia and damping forces, respectively. The right part of the equation reflects the action of the primary dynamic component of the cutting force.

Since the blade’s cross-sectional area varies along the length, the following expressions give the specific mass and the cross-sectional moment of inertia:

$$\mu \left(z\right)={\mu }_0\left(1-\alpha \frac{z}{L}\right); I\left(z\right)=I_0{\left(1-\alpha \frac{z}{L}\right)}^3; {\mu }_0=\rho b{h}_0; I_0=\frac{b{h}_0}{12};\alpha =\frac{\mathsf{\Delta }h}{h_0},$$

(2)

where L—length, m; μ₀—specific mass at the initial cross-section z = 0, kg/m; I₀—initial cross-sectional moment of inertia, m⁴; ρ—material density, kg/m³; b, h₀—width and thickness of the blade, respectively, m; and α—tangent of the change in blade’s thickness Δh, m, through the length.

To evaluate the eigenfrequencies of the blade’s oscillations for the complex mode shape:

$$W_0\left(z,t\right)=U_k\left(z\right)e^{i{\omega }_{k}t},$$

(3)

the initial equation should be rewritten to set the right part to zero and without external damping:

$$\frac{d^2}{d{z}^2}\left\{EI\left(z\right)\frac{d^2}{d{z}^2}\left[U_k\left(z\right)e^{i{\omega }_{k}t}\right]\right\}+\mu \left(z\right)\frac{d^2}{d{t}^2}\left[U_k\left(z\right)e^{i{\omega }_{k}t}\right]=0,$$

(4)

where k—number of the eigenfrequency ω_k, rad/s (k ∊ 1, 2, …) and U_k(z)—k-th dimensionless mode shape.

After equal transformations, the following fourth-order differential equation is obtained:

$$\frac{d^2}{d{z}^2}\left[EI\left(z\right)\frac{d^2{U}_k\left(z\right)}{d{z}^2}\right]={\omega }_k^2 \mu \left(z\right)U_k\left(z\right).$$

(5)

This expression will allow further significant reductions in the complexity of the model of forced oscillations. According to the theory of plates and shells [26], mainly using the Galerkin variational approach [27], it also allows evaluating the mode shape using Equation (6):

$$\delta \underset{0}{\overset{L}{{\displaystyle \int }}}\left\{\frac{d^2}{d{z}^2}\left[EI\left(z\right)\frac{d^2{U}_k\left(z\right)}{d{z}^2}\right]-{\omega }_k^2 \mu \left(z\right)U_k\left(z\right)\right\}{U}_k\left(z\right)dz=0.$$

(6)

Within the variational approach, the following mode shape U_k(z) and the corresponding kinematic boundary conditions are considered:

$$U_k\left(z\right)={\left(\frac{z}{L}\right)}^{s_k};U_k\left(0\right)=0;{\left[\frac{d{U}_k\left(z\right)}{dz}\right]}_0=0,$$

(7)

where s_k—power index for the k-th eigenfrequency.

After the substitution of this mode shape in the variational equation, the following higher-order equation is obtained to evaluate the unknown parameter s_k:

$$R\left(s_k\right)=\underset{0}{\overset{1}{{\displaystyle \int }}}\left\{\frac{d^2}{d{\zeta }^2}\left[{\left(1-\alpha \zeta \right)}^3\frac{d^2}{d{\zeta }^2}\left({\zeta }^{s_k}\right)\right]-{\gamma }_k^2\left(1-\alpha \zeta \right){\zeta }^{s_k}\right\}{\zeta }^{s_k}dz\to min,$$

(8)

where R(s_k)—target function; ζ—dimensionless coordinate; and γ_k—the dimensionless k-th eigenfrequency:

$$\zeta =\frac{z}{L}; {\gamma }_k=\frac{{\omega }_k}{{\omega }_e}; {\omega }_e=\frac{1}{L^2}\sqrt{\frac{E{I}_0}{{\mu }_0}},$$

(9)

and ω_e—equivalent frequency, rad/s.

The following infinite decomposition is considered to obtain dynamic characteristics of forced oscillations:

$$W\left(z,t\right)={\displaystyle \sum }_{k=1}^{\infty }{C}_k{U}_k\left(z\right)e^{i\omega t},$$

(10)

where C_k—amplitudes, m, as weight factors, which characterize the superposition of deflected modes by each mode shape.

By considering this expression and Formula (5), Equation (1) after equal transformations takes the following form:

$${\displaystyle \sum }_{k=1}^{\infty }{C}_k\left[\left({\omega }_k^2-{\omega }^2\right)\mu \left(z\right)+i\omega \beta \left(z\right)\right]U_k\left(z\right)=f_0.$$

(11)

Additionally, due to the normalizing identities [28]:

$$\underset{0}{\overset{L}{{\displaystyle \int }}}\mu \left(z\right)U_k\left(z\right)U_p\left(z\right)dz=0; \underset{0}{\overset{L}{{\displaystyle \int }}}\beta \left(z\right)U_k\left(z\right)U_p\left(z\right)dz=0,$$

(12)

for each numbers p ≠ k, the previous equation is equally modified:

$$C_p\left({\omega }_k^2-{\omega }^2\right)\underset{0}{\overset{L}{{\displaystyle \int }}}\mu \left(z\right)U_k\left(z\right)U_p^2\left(z\right)dz+i\omega C_p\underset{0}{\overset{L}{{\displaystyle \int }}}\beta \left(z\right)U_p^2\left(z\right)dz=f_0\underset{0}{\overset{L}{{\displaystyle \int }}}{U}_p\left(z\right)dz.$$

(13)

Finally, after considering Expressions (2) and (7) and the constant damping factor β(z) = β₀, the k-th complex amplitude can be obtained as follows:

$$C_k\left(\omega \right)=\frac{{\displaystyle \frac{2s_k+1}{s_k+1}}{f}_0}{\left[1-{\displaystyle \frac{2s_k+1}{2\left(s_k+1\right)}}\alpha \right]\left({\omega }_k^2-{\omega }^2\right){\mu }_0+i\omega {\beta }_0}=\left|C_k\left(\omega \right)\right|e^{i\phi },$$

(14)

where |C_k(ω)|—the magnitude of the complex amplitude as an amplitude frequency response and m; φ—phase shift, rad:

$$\left|C_k\right|=\frac{{\displaystyle \frac{2s_k+1}{s_k+1}}{f}_0}{\sqrt{{\left\{\left[1-{\displaystyle \frac{2s_k+1}{2\left(s_k+1\right)}}\alpha \right]\left({\omega }_k^2-{\omega }^2\right){\mu }_0\right\}}^2+{\left(\omega {\beta }_0\right)}^2}}; \phi =atan\left\{\frac{\omega {\beta }_0}{\left[1-{\displaystyle \frac{2s_k+1}{2\left(s_k+1\right)}}\alpha \right]\left({\omega }_k^2-{\omega }^2\right){\mu }_0}\right\}.$$

(15)

The amplitude of the specific cutting force, f₀, can be evaluated at operating speed ω₀ (where the amplitude equals A₀, m) from the condition |C_k| = A₀. In this case, if the operating speed is much less than the first eigenfrequency (ω << ω₁), that is based on the assumptions that operating speed and number of teeth are within the common range for steel finishing operations in the conventional and high speed modes for steam and gas turbine blades up to 100 mm long. Then, the cutting force is as follows:

$$f_0=\frac{s_k+1}{2s_k+1}\left[1-\frac{2s_k+1}{2\left(s_k+1\right)}\alpha \right]\left({\omega }_1^2-{\omega }_0^2\right){\mu }_0{A}_0.$$

(16)

Additionally, in the case of significant external damping in the considered system, the specific damping factor, β₀, can be evaluated by the highest amplitude, A_max, m. In this case, ω → ω₁ and |C_k| = A_max. Therefore, the damping factor is as follows:

$${\beta }_0=\left[1-\frac{2s_k+1}{2\left(s_k+1\right)}\alpha \right]\frac{\left({\omega }_1^2-{\omega }_0^2\right){\mu }_0}{{\omega }_1}\frac{A_0}{A_{max}}.$$

(17)

Overall, Formulas (15)–(17) allow for predetermining the vibration state of the blade during its milling. The practical implementation and verification of the proposed methodology are presented in Section 3 and Section 4.

3. Results of Blade Milling Simulations, Harmonic Response Analyses, and Analytical Calculations

The following case study for the axial turbine blade (Figure 2) has been considered to implement the developed methodology. Particularly, the parameters of Formula (2) are as follows: blade length L, width b, and initial thickness h₀. Additionally, the thickness of the narrowest part is h₁, the thickness difference is Δh, and the tangent of the change in the blade’s thickness, α, is Δh/h₀ = 0.571.

The maximum tolerances for the dimensions of the working part of the gas turbine blade in the tangential direction to the base (Figure 2a), with the length of the working part up to 100 mm (inclusive), are in the range of a_root = ±0.2 mm in the first root control section and a_tip = ±0.3 mm in the last tip complete control section.

For variable profile thickness turbine blades, the profile shape distortion c_prof in the control section is assigned according to the most significant chord value. Limits of distortions of the trailing edge, δ_edge, thickness extend over a section of 0.1 chord length of b (Figure 2b).

The blade material is steel AISI 4340 with a material density of ρ = 7850 kg/m³ and a Young’s modulus of E = 2.1 × 10¹¹ Pa. The initial specific mass is μ₀ = ρbh₀ = 2.748 kg/m and the initial cross-sectional moment of inertia I₀ = bh₀³/12 = 2.233 × 10^–9 m⁴. According to Formula (9), the specific eigenfrequency is ω_e = 2666 rad/s. The machining speed is n₀ = 5593 rpm and the corresponding rotational speed is ω₀ = πn₀/30 = 586 rad/s. The end mill diameter is D_mill = 8 × 10^–3 m; the number of flutes is n_f = 4; the axial depth of cut is a_p = 2 × 10^–3 m; the radial depth of cut is a_e = 0.3 × 10^–3 m; the feed is V_f = 129 × 10^–3 m/s; and the feed per tooth is f_z = 0.346 × 10⁻³ m.

Numerical simulations were performed using the Abaqus/Explicit for machining operation and ANSYS software for harmonic response analysis. Two case studies were considered. The first case study considers the application of a conventional machining strategy (Figure 3a). The second one considers a strategy with an overlap of inter-line scallops at a 30° angle to the surface (Figure 3b).

The fracture behavior of the AISI 4340 steel is based on plasticity and damage empirical constants that are predicted using the empirical Johnson–Cook model [29].

In order to perform a harmonic response analysis, the force component, F_y, which has a major cyclic impact as an excitation force of the working part of the turbine blade vibration in the profile machining, was determined and is included in

Table 1. Output values of peak force component, F_y, and stress according to the machining time and selected strategy.

Strategy	Indicator	Time, ms
		0.6	3.0	5.0	8.0	11.0	14.0
Conventional	Fy, N	101	831	463	658	508	605
	Stress, MPa	548	551	680	703	711	747
Overlapping	Fy, N	55.0	153	296	424	541	466
	Stress, MPa	348	440	506	509	515	523

.

After considering the simulation results, the peak force values are lower in the case of an overlapping strategy. This is explained by the remaining geometry after the previous pass and an additional 30° inclination angle in the case of that strategy, which all leads to lower stress and a different material removal rate.

The eigenfrequency of the turbine blade is an essential factor in establishing the resonance mode. If the cyclic load frequency impact matches the eigenfrequency of the turbine blade, resonance mode occurs, and the amplitude of the vibrations increases. The dominant oscillation generator is a milling cutter in the mechanical milling system. Each time the cutter cuts into the material, the low stiffness working part of the turbine blade loses its stability. As a result of modal numerical simulations, the eigenfrequencies are obtained and presented in Figure 4.

Figure 5 shows that the smallest cumulative mass fraction in the excitation of vibrations at obtained eigenfrequencies is required in the Y direction. This fact corresponds to the direction of the minimum stiffness of the turbine blade. It also confirms the reliability of the chosen simplified design scheme presented in Figure 1.

The milling cutter and turbine blade interaction can be considered as a mechanical system, with a response to a dynamic force similar to the following: the response can converge, the systems can oscillate constantly, and the response of the system can diverge. The third type of response should be avoided, since the system could be distracted by a resonance.

Harmonic response analysis allows for predicting the stable dynamic functioning of structures, checking whether the structures will successfully overcome resonance mode and consequently fatigue failure, non-compliance with technical requirements, and other undesirable effects of the forced vibrations.

The harmonic response analysis setup for ANSYS for each strategy is as follows: previously defined modal values are used, the fixed support is the fir tree surfaces of the turbine blade 3D model, the analyzed frequency is in the range from 0 to 2760 Hz, and the input nodal force load is placed at the critical point at the free end of the blade. The frequency response is taken from the same point as where the force is applied.

As a result of harmonic response numerical simulations, the experimental data of the dynamics of high-speed milling for turbine blades in two machining strategies are summarized in

Table 2. Experimental result data for two different machining strategies.

Frequency, Hz	Amplitude, mm
	1	2
120	0.167	0.098
240	0.170	0.101
360	0.174	0.104
480	0.180	0.107
600	0.189	0.113
720	0.202	0.120
840	0.218	0.130
960	0.241	0.144
1080	0.374	0.163
1200	0.324	0.193
1320	0.404	0.240
1440	0.554	0.330
1560	0.926	0.551
1680 *	2.837 *	1.689 *
1800	1.542	0.918
1920	0.649	0.387
2040	0.398	0.237
2160	0.281	0.168
2280	0.215	0.128
2400	0.172	0.102
2520	0.142	0.084
2640	0.120	0.071
2760	0.103	0.061

* Resonance mode.

.

The maximum amplitude of forced oscillations occurs at the first critical frequency f₁ = 1680 Hz, which is the resonance mode. It is expected that, at this frequency, the working part of the turbine blade oscillates in the direction of its minimum stiffness during the machining of the outer and inner profiles. The critical frequency is ω₁ = 2πf₁ = 10,560 rad/s.

According to the variational approach, the power index, s_k, of Formula (7) for k = 1 should be evaluated by minimizing the target function (8). The corresponding result is presented in Figure 6.

The value s₁ = 6.05 minimizes the target function, R(s₁). Therefore, the mode shape (7) in dimensionless coordinates, ζ = z/L = 14.286·z, can be set as U₁(ζ) = ζ^6.05.

According to Formula (16), the evaluated specific force, f₀, is equal to 12.8 kN/m for the 1st scenario and 7.7 kN/m for the 2nd scenario. Additionally, according to Expression (17), the evaluated value of the specific damping factor, β₀, is in the range of (0.8 ± 0.02) kN·s/m² for both scenarios.

The normalized root mean square error (NRMSE) is used measure of the discrepancies between amplitudes A_m,i predicted by a model and the amplitudes A_e,i obtained from experiment. The NRMSE facilitates the comparison between strategies and is obtained as follows:

$$NRMSE=\frac{\sqrt{{{\displaystyle \sum }}_{i=1}^n{\left(A_{e,i}-A_{m,i}\right)}^2/n }}{{\overline{A}}_{e,i}},$$

(18)

where i—the number of data points; n—the total number of points; and ${\overline{A}}_{e,i}$—the average value of experimentally obtained amplitudes, mm.

The overall amplitude–frequency responses, obtained by Formula (15) for the considered machining modes, are presented in Figure 7. Notably, for both scenarios, the NRMSE is less than 0.1.

For the operating frequency of 372.9 Hz, the amplitudes of the forced oscillations are in the range between 0.174 mm and 0.104 mm during the axial turbine blade machining, applying conventional and overlapping strategies, respectively. The amplitudes do not exceed the acceptable accuracy level of ±0.24 mm for the turbine blade profile shape in the control section. The control section is set in the region of the largest value of chord b (Figure 2b).

4. Discussion

The relatively low value of the normalized root mean square, which for both case studies does not exceed 0.1, proves the reliability of the developed approach. The acceptable range of 0.104–0.174 mm for the amplitude of forced oscillations also proves the vibration reliability of the case studies for machining turbine blades.

By using a conventional strategy, an accuracy level of 0.25 mm (from −0.01 mm to +0.15 mm) for the trailing edge can theoretically be exceeded by 0.02 mm. This fact is explained by the fact that during the study, the speed and feed and consequently the frequency of processing were constant along the entire blade profile, which implies the application of variable operating conditions depending on the end mill cutter position relative to the turbine blade profile.

Undesirable vibrations, which lead to large oscillation amplitudes or even resonance modes, can be bypassed by additionally developed turbine blade dumping fixtures, variation of the milling process conditions, milling cutter parameters, etc.

5. Conclusions

Mathematical models of free and forced oscillations of turbine blades during machining were developed. Critical frequencies and corresponding mode shapes of free oscillations were identified using regression procedures.

In order to verify the models, numerical simulations have been performed for machining operations and harmonic response analysis for the two case studies, i.e., conventional and overlapping strategies. The peak values of the F_y force component, stress, eigenfrequencies, and vibration amplitudes were obtained, confirming the mathematical models’ adequacy.

An operating frequency of 372.9 Hz was defined. The forced oscillations range was between 0.174 mm and 0.104 mm during the axial turbine blade machining when applying conventional and overlapping strategies, respectively, which does not exceed the acceptable accuracy level of ±0.24 mm for the turbine blade profile shape in the in the region of the largest value of the chord.