Introduction. Natural vibration frequency is the physical property of structures, which is related to their quality, stiffness, and shape [1]. Cracks affect the structure stiffness, which in turn changes the natural vibration frequency of the structure. As key components of aero-engines, the working environment of rotor blades is severe, in which cracks are often initiated and propagated by vibration. Cracks will change the natural vibration frequency of blades. Sympathetic vibration may occur as soon as the frequency of the external exciting force becomes equal or close to the natural vibration frequency, which promotes rapid growth of cracks, leading to blade fracture, affecting engine performance, and even causing major accidents of aircraft failure. In order to prevent blade fracture, it is necessary to study the vibration performance of blades and take measures to avoid sympathetic vibration. On the other hand, it is necessary to study the influence of cracks on blade vibration frequency, in order to diagnose blade fracture failure in time.

For blades without cracks, the vibration characteristics have been not only analyzed by finite element method, but also have been studied deeply and extensively, according to some specific factors such as fixed method [2, 3], ambient temperature [4], load type [5], and so on. According to the fracture mechanics and the plane stress theory, the calculation method of cross-sectional equivalent bending stiffness of cracked blades was established using the beam model in [6]. In [7], the influence of cracks on blade vibration modes was studied using the Galerkin method. In [8], the characteristics of free and forced vibrations of cracked cantilever beams were studied using the finite element method. However, quite a few studies have been conducted on the variation regulation of blade natural frequencies with different crack locations and lengths.

In this paper, the natural vibration frequency of the equivalent section of TC4 titanium alloy plate blade is studied to find out the effect of cracking on the natural vibration frequency of blade. Under conditions of free or fixed blade roots, the first-order natural vibration frequency of the blade with different crack lengths in the same location or the same crack length in different locations is obtained by the finite element calculation. The theoretical guidance is provided for the diagnosis of blade crack failures using the variation of vibration frequency to predict crack location or crack length.

1. Theoretical Details of Vibration Differential Equation. The vibration frequency of a system can be obtained by solving the dynamic differential equation of the system [9]

$$ M\ddot{X}(t)+{M}_G\dot{X}(t)+ KX(t)=F, $$
(1)

where X (t ), \( \dot{X}(t) \), and \( \ddot{X}(t) \) are the nodal displacement vector, velocity vector, and acceleration vector, respectively, M, MG , and K are the mass, damping matrix, and stiffness matrices, while F is the external force of the system.

The natural vibration frequency is frequency of the system with free vibration. In this case, the external force is zero, and the damping of the system can be neglected. Then, the above linear differential equation can be reduced to the following homogeneous equation:

$$ M\ddot{X}(t)+ KX(t)=0. $$
(2)

When the system is fixed on boundary S, the differential equation have zero displacement boundary

$$ {X}_S=0. $$
(3)

The mass matrix of the system is determined by the spatial distribution of mass, i.e., the spatial shape and material density of the system. The stiffness matrix is mainly related to the spatial shape and elastic modulus of the system. For the structure with cracks, the shape of the vibration process will be different from that without cracks. Under this condition, the mass M and stiffness K matrices will be changed. Accordingly, the natural vibration frequency of the system will also be changed.

2. Computational Model of Natural Vibration Frequency of Plate Blades with Crack. The research object of this paper is TC4 titanium alloy plate blade with a uniform cross section of thickness d = 8 mm, width b = 80 mm, and length l =160 mm, as shown in Fig. 1. The material properties at room temperature are as follows: elastic modulus E = 1 12 ·1011. N/m2, mass density ρ = 4 44 ·103. kg/m3, and Poisson’s ratio μ = 0.33.

Fig. 1.
figure 1

Schematic diagram of the blade and crack location and crack length.

Full size image

It is assumed that cracks with lengths a of 10, 20, 30, 40, 50, 60, and 70 mm are located at the distance (crack height l0) of 20, 40, 60, 80, 100, 120, and 140 mm from the root of the blade, respectively. The natural vibration frequencies of blades with and without cracks are calculated under conditions of complete freedom and fixed blade root, respectively. Considering that the cracks in the blade are completely closed in the natural state, the crack width is assumed to be zero.

The finite element simulation software ANSYS 14.5 [10] was used in the present calculation. The 20-node 3-D isoparametric element SOLID186 was selected, and the unit size was controlled within 1 mm. A sweeping method was used to partition the blades, since the shape of the model was regular, and regular mesh generation was feasible. After meshing, the whole blade contained 102,400 single elements and 454,257 nodes, as shown in Fig. 2. It should be noted that the stress at the crack tip is singular. During the stress analysis, singular elements must be used at the crack tip. The effect of the singular stress at the crack tip on the natural vibration frequency is minimal, so the conventional element can be used in this analysis [11].

Fig. 2.
figure 2

The finite element model of blade root fixed.

Full size image

3. Calculation Results and Discussion. According to the finite element model mentioned above, the first-order natural vibration frequencies can be obtained for the blades without cracks and blades with different cracks of lengths at different locations under complete freedom and fixed blade root, respectively. The results for free blades and fixed blade root are shown in Tables 1 and 2, respectively. In these tables, a is the length of the crack, l0 is the height of the crack, f is the natural frequency of the blade, and δ is the percentage of frequency reduction, which can be determined by the following equation:

$$ \updelta =\frac{f-{f}_0}{f_0}, $$
(4)
Table 1. First-Order Natural Vibration Frequencies and Its Relative Variation in the Free State
Full size table
Table 2. First-Order Natural Vibration Frequencies and Its Relative Variation in Blade Root Fixed State
Full size table

where f0 is the natural frequency of the blade without cracks.

According to the data in Tables 1 and 2, the curves of natural vibration frequency varying with crack length and crack height are drawn in Figs. 3–6.

Fig. 3.
figure 3

Curve of natural vibration frequency with crack length in the free state.

Full size image
Fig. 4.
figure 4

Curve of natural vibration frequency with crack height in the free state.

Full size image
Fig. 5.
figure 5

Curve of natural vibration frequency with crack length in root fixed state.

Full size image
Fig. 6.
figure 6

Curve of natural vibration frequency with crack height in root fixed state.

Full size image

From Tables 1 and 2 and Figs. 3–6, it can be seen that the natural vibration frequencies of blades vary with changes in the crack location, crack length, and fixing mode. That is to say, once the system changes, the natural vibration frequency will change, which verifies the concept that the natural vibration frequency is the natural physical property of the system itself.

As can be seen in Figs. 3 and 5, curves of the natural frequency with different crack heights will meet at one point when the crack lengths approach zero. Because the blade tends to be in intact condition, the natural frequency must tend to the value of intact blade. As can be seen in Figs. 4 and 6, curves of the natural vibration frequency with different crack locations will also meet at one point when crack location approaches the blade tip in root fixed state, or approaches both ends in the free state. Because the crack location tends to the free end, it also means that the blade tends to the intact condition, and the natural vibration frequency will inevitably tend to the value of the intact blade.

In the same location, when the crack length increases, the natural vibration frequency decreases gradually. The degree of decrease in the natural frequency is gradually increasing compared to the intact state. It shows that the longer the crack, the greater the influence on the stiffness matrix and, thus, the greater the impact on the natural vibration frequency of the blade. The crack presence changes the stiffness of the blade, thus affecting the natural vibration frequency. When the location and length of the crack are determined, the degree of stiffness variation will be known, as well as changes in the natural frequency. Therefore, if the crack length or its location are known, the corresponding crack position or its length can be inferred by measuring the natural vibration frequency and calculating its relative variation rate using the calculation results in Tables 1 or 2.

In the root fixed state, the closer the crack is to the leaf root, the greater the effect on the natural vibration frequency. In this case, the first-order bending vibration is only one vibration node line located in the blade root. The closer to the vibration node line, the greater the influence of the crack on the stiffness matrix. For example, when the crack is in seven eighths of a blade height (l0 = 140 mm), it has a weak effect on the natural vibration frequency. The natural frequency just drops by 2.12% when the crack length reaches 70 mm. Comparatively, when the crack is in one eighths of a blade height (l0 = 20 mm), the natural vibration frequency drops by 50.1% when the crack length reaches 70 mm. The rotor blades in the actual engine can be approximated as fixed blade root, so the location or length of the crack can be judged according to this rule.

In the free state, cracks located in the middle half of the blade body have a greater impact on the natural frequency, while cracks located in the upper and lower quarters of the blade body have less effect on the natural frequency. And the closer to the free end, the less impact. This is because the two vibration nodal lines of the first-order bending vibration in the free state are located at one quarter and three quarters of the blade height. The influence of the crack on the stiffness matrix is related to the distances from the two lines.

When the crack is located between these two lines, the effect of the same length crack on the natural vibration frequency is basically the same. It shows that between two lines the crack position has less influence on stiffness. When the crack length is 70 mm, the degree of reduction of the natural vibration frequency is between 53.54 and 55.36% within one quarter and three quarters of the blade height, and the maximum difference is only 1.82%. At the height of one eighth of the blade, the natural frequency decreased by 48.26%, which differs by 7.1% from that in the blade middle.

In the free state, with an increase in the crack length in the middle of the blade, the influence on the natural vibration frequency increases rapidly. When the crack length reaches 70 mm, the natural vibration frequency is reduced by more than 50%. The crack effect on the natural frequency at the height of one-half blade in the free state is very close to that of the same length crack at the height of one-eighth blade in fixed state, indicating that the crack effect on the stiffness matrix is approximately equal in these two cases.

Conclusions. The calculation results show that the natural vibration frequency is related to quality, shape, and material of the structure. Any changes in the material, shape, and external constraints of the system will cause a change in the mass matrix, stiffness matrix, or boundary conditions, which will lead to the natural vibration frequency variation. Specifically, the main conclusions are as following:

  1. 1.

    Cracks reduce the blade stiffness and affect its natural vibration frequency. The longer the crack, the greater the change of stiffness matrix and, thus, the greater the impact on the natural vibration frequency of the blade. When the crack length tends to zero, or the crack position tends to the free end, the crack effect on the natural vibration frequency of the system gradually decreases until zero. Therefore, if the crack length or its location is known, the corresponding crack location or its length can be inferred by measuring the natural vibration frequency and calculating its relative variation rate.

  2. 2.

    In terms of the blade studied, when the crack is located between two vibration nodal lines of the first-order bending vibration in the free state, the effect of the same length crack on the natural vibration frequency is basically the same. It shows that the crack location in-between two lines has a weak effect on stiffness. When the crack length is 70 mm, the degree of reduction of the natural vibration frequency is between 53.54 and 55.36% within one quarter and three quarters of the blade height, and the maximum difference is only 1.82%.

  3. 3.

    The crack effect at the height of one-half blade in the free state on natural frequencies is very close to that of the same length crack at the height of one-eighth blade in blade root fixed state, indicating that the crack effect on stiffness matrix is approximately equal in these two cases.