Dynamics of Blade With Geometrical Nonlinearity
of Nonstationary Vibrations, A.N. Podgorny Institute for Mechanical Engineering
The compressor blade is described by pre-twisted, variable thickness, double curvature shallow shell with geometrical nonlinearity. The R-function method and the Rayleigh- Ritz approach are used collectively to calculate eigenmodes of linear vibrations. Nonlinear vibrations of shell are approximated by using these eigenmodes. The obtained finite degrees-of-freedom system has three internal resonances. The free nonlinear vibrations are studied by nonlinear modes.
Keywords: shallow shells, finite degree-of-freedom system, internal resonances, nonlinear normal modes
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Significant dynamics loads act on the blades of turbo machine during their operation. Therefore, blades can perform vibrations with amplitudes commensurable with their thickness. Such blade deformations are described by geometrical nonlinear theory. Many efforts have been done to study blade vibrations. Venkatsan, Nagaraj (1982) studied nonlinear vibrations of rotating blades. They came to the conclusions, that the frequency response can be hard or soft. The vibrations of turbine blades under the action of longitudinal periodic force have been considered by Chen, Peng (1995). Using geometrically nonlinear theory and finite element method, the blade nonlinear model is obtained. The vibrations of shallow anisotropic blades are treated by Abe et al. (2000). They used the Rayleigh-Ritz method to analyze linear vibrations. Nonlinear vibrations are expanded into two linear modes. The finite degrees-of- freedom model is obtained by the Galerkin procedure. Liew, Lim (1996) used energetic approach to study linear vibrations of shallow shells with different Gaussian curvature and rectangular base. Nonlinear vibrations of hydraulic turbine blades, which are modeled by pre-twisted shell with variable thickness and ring sector-shaped base, are treated by Hu, Tsuiji (1999), Sakiyama et al. (2002). The dependence of eigenfrequencies and eigenmodes on the pre-twisted angle and thickness are investigated. The papers of Hoa (1981) and Ross (1975) are devoted to linear vibrations analysis of blades, which are modeled by shells. The results of finite element analysis of turbo machine blades are compared with the experimental data of Mindle and Torvik (1987). Didkovskii (1974) analyzed the parametric vibrations of turbo machine blades in gas flow. The sufficient conditions of dynamical stability are obtained. Nabi and Ganesan (1996) compared the beam and plate models of turbo machine blades. They came to the conclusion that the plate models are better. Pisarenko, Vorob’ev (2000) concluded that the theory of pre-twisted beams gives the adequate eigenfrequencies of blade vibrations. However, the more accurate models should be used for detailed stresses analysis. Choi, Chou (2001) analyze the blade vibrations with account of shear. The influence of shroud on vibrations is considered. Pak et al. (1992) modeled the turbo machine blades by thin elastic beams. Flexural, torsion and flexural-torsion vibrations are analyzed in detail. Breslavsky et al. (2011) studied nonlinear multi-mode vibrations of hydraulic turbine blade, both in vacuo and immersed in the fluid. The blade is modeled by shallow shell with ring sector base.
nonlinear vibrations of compressor blades are treated in this paper.
2. The problem formulation
The vibrations of the compressor blades (Figure 1) are considered. The blades are modeled by pre-twisted shallow shell with trapezoidal base and variable thickness. The sketch of the shell base is presented in Figure 2. The shell is clamped on one side and it is free on others three sides. The coordinate plane () passes through the chord of the root section and the central line of the blade. Moreover, the axis coincides with the central line of the blade. The geometry of the shell middle surface is complex, therefore, the shell is not considered in principal coordinates.
The compressor blade previously studied by Meerovich (1956, 1961) is treated. The shape of the untwisted shell middle surface can be described by the following function:
where is the distance between the middle surface and the plane xy in the root section; is the length of the root section chord; ; is the distance between the middle surface and the plane () on the peripheral edge; is the length of the blade. The thickness of the untwisted blade can be presented in the following analytical form (Meerovich 1961, 1956):
where is the maximal thickness of the root section; is the maximal thickness of the peripheral edge.
It is assumed, that the blade is pre-twisted uniformly and pre-twisted angle is denoted by . Then the thickness and the shape of middle surface can be presented as:
The boundary conditions on the clamped edge are the following:
where , are displacements of the middle surface points in x, y, z directions, respectively.
As thin shell is considered, the shear is not taken into account. Then the potential energy of the shell can be presented as (Grigoluk, Kabanov, 1978; Goldenveizer et al. 1979):
where are the Young’s modulus and the Poisson’s ratio of the shell material. The parameters of shell deformations are determined in the following form (Grigoluk, Kabanov, 1978):
; ; ; (3)
; ; ;
; ; ,
where are linear components of membrane strains of shell middle surface; are components of the bending deformations of the middle surface; and are rotation angles about the tangent vectors to the lines x and y, respectively; is angle of rotation about normal to the middle surface. Curvatures of the middle surface are determined as:
; ; ,
where , are coefficients of the first and the second quadratic form of the middle surface.
The kinetic energy of the shell has the following form (Amabili, 2008):
where is the shell material density.
3. Linear vibrations analysis
In this section the linear vibrations of the shell are considered and nonlinear terms are not taken into account in the expressions (3). The Rayleigh-Ritz method is used to determine eigenfrequencies and eigenmodes of shell, so only the kinematic boundary conditions are taken into account. The boundary conditions on the shell free edges are natural. Therefore, they are not taken into account in linear analysis.
The periodic vibrations of the shell are presented in the following form:
R-function (Rvachev, Sheiko, 1995) is used to satisfy the boundary conditions (1).
R-functions are very useful for shallow shells with complex base analysis (
The function can be derived for arbitrary domains with analytical boundaries. The methods for these functions construction are considered by Rvachev, Sheiko (1995).
According to the R-function theory, the clamped part of the shell boundary can be presented by the following function:
where is the operation of R-conjunction. The functions and describe axis and the domain , respectively. These functions can be presented in the following form:
The steps of the function construction are shown in Figure 3.
Following the Rayleigh-Ritz method, the vibrations modes are presented as:
where , , are trial functions. The expressions (5) satisfy boundary conditions (1). The following trial functions are used in this paper:
where describes the trapezoidal base of the shallow shell.
The expressions (5) are substituted into the Lagrange functional and the integration is carried out. Then the functional is obtained as quadratic form of the parameters . In general, can be presented in the following form:
The minimum of the Lagrange functional is determined from the following equations: ;. These equations can be presented as the eigenvalue problem:
4. Nonlinear finite degrees-of-freedom model of vibrations
Nonlinear vibrations of the blade are approximated by the eigenmodes of linear vibrations. Then the shell bending vibrations can be presented as:
where are normalized eigenmodes. The displacements and are presented as:
where and are eigenmodes of in-plane vibrations. Now the equations (9, 10) are substituted into the kinetic and potential energy (2, 4) and the integration is carried out. As a result, the potential and kinetic energies are obtained as functions of the generalized coordinates and the generalized velocities in the following form:
Using the kinetic and potential energy, the Lagrange equations with respect to the generalized coordinates are derived.
Due to the pre-twist of the shell, the displacements orthogonal to the blade middle surface consist of both displacements and . The displacements have moderate amplitudes. Therefore, the inertial terms only in the direction are not taken into account. Then the system of ordinary differential Lagrange equations is transformed into the system of linear algebraic equations:
and the system of ordinary differential equations. The solutions of linear algebraic equations are substituted into the system of ordinary differential equations. The following dimensionless variables and parameters are used in future analysis:
, , , (11)
where is the first (minimal) eigenfrequency. Then the finite degrees-of-freedom dynamical system with respect to dimensionless generalized coordinates and parameters has the following form:
where are known system parameters. The matrix can be presented as
where is matrix of dimensionless eigenfrequencies. By means of the coordinate transformations
the system (12) is rewritten with respect to the modal coordinates:
5. Nonlinear normal modes of vibrations
Shaw-Pierre nonlinear normal modes are used to analyze the dynamical system
(13) (Shaw, Pierre, 1993; Jiang et al. 2005; Avramov, 2008;
Due to these internal resonances, the shell motions have three active generalized coordinates . Then the invariant manifold of the system can be presented as (Jiang et al. 2005):
This invariant manifold satisfies the following partial differential equations:
The functions can be presented as truncated power series as:
where , , are unknown coefficients. As the system (13) is presented with respect to the modal coordinates, the coefficients of summands (17) with degrees lower than two are equal to zero (Jiang et al. 2005). The solutions (17) are substituted into (16) and the summands with are equated. As a result, the set of the systems of linear algebraic equations is obtained. The first system is linear with respect to the coefficients of the second degrees (i.e. the summands with ) and the second system is linear with respect to the coefficients of the third degrees (i.e. summands with ). The coefficients of the series (17) are obtained by solving sequentially these systems of linear algebraic equations. Thus, the nonlinear mode (15) can be obtained.
The equations (17) are substituted into the third, fourth and fifth equations of the system (13). As a result, the system of three ordinary differential equations describing the motions on nonlinear modes is obtained in the following form:
, . (18)
The harmonic balance method is used to study the motions on the nonlinear mode (18). The solutions of the system (18) are presented as:
, . (19)
Substituting the expressions (19) into the equations (18), the system of nonlinear algebraic equations with respect to ; is derived. This system is solved by the continuation technique (Seydel, 1997).
To analyze stability of periodic motions, the system (13) is rewritten in the following vector form:
where is phase vector, is vector function of . The stability of periodic motion is analyzed. Following the book (Yakubovich, Starzhinskii, 1975), small perturbation is added to periodic motions. The evolution of small perturbations in time is described by the following system of ordinary differential equations:
where is the Jacobi matrix.
Following the book (Parker, Chua, 1989), the fundamental matrix of the system (21) is calculated to analyze stability numerically. This matrix is a solution of the system (21) with initial conditions in the form of the identity matrix . The fundamental matrix at (where is vibrations period) is called monodromy matrix (Yakubovich, Starzhinskii, 1975). The conclusions about the stability of periodic motions are made according to the eigenvalues of monodromy matrix.
6. Numerical results
In order to perform numerical analysis the steel compressor blade is considered with the following parameters (Meerovich, 1961): m, m, m, , m, , , N/m2, , kg/m3.
The calculations of linear vibrations are carried out by the Rayleigh- Ritz method. The results of the eigenfrequency calculations are presented in the first row of Table 1. The experimental and numerical data of the previous research, which are obtained by Meerovich (1961, 1956), are presented in the second and the third row of Table 1.
Figure 4 shows the nodal lines of the first six eigenmodes of the blade vibrations. These results are close to the data from (Meerovich, 1961, 1956). However, the eigenfrequency, which is close to the fourth one, is leaved out in (Meerovich, 1961).
first five eigenmodes are used in the expansions (9, 10) to obtain the finite
degrees-of-freedom model of the blade nonlinear vibrations. Excluding the
generalized coordinates , the system with ten degrees-of-freedom in the form of (13)
is derived. The
The vibrations with dominant generalized coordinate , which is excited due to the internal resonance (14), are analyzed. In this case, the generalized coordinate is close to zero. The backbone curve of these motions is presented in Figure 5, which shows the vibrations amplitudes versus . The stable and unstable motions are shown on this figure by solid and dotted lines, respectively. The eigenfrequencies of linear vibrations are shown by vertical lines in Figure 5. The free vibrations (Figure 5) are stable until the amplitude of is small. If the frequency is increased, the vibrations amplitude decrease quickly. Note that the vibrations and have the dominant harmonics with the frequency . As follows from Figure 5, the free nonlinear vibrations become unstable at certain values of the vibration amplitudes. The frequencies of nonlinear vibrations (Figure 5) are not close to the frequencies of linear ones. Therefore, the geometrical nonlinearity affects essentially on the shell vibrations.
the vibrations with dominant generalized coordinate are analyzed. Such
motions are denoted by the number
Now another type of motions on the nonlinear mode (15) is considered. In this case the vibrations have significant amplitudes; the rest generalized coordinates are not excited. These vibrations are shown in Figure 7. The free nonlinear vibrations become unstable at some value of vibrations amplitudes.
The harmonic balance method is used directly to the system of nonlinear ordinary differential equations (13) to obtain the motions, which differ from the vibrations on the considered nonlinear normal mode. As a result, additional type of periodic motions is obtained. The generalized coordinate in this regime is active too. These vibrations are shown by the branch 2 on Figure 6. This branch describes the periodic motions, which split off the motions 1 due to the period doubling bifurcation.
In order to verify the obtained results the direct numerical integration of the system (13) is carried out. The initial conditions are obtained from the solution (19). The results are shown in Figure 5 and Figure 7 by dots. The data of the direct numerical integration are very close to the results of the normal mode analysis. So, the method suggested in the previous section adequately describes the system vibrations.
Now the projections of the middle surface points vibrations on the normal to this surface are treated. The corner points with the coordinates and have maximal displacements. However, the shell in these points is very thin. Therefore, the displacements of the shell points at small distance from the corner points are considered. Due to pre-twist of the blade, the displacements in the orthogonal direction to the middle surface contain both and . These displacements can be presented as
The backbone curves are shown in Figure 8, where the vibrations amplitudes versus the frequency are plotted. Figure 8 shows the backbone curves of the different vibration modes. The horizontal line shows the thickness of the shell in the point, where the vibrations are determined. The motions with dominant generalized coordinate and with dominant coordinate are shown on Figure 8a and Figure 8b, respectively.
The compressor blade is described by cantilever, pre-twisted, double curved, variable thickness shallow shell with trapezoidal base. At the first phase of the analysis the eigenfrequencies and eigenmodes are determined. The three internal resonances between the third, the fourth and the fifth vibrations eigenfrequencies are observed in the system. The different modes of the compressor blade nonlinear vibrations are described by hard backbone curves. It is shown that the amplitudes of stable vibrations of peripheral edge are commeasurable with the blade thickness in this region.
The free nonlinear vibrations become unstable at certain values of the vibrations amplitudes. Moreover, the frequencies of nonlinear vibrations are not close to the frequencies of linear ones. Therefore, geometrical nonlinearity affects essentially on the shell vibrations.
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Fig. 1. Sketch of the blade
Fig. 2. Sketch of the shell base
Fig.3. The steps of the construction of the function , which describes the clamped part of the shell boundary. The regions, where the function is positive, are shown by grey color. The black lines show the zero of the function.
a. b. c. d. e. f.
Fig.4. The nodal lines of eigenmodes corresponding to the following eigenfrequencies in Hz:
a. b. c. d.
Fig. 5. The backbone curves of the vibrations with dominant general coordinate
Fig. 6. The backbone curves of the free unstable nonlinear vibrations
Fig. 7. The vibrations of the shell with dominant general coordinate
Fig.8. The backbone curves of shell vibrations with dominant general coordinates: a. ; b. .
Table 1. Eigenfrequencies of vibrations
Eigenfrequencies in Hz
The results of the calculations
The results of previous calculations
The experimental data (Meerovich, 1961)