Dynamics of Blade With Geometrical Nonlinearity

Department of Nonstationary Vibrations, A.N. Podgorny Institute for Mechanical Engineering Problems, National Academy of Sciences of Ukraine, 2/10 Dm. Pozharskoho St., 61046, Kharkiv, Ukraine

1e-mail: id.breslavsky@gmail.com

2e-mail: kvavr@kharkov.ua

Abstract:

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

*Corresponding author. Tel.: +38 057 3494783;  Fax: +38  0572 944635.
1.      Introduction

The nonlinear vibrations of compressor blades are treated in this paper. Nonlinear dynamics are expanded into eigenmodes of vibrations. The finite degrees-of-freedom model is obtained. Shaw-Pierre nonlinear normal modes are used to study the vibrations accounting internal resonances.

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:

,                                                                 (1)

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):

(2)

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):

,                                                                  (4)

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:

;

The 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 (Kurpa et al. 2007; Rvachev, Sheiko, 1995; Avramov et al. 2010, Breslavsky, Avramov). The R-function  satisfies the following conditions:

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:

;

;                                                                                (5)

,

where , ,  are trial functions. The expressions (5) satisfy boundary conditions (1). The following trial functions are used in this paper:

(6)

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:

.                                                                                                   (7)

The minimum of the Lagrange functional is determined from the following equations: ;. These equations can be presented as the eigenvalue problem:

(8)

where .

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:

,                                                                                    (9)

where  are normalized eigenmodes. The displacements  and  are presented as:

 , , (10)

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:

(12)

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:

(13)

5.      Nonlinear normal modes of vibrations

The Shaw-Pierre nonlinear normal modes are used to analyze the dynamical system (13) (Shaw, Pierre, 1993; Jiang et al. 2005; Avramov, 2008; Mikhlin, Avramov). As follows from the numerical results, which are treated in the next section, the dynamical system (13) has three internal resonances:

.                                                                           (14)

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):

(15)

This invariant manifold satisfies the following partial differential equations:

.

The functions  can be presented as truncated power series as:

 , , (17)

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:

,                                                                                                               (20)

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:

(21)

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).

The 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 nonlinear dynamics of this system is analyzed by nonlinear normal mode, using the results of the previous section. As a result of the nonlinear mode analysis, three-degree-of-freedom system (18) is derived. This system is analyzed by harmonic balance method.

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.

Now the vibrations with dominant generalized coordinate  are analyzed. Such motions are denoted by the number 1 in Figure 6. This figure shows the dependence of the vibrations amplitudes  on . All motions, which are shown on Figure 6, are unstable.

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.

7.      Conclusions

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.

References

Abe, A., Kobayashi, Y., and Yamada, G., 2000. Non-linear vibration characteristics of clamped laminated shallow shells. J. Sound Vib. 234, 405-426.

Amabili, M., 2008. Nonlinear vibrations and stability of shells and plates. Cambridge University Press, New York.

Avramov, K.V., 2008. Analysis of forced vibrations by nonlinear modes. Nonlinear Dyn. 53, 117-127.

Avramov, K.V., Tyshkovets, O., Maksymenko-Sheyko, K.V., 2010. Nonlinear dynamics of circular plates with cutouts. R-function method. ASME J. Vib. Acoust. 132, ¹5, 110-135.

Breslavsky, I., Avramov, K.V., Nonlinear modes of cylindrical panels with complex boundaries. R-function method. Mecc., in press.

Breslavsky, I.D., Strel’nikova, E.A., Avramov, K.V., 2011. Dynamics of shallow shells with geometrical nonlinearity interacting with fluid. Comput. Struct. 89, 496-506.

Chen, L.W., Peng, W.K., 1995. Dynamic stability of rotating blades with geometrical non-linearity. J. Sound Vib. 187, 421-433.

Choi, S.-T., Chou, Y.-T., 2001. Vibration analysis of elastically supported turbomachinery blades by the modified differential quadrature method. J. Sound Vib. 240, 937-953.

Didkovskii, V.N., 1974. Parametric vibrations of turbomachines in a flow. Strength Mater. 8, 9-13.

Goldenveizer, A.L., Lidskii, V.B., Tovstik, P.E., 1979. Free vibrations of thin elastic shells. Nauka, Moscow (in Russian).

Grigoluk, E.I., Kabanov, V.V., 1978. Stability of shells. Nauka, Moscow (in Russian).

Hoa, S.V., 1981. Vibration frequency of a curved blade with weighted edge. J. Sound Vib. 79, 107-119.

Hu, X.X., Tsuiji, T., 1999. Free vibration analysis of curved and twisted cylindrical thin panels. J. Sound Vib. 219, 63-68.

Jiang, D., Pierre, C., Shaw, S.W., 2005. The construction of non-linear modes for systems with internal resonance. Int. J. Non-Linear Mech. 40, 729-746.

Kurpa, L., Pilgun, G., Amabili, M., 2007. Nonlinear vibrations of shallow shells with complex boundary: R- functions methods and experiments. J. Sound Vib. 306, 580-600.

Liew, K.M., Lim, C.W., 1996. Vibration of doubly-curved shallow shells, Acta Mech. 114, 95-119.

Meerovich, I.I., 1956. Vibrations of slightly curved and pre-twisted blades. Trudy CIAM 271. Oborongiz, Moscow (in Russian).

Meerovich, I.I., 1961. Stress distribution in vibrating compressor blades. Oborongiz, Moscow (in Russian).

Mikhlin, Yu.V., Avramov, K.V., Nonlinear normal modes of mechanical systems. Part I: Theory. Appl. Mech. Rev., in press.

Mindle, W.L., Torvik, P.J., 1987. Multiple modes in the vibration of cantilevered shells. J. Sound Vib. 115, 289-301.

Mohamed Nabi, S., Ganesan, N., 1996. Comparison of beam and plate theories for free vibrations of metal matrix composite pre-twisted blades. J. Sound Vib. 189, 149-160.

Pak, C.H., Rand, R.H., Moon, F.C., 1992. Free vibrations of a thin elastica by normal modes. Nonlinear Dyn. 3, 347-364.

Parker T.S., Chua L.O., 1989. Practical numerical algorithms for chaotic systems. Springer-Verlag, New York.

Pisarenko, G.S., Vorob’ev, Yu.S., 2000. Issues of simulation of turbomachine blade vibration. Strength Mater. 32, 487-489.

Ross, C.T.F., 1975. Free vibration of thin shells. J. Sound Vib. 39, 337-344.

Rvachev, V.L., Sheiko, T.I., 1995. R-function in boundary value problem in mechanics. Appl. Mech. Rev. 48, 305-316.

Sakiyama, T., Hu, X.X., Matsuda, H., Morita, C., 2002. Vibration of twisted and curved cylindrical panels with variable thickness. J. Sound Vib. 254, 481-502.

Seydel, R., 1997. Nonlinear computation. Int. J. Bifurc. Chaos 7, 2105-2126.

Shaw, S.W., Pierre, C., 1993. Normal modes for nonlinear vibratory systems. J. Sound Vib. 164, 58-124.

Venkatsan, C., Nagaraj, V.T., 1982. Non-linear flapping vibrations of rotating blades. J. Sound Vib. 84, 549-556.

Yakubovich, V.A., Starzhinskii, V.M., 1975. Linear Differential Equations with Periodic Coefficients, Wiley, New York.

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.

e.  f.

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

a.                                                             b.

Fig.8. The backbone curves of shell vibrations  with dominant general coordinates: a. ; b. .

Table 1. Eigenfrequencies of vibrations

 Eigenfrequencies in Hz 1 The results of the calculations 275.35 1020.14 1238.08 2428.35 2628.88 3153.8 2 The results of previous calculations (Meerovich, 1961) 285 1000 1215 2560 3280 3 The experimental data (Meerovich, 1961) 273 1000 1223 2457 2946