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Linear vibration analysis of rotating wind-turbine blade
Current Applied Physics 10 (2010) S332–S334
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
Linear vibration analysis of rotating wind-turbine blade Jung-Hun Park *, Hyun-Yong Park, Seok-Yong Jeong, Sang-Il Lee, Young-Ho Shin, Jong-Po Park Wind Power Engineering Team, Doosan Heavy Industries and Construction, Daejeon 305-811, Republic of Korea
a r t i c l e
i n f o
Article history: Received 12 December 2008 Accepted 8 June 2009 Available online 11 November 2009 Keywords: Rotating blade Natural frequency variation Multi-body dynamics Centrifugal effect Wind turbine
a b s t r a c t In the wind-turbine design, linear vibration analysis of the wind-turbine blade should be performed to get vibratory characteristics and to avoid structural resonance. EOM (equations of motion) for the blade are derived and vibratory characteristics of a rotating blade are observed and discussed in this work. Linear vibration analysis requires the linearized EOM with DOF (degree of freedom). For the system with large DOF, the derivation and linearization of EOM are very tedious and difficult. Constrained multi-body technique is employed to derive EOM’s to alleviate this burden. It is well known that natural frequencies and corresponding modes vary as rotating velocity changes. At the operating condition with relatively high rotating velocity, almost all commercial programs cannot predict the blade frequencies accurately. Numerical problems were solved to verify the accuracy of the proposed method. Through the numerical problems, this work shows that the proposed method is useful to predict the vibratory behavior of the rotating blade. Furthermore, a numerical problem was solved to check the numerical accuracy of commercial program results within operating region. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Rotating cantilever structures are commonly used for turbine blades, aircraft blades, etc. When the blades rotate, their stiffnesses change due to the stretching caused by centrifugal inertia forces. Their variations are important in these rotating machineries. Many researchers have focused on developing the methods to predict natural frequency variations accurately. An analytical method to calculate the natural frequencies of a rotating beam was presented by Southwell and Gough [1]. They proposed an equation which related the natural frequency to thee rotating frequency of a beam. This equation, which is frequently called the Southwell equation, has been widely used because it is simple and easy to use. Putter and Manor [2] applied the assumed mode approximation method for the modal analysis of a rotating beam. Yoo et al. [3–5] derived linearized EOM. They used hybrid coordinates instead of Cartesian coordinate to represent geometric nonlinearity in the linearized equation. These conventional methods have some drawbacks. First, the derivation and linearization of EOM are very tedious and difficult. Second, general method to derive EOM for various mechanisms does not exist. Constrained multi-body technique can be employed to overcome these drawbacks. Mechanical systems can be modelled as constrained multibody systems that consist of rigid and flexible bodies, joints, springs, dampers, forces and so on. In general,
* Corresponding author. Fax: +82 42 712 2299. E-mail address: [email protected] (J.-H. Park). 1567-1739/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2009.11.036
the EOM governing constrained multibody systems consist of non-linear differential and algebraic equations. To obtain the response of a constrained multibody system, several computational methods were introduced [6–10]. Several commercial programs for multibody system analysis (see, for instance, Refs. [11–13]) are available nowadays. Through these programs, kinematic, dynamic, and linear vibration analyses of constrained multibody systems can be performed for various mechanical systems. Unfortunately, these programs cannot predict modal characteristics for rotating blades accurately. Choi et al. [10] proposed the basic formulation to predict modal characteristics for mechanical systems including rotating blades. In this paper, a systematic formulation method is derived and natural frequency variations are investigated. This method is based on the previous study [10] and recursive formulations [6,7]. To verify the effectiveness and the accuracy of the proposed method, one numerical example was solved and the results were compared with the previous study. The frequency variations of the wind-turbine blade are examined and the variations are compared to wind turbine regulation [14]. 2. Derivation of EOMs By employing the Cartesian co-ordinate set, the equations of motion of a constrained multibody system can be derived (see Refs. [6,7]) as follows:
MY_ þ UTZ k ¼ Q :
ð1Þ
S333
J.-H. Park et al. / Current Applied Physics 10 (2010) S332–S334
where
Time derivative of Eq. (18) gives
h
UT ¼ UTJ
i
UTC
UTM :
ð2Þ
In the above equation, J U, C U, and M U denotes constraints due to joint in open tree, cut joints, and pre-defined motion. Cartesian velocity, Y can be expressed as relative velocity,v :
Y ¼ Bv:
ð3Þ
In the above equation, B denote velocity transformation matrix. Cartesian acceleration can be obtained by the time differentiation of Eq. (3).
Y_ ¼ B_ v þ B_ v :
ð4Þ
€
€ D þ Uq q € I þ c ¼ 0: U ¼ UqD q I € D can be obtained as From the above equation, q 1 € D ¼ U1 € q qD UqI qI UqD c :
T J
T
B MðB_ v þ Bv_ Þ þ B ð UTZ kJ þ C UTZ kC þ M UTZ kM Þ ¼ B Q :
ð5Þ
In thee above equation, BT ðJ UTZ Þ becomes a null matrix since B is the null space of J UZ . BT ðC UTZ Þ and BT ðM UTZ Þ can be re-written as:
BT ðC UTZ Þ ¼ ðC UZ BÞT ¼ C UTq : T M
T ZÞ
B ð U
T
M
v_ ¼
¼ ð UZ BÞ ¼ U
M v_ þ UT q k ¼ Q : M
ð8Þ
T UT ;
ð9Þ
M ¼ BT MB;
ð10Þ
Q ¼ BT Q BT MB_ v :
ð11Þ
Relative coordinates can be partitioned into dependent and independent coordinates.
T
qTD
; ¼ Rv_ I þ c
ð21Þ
where
"
c ¼
0 U1 qD c
# ð22Þ
:
Substitution of Eq. (22) into Eq. (8) gives the following equation: T
Þ ¼ R T Q R M ðRv_ I þ c
: v_ I ¼ Q or M
ð23Þ
where
¼ RT M R: M
ð24Þ
¼ Q R T M c : Q
ð25Þ
3. Results and discussion
where
q ¼ qTI
v_ I v_ D
ð7Þ
Substitution of Eqs. (6) and (7) into Eq. (5) gives
U ¼ ½ C UT
ð6Þ
T q:
M
ð20Þ
Relative velocity can be represented as
Substitution of Eqs. (3) and (4) into Eq. (1) and pre-multiplication of BT give the following equations: T
ð19Þ
For the proposed multibody formulation, the beam is discretized into 51 rigid bodies that are connected through 50 beam elements as shown in Fig. 1. Body 1 is the rotating hub and bodies 2–52 are the discretized rigid bodies. The hub and the ground are connected by a revolute joint; hub and body 2 are connected by a fixed joint; and the rest contiguous bodies are connected by beam elements. Table 1 shows the lowest three natural frequencies of the rotating cantilever beam with 10 rad/s. The results obtained by the
ð12Þ
:
In the above equation, qD and qI denote dependent coordinates and independent coordinates respectively. Several methods of selecting independent co-ordinate sets are known (see, for instance, Ref. [8]) Variation of the constraint equations U yields the following equations:
dU ¼ UqD dqD þ UqI dqI ¼ 0;
ð13Þ
dqD ¼ U1 qD UqI dqI :
ð14Þ
From the above equation, dq can be written as
" dq ¼ RdqI ¼
I
#
U1 qD UqI
dqI ;
ð15Þ
where
" R¼
#
I U1 qD UqI
:
ð16Þ
Time derivative of the constraint equations U yields the following equations:
U_ ¼ UqD q_ D þ UqI q_ I þ Ut ¼ 0:
q_ D ¼ U1 qD
UqI q_ I þ Ut :
Table 1 Natural frequency variations.
ð17Þ
From the above equation,
Fig. 1. Structural system example (cantilever beam): (a) FE model, (b) compressed model and (c) FE condensation.
ð18Þ
1st Freq. 2nd Freq. 3rd Freq.
Present
Ref. [3]
Error (%)
11.71 70.37 189.7
11.71 70.44 189.9
0 0.1 0.1
S334
J.-H. Park et al. / Current Applied Physics 10 (2010) S332–S334
Fig. 2. FE condensation for blade.
wind-turbine blade designer should consider 10% natural frequency variations to avoid vibratory resonance (fig. 2). 4. Conclusions In this paper, modal characteristics of rotating blade were investigated and a computational algorithm was proposed to find the modal characteristics. When blades rotate, their stiffnesses change due to the stretching caused by centrifugal inertia forces. Their variations are important in wind-turbine blades. From the numerical example, frequency variations are examined. Furthermore, the actual frequency variations within operating speed can have larger ranger than those of the GL regulation. Acknowledgments
Fig. 3. Natural frequency variations of blade.
This research has been supported by the Renewable Energy Program (Offshore Site Demonstration of Domestic 3 MW Wind Turbine) funded by the MKE. References
present multibody formulation were compared to those obtained by the method in Ref. [3]. The two results are in good agreement. The relatively small differences originate from the use of consistent mass in Ref. [3]. In Ref. [3], a non-Cartesian stretch variable along with the Rayleigh–Ritz assumed mode method is employed to derive the equations of motion. From the numerical results, the numerical accuracy of the proposed method is obtained. The turbine blade is discretized into 21 rigid bodies that are connected through 20 complex stiffness elements as shown in Fig. 3. The rotating hub and remaining bodies are the discretized rigid bodies. The hub and the ground are connected by a revolute joint; hub and adjacent body are connected by a fixed joint; and the rest contiguous bodies are connected by complex spring elements. Fig. 3 shows the lowest five natural frequencies of the rotating blade according to the rotating speed. The analysis results say that maximum frequency variations have 8% and 16.5% at 20 RPM and 30 RPM respectively. GL guide line (Ref. [14]) regulates that design natural frequencies should be apart from the exciting frequencies. Generally, maximum operating speed of 2 or 3 MW machine is near 20 RPM. From the numerical results, it is known that the
[1] R.V. Southwell, F. Gough, The free transverse Vibration of Airscrew Blades, British A.R.C. Reports and Memoranda, no. 766, 1921. [2] S. Putter, H. Manor, Natural frequencies of radial rotating beams, J. Sound Vib. 56 (1978) 175–185. [3] H.H. Yoo, S.H. Shin, Vibration analysis of rotating cantilever beams, J. Sound Vib. 212 (5) (1998) 807–828. [4] H.H. Yoo, J.H. Park, J.H. Park, Vibration analysis of rotating pre-twisted blades, J. Comput. Struct. 79 (2001) 1811–1819. [5] S.H. Lee, S.H. Shin, H.H. Yoo, Flapwise bending vibration analysis of rotating composite cantilever beams, KSME Int. J. 18 (2) (2004) 240–245. [6] D.S. Bae, E.J. Haug, A recursive formulation for constrained mechanical system dynamics: Part I. Open loop systems, Mech. Struct. Mach. 15 (3) (1987) 359– 382. [7] D.S. Bae, E.J. Haug, A recursive formulation for constrained mechanical system dynamics: Part II. Closed loop systems, Mech. Struct. Mach. 15 (4) (1987) 481– 506. [8] P.E. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1988. [9] S.S. Kim, M.J. Vanderploeg, A general and efficient method for dynamic analysis of mechanical systems using velocity transformations, American society of mechanical engineers, J. Mech. Trans. Autom. Des. 108 (1986) 176–182. [10] D.H. Choi, J.H. Park, H.H. Yoo, Modal analysis of constrained multibody systems undergoing rotational motion, J. Sound Vib. 280 (2005) 63–76. [11] ADAMS User’s Guide, MSC Software Corporation, 2003. [12] DADS User’s Manual, LMSCADSI Inc., 2002. [13] RecurDyn User’s Manual Version 4, FunctionBay Inc., 2002. [14] Guideline for the Certification of Offshore Wind Turbines, 2005.
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
Linear vibration analysis of rotating wind-turbine blade Jung-Hun Park *, Hyun-Yong Park, Seok-Yong Jeong, Sang-Il Lee, Young-Ho Shin, Jong-Po Park Wind Power Engineering Team, Doosan Heavy Industries and Construction, Daejeon 305-811, Republic of Korea
a r t i c l e
i n f o
Article history: Received 12 December 2008 Accepted 8 June 2009 Available online 11 November 2009 Keywords: Rotating blade Natural frequency variation Multi-body dynamics Centrifugal effect Wind turbine
a b s t r a c t In the wind-turbine design, linear vibration analysis of the wind-turbine blade should be performed to get vibratory characteristics and to avoid structural resonance. EOM (equations of motion) for the blade are derived and vibratory characteristics of a rotating blade are observed and discussed in this work. Linear vibration analysis requires the linearized EOM with DOF (degree of freedom). For the system with large DOF, the derivation and linearization of EOM are very tedious and difficult. Constrained multi-body technique is employed to derive EOM’s to alleviate this burden. It is well known that natural frequencies and corresponding modes vary as rotating velocity changes. At the operating condition with relatively high rotating velocity, almost all commercial programs cannot predict the blade frequencies accurately. Numerical problems were solved to verify the accuracy of the proposed method. Through the numerical problems, this work shows that the proposed method is useful to predict the vibratory behavior of the rotating blade. Furthermore, a numerical problem was solved to check the numerical accuracy of commercial program results within operating region. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Rotating cantilever structures are commonly used for turbine blades, aircraft blades, etc. When the blades rotate, their stiffnesses change due to the stretching caused by centrifugal inertia forces. Their variations are important in these rotating machineries. Many researchers have focused on developing the methods to predict natural frequency variations accurately. An analytical method to calculate the natural frequencies of a rotating beam was presented by Southwell and Gough [1]. They proposed an equation which related the natural frequency to thee rotating frequency of a beam. This equation, which is frequently called the Southwell equation, has been widely used because it is simple and easy to use. Putter and Manor [2] applied the assumed mode approximation method for the modal analysis of a rotating beam. Yoo et al. [3–5] derived linearized EOM. They used hybrid coordinates instead of Cartesian coordinate to represent geometric nonlinearity in the linearized equation. These conventional methods have some drawbacks. First, the derivation and linearization of EOM are very tedious and difficult. Second, general method to derive EOM for various mechanisms does not exist. Constrained multi-body technique can be employed to overcome these drawbacks. Mechanical systems can be modelled as constrained multibody systems that consist of rigid and flexible bodies, joints, springs, dampers, forces and so on. In general,
* Corresponding author. Fax: +82 42 712 2299. E-mail address: [email protected] (J.-H. Park). 1567-1739/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cap.2009.11.036
the EOM governing constrained multibody systems consist of non-linear differential and algebraic equations. To obtain the response of a constrained multibody system, several computational methods were introduced [6–10]. Several commercial programs for multibody system analysis (see, for instance, Refs. [11–13]) are available nowadays. Through these programs, kinematic, dynamic, and linear vibration analyses of constrained multibody systems can be performed for various mechanical systems. Unfortunately, these programs cannot predict modal characteristics for rotating blades accurately. Choi et al. [10] proposed the basic formulation to predict modal characteristics for mechanical systems including rotating blades. In this paper, a systematic formulation method is derived and natural frequency variations are investigated. This method is based on the previous study [10] and recursive formulations [6,7]. To verify the effectiveness and the accuracy of the proposed method, one numerical example was solved and the results were compared with the previous study. The frequency variations of the wind-turbine blade are examined and the variations are compared to wind turbine regulation [14]. 2. Derivation of EOMs By employing the Cartesian co-ordinate set, the equations of motion of a constrained multibody system can be derived (see Refs. [6,7]) as follows:
MY_ þ UTZ k ¼ Q :
ð1Þ
S333
J.-H. Park et al. / Current Applied Physics 10 (2010) S332–S334
where
Time derivative of Eq. (18) gives
h
UT ¼ UTJ
i
UTC
UTM :
ð2Þ
In the above equation, J U, C U, and M U denotes constraints due to joint in open tree, cut joints, and pre-defined motion. Cartesian velocity, Y can be expressed as relative velocity,v :
Y ¼ Bv:
ð3Þ
In the above equation, B denote velocity transformation matrix. Cartesian acceleration can be obtained by the time differentiation of Eq. (3).
Y_ ¼ B_ v þ B_ v :
ð4Þ
€
€ D þ Uq q € I þ c ¼ 0: U ¼ UqD q I € D can be obtained as From the above equation, q 1 € D ¼ U1 € q qD UqI qI UqD c :
T J
T
B MðB_ v þ Bv_ Þ þ B ð UTZ kJ þ C UTZ kC þ M UTZ kM Þ ¼ B Q :
ð5Þ
In thee above equation, BT ðJ UTZ Þ becomes a null matrix since B is the null space of J UZ . BT ðC UTZ Þ and BT ðM UTZ Þ can be re-written as:
BT ðC UTZ Þ ¼ ðC UZ BÞT ¼ C UTq : T M
T ZÞ
B ð U
T
M
v_ ¼
¼ ð UZ BÞ ¼ U
M v_ þ UT q k ¼ Q : M
ð8Þ
T UT ;
ð9Þ
M ¼ BT MB;
ð10Þ
Q ¼ BT Q BT MB_ v :
ð11Þ
Relative coordinates can be partitioned into dependent and independent coordinates.
T
qTD
; ¼ Rv_ I þ c
ð21Þ
where
"
c ¼
0 U1 qD c
# ð22Þ
:
Substitution of Eq. (22) into Eq. (8) gives the following equation: T
Þ ¼ R T Q R M ðRv_ I þ c
: v_ I ¼ Q or M
ð23Þ
where
¼ RT M R: M
ð24Þ
¼ Q R T M c : Q
ð25Þ
3. Results and discussion
where
q ¼ qTI
v_ I v_ D
ð7Þ
Substitution of Eqs. (6) and (7) into Eq. (5) gives
U ¼ ½ C UT
ð6Þ
T q:
M
ð20Þ
Relative velocity can be represented as
Substitution of Eqs. (3) and (4) into Eq. (1) and pre-multiplication of BT give the following equations: T
ð19Þ
For the proposed multibody formulation, the beam is discretized into 51 rigid bodies that are connected through 50 beam elements as shown in Fig. 1. Body 1 is the rotating hub and bodies 2–52 are the discretized rigid bodies. The hub and the ground are connected by a revolute joint; hub and body 2 are connected by a fixed joint; and the rest contiguous bodies are connected by beam elements. Table 1 shows the lowest three natural frequencies of the rotating cantilever beam with 10 rad/s. The results obtained by the
ð12Þ
:
In the above equation, qD and qI denote dependent coordinates and independent coordinates respectively. Several methods of selecting independent co-ordinate sets are known (see, for instance, Ref. [8]) Variation of the constraint equations U yields the following equations:
dU ¼ UqD dqD þ UqI dqI ¼ 0;
ð13Þ
dqD ¼ U1 qD UqI dqI :
ð14Þ
From the above equation, dq can be written as
" dq ¼ RdqI ¼
I
#
U1 qD UqI
dqI ;
ð15Þ
where
" R¼
#
I U1 qD UqI
:
ð16Þ
Time derivative of the constraint equations U yields the following equations:
U_ ¼ UqD q_ D þ UqI q_ I þ Ut ¼ 0:
q_ D ¼ U1 qD
UqI q_ I þ Ut :
Table 1 Natural frequency variations.
ð17Þ
From the above equation,
Fig. 1. Structural system example (cantilever beam): (a) FE model, (b) compressed model and (c) FE condensation.
ð18Þ
1st Freq. 2nd Freq. 3rd Freq.
Present
Ref. [3]
Error (%)
11.71 70.37 189.7
11.71 70.44 189.9
0 0.1 0.1
S334
J.-H. Park et al. / Current Applied Physics 10 (2010) S332–S334
Fig. 2. FE condensation for blade.
wind-turbine blade designer should consider 10% natural frequency variations to avoid vibratory resonance (fig. 2). 4. Conclusions In this paper, modal characteristics of rotating blade were investigated and a computational algorithm was proposed to find the modal characteristics. When blades rotate, their stiffnesses change due to the stretching caused by centrifugal inertia forces. Their variations are important in wind-turbine blades. From the numerical example, frequency variations are examined. Furthermore, the actual frequency variations within operating speed can have larger ranger than those of the GL regulation. Acknowledgments
Fig. 3. Natural frequency variations of blade.
This research has been supported by the Renewable Energy Program (Offshore Site Demonstration of Domestic 3 MW Wind Turbine) funded by the MKE. References
present multibody formulation were compared to those obtained by the method in Ref. [3]. The two results are in good agreement. The relatively small differences originate from the use of consistent mass in Ref. [3]. In Ref. [3], a non-Cartesian stretch variable along with the Rayleigh–Ritz assumed mode method is employed to derive the equations of motion. From the numerical results, the numerical accuracy of the proposed method is obtained. The turbine blade is discretized into 21 rigid bodies that are connected through 20 complex stiffness elements as shown in Fig. 3. The rotating hub and remaining bodies are the discretized rigid bodies. The hub and the ground are connected by a revolute joint; hub and adjacent body are connected by a fixed joint; and the rest contiguous bodies are connected by complex spring elements. Fig. 3 shows the lowest five natural frequencies of the rotating blade according to the rotating speed. The analysis results say that maximum frequency variations have 8% and 16.5% at 20 RPM and 30 RPM respectively. GL guide line (Ref. [14]) regulates that design natural frequencies should be apart from the exciting frequencies. Generally, maximum operating speed of 2 or 3 MW machine is near 20 RPM. From the numerical results, it is known that the
[1] R.V. Southwell, F. Gough, The free transverse Vibration of Airscrew Blades, British A.R.C. Reports and Memoranda, no. 766, 1921. [2] S. Putter, H. Manor, Natural frequencies of radial rotating beams, J. Sound Vib. 56 (1978) 175–185. [3] H.H. Yoo, S.H. Shin, Vibration analysis of rotating cantilever beams, J. Sound Vib. 212 (5) (1998) 807–828. [4] H.H. Yoo, J.H. Park, J.H. Park, Vibration analysis of rotating pre-twisted blades, J. Comput. Struct. 79 (2001) 1811–1819. [5] S.H. Lee, S.H. Shin, H.H. Yoo, Flapwise bending vibration analysis of rotating composite cantilever beams, KSME Int. J. 18 (2) (2004) 240–245. [6] D.S. Bae, E.J. Haug, A recursive formulation for constrained mechanical system dynamics: Part I. Open loop systems, Mech. Struct. Mach. 15 (3) (1987) 359– 382. [7] D.S. Bae, E.J. Haug, A recursive formulation for constrained mechanical system dynamics: Part II. Closed loop systems, Mech. Struct. Mach. 15 (4) (1987) 481– 506. [8] P.E. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1988. [9] S.S. Kim, M.J. Vanderploeg, A general and efficient method for dynamic analysis of mechanical systems using velocity transformations, American society of mechanical engineers, J. Mech. Trans. Autom. Des. 108 (1986) 176–182. [10] D.H. Choi, J.H. Park, H.H. Yoo, Modal analysis of constrained multibody systems undergoing rotational motion, J. Sound Vib. 280 (2005) 63–76. [11] ADAMS User’s Guide, MSC Software Corporation, 2003. [12] DADS User’s Manual, LMSCADSI Inc., 2002. [13] RecurDyn User’s Manual Version 4, FunctionBay Inc., 2002. [14] Guideline for the Certification of Offshore Wind Turbines, 2005.