This document discusses methods for identifying modal parameters like natural frequencies, damping coefficients, and mode shapes from experimental vibration data. It focuses on using continuous wavelet transform (CWT) which can identify these parameters more accurately than Fourier-based methods, especially in noisy environments. The document presents a new method to determine mode shapes for linear systems with proportional damping excited by an impact force, based on response time signals and known natural frequencies. A Morlet wavelet with an adjustment parameter is used to identify natural frequencies and damping ratios. A numerical case study confirms the accuracy of the proposed mode shape identification method based on results from the wavelet analysis.
Hybrid Algorithm for Dose Calculation in Cms Xio Treatment Planning SystemIOSR Journals
This study aimed at designing an improved hybrid algorithm by explicitly solving the linearized Boltzmann transport equation (LBTE) which is the governing equation that describes the macroscopic behaviour of radiation particles (neutrons, photons, electrons, etc). The algorithm accuracy will be evaluated using a newly designed in-house verification phantom and its results will be compared to those of the other XiO photon algorithms. The LBTE was solved numerically to compute photon transport in a medium. A programming code (algorithm) for the LBTE solution was developed and applied in the treatment planning system (TPS). The accuracy of the algorithm was evaluated by creating several plans for both the designed phantom and solid water phantom using the designed algorithm and other Xio photon algorithms. The plans were sent to a pre-calibrated Eleckta linear accelerator for measurement of absorbed dose.The results for all treatment plans using the hybrid algorithm compared to the 3 Xio photon algorithms were within 4 % limit. Calculation time for the hybrid algorithm was less in plans with larger number of beams compared to the other algorithms; however, it is higher for single beam plans. The hybrid algorithm provides comparable accuracy in treatment planning conditions to the other algorithms. This algorithm can therefore be employed in the calculation of dose in advance techniques such as IMRT and Rapid Arc by a radiotherapy centres with cmsxio treatment planning system as it is easy to implement.
Character recognition is a new research field in the domain of pattern recognition which deals with the
style of writing. Some of the challengeable problems in character identification are changing in the style of
writing, font and turns of words and etc. In this paper, the goal is Persian character identification using
independent orthogonal moment as the feature extraction technique.The proposed feature extraction
method is the combination of Pseudo-Zernike Moment and Fourier-Mellin Moment called Pseudo-Zernike-
Mellin Moment to extract feature vector from Persian characters. The proposed character identification
system is evaluated on the HODA dataset and obtained 97.76% acceptance rate.
Correlation Analysis of Tool Wear and Cutting Sound SignalIJRES Journal
With the classic signal analysis and processing method, the cutting of the audio signal in time
domain and frequency domain analysis. We reached the following conclusions: in the time domain analysis,
cutting audio signals mean and the variance associated with tool wear state change occurred did not change
significantly, and tool wear is not high degree of correlation, and the mean-square value of the audio signal
changes in the size and tool wear the state has a good relationship.
Hybrid Algorithm for Dose Calculation in Cms Xio Treatment Planning SystemIOSR Journals
This study aimed at designing an improved hybrid algorithm by explicitly solving the linearized Boltzmann transport equation (LBTE) which is the governing equation that describes the macroscopic behaviour of radiation particles (neutrons, photons, electrons, etc). The algorithm accuracy will be evaluated using a newly designed in-house verification phantom and its results will be compared to those of the other XiO photon algorithms. The LBTE was solved numerically to compute photon transport in a medium. A programming code (algorithm) for the LBTE solution was developed and applied in the treatment planning system (TPS). The accuracy of the algorithm was evaluated by creating several plans for both the designed phantom and solid water phantom using the designed algorithm and other Xio photon algorithms. The plans were sent to a pre-calibrated Eleckta linear accelerator for measurement of absorbed dose.The results for all treatment plans using the hybrid algorithm compared to the 3 Xio photon algorithms were within 4 % limit. Calculation time for the hybrid algorithm was less in plans with larger number of beams compared to the other algorithms; however, it is higher for single beam plans. The hybrid algorithm provides comparable accuracy in treatment planning conditions to the other algorithms. This algorithm can therefore be employed in the calculation of dose in advance techniques such as IMRT and Rapid Arc by a radiotherapy centres with cmsxio treatment planning system as it is easy to implement.
Character recognition is a new research field in the domain of pattern recognition which deals with the
style of writing. Some of the challengeable problems in character identification are changing in the style of
writing, font and turns of words and etc. In this paper, the goal is Persian character identification using
independent orthogonal moment as the feature extraction technique.The proposed feature extraction
method is the combination of Pseudo-Zernike Moment and Fourier-Mellin Moment called Pseudo-Zernike-
Mellin Moment to extract feature vector from Persian characters. The proposed character identification
system is evaluated on the HODA dataset and obtained 97.76% acceptance rate.
Correlation Analysis of Tool Wear and Cutting Sound SignalIJRES Journal
With the classic signal analysis and processing method, the cutting of the audio signal in time
domain and frequency domain analysis. We reached the following conclusions: in the time domain analysis,
cutting audio signals mean and the variance associated with tool wear state change occurred did not change
significantly, and tool wear is not high degree of correlation, and the mean-square value of the audio signal
changes in the size and tool wear the state has a good relationship.
KMEM4212_Applied Vibration_Group Assignment_Report_CL 3Max Lee
KMEM4212 Applied Vibration (University of Malaya)
Co-operative Learning 3 (CL 3)
This report contains a clear methodology that may be helpful for those who wish to run the experiment by themselves.
Sharing with you (Mechanical Engineering students) who may be benefited from it.
Feel free to connect with me at maxermesilliam@gmail.com
Recovery of low frequency Signals from noisy data using Ensembled Empirical M...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Ensemble Empirical Mode Decomposition: An adaptive method for noise reductionIOSR Journals
Abstract:Empirical mode decomposition (EMD), a data analysis technique, is used to denoise non-stationary and non-linear processes. The method does not require any pre & post processing of signal and use of any specified basis functions. But EMD suffers from a problem called mode mixing. So to overcome this problem a new method known as Ensemble Empirical mode decomposition (EEMD) has been introduced. The presented paper gives the detail of EEMD and its application in various fields. EEMD is a time–space analysis method, in which the added white noise is averaged out with sufficient number of trials; and the averaging process results in only the component of the signal (original data). EEMD is a truly noise-assisted data analysis (NADA) method and represents a substantial improvement over the original EMD. Keywords –Data analysis, Empirical mode decomposition, intrinsic mode function, mode mixing, NADA,
Ensemble Empirical Mode Decomposition: An adaptive method for noise reduction IOSR Journals
Empirical mode decomposition (EMD), a data analysis technique, is used to denoise non-stationary
and non-linear processes. The method does not require any pre & post processing of signal and use of any
specified basis functions. But EMD suffers from a problem called mode mixing. So to overcome this problem a
new method known as Ensemble Empirical mode decomposition (EEMD) has been introduced. The presented
paper gives the detail of EEMD and its application in various fields. EEMD is a time–space analysis method, in
which the added white noise is averaged out with sufficient number of trials; and the averaging process results
in only the component of the signal (original data). EEMD is a truly noise-assisted data analysis (NADA)
method and represents a substantial improvement over the original EMD.
Applications of Artificial Neural Network and Wavelet Transform For Conditio...IJMER
The vibration analysis of rotating machinery indicates of the condition of potential faults such
as unbalance, bent shaft, shaft crack, bearing clearance, rotor rub, misalignment, looseness, oil whirl
and whip and other malfunctions. More than one fault can occur in a rotor. This paper describes the
application of Artificial Neural Network (ANN) and Wavelet Transform (WT) for the prediction of the
effect of the combined faults of unbalance and bearing clearance on the frequency components of
vibration signature of the rotating machinery. The experimental data of frequency components and
corresponding Root Mean Square (RMS) velocity (amplitude) data are used as inputs to train the ANN,
which consists of a three-layered network. The ANN is trained using an improved multilayer feed forward
back propagation Levenberg-Marquardt algorithm. In particular, an overall success rates achieved were
99.78% for unbalance, 99.81% bearing clearance, and 99.45% for the combined faults of unbalance and
bearing clearance. The wavelet transform approach enables instant to instant observation of different
frequency components over the full spectrum. A new technique combining the WT with ANN performs
three general tasks data acquisition, feature extraction and fault identification. This method is tested
successfully for individual and combined faults of unbalance and bearing clearance at a success rate of
99.99%.
Hybrid Algorithm for Dose Calculation in Cms Xio Treatment Planning SystemIOSR Journals
This study aimed at designing an improved hybrid algorithm by explicitly solving the linearized Boltzmann transport equation (LBTE) which is the governing equation that describes the macroscopic behaviour of radiation particles (neutrons, photons, electrons, etc). The algorithm accuracy will be evaluated using a newly designed in-house verification phantom and its results will be compared to those of the other XiO photon algorithms. The LBTE was solved numerically to compute photon transport in a medium. A programming code (algorithm) for the LBTE solution was developed and applied in the treatment planning system (TPS). The accuracy of the algorithm was evaluated by creating several plans for both the designed phantom and solid water phantom using the designed algorithm and other Xio photon algorithms. The plans were sent to a pre-calibrated Eleckta linear accelerator for measurement of absorbed dose.The results for all treatment plans using the hybrid algorithm compared to the 3 Xio photon algorithms were within 4 % limit. Calculation time for the hybrid algorithm was less in plans with larger number of beams compared to the other algorithms; however, it is higher for single beam plans. The hybrid algorithm provides comparable accuracy in treatment planning conditions to the other algorithms. This algorithm can therefore be employed in the calculation of dose in advance techniques such as IMRT and Rapid Arc by a radiotherapy centres with cmsxio treatment planning system as it is easy to implement.
Condition Monitoring of Rotating Equipment Considering the Cause and Effects ...IJMERJOURNAL
ABSTRACT: This paper attempts to summarise and review the recent research and developments in diagnostics and prognostics of mechanical systems implementing Condition Monitoring with emphasis on models, algorithms and technologies for data processing and maintenance decision-making. Realising the increasing trend of using multiple sensors in condition monitoring, the authors also discuss different techniques for multiple sensor data fusion. The paper concludes with a brief discussion on current practices, possible future trends of Condition Monitoring with a brief outline on the novelty of the current research work.
Comparison of signal smoothing techniques for use in embedded system for moni...Dalton Valadares
Paper about the comparison between some signal smoothing techniques for use in an embedded system responsible for monitoring the biofuels quality, specificaly the oxidative stability.
Trend removal from raman spectra with local variance estimation and cubic spl...csijjournal
Trend removal is an important problem in most communication systems. Here, we show a proposed
algorithm for trend (background) removal from Raman Spectra by merging local variance estimation and
cubic spline interpolation methods. We found that Raman spectrum does not need a smoothing process to
remove trend from noisy signals. Employing this technique results in more speedy and noiseless systems
than other techniques that use wavelet transformation to suppress noise.
TREND REMOVAL FROM RAMAN SPECTRA WITH LOCAL VARIANCE ESTIMATION AND CUBIC SPL...csijjournal
Trend removal is an important problem in most communication systems. Here, we show a proposed algorithm for trend (background) removal from Raman Spectra by merging local variance estimation and cubic spline interpolation methods. We found that Raman spectrum does not need a smoothing process to remove trend from noisy signals. Employing this technique results in more speedy and noiseless systems than other techniques that use wavelet transformation to suppress noise
Trend Removal from Raman Spectra With Local Variance Estimation and Cubic Spl...csijjournal
Abstract
Trend removal is an important problem in most communication systems. Here, we show a proposed algorithm for trend (background) removal from Raman Spectra by merging local variance estimation and cubic spline interpolation methods. We found that Raman spectrum does not need a smoothing process to remove trend from noisy signals. Employing this technique results in more speedy and noiseless systems than other techniques that use wavelet transformation to suppress noise.
Keywords
Raman spectroscopy, Background correction method, Local variance, Cubic spline interpolation.
TREND REMOVAL FROM RAMAN SPECTRA WITH LOCAL VARIANCE ESTIMATION AND CUBIC SPL...csijjournal
Trend removal is an important problem in most communication systems. Here, we show a proposed algorithm for trend (background) removal from Raman Spectra by merging local variance estimation and cubic spline interpolation methods. We found that Raman spectrum does not need a smoothing process to remove trend from noisy signals. Employing this technique results in more speedy and noiseless systems than other techniques that use wavelet transformation to suppress noise.
KMEM4212_Applied Vibration_Group Assignment_Report_CL 3Max Lee
KMEM4212 Applied Vibration (University of Malaya)
Co-operative Learning 3 (CL 3)
This report contains a clear methodology that may be helpful for those who wish to run the experiment by themselves.
Sharing with you (Mechanical Engineering students) who may be benefited from it.
Feel free to connect with me at maxermesilliam@gmail.com
Recovery of low frequency Signals from noisy data using Ensembled Empirical M...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Ensemble Empirical Mode Decomposition: An adaptive method for noise reductionIOSR Journals
Abstract:Empirical mode decomposition (EMD), a data analysis technique, is used to denoise non-stationary and non-linear processes. The method does not require any pre & post processing of signal and use of any specified basis functions. But EMD suffers from a problem called mode mixing. So to overcome this problem a new method known as Ensemble Empirical mode decomposition (EEMD) has been introduced. The presented paper gives the detail of EEMD and its application in various fields. EEMD is a time–space analysis method, in which the added white noise is averaged out with sufficient number of trials; and the averaging process results in only the component of the signal (original data). EEMD is a truly noise-assisted data analysis (NADA) method and represents a substantial improvement over the original EMD. Keywords –Data analysis, Empirical mode decomposition, intrinsic mode function, mode mixing, NADA,
Ensemble Empirical Mode Decomposition: An adaptive method for noise reduction IOSR Journals
Empirical mode decomposition (EMD), a data analysis technique, is used to denoise non-stationary
and non-linear processes. The method does not require any pre & post processing of signal and use of any
specified basis functions. But EMD suffers from a problem called mode mixing. So to overcome this problem a
new method known as Ensemble Empirical mode decomposition (EEMD) has been introduced. The presented
paper gives the detail of EEMD and its application in various fields. EEMD is a time–space analysis method, in
which the added white noise is averaged out with sufficient number of trials; and the averaging process results
in only the component of the signal (original data). EEMD is a truly noise-assisted data analysis (NADA)
method and represents a substantial improvement over the original EMD.
Applications of Artificial Neural Network and Wavelet Transform For Conditio...IJMER
The vibration analysis of rotating machinery indicates of the condition of potential faults such
as unbalance, bent shaft, shaft crack, bearing clearance, rotor rub, misalignment, looseness, oil whirl
and whip and other malfunctions. More than one fault can occur in a rotor. This paper describes the
application of Artificial Neural Network (ANN) and Wavelet Transform (WT) for the prediction of the
effect of the combined faults of unbalance and bearing clearance on the frequency components of
vibration signature of the rotating machinery. The experimental data of frequency components and
corresponding Root Mean Square (RMS) velocity (amplitude) data are used as inputs to train the ANN,
which consists of a three-layered network. The ANN is trained using an improved multilayer feed forward
back propagation Levenberg-Marquardt algorithm. In particular, an overall success rates achieved were
99.78% for unbalance, 99.81% bearing clearance, and 99.45% for the combined faults of unbalance and
bearing clearance. The wavelet transform approach enables instant to instant observation of different
frequency components over the full spectrum. A new technique combining the WT with ANN performs
three general tasks data acquisition, feature extraction and fault identification. This method is tested
successfully for individual and combined faults of unbalance and bearing clearance at a success rate of
99.99%.
Hybrid Algorithm for Dose Calculation in Cms Xio Treatment Planning SystemIOSR Journals
This study aimed at designing an improved hybrid algorithm by explicitly solving the linearized Boltzmann transport equation (LBTE) which is the governing equation that describes the macroscopic behaviour of radiation particles (neutrons, photons, electrons, etc). The algorithm accuracy will be evaluated using a newly designed in-house verification phantom and its results will be compared to those of the other XiO photon algorithms. The LBTE was solved numerically to compute photon transport in a medium. A programming code (algorithm) for the LBTE solution was developed and applied in the treatment planning system (TPS). The accuracy of the algorithm was evaluated by creating several plans for both the designed phantom and solid water phantom using the designed algorithm and other Xio photon algorithms. The plans were sent to a pre-calibrated Eleckta linear accelerator for measurement of absorbed dose.The results for all treatment plans using the hybrid algorithm compared to the 3 Xio photon algorithms were within 4 % limit. Calculation time for the hybrid algorithm was less in plans with larger number of beams compared to the other algorithms; however, it is higher for single beam plans. The hybrid algorithm provides comparable accuracy in treatment planning conditions to the other algorithms. This algorithm can therefore be employed in the calculation of dose in advance techniques such as IMRT and Rapid Arc by a radiotherapy centres with cmsxio treatment planning system as it is easy to implement.
Condition Monitoring of Rotating Equipment Considering the Cause and Effects ...IJMERJOURNAL
ABSTRACT: This paper attempts to summarise and review the recent research and developments in diagnostics and prognostics of mechanical systems implementing Condition Monitoring with emphasis on models, algorithms and technologies for data processing and maintenance decision-making. Realising the increasing trend of using multiple sensors in condition monitoring, the authors also discuss different techniques for multiple sensor data fusion. The paper concludes with a brief discussion on current practices, possible future trends of Condition Monitoring with a brief outline on the novelty of the current research work.
Comparison of signal smoothing techniques for use in embedded system for moni...Dalton Valadares
Paper about the comparison between some signal smoothing techniques for use in an embedded system responsible for monitoring the biofuels quality, specificaly the oxidative stability.
Trend removal from raman spectra with local variance estimation and cubic spl...csijjournal
Trend removal is an important problem in most communication systems. Here, we show a proposed
algorithm for trend (background) removal from Raman Spectra by merging local variance estimation and
cubic spline interpolation methods. We found that Raman spectrum does not need a smoothing process to
remove trend from noisy signals. Employing this technique results in more speedy and noiseless systems
than other techniques that use wavelet transformation to suppress noise.
TREND REMOVAL FROM RAMAN SPECTRA WITH LOCAL VARIANCE ESTIMATION AND CUBIC SPL...csijjournal
Trend removal is an important problem in most communication systems. Here, we show a proposed algorithm for trend (background) removal from Raman Spectra by merging local variance estimation and cubic spline interpolation methods. We found that Raman spectrum does not need a smoothing process to remove trend from noisy signals. Employing this technique results in more speedy and noiseless systems than other techniques that use wavelet transformation to suppress noise
Trend Removal from Raman Spectra With Local Variance Estimation and Cubic Spl...csijjournal
Abstract
Trend removal is an important problem in most communication systems. Here, we show a proposed algorithm for trend (background) removal from Raman Spectra by merging local variance estimation and cubic spline interpolation methods. We found that Raman spectrum does not need a smoothing process to remove trend from noisy signals. Employing this technique results in more speedy and noiseless systems than other techniques that use wavelet transformation to suppress noise.
Keywords
Raman spectroscopy, Background correction method, Local variance, Cubic spline interpolation.
TREND REMOVAL FROM RAMAN SPECTRA WITH LOCAL VARIANCE ESTIMATION AND CUBIC SPL...csijjournal
Trend removal is an important problem in most communication systems. Here, we show a proposed algorithm for trend (background) removal from Raman Spectra by merging local variance estimation and cubic spline interpolation methods. We found that Raman spectrum does not need a smoothing process to remove trend from noisy signals. Employing this technique results in more speedy and noiseless systems than other techniques that use wavelet transformation to suppress noise.
1. 1 INTRODUCTION
Estimation of the modal parameters in terms of natural frequencies, damping coefficients and
mode shapes from experimental data is a fundamental problem in structural dynamics. The mo-
dal parameter identification methods can be categorized in to the Single Degree Of Freedom
(SDOF) methods and the Multi Degrees Of Freedom (MDOF) methods. Pick peaking method,
circle fit method and line fit method are the classical methods for modal parameter identifica-
tion (Ewins 2000). The recent method of three point finite difference method (Yin and Du-
hamel 2000) gives more accurate results compared to the traditional methods. Least square
complex exponential method (Smith 1981), poly-reference time domain method (Zhang 1987),
Ibrahim time domain method (Ibrahim and Mikulcik 1977), automated parameter identification
and order reduction for discrete time series (Hollkamp and Batill 1991) are among the MDOF
methods for modal parameter determination (Feng et al. 1998). The basis of most of these
methods is Fourier analysis which transforms the time data to the frequency data. However,
Fourier analysis cannot determine the modal parameters accurately in the noisy environments.
Some methods consist of pre-filtering of the input signals can improve the results. Moreover,
close modes may hardly be identified using the techniques based on the Fourier analysis. In
contrast to the Fourier transform which has a uniform resolution in frequency domain, the
wavelet transform has the property of double resolution in both the time and frequency domain
(Miranda 2008). By using this property, the wavelet transform can be adjusted to analyze the
non-stationary signals. Also, the strongly coupled modes can be identified by tuning the wave-
lets. Moreover, the inherent ability of wavelet transform in filtering out the noise contaminating
a signal is an important advantage for identifying the modal parameters.
Three methods for estimating the damping ratios based on the Continuous Wavelet Trans-
form (CWT) were proposed in (Staszewski 1997). A procedure of identification of natural fre-
quencies and damping ratios of the system from its free decays using wavelet transform was
presented in (Ruzzene et al. 1997). In order to improve the accuracy of the modal parameters
identification a modified Morlet wavelet function with an adjusting parameter was proposed by
Mode shapes identification from the results of wavelet transform:
Theory and simulation
M.R. Ashory, M. Jafari
Modal Analysis Lab., School of Mechanical Engineering, Semnan University, Semnan, Iran
M.M. Khatibi
Department of Mechanical Engineering, Islamic Azad University - Semnan Branch, Semnan, Iran
A. Malekjafarian
Palande-Saf Center; University of Applied Science and Technology, Semnan, Iran
ABSTRACT: In this article a new method is presented to determine the mode shapes of linear
dynamic systems with proportional viscous damping excited by an impact force. The time sig-
nals of responses and a priori knowledge of the natural frequencies are required. The method is
particularly suitable for the wavelet techniques which can estimate the natural frequencies and
damping ratios. A previously proposed method based on a modified Morlet wavelet function
with an adjusting parameter is used to identify the natural frequencies and damping ratios of
system. Then the mode shapes are estimated using the new method. It is shown that the ex-
tracted mode shapes are not scaled. Therefore, mass change method is used for scaling the
mode shapes. Also the effect of noise on the extracted modal parameters is investigated. The
validity of method is demonstrated by a numerical case study.
2. 2 IOMAC'11 – 4th
International Operational Modal Analysis Conference
Lardies and Gouttebroze (2002). A modal parameter identification procedure using continuous
wavelet transform including the mode shape identification was presented by Le and Argoul
(2004). There is no complete procedure for mode shape identification in most of the wavelet
based identification methods.
The contribution of this paper consists essentially of introducing a mode shape identification
method from the wavelet analysis of response data for the systems with proportional viscous
damping excited by an impact force. A Morlet wavelet with an adjusting parameter is used for
identification of the natural frequencies and damping coefficients. A numerical case study con-
firms the accuracy of method.
2 THEORY
2.1 Mode shape identification method
For a MDOF system with proportional viscous damping, the mode shapes are identical to those
of the undamped system (Ewins 2000). The response for each degree of freedom for viscously
damped system can be calculated by (Ewins 2000):
∑
−
−
n
r
rrr
ti
ri tety rr
1
2
))1(cos(.)( θζωωζ
A (1)
when the system is undamped:
nrr ,...,2,10 ζ (2)
Eq. (1) can be formulated in the matrix form as:
∑
Υ
n
r
rrr tt
1
)cos(. θωA (3)
where rA is the unscaled mode shapes of system. If the system is excited by an impact force,
the initial conditions are:
0(0 Υ ) (4)
0V(0) Υ& (5)
Inserting Eq. (4) into Eq. (3); gives:
0cos.
1
∑
r
n
r
r θA (6)
As the mode shape vectors are linearly independent, it is concluded that:
nrr ,...,2,10cos θ (7)
which results in:
,...2,1,0
2
2 kkr
π
πθ (8)
By inserting Eq. (8) in to Eq. (3), the following equation is obtained:
∑
Υ
n
r
rr tt
1
sin. ωA (9)
In which the elements of rA can also be negative to compensate the negative sign of
)cos( rr t θω when ,...2,1,0,
2
2 kkr
ππθ Eq. (9) can be rewritten as: