**7. Appendix A**

As Shu presents in his book (Shu, 2000), the differential quadrature method, DQM, is a numerical technique for solving differential equations.

In order to obtain the DQM analog equations to the governing equations of the rotating beam and its boundary conditions, the beam domain is discretized in a grid of points using the Chebyshev – Gauss - Lobato expression, (Shu & Chen, 1999):

$$\mathbf{x}\_{i} = \frac{1 - \cos\left[\left(i - 1\right)\pi \;/\left(n - 1\right)\right]}{2} \; ; \; i = 1, 2, ..., n$$

where *n* is the number of discrete points or nodes and *<sup>i</sup> x* is the coordinate of node *i*.

Fig. A1. Grid of *n* points

The weighting coefficients (1) *Ai <sup>j</sup>* and (2) *Ai <sup>j</sup>* , which appeared in the linear algebraic equations of quadrature (28-35), were determined using the explicit expressions cited by (Bert & Malik, 1996).

The coefficients (1) *Ai <sup>j</sup>* correspond to first order derivatives and can be arranged in a square matrix of order *n*.

The matrix elements (1) *Ai <sup>j</sup>* with *i* <sup>≠</sup> *<sup>j</sup>* , are determined by:

Free Vibration Analysis of Centrifugally

ISSN 0022-460X

ISSN 1225-4568

578-588, ISSN 0020-7403

*Journal*. Vol.19, No.11, pp. 1459-1466

pp.461-481, ISSN 0022-460X

*Sound and Vibration.* Vol. 240, pp. 303-322.

Vol.270, pp.1045-1055, ISSN 0022-460X

*and Vibration.* Vol.289, pp.413-420, ISSN 0022-460X

Holland

*Anal*. Vol.34, pp. 235-238, ISSN 0022-247X

Vol.38, pp. 51-856, ISSN 0001-1452

Stiffened Non Uniform Timoshenko Beams 313

Banerjee, J.; Su, H, & Jackson, D. (2006). Free vibration of rotating tapered beams using the

Bellman, R. & Casti, J. (1971). Differential quadrature and long-term Integration*. J. Math.*

Bellman, R.E. & Roth, R.S. (1986). *Methods in approximation: techniques for mathematical* 

Bert, C. & Malik, M. (1996). Differential quadrature method in computational mechanics: A

Choi, S.; Wu J. & Chou Y. (2000). Dynamic analysis of a spinning Timoshenko beam by the

Felix, D.H.; Rossi, R. E. & Bambill, D. V. (2008). Vibraciones transversales por el método de

Ganesh, R and Ganguli, R. (2011). Physics based basis function for vibration analysis of high

Gunda, J. B. & Ganguli R. (2008). New rational interpolation functions for finite element

Gunda, J.B.; Singh, A.P.; Chhabra, P.S. & Ganguli, R. (2007). Free vibration analysis of

Kumar A. & Ganguli R. (2009). Rotating Beams and Nonrotating Beams with Shared Eigenpair, *Journal of Applied Mechanics*. Vol.76. No.5, pp. 1-14, ISSN: 0021-8936 Hodges, D. H. & Rutkowski, M. J. (1981). Free vibration analysis of rotating beams by a

Lin, S. C. & Hsiao, K. M. (2001). Vibration analysis of a rotating Timoshenko beam. *Journal of* 

Liu, G. R. & Wu, T. Y. (2001). Vibration analysis of beams using the generalized differential

Naguleswaran, S. (2004). Transverse vibration and stability of an Euler–Bernoulli beam with

Özdemir, Ö. & Kaya, M.O. (2006). Flapwise bending vibration analysis of a rotating tapered

review. *Applied Mechanics Review* Vol.49, pp. 1-28, ISSN 0008-6900

*Diseño en Ingeniería.* Vol. 25, No. 2, pp. 111-132, ISSN 0213-1315

*Mechanics*, Vol.27, No.2, pp. 243-257, ISSN 1225-4568

dynamic stiffness method. *Journal of Sound and Vibration*, Vol. 298, pp. 1034-1054,

*modelling*, Editorial D. Reidel Publishing Company, ISBN 9-027-72188-2, Dordrecht,

differential quadrature method. *American Institute of Aeronautics and Astronautics*

cuadratura diferencial de una viga Timoshenko rotante, escalonada y elásticamente vinculada, *Mecánica Computacional* Vol. XXVII, pp.1957-1973, ISBN 1666-6070 Felix, D. H.; Bambill, D. V. & Rossi, R. E. (2009). Análisis de vibración libre de una viga

Timoshenko escalonada, centrífugamente rigidizada, mediante el método de cuadratura diferencial, *Revista Internacional de Métodos Numéricos para Cálculo y* 

speed rotating beams. *Structural Engineering and Mechanics*, Vol.39, No.1, pp. 21-46,

analysis of rotating beams. *International Journal of Mechanical Sciences*; Vol. 50, pp.

rotating tapered blades using Fourier-p superelement, *Structural Engineering and* 

variable order finite method, *American Institute of Aeronautics and Astronautics* 

quadrature rule and domain decomposition. *Journal of Sound and Vibration.* Vol.246,

step change in cross-section and in axial force. *Journal of Sound and Vibration.* 

cantilever Bernoulli–Euler beam by differential transform method. *Journal of Sound* 

$$A\_{ij}^{(1)} = \frac{\prod \left( {x\_i} \right)}{\left( {x\_i} - {x\_j} \right) \prod \left( {x\_j} \right)}$$

where

 1 *n i i i x xx* ; 1 *n j i x xx j* ;

The coefficients (1) *Ai <sup>j</sup>* with *i* = *j* , will tend to infinity and need to be calculated in another way.

The coefficients (2) *Ai <sup>j</sup>* correspond to second-order derivatives and are obtained from

$$A\_{ij}^{(2)} = 2\left[A\_{ii}^{(1)} \ast A\_{ij}^{(1)} - \frac{A\_{ij}^{(1)}}{\varkappa\_i - \varkappa\_j}\right],$$

with *i* ≠ *j* and *i, j* = 1, 2, 3, …, *n*.

Because the sum of the weighting coefficients of a row of the matrix is zero, it is easy to calculate the diagonal terms of derivatives of any order *q*, using the following expression:

$$\mathcal{A}\_{ii}^{(q)} = -\sum\_{\substack{j=1 \ j \neq i}}^n A\_{ij}^{(q)}.$$

And the equations for *q* equal to 1 and 2, corresponding to first and second order derivatives, are:

$$A\_{ii}^{(1)} = -\sum\_{\substack{j=1 \ j \neq i}}^n A\_{ij}^{(1)} \; ; \; \; A\_{ii}^{(2)} = -\sum\_{\substack{j=1 \ j \neq i}}^n A\_{ij}^{(2)} \; .$$

#### **8. Acknowledgment**

The authors gratefully acknowledge the support of the Universidad Nacional del Sur (UNS) and the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina.

#### **9. References**


(1) *<sup>x</sup> <sup>i</sup>*

The coefficients (2) *Ai <sup>j</sup>* correspond to second-order derivatives and are obtained from

*A AA*

(1) (1) 1

*Mechanics*, Vol. 34, No. 2, pp. 231-245, ISSN 12254568

*n ii ij j ji A A* 

*x xj* 

;

The coefficients (1) *Ai <sup>j</sup>* with *i* = *j* , will tend to infinity and need to be calculated in another

(2) (1) (1) 2 \* *ij ij ii ij*

Because the sum of the weighting coefficients of a row of the matrix is zero, it is easy to calculate the diagonal terms of derivatives of any order *q*, using the following expression:

> ( ) ( ) 1 *<sup>n</sup> q q ii ij j ji A A*

And the equations for *q* equal to 1 and 2, corresponding to first and second order

; (2) (2)

The authors gratefully acknowledge the support of the Universidad Nacional del Sur (UNS) and the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina.

Bambill, D.V.; Felix, D.H. & Rossi, R. E. (2010). Vibration analysis of rotating Timoshenko

Banerjee, J. (2000). Free vibration of centrifugally stiffened uniform and tapered beams using

Banerjee, J. (2001). Dynamic stiffness formulation and free vibration analysis of centrifugally

beams by means of the differential quadrature method. *Structural Engineering and* 

the dynamic stiffness method. *Journal of Sound and Vibration,* Vol.233, No.5, pp. 857-

stiffened Timoshenko beam. *Journal of Sound and Vibration,* Vol.247, pp. 97-115,

*ij*

*A*

 1

*n i i i x xx*

 

where

way.

with *i* ≠ *j* and *i, j* = 1, 2, 3, …, *n*.

derivatives, are:

**8. Acknowledgment** 

875, ISSN 0022-460X

ISSN 0022-460X

**9. References** 

 

*<sup>x</sup> <sup>i</sup> <sup>j</sup>*

*j*

1

(1)

*A*

 

*x x*

*i j*

1

*n ii ij j ji A A* 

 ;

*i x xx j*

*n*


**14** 

*Poland* 

Tomasz Gałka

*Institute of Power Engineering,* 

**Vibration-Based Diagnostics of Steam Turbines** 

Of three general maintenance strategies – run-to-break, preventive maintenance and predictive maintenance – the latter, also referred to as condition-based maintenance, is becoming widely recognized as the most effective one (see e.g. Randall, 2011). To exploit its potential to the full, however, it has to be based on reliable condition assessment methods and procedures. This is particularly important for critical machines, characterized by high unit cost and serious

In general, technical diagnostics may be defined as determining technical condition on the basis of objective methods and measures. The objectivity implies that technical condition assessment is based on measurable physical quantities. These quantities are sources of diagnostic symptoms. For any given class of objects, the development of technical

At the *measurement* stage we are able to measure physical quantities relevant to the object technical condition. On the basis of measurement data, at the *qualitative diagnostics* stage faults and malfunctions are identified and located with the aid of an appropriate diagnostic model. *Quantitative diagnostics* consists in estimating damage degree (advancement), for which a reference scale is necessary. Finally, *prognosis* is an estimation of the period remaining until an intervention is needed. Qualitative diagnostics may be viewed as being aimed at detecting hard (random) failures, while the aim of the quantitative diagnosis is to

Complex objects, like steam turbines, are characterized by a number of residual processes (such as vibration, noise, heat radiation etc.) that accompany the basic process of energy transformation, and hence a number of condition symptom types. For all rotating machines, vibration-based symptoms are the most important ones for technical condition assessment,


consequences of a potential failure. Steam turbines provide here a good example.

diagnostics essentially involves four principal stages (Crocker, 2003), namely:

**1. Introduction** 



due to at least three reasons:


trace the soft (natural) fault evolution (Martin, 1994).


