Vibration of circular plate with multiple eccentric circular perforations by the Rayleigh-Ritz method

Abstract

The free vibration of a circular plate with multiple perforations is analyzed by using the Rayleigh-Ritz method. Admissible functions are assumed to be separable functions of radial and tangential coordinates. Trigonometric functions are assumed in the circumferential direction. The radial shape functions are the boundary characteristic orthogonal polynomials generated following the Gram-Schmidt recurrence scheme. The assumed functions are used to estimate the kinetic and the potential energies of the plate depending on the number and the position of the perforations. The eigenvalues, representing the dimensionless natural frequencies, are compared with the results obtained using Bessel functions, where the exact solution is available. Moreover, the eigenvectors, which are the unknown coefficients of the Rayleigh-Ritz method, are used to present the mode shapes of the plate. To validate the analytical results of the plates with multiple perforations, experimental investigations are also performed. Two unique case studies that are not addressed in the existing literature are considered. The results of the Rayleigh-Ritz method are found to be in good agreement with those from the experiments. Although the method presented can be employed in the vibration analysis of plates with different boundary conditions and shapes of the perforations, circular perforations that are free on the edges are studied in this paper. The results are presented in terms of dimensionless frequencies and mode shapes.[PUBLICATION ABSTRACT]

Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448

www.springerlink.com/content/1738-494x DOI 10.1007/s12206-012-0325-7

Vibration of circular plate with multiple eccentric circular perforations by the Rayleigh-Ritz method

Khodabakhsh Saeedi1,*, Alfin Leo2, Rama B. Bhat1 and Ion Stiharu1

1CONCAVE Research Center, Mechanical and Industrial Engineering Department, Concordia University, Montreal, QC, Canada

2Parker Filtration Canada, Laval, QC, Canada

(Manuscript Received July 28, 2011; Revised December 19, 2011; Accepted February 6, 2012) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract

The free vibration of a circular plate with multiple perforations is analyzed by using the Rayleigh-Ritz method. Admissible functions are assumed to be separable functions of radial and tangential coordinates. Trigonometric functions are assumed in the circumferential direction. The radial shape functions are the boundary characteristic orthogonal polynomials generated following the Gram-Schmidt recurrence scheme. The assumed functions are used to estimate the kinetic and the potential energies of the plate depending on the number and the position of the perforations. The eigenvalues, representing the dimensionless natural frequencies, are compared with the results obtained using Bessel functions, where the exact solution is available. Moreover, the eigenvectors, which are the unknown coefficients of the Rayleigh-Ritz method, are used to present the mode shapes of the plate. To validate the analytical results of the plates with multiple perforations, experimental investigations are also performed. Two unique case studies that are not addressed in the existing literature are considered. The results of the Rayleigh-Ritz method are found to be in good agreement with those from the experiments. Although the method presented can be employed in the vibration analysis of plates with different boundary conditions and shapes of the perforations, circular perforations that are free on the edges are studied in this paper. The results are presented in terms of dimensionless frequencies and mode shapes.

Keywords: Boundary characteristic orthogonal polynomials; Circular perforation; Circular plate; Rayleigh-Ritz method; Vibration ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction

Many applications need circular plates with multiple perfo-rations. One such example would be the turbo screen used to protect the turbo charger in locomotives. The turbo screen has an important role in protecting the turbo charger from broken engine parts. Potential hazard could be caused by parts of piston rings or broken valves. The perforations in the turbo screen allow the flow of exhaust gas from the exhaust manifold of the engine to the turbine section of the turbo charger, thereby running the compressor of the turbo charger. Circular plates could be classified into three groups according to the location and size of the perforations: plates may have no perforations (full), one centered perforation (annular), or plates with multiple perforations. Indeed, there are many other engineering examples of perforated diaphragms, such as automobile wheels, which are perforated not only for aesthetics and weight reduction but also for the air cooling of brake shoes. [1, 2]. Of these three types of plates, full and annular plates have

been extensively studied by many researchers. These two have been frequently used in the construction of aeronautical structures, ship structures, and in several other industrial areas [3, 4].

Compared with the studies of annular plates, the studies on the plates with multiple eccentric perforations are rare. There are some studies where finite element method [4] and Bessel functions satisfying the boundary conditions [5, 6] have been used for the free vibration analysis of plates with eccentric perforations. The finite element method has been used [7] to study the effects of the eccentricity, the size of the perforation, and the boundary conditions on the natural frequencies and the mode shapes of a circular plate with eccentric perforation. Polynomials are employed as radial admissible functions in the Rayleigh-Ritz method in order to determine the frequencies of circular plates with circular [8, 9] and rectangular [10] perforations. The boundary element method (BEM) has been used to study the natural vibration of circular plates with various boundary conditions such as clamped and simply-supported [11, 12]. Also the boundary integral equation method (BIEM) has been proposed as a semi-analytical solution to study the free vibration of a circular plate with circular perforations [13-16].

The Rayleigh-Ritz method has extensively gained the atten-

*Corresponding author. Tel.: [phone] ext. 7268, Fax.: [phone] E-mail address: [email protected]

Recommended by Associate Editor Cheolung Cheong. KSME & Springer 2012

1440 K. Saeedi et al. / Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448

tion of researchers [17, 18] because the assumed deflection functions are required to satisfy the geometrical boundary conditions only. The boundary characteristic orthogonal polynomials, which have been proposed by Bhat [19] and used by Rajalingham et al. [20, 21] and Chakraverthy et al. [22], not only provide diagonalized mass matrix, but also satisfy the geometrical boundary conditions. In the present study, the natural frequencies and the mode shapes of plates with multiple circular perforations are calculated, using the product of the boundary characteristic orthogonal polynomials and the trigonometric functions as the assumed deflection functions in the Rayleigh-Ritz method. The results are first validated for clamped circular and clamped annular plates without perforations by comparing the results with the Bessel functions solution. The results for the perforated plates are experimentally validated. Two cases that are not found elsewhere in the literature are studied:

A clamped circular plate with one circular perforation at the center and eight equally spaced circular perforations close to the outer edge.

A clamped circular plate with one circular perforation at the center, eight equally spaced circular perforations close to the center and sixteen 16 equally spaced circular perforations close to the outer edge.

2. Methodology

Denoting W(r,) as the deflection shape expressed in the polar coordinates and as the natural frequency of the plate, the maximum kinetic energy Tmax can be written as:

2 2 max

( , ) ( ) cos sin

= +

(4)

where m

M N

m mn mn m n

W r r A n B n

( )

= =

1 0

the mth Boundary Characteristic Orthogonal Polynomial in the radial direction, and substituting it into the preceding expressions for maximum kinetic and potential energies results in:

( ) ( )

1 ( ) cos sin ( ) cos sin

T h r A j B j r A n B n dA

= + +

2 ,

(5) ( ) ( )( ) ( ) ( )

( ) ( )

D

U V A j B j V A n B n

r A j B j V A n B n

V A j B j

= + +

+ +

+

( )( ) ( ) ( )

cos sin cos sin

2

1 ( ) cos sin cos sin

1 cos sin

+

( ) cos sin

2 1 sin cos sin cos

r A n B n

V A j B j V A n B n dA

(6)

where

+ + +

2 2 2

+

n r r r r n r

V r r r

r r r

n r r n r

V r r

r r r

n n nr r n r

V r r

r r r

= + =

2 2

2 2

= =

1 ( ) ( ) ( )

(1) ( ) ( ) ( )

1 ( ) ( )

( ) ( ) and

( ) ( )

( ) ( )

m m m mn m m m

m m mn m m

m m mn m m

(2)

2 2

= =

(3)

.

2 2

(7)

The Rayleigh quotient is given by

2 max

* max

T h W dA

1

2 A

= (1)

where is the mass density per unit volume, h is the thickness of the plate, and A represents the area over the plate. Also the maximum strain energy of the deformed circular plate Umax is

given by:

U

T

= (8)

where

2 * max max

T T

= . Applying the condition of stationarity of the natural frequencies with respect to the arbitrary constants Aij and Bij, we have:

2 2

0 and 0

2 2

max 2 2 2

2 2

2 2 2

2

1 1

2

1 1

= =

(9)

which results in an eigen-value problem as follows:

[ ] { }

{ } [ ]

2

D W W W

U r r r r

W W W r r r r

1 W dA

r r

= + +

+

+

A B

ij ij

A

( ) ( )

2 1

2 1

(2)

=

A A

B B

mn mn

mn mn

2 0 .

{ }

{ }

K M (10)

The elements of matrices [K] and [M] are obtained from the following equations:

( )

1 0

where is Poissons ratio, and D the flexural rigidity is:

= +

T r j r A n B n dA

A

= =

( )cos ( ) cos sin

i m mn mn m n

3

2 . 12 1

Eh

ij A

( )

D

= (3)

In the equation above, E is Youngs modulus. Assuming the deflection function as:

,

(11)

( )

1 0

= +

T r j r A n B n dA

B

= =

( )sin ( ) cos sin

i m mn mn m n

ij A

,

(12)

K. Saeedi et al. / Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448 1441

= + +

+

+

U V j V A n B n r j V A n B n

A

V j r A n B n

V j V

(1) (1) (2)

1 0 1 0

(2)

1 0

(3) (3

cos cos sin 1 ( )cos cos sin

1 cos ( ) cos sin

2 1 sin

ij mn mn mn i mn mn mn m n m n

( ) ( ) ( ) ( ) ( )( )

ij A

= = = =

= =

(13)

ij m mn mn m n

ij mn

) ( )

A n B n dA

sin cos

mn mn m n

= =

1 0

= + +

+

+

U V j V A n B n r j V A n B n

B

V j r A n B n

V j V

( ) ( ) ( ) ( ) ( )( )

(1) (1) (2)

1 0 1 0

(2)

1 0

(3) (3)

sin cos sin 1 ( )sin cos sin

1 sin ( ) cos sin

2 1 cos

ij mn mn mn i mn mn mn m n m n

ij A

= = = =

= =

(14)

+

ij m mn mn m n

ij mn

( )

A n B n dA

sin cos .

mn mn m n

= =

1 0

3. Generation of boundary characteristic orthogonal polynomials

The members of the boundary characteristic orthogonal polynomials set must satisfy the geometrical boundary conditions of the plate. The boundary conditions are ( ) ( ) 0

R R

= = for an outer edge-clamped plate. To satisfy the boundary conditions at the perimeter of the plate, the first member is considered as follows:

2 2 2 1 1

( ) ( )

r d r R

= . (15)

The higher order polynomials are created by following the Gram-Schmidt process as:

2 2 1 1

( ) ( ) ( )

r d r b r

= , (16)

[ ]

1 1 1 2

( ) ( ) ( ) ( ) 3

m m m m m m

r d r b r c r m

= (17)

where the coefficients bm and cm are defined such that the polynomials become orthogonal.

2 21 1( )

Fig. 1. Schematic layout of experimental setup.

R

m m

R

b r r dr

= , (18)

= . (19)

Also, the constant dm can be identified by normalizing the boundary characteristic orthogonal polynomials as:

R 2 ( ) 1

m

R r rdr

=

. (20)

Ri refers to the inner radius of the plate, and in the case where there is no hole at the center, it is equal to zero.

4. Experimental investigation

An exact solution using Bessel functions is not possible for disks with axisymmetrical multiple perforations located at

different distances from the center. Approximate solutions are obtained using the Rayleigh-Ritz method, and in order to validate these results, experimental investigations on perforated disks are performed. The objective of the experiment is to find the frequency response of plates with circular perforations positioned at locations corresponding to the various configurations addressed in the current study. Natural frequencies below 1000 Hz only are obtained. The setup consists of the disk assembly, the shaker, a signal conditioner, and an accelerometer. The disk assembly consists of a thick base-plate, two rings, and the disk. The rings sandwich the disk and are used to clamp the disk using a nut and bolt assembly. The shaker is excited from frequency 10 Hz to 1000 Hz. The vibration is picked up by an accelerometer, amplified, and fed to the signal analyzer. The FFT analysis of the amplified signal shows the peaks corresponding to the resonant frequencies.

The schematic of the experimental setup is shown in Fig. 1. An accelerometer is placed on the shaker and the disk assembly. The signal is amplified through an amplifier and fed to the

R

c r r r dr

2 ( ) ( )

m m m

R

1442 K. Saeedi et al. / Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448

signal conditioner (FFT Analyzer). The accelerometer can be used for measurements in the frequencies between 10 Hz to 8 KHz. The position of the accelerometer is crucial since care has to be taken to avoid positioning the accelerometer on a nodal line. Four disks with different perforation patterns are considered for the experimental investigation. Full circular disk Annular disk Disk with one perforation at its center and eight axisym-metric perforations close to the outer edge

Disk with one perforation at its center and eight axisym-metric perforations close to the center and 16 axisymmetric perforations close to the outer edgeThe experimental investigations were performed on four different disks made of aluminum sheet metal of the same thickness. The outer radius of the disks was 150 mm. All the perforations made in the disks had the same radius of 12.7 mm. The disks were clamped at the periphery and the edges of the perforations had free boundary conditions.

5. Numerical results

The results obtained using the Rayleigh-Ritz method are validated for some known cases with data from the open literature and also are validated experimentally. In the following, two popular case studies using a full circular plate and an annular plate are considered. The integrals in Eqs. (11) - (14) have been calculated for 0 2

Fig. 2. Geometry and coordinates of the hole vs. the reference as the

center of the plate.

Fig. 3. Integral boundaries of the hole.

and 0 r R

for the full circular plate. The boundaries for the annular plate are 0 2

and i

R r R

. Since these case studies are axi-symmetric, the deflection shape function can be reduced to

( )

1 0

W r r A n

( , ) ( ) cos

m mn m n

= . (21)

Furthermore, the following identity significantly reduces the amount of the calculation:

= =

= = . (26)

5.1 Full circular plate with clamped outer edge

The origin of the polar coordinate system is at the center of the circular plate. The boundary conditions possess symmetry with respect to the diameter of the circular plate. The deflection function in terms of Bessel functions and trigonometric functions is written as:

0 cos cos 0

2 n m d m n

=

. (22)

When there is an eccentric hole at the position (e, 0) on the plate (see Fig. 2), the amounts of kinetic and potential energies corresponding to the hole are calculated and subtracted from the total kinetic and potential energies of the plate, respectively. According to Fig. 3, kinetic and potential energies of a circular perforation off the center are calculated in the region

1 2

( ) ( )

r r r

and 0 0 .

+ It should be noted

that:

sin a

e

= , (23)

(27)

where the coefficients n

A

r r

W r A J C I n

R R

( ) ( ) ( ) ( )

, cos

n n n n n

= +

and n

C are determined from the boundary conditions. By applying the boundary conditions for a clamped plate and by using recurrence relations, the Bessel function of a higher order can be converted to the Bessel func-

2 2 21 0 0

( ) cos( ) sin ( )

r e a e

= , (24)

2 2 22 0 0

( ) cos( ) sin ( )

r e a e

= + . (25)

With R as the outer radius of the plate, the following dimensionless values are defined to help one compare the results obtained for the present research with the results given in the available literature.

4 2 4 2 hR

D

K. Saeedi et al. / Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448 1443

Table 1. Comparison of the dimensionless natural frequencies of a circular plate with clamped boundary.

n 0 1 2 3 Number

of

radial

nodes

Bessel R.R.* Exp.** Bessel RR Exp. Bessel R.R. Exp. Bessel R.R. Exp.

0 10.213 10.216 9.541 21.254 21.260 21.011 34.884 34.897 34.073 51.034 51.050 51.361 1 39.775 39.777 37.637 60.828 60.829 N/A 84.589 84.593 N/A 111.023 111.028 N/A 2 89.090 89.104 N/A*** 120.083 120.097 N/A 153.828 153.836 N/A 190.295 190.329 N/A 3 158.175 158.185 N/A 199.043 199.093 N/A 242.729 242.942 N/A 289.162 289.402 N/A * R.R.: Rayleigh-Ritz method results. M = 10 orthogonal polynomials were used in the Rayleigh-Ritz method.

** Exp.: Experiment results

*** Not available

Table 1. Comparison of the dimensionless natural frequencies of a circular annular plate (Ri /R = 0.085*) with clamped outer edge and free inner

edge.

n 0 1 2 3 Number

of

radial

nodes

Bessel R.R. Exp. Bessel R.R. Exp. Bessel R.R. Exp. Bessel R.R. Exp.

0 10.165 10.166 9.338 21.227 21.239 20.948 34.866 34.927 34.451 51.033 51.086 51.740 1 39.416 39.425 37.547 60.385 60.399 N/A 84.534 83.750 N/A 111.0231 111.8920 N/A 2 89.458 89.467 N/A 118.184 118.207 N/A 153.456 153.960 N/A 190.254 191.058 N/A 3 161.398 161.399 N/A 194.289 194.315 N/A 241.235 242.35 N/A 289.008 290.789 N/A * The value of Ri /R = 0.085 is chosen on the basis of the limitation offered by the experimental facility.

Fig. 4. Circular plate with 25 perforations in clamped free boundary

condition a/R = 0.085, e1 = 50 mm, e2 = 100 mm.

. (29)

Applying boundary conditions to both the clamped outer edge and the free inner edge, and using the identities of Bessel functions, the dimensionless natural frequencies can be obtained [1].

Table 2 shows the comparison of the results obtained by the Rayleigh-Ritz method and those from the experimental inves

tion of a lower order. The frequency equation is obtained as follows:

( ) ( ) ( ) ( )

1 1 0

n n n n

J I I J

,

r r

W r A J B Y

R R

r r

C I D K n

R R

( ) ( ) ( ) ( ) ( )

= +

n n n n n

n n n n

cos

+ = . (28)

Table 1 compares the results of the exact solution using Bessel functions with those obtained using the Rayleigh-Ritz method and the experiment results. The results using the Rayleigh-Ritz method are in good agreement with the exact

solution. Of course, a perfect clamping condition is not possible in an experimental setup. The disk is sandwiched between two rings made of Plexiglas. The entire setup is clamped to the base plate using 16 equally spaced clamps. The angular distance between these clamps is approximately 62.5 mm. The restriction offered by the clamping condition is not perfect as in the analytical model. This boundary condition, even though not a perfect clamping condition, provides a close approximation to the results provided by perfectly clamped boundary conditions.

5.2 Annular plate with outer edge clamped and inner edge free

A plate with an outer radius R and a concentric hole of radius Ri is considered. The deflection function corresponding to an annular plate can be written as follows:

+ +

1444 K. Saeedi et al. / Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448

(a) First mode shape (b) Second mode shape

(c) Third mode shape (d) Fourth mode shape

(e) Fifth mode shape

Fig. 5. First five modes of the annular plate with outer edge clamped and inner edge free.

tigation along with those from the Bessel functions formulation. The results are in good agreement. However, there are some deviations in experimental results caused by the imperfect clamping along the periphery of the disk. In addition, the mass of the accelerometer used to pick up the vibration, even though it is low (5grams), may also contribute to the deviation. Because of experimental limitations, it was impossible to per-

form experimental investigations for all the frequencies tabulated in Table 1. For validation purposes, comparing the first few frequencies is considered sufficient. Hence, only the first five frequencies are compared. Fig. 5 shows the mode shapes of these five frequencies, respectively. The first and fourth modes are axisymmetric, whereas the second, third, and fifth are antisymmetric.

K. Saeedi et al. / Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448 1445

Table 3. Comparison of the dimensionless natural frequencies of a

plate with one perforation in the center and eight eccentric perforations

close to the center.

Present method Experiment 1 10.1648 10.06992 21.5376 21.20073 34.9697 34.70364 39.9376 40.71755 51.7232 51.88616 60.8489 N/A7 69.2284 N/A8 73.1309 N/A9 84.0140 N/A10 89.4748 N/A

Table 4. Dimensionless natural frequencies of a circular plate with one

perforation in the center, eight eccentric perforations at e1 = 50 mm

and sixteen eccentric perforations at e2 = 100 mm.

Present method Experiment 1 10.5243 10.47022 21.8869 21.45313 35.5827 35.02034 39.8545 40.88715 51.7732 50.96846 61.1589 N/A7 71.1655 N/A8 85.5563 N/A9 86.7500 N/A10 93.9313 N/A

Fig. 6. Circular plate with 25 perforations in clamped free boundary

condition a/R = 0.085, e1 = 50 mm, e2 = 100 mm.

Fig. 7. Circular plate with 25 perforations in clamped free boundary

condition a/R = 0.085, e1 = 50 mm, e2 = 100 mm.

5.3 Disk with one perforation at its center and eight axi-symmetric perforations closer to the outer edge

This case study has nine perforations as shown in Fig. 6. Most of the perforations are placed close to the periphery of the plate. The presence of perforations provides a combined reduction of mass and stiffness. The results are shown in Table 3. As there is no published work on this case with which the results could be compared, the previously validated Rayleigh-Ritz method is used as a standard. As can be seen in Table 3, results from both the experiment and the Rayleigh-Ritz method are in good agreement. Comparison of the natural frequencies with those of the annular plate shows that the first five frequencies of both case studies are similar. However, the higher frequencies are smaller than the frequencies of the annular plate.

5.4 Disk with one perforation at its center and eight perforations at a distance of 50 mm from the center and 16 perforations at a distance of 100 mm from the center

Fig. 7 shows a plate with 25 perforations. This case study corresponds to the extension of the previous case study. The

plate has eight perforations distributed evenly at a distance of 50 mm from the center and 16 perforations distributed evenly at a distance of 100 mm from the center. This causes a change in mass as well as stiffness. The experimental results and the Rayleigh Ritz method results, tabulated in Table 4, are in good agreement. Fig. 8 shows the first five vibration modes of this case study obtained by the Rayleigh-Ritz method. Whereas the first frequencies which are similar to the frequencies of previous case studies, the higher frequencies are between the frequencies of the annular plate and those of the plate with nine perforations.

6. Conclusions

The approximate analysis using the Rayleigh-Ritz method is able to predict the natural frequencies of circular disks with perforations that are axisymmetrically arranged on the disk. Even though there are studies based on eccentric perforations, there is no study based on multiple perforations as addressed in the present work. Experiments are performed to validate the

1446 K. Saeedi et al. / Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448

(a) First mode shape (b) Second mode shape

(c) Third mode shape (d) Fourth mode shape

(e) Fifth mode shape

Fig. 8. First five modes of the plate with 25 perforations with outer edge clamped and inner edge free.

results of the Rayleigh-Ritz method.

The perforations perform a dual role of mass reduction and stiffness reduction, thereby influencing the frequencies. Although extending the number of perforations to a higher scale is possible with the presented method, it is not pursued with a

higher number of holes because of the complexity in the calculation of natural frequencies.

Further, if the application has some constraints in terms of frequency, the fundamental frequency and its higher harmonics could be trimmed as required by the application by adjust-

K. Saeedi et al. / Journal of Mechanical Science and Technology 26 (5) (2012) 1439~1448 1447

ing the number and location of the perforations. The same number of perforations can produce lower or higher resonant frequency based on radial distance.

The clamping points in the experiment have a significant influence on the resonant frequencies. The disk is clamped at 16 points by using nuts and bolts, and the rings provide additional support. It was confirmed that the non-compliance with the perfect clamping conditions has a more prominent impact on the higher modes than on the lower modes. The experimental results for the lower modes are in good agreement with the approximate analytical results when compared with that of the higher modes.

List of symbols

m

: mth Orthogonal polynomial : Mass density : Poissons ratio0 : Angular position of hole : Natural frequency of plate : Dimensionless frequencya : Radius of hole (except annular plate)e : Radial distance between center of plate and center of perforationsh : Thickness of plateA : Area of plateD : Flexural rigidityE : Youngs modulus [K] : Stiffness matrix [M] : Mass matrixM : Number of polynomialsN : Number of trigonometric functionsR : Outer radius of plateRi : Inner radius of annular plateTmax : Maximum kinetic energyUmax : Maximum strain energy of the deformed plate W(r,) : Deflection shape defined in polar coordinates

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Khodabakhsh Saeedi is a Ph.D student in the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Canada. He received his M.Sc. in Applied Mechanics from Sharif University of Technology, Tehran, Iran. His research interests include, vehicle dynamics, vibration of structures, and composite materials.

Alfin Leo is a Mechanical Engineer with a Doctoral degree in Microsystems Engineering for high temperature applications from Concordia University, Montreal, Canada. He is currently involved in R&D activities in the railroad division of Parker filtration Canada and in developing filtration systems for locomotive engines. Along with his publications he has filed a Patent in the area of locomotive exhaust filtration system. His specialization includes vibrations, high temperature microsystems, engine intake and exhaust filtration systems for rail road applications, fluid power systems and automation.

Rama Bhat is a Professor of Mechanical and Industrial Engineering at Concordia University, Montreal, Canada. He completed his Ph.D in Mechanical Engineering from IIT Madras, India, in 1972. Dr. Bhats research area covers mechanical vibrations, vehicle dynamics, structural acoustics, rotor dynamics, and dynamics of micro-electro-mechanical systems. He served as the president of Canadian Society for Mechanical Engineering in 2004-2006. He has been awarded the prestigious NASA Award for Technical Innovation for his contribution in developing PROSSSProgramming Structured Synthesis System. Dr. Bhat proposed the use of Boundary Characteristic Orthogonal Polynomials for use in the Rayleigh-Ritz method in 1985.

Ion Stiharu received Dipl. Eng. and Ph.D degrees from the Polytechnic University of Bucharest, Bucharest, Romania, in 1979 and 1989, respectively. He is currently a Professor and the Director of the CONCAVE Research Centre with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada, where he founded the micro-systems research back in 1990s. His research covers mainly micro-system technologies and MEMS. He has more than 350 publications along with few patents on the applications of MEMS to his credit Dr. Stiharu is a Fellow of the Canadian Society of Mechanical Engineers and the American Society of Mechanical Engineers ASME International. He has been part of many conference organizing committees and has been the organizer of a number of lecture series in MEMS and NEMS.