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Lowest natural frequency of clamped circular arcs of linearly tapered width
Journal
LOWEST
of Sound and Vibration
NATURAL
(1991)
FREQUENCY
M.
J. MAURIZI.
Department of Engineering,
144(2),
357-361
OF CLAMPED TAPERED WIDTH R. E. Rossr
CIRCULAR
AND
P. M.
ARCS
OF
LINEARLY
BELLES
Universidad National de1 Sur, 8000 Bahia Blanca, Argentina
(Receiued 30 March 1990) 1. INTRODUCTION
In a now classic paper [l] Den Hartog obtained approximate expressions for the first and second natural frequency coefficients of pinned and clamped circular arcs of uniform cross section subjected to in-plane vibrations. Since this was probably the first study in which the Rayleigh-Ritz method was applied in the dynamic analysis of arcs, one can say that Den Hartog paved the way to the solution of more complex dynamic problems of arch-type structures. In a recently published paper [2] Filipich and Laura discussed Den Hat-tog’s results at the light of the exact eigenvalues obtained by Filipich in his doctoral thesis [3]. The purpose of the present letter is to obtain the lowest frequency coefficients of symmetric and antisymmetric modes of in-plane vibrations of clamped, circular arcs of continuously varying cross-section (see Figure 1). It is assumed that the width of the
(a)
Figure 1. Structural system considered
in the present investigation
and executing
in-plane v&rations.
357 0022-460X/91/020357+05
%03.00/O
0
1991 Academic Press Limited
358
LETTERS
TO THE
EDITOR
cross-section of the structural element varies linearly with the angular co-ordinate while the thickness remains constant. Numerical values of the frequency coefficients are determined as a function of the governing geometric parameters: slenderness ratio, taper of the cross-sectional width and total arc angle. Two independent solutions are obtained, by using (a) the Rayleigh method coupled with Den Hartog’s procedure [l] in the case of extensional vibrations and the Rayleigh-Ritz approach when studying the antisymmetric mode, and (b) the finite element algorithmic procedure. 2.
ANALYTICAL
SOLUTION
Consider first the case of extensional vibrations. The strain and kinetic energy expressions for extensional vibrations may be written as
where E is the Young’s modulus, cy is the total angle of the arc, R is the middle line radius, S( &), is the cross-sectional area, Z(4) is the second moment of area of cross-section and ~(4) is the mass per unit length. The boundary conditions are u=w=dw/&p=O I$ = *ff/2, for where u and w are the tangential and radial displacement components, The following approximations will be used [l]:
(3) respectively.
u = u, = {a, sin (27rd~/c~)+ a2 sin (67r+/(~)} sin (of), w 2: w, = A(1 +cos (2@/cr)}
sin (wt).
(4a) (4b)
These satisfy the governing boundary conditions (3); w is the angular frequency, and A, a, and a, are constants. It is assumed that the cross-sectional width varies in accordance with the functional relation (see Figure 1) b(4) = b,f(+),
(5)
where_/-(+)= 1+2(b’/b,J(l-2(4I/a). Substituting expressions (4) and (5) into equations (1) and (2) yields U and T. As in reference [l], the constants a, and a2 are determined by minimizing the strain energy U. Equating the maximum values of the strain and kinetic energies one obtains a frequency equation of the type w = (C,I~*)(W
po)“‘,
(6)
where C, is a function of the governing mechanical and geometrical parameters. In the case of non-extensional vibrations of the arc one takes [ 11
0)
LETTERS
TO
THE
359
EDITOR
~=[(~-~)(9-(*-1,“)(~)+(2,~-~)(~)~+2s($)’]sin(ol),
(7b)
and obtains the frequency parameter (8)
w = (C.;/~2)(WP”PZ,
by making use of the Rayleigh-Ritz method. Obviously, C,>is a function of the mechanical and geometric parameters which come into play. 3.
NUMERICAL
RESULTS
AND
CONCLUSIONS
In order to apply the finite element algorithm the semi-arc was divided into 12 elements. Shear and rotatory inertia effects were neglected. In Figures 2(a) and 2(b) are depicted the values of the frequency coefficients C, and C; corresponding to the symmetric and antisymmetric modes of vibration for b’/ b, = +0.4 and b’/ b. = -0.4. The values of both frequency coefficients of arcs of non-uniform width are smaller than the corresponding frequency parameters of uniform arcs when b’/ 6, > 0. On the other hand, they are larger when b’/ b, -C 0. In a similar manner as was shown by Den Hartog [ 11, depending upon the mechanical parameters R/k and (Y,the numerical values of the frequency coefficients will define which configuration corresponds to the fundamental mode and which to the second normal mode. 70. (0)
R/k
10 20
40
60
80
(symmetric
100
120
mode)
140
160
180
a (degrees) Figure 2. Frequency f0.4; (b) b’/ b, = -0.4.
coefficients
corresponding
to the lowest symmetric
and antisymmetric
mode. (a) b’/ b, =
360
LETTERS
TO
TABLE
Values
EDITOR
THE
1
of acrdetermined by using the analytical solution ( CX~,is the total angle qf the arc and for which C, = C.;)
b’l b,
20
40
60
80
100
-0.4 -0.3
98.2 99.3
70.9
58.3
50.7
45.5
71.8 72.5 73.1 73.5 74.0 74.3 74.6 74.9
59.1 59.7 60.2 60.6 61.0 61.3 61.6 61.8
51.4 51.9 52.4 52.8 53.1 53.4 53.6 53.8
46.1 46,6 47.0 47.4 47.7 47.9 48.2 48.3
-0.2 -0.1 0.0 +0*1 +0.2 $0.3 +0.4
100.1 100.7 101.3 101.8 102.2 102.5 102.9
Table 1 defines the value of CYsuch that, for prescribed values of values of both frequency coefficients coincide. In view of the peculiar of the total angle, it is suggested to define it as a “critical” value of A close inspection of Table 1 reveals that as b’/b, varies from parameter (Y,,experiences a very small increment (approximately 3% of the order of 6% when R/k = 100).
b’/b, and R/k, the nature of this value (Y,(Y,,. -0.4 to +0.4, the when Rl k = 20 and
300
250
,A--
R/k = 80,/1’
/
/
I
I/
Symmetric
Antisymmeh
/ 0
mode
mode
I
I
1
I
I
I
I
/
20
40
60
80
100
120
140
160
1
a (degrees) Figure 3. Comparison of the analytical (---) and finite element (---) solutions for the lowest frequencies corresponding to the symmetric and antisymmetric modes (constant width arc). Note: in the case of the antisymmetric mode both solutions coincide.
LETTERS
TO THE
361
EDITOR
Interesting features of the dynamic behavior of the arcs investigated in the present study and of the methodologies employed can be observed in Figure 3 where the case of an arc of uniform width is considered. In the case of the antisymmetric mode both methodologies, the analytical and the finite element algorithmic solution, coincide from a practical engineering viewpoint (see Table 2), while when considering the symmetric, extensional mode one observes that the approximate analytical solution yields results which are extremely high for R/k = 80 when (Y> 60”. On the other hand, both sets of results deviate considerably for LY> 100” when R/k = 20.
TABLE
Comparison
of values of the frequency
the Rayleigh-Ritz
method
2
coeficient
(RRM)
C’j for b’/ b,, = 0 as obtained
and (b) the jinite element
method
by using (a) (FEM)
a (in degrees) Methods
20
40
60
80
100
120
140
160
180
RRM FEM
60.75 60.74
58.03 58.01
53.76 53.75
48.30 48.31
42.09 42.11
35.55 35.58
29.08 29.11
23.00 23.03
17.54 17.58
An important practical conclusion of the present study is the fact that it has been possible to define the range of validity, from the point of view of a reasonable engineering accuracy, of the approximating co-ordinate functions first used by Den Hartog to represent the displacement field in the case of the lowest symmetric (extensional) mode of vibration of a circular arc and which have also been used, in the present study, in the case of a vibrating arc of non-uniform width.
ACKNOWLEDGMENTS
has been sponsored by CONICET Research and Development The present investigation Program (PID 3000500/88). The authors are indebted to Dr Patricia A. A. Laura for his helpful suggestions and constructive advice. The work of tile third author was supported by a grant from the CIC (Comisi6n de Investigaciones Cientificas de la Provincia de Buenos Aires).
REFERENCES
J. P. DEN HARTOG 1928 Philosophical Magazine Series 7, 5, 400-408. The lowest natural frequency of circular arcs. C. P. FILIPICH and P. A. A. LAURA 1988Journal of Sound and Vibration 125, 393-396. First and second natural frequencies of hinged and clamped circular arcs: a discussion of a classical paper. C. P. FILIPICH 1988Ph.D. Thesis, Unioersidad Nacionalde Cbrdoba, Grdoba, Argentina. In-plane vibrations of arches and rings taking into account shear and rotatory inertia effects.
LOWEST
of Sound and Vibration
NATURAL
(1991)
FREQUENCY
M.
J. MAURIZI.
Department of Engineering,
144(2),
357-361
OF CLAMPED TAPERED WIDTH R. E. Rossr
CIRCULAR
AND
P. M.
ARCS
OF
LINEARLY
BELLES
Universidad National de1 Sur, 8000 Bahia Blanca, Argentina
(Receiued 30 March 1990) 1. INTRODUCTION
In a now classic paper [l] Den Hartog obtained approximate expressions for the first and second natural frequency coefficients of pinned and clamped circular arcs of uniform cross section subjected to in-plane vibrations. Since this was probably the first study in which the Rayleigh-Ritz method was applied in the dynamic analysis of arcs, one can say that Den Hartog paved the way to the solution of more complex dynamic problems of arch-type structures. In a recently published paper [2] Filipich and Laura discussed Den Hat-tog’s results at the light of the exact eigenvalues obtained by Filipich in his doctoral thesis [3]. The purpose of the present letter is to obtain the lowest frequency coefficients of symmetric and antisymmetric modes of in-plane vibrations of clamped, circular arcs of continuously varying cross-section (see Figure 1). It is assumed that the width of the
(a)
Figure 1. Structural system considered
in the present investigation
and executing
in-plane v&rations.
357 0022-460X/91/020357+05
%03.00/O
0
1991 Academic Press Limited
358
LETTERS
TO THE
EDITOR
cross-section of the structural element varies linearly with the angular co-ordinate while the thickness remains constant. Numerical values of the frequency coefficients are determined as a function of the governing geometric parameters: slenderness ratio, taper of the cross-sectional width and total arc angle. Two independent solutions are obtained, by using (a) the Rayleigh method coupled with Den Hartog’s procedure [l] in the case of extensional vibrations and the Rayleigh-Ritz approach when studying the antisymmetric mode, and (b) the finite element algorithmic procedure. 2.
ANALYTICAL
SOLUTION
Consider first the case of extensional vibrations. The strain and kinetic energy expressions for extensional vibrations may be written as
where E is the Young’s modulus, cy is the total angle of the arc, R is the middle line radius, S( &), is the cross-sectional area, Z(4) is the second moment of area of cross-section and ~(4) is the mass per unit length. The boundary conditions are u=w=dw/&p=O I$ = *ff/2, for where u and w are the tangential and radial displacement components, The following approximations will be used [l]:
(3) respectively.
u = u, = {a, sin (27rd~/c~)+ a2 sin (67r+/(~)} sin (of), w 2: w, = A(1 +cos (2@/cr)}
sin (wt).
(4a) (4b)
These satisfy the governing boundary conditions (3); w is the angular frequency, and A, a, and a, are constants. It is assumed that the cross-sectional width varies in accordance with the functional relation (see Figure 1) b(4) = b,f(+),
(5)
where_/-(+)= 1+2(b’/b,J(l-2(4I/a). Substituting expressions (4) and (5) into equations (1) and (2) yields U and T. As in reference [l], the constants a, and a2 are determined by minimizing the strain energy U. Equating the maximum values of the strain and kinetic energies one obtains a frequency equation of the type w = (C,I~*)(W
po)“‘,
(6)
where C, is a function of the governing mechanical and geometrical parameters. In the case of non-extensional vibrations of the arc one takes [ 11
0)
LETTERS
TO
THE
359
EDITOR
~=[(~-~)(9-(*-1,“)(~)+(2,~-~)(~)~+2s($)’]sin(ol),
(7b)
and obtains the frequency parameter (8)
w = (C.;/~2)(WP”PZ,
by making use of the Rayleigh-Ritz method. Obviously, C,>is a function of the mechanical and geometric parameters which come into play. 3.
NUMERICAL
RESULTS
AND
CONCLUSIONS
In order to apply the finite element algorithm the semi-arc was divided into 12 elements. Shear and rotatory inertia effects were neglected. In Figures 2(a) and 2(b) are depicted the values of the frequency coefficients C, and C; corresponding to the symmetric and antisymmetric modes of vibration for b’/ b, = +0.4 and b’/ b. = -0.4. The values of both frequency coefficients of arcs of non-uniform width are smaller than the corresponding frequency parameters of uniform arcs when b’/ 6, > 0. On the other hand, they are larger when b’/ b, -C 0. In a similar manner as was shown by Den Hartog [ 11, depending upon the mechanical parameters R/k and (Y,the numerical values of the frequency coefficients will define which configuration corresponds to the fundamental mode and which to the second normal mode. 70. (0)
R/k
10 20
40
60
80
(symmetric
100
120
mode)
140
160
180
a (degrees) Figure 2. Frequency f0.4; (b) b’/ b, = -0.4.
coefficients
corresponding
to the lowest symmetric
and antisymmetric
mode. (a) b’/ b, =
360
LETTERS
TO
TABLE
Values
EDITOR
THE
1
of acrdetermined by using the analytical solution ( CX~,is the total angle qf the arc and for which C, = C.;)
b’l b,
20
40
60
80
100
-0.4 -0.3
98.2 99.3
70.9
58.3
50.7
45.5
71.8 72.5 73.1 73.5 74.0 74.3 74.6 74.9
59.1 59.7 60.2 60.6 61.0 61.3 61.6 61.8
51.4 51.9 52.4 52.8 53.1 53.4 53.6 53.8
46.1 46,6 47.0 47.4 47.7 47.9 48.2 48.3
-0.2 -0.1 0.0 +0*1 +0.2 $0.3 +0.4
100.1 100.7 101.3 101.8 102.2 102.5 102.9
Table 1 defines the value of CYsuch that, for prescribed values of values of both frequency coefficients coincide. In view of the peculiar of the total angle, it is suggested to define it as a “critical” value of A close inspection of Table 1 reveals that as b’/b, varies from parameter (Y,,experiences a very small increment (approximately 3% of the order of 6% when R/k = 100).
b’/b, and R/k, the nature of this value (Y,(Y,,. -0.4 to +0.4, the when Rl k = 20 and
300
250
,A--
R/k = 80,/1’
/
/
I
I/
Symmetric
Antisymmeh
/ 0
mode
mode
I
I
1
I
I
I
I
/
20
40
60
80
100
120
140
160
1
a (degrees) Figure 3. Comparison of the analytical (---) and finite element (---) solutions for the lowest frequencies corresponding to the symmetric and antisymmetric modes (constant width arc). Note: in the case of the antisymmetric mode both solutions coincide.
LETTERS
TO THE
361
EDITOR
Interesting features of the dynamic behavior of the arcs investigated in the present study and of the methodologies employed can be observed in Figure 3 where the case of an arc of uniform width is considered. In the case of the antisymmetric mode both methodologies, the analytical and the finite element algorithmic solution, coincide from a practical engineering viewpoint (see Table 2), while when considering the symmetric, extensional mode one observes that the approximate analytical solution yields results which are extremely high for R/k = 80 when (Y> 60”. On the other hand, both sets of results deviate considerably for LY> 100” when R/k = 20.
TABLE
Comparison
of values of the frequency
the Rayleigh-Ritz
method
2
coeficient
(RRM)
C’j for b’/ b,, = 0 as obtained
and (b) the jinite element
method
by using (a) (FEM)
a (in degrees) Methods
20
40
60
80
100
120
140
160
180
RRM FEM
60.75 60.74
58.03 58.01
53.76 53.75
48.30 48.31
42.09 42.11
35.55 35.58
29.08 29.11
23.00 23.03
17.54 17.58
An important practical conclusion of the present study is the fact that it has been possible to define the range of validity, from the point of view of a reasonable engineering accuracy, of the approximating co-ordinate functions first used by Den Hartog to represent the displacement field in the case of the lowest symmetric (extensional) mode of vibration of a circular arc and which have also been used, in the present study, in the case of a vibrating arc of non-uniform width.
ACKNOWLEDGMENTS
has been sponsored by CONICET Research and Development The present investigation Program (PID 3000500/88). The authors are indebted to Dr Patricia A. A. Laura for his helpful suggestions and constructive advice. The work of tile third author was supported by a grant from the CIC (Comisi6n de Investigaciones Cientificas de la Provincia de Buenos Aires).
REFERENCES
J. P. DEN HARTOG 1928 Philosophical Magazine Series 7, 5, 400-408. The lowest natural frequency of circular arcs. C. P. FILIPICH and P. A. A. LAURA 1988Journal of Sound and Vibration 125, 393-396. First and second natural frequencies of hinged and clamped circular arcs: a discussion of a classical paper. C. P. FILIPICH 1988Ph.D. Thesis, Unioersidad Nacionalde Cbrdoba, Grdoba, Argentina. In-plane vibrations of arches and rings taking into account shear and rotatory inertia effects.