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Modes for the coupled Timoshenko model with a restrained end
ARTICLE IN PRESS JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 296 (2006) 1053–1058 www.elsevier.com/locate/jsvi
Short Communication
Modes for the coupled Timoshenko model with a restrained end J.R. Claeyssena,, S.N.J. Costab a
Instituto de Matema´tica-Promec, Universidade Federal do Rio Grande do Sul, P.O. Box 10673, 90.001-970 Porto Alegre, RS-Brazil b Instituto de Matema´tica-PPGMAp, Universidade Federal do Rio Grande do Sul, 90.001-970 Porto Alegre, RS-Brazil Received 4 October 2004; received in revised form 23 January 2006; accepted 21 February 2006 Available online 14 June 2006
Abstract The modes of the second-order Timoshenko system for the displacement and rotation of a fixed beam with a restrained end at the left are formulated in terms of a fundamental spatial response. This is done without decoupling the system into fourth-order scalar equations. The restrained end leads to time–space boundary conditions which introduce the frequency as a parameter into the system of equations for determining the modes. These equations involve first-order derivatives and, consequently, the modes are determined by solving a non-conservative differential system. This modal differential equation is discussed in terms of a fundamental matrix response. It is determined by applying a closed formula that was obtained by the first author and involves the characteristic polynomial of the modal differential equation. r 2006 Elsevier Ltd. All rights reserved.
1. Introduction Vibrations of beams according to the Timoshenko theory has been considered by several authors [1–4]. Here, we characterize the modes of the Timoshenko model for the displacement and rotation of a fixed beam with a mass welded to the other end and attached to a translational spring [5]. In the literature, the Timoshenko model is usually decoupled into two fourth-order equations of the same type [6,7]. The frequency equation and mode shapes are then formulated in terms of the classical Euler basis. This later is constructed in terms of the roots of the associated characteristic polynomial of a fourth-order differential equation. The involved constants in the solution modes are then determined by substitution into the original coupled system and boundary conditions. Another approach has been given in Refs. [8–11]. It consists in using a basis generated by a fundamental matrix response which is characterized by given initial conditions. This avoids the need of decoupling the second-order Timoshenko system. It is kept as a system formed by two coupled equations of second order. The eigenanalysis is studied in terms of such fundamental response. It turns out that the amplitude vector of a mode satisfies a non-conservative second-order differential equation. As a basis of solutions, we choose the one that is generated by a fundamental matrix response and its derivative. This response can be computed in closed-form through a formula derived in Ref. [8]. See also Ref. [9] or Ref. [10] for mth-order systems in continuous and discrete time. Corresponding author.
E-mail addresses: [email protected] (J.R. Claeyssen), [email protected] (S.N.J. Costa). 0022-460X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2006.02.025
ARTICLE IN PRESS 1054
J.R. Claeyssen, S.N.J. Costa / Journal of Sound and Vibration 296 (2006) 1053–1058
After determining the modes for the coupled Timoshenko equations, we observe that the components of the modes satisfy the fourth-order equation that is obtained by decoupling the system. 2. Formulation of the problem In Clark [1] or Ginsberg [5], the unforced coupled equations of the Timoshenko model f or the total deflection u and the bending slope c of a uniform beam are derived and given by q2 uðt; xÞ q quðt; xÞ rA cðt; xÞ ¼ 0, kGA qt2 qx qx q2 cðt; xÞ q2 cðt; xÞ quðt; xÞ cðt; xÞ ¼ 0. rI EI kGA qt2 qx2 qx
(1)
Here r is the mass density, A the cross-sectional area, rA the mass per unit length, rI the mass moment of inertia per unit length about the neutral axis which passes through the center, E the modulus of elasticity, G the shear modulus, and k a numerical factor depending on the shape of the cross-section. When a cube of mass m and side dimensions b is welded to the end of a left end clamped beam of length L, and a spring of stiffness k is attached to the cube, the boundary conditions, that include translational and rotatory inertia of the cube as shown in (Fig. 1), are given by uðt; 0Þ ¼ 0, cðt; 0Þ ¼ 0, qu b b€ kGA c ðt; LÞ þ k u þ c þ m u€ þ c ðt; LÞ ¼ 0, qx 2 2 qc b qu 1 € LÞ ¼ 0. EI kGA c þ mb2 cðt; qx 2 qx 6 3. Eigenanalysis In order to determine the eigenfunctions of above system, we substitute the functions u ¼ eiot UðxÞ, c ¼ e CðxÞ into the equations given in Eq. (1). This results in the system iot
kGAU 00 kGAC0 þ rAo2 U ¼ 0, EIC00 þ kGAU 0 þ ðrIo2 kGAÞC ¼ 0.
Fig. 1. A uniform cantilever beam with a restrained end.
(2)
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By substituting the modal functions into the boundary conditions, we have Uð0Þ ¼ 0, Cð0Þ ¼ 0, kb mbo2 0 2 kGA CðLÞ ¼ 0, kGAU ðLÞ þ ðk o mÞUðLÞ þ 2 2 kGAb 0 kGAb mb2 o2 U ðLÞ þ EIC0 ðLÞ þ CðLÞ ¼ 0. 2 2 6
(3)
ð4Þ
In matrix form, it follows that MX00 ðxÞ þ CX0 ðxÞ þ KðoÞXðxÞ ¼ 0,
(5)
where X denotes the vector of the amplitudes U, C for the deflection and the bending slope, respectively, and kGA 0 0 kGA M¼ ; C¼ , 0 EI kGA 0 " # rAo2 0 KðoÞ ¼ . 0 rIo2 kGA We observe that the effect of the shear stress introduces the matrix coefficient C. The frequency appears in the matrix coefficient K as it would be with an Euler–Bernoulli beam. In matrix form, the boundary conditions are written as A11 Xð0Þ þ B11 X0 ð0Þ ¼ 0, A21 XðLÞ þ B21 X0 ðLÞ ¼ 0, where A11 is the 2 2 identity matrix, B11 is the 2 2 zero matrix and 2 3 b 2 2 2 ðk mo m kGA þ Þ k o kGA 6 7 2 6 7 4 kGAb ; B ¼ A21 ¼ 6 7 2 21 b o mb 4 5 kGA 0 2 2 3
(6)
0
3
EI
5.
3.1. Solution basis The general solution of Eq. (5) in terms of initial values can be written as X ðxÞ ¼ h0 ðxÞX ð0Þ þ h1 ðxÞX 0 ð0Þ, h0 ðxÞ ¼ h0 ðxÞM þ hðxÞC, h1 ðxÞ ¼ hðxÞM,
(7)
X ðxÞ ¼ hðxÞc1 þ h0 ðxÞc2
(8)
or for arbitrary vectors c1 and c2 . Here hðxÞ is the solution of the initial value problem Mh00 ðxÞ þ Ch0 ðxÞ þ KðoÞhðxÞ ¼ 0, hð0Þ ¼ 0;
Mh0 ð0Þ ¼ I,
where 0 denotes the 2 2 null matrix and I the 2 2 identity matrix.
(9) (10)
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For determining hðxÞ we use the formula derived in Ref. [8]. We first consider the characteristic polynomial PðsÞ ¼ det½s2 M þ sC þ K ¼
4 X
bk s4k .
k¼0
It turns out that b0 ¼ ab;
b1 ¼ 0;
b2 ¼ aeo2 þ co2 b;
b3 ¼ 0;
b4 ¼ co4 e co2 a,
where a ¼ kGA;
b ¼ EI;
c ¼ rA;
e ¼ rI.
Then we find the solution hðxÞ of the initial value problem 4 X
bk hð4kÞ ðxÞ ¼ b0 hðivÞ ðxÞ þ b2 h00 ðxÞ þ b4 hðxÞ ¼ 0,
k¼0
hð0Þ ¼ h0 ð0Þ ¼ h00 ð0Þ ¼ 0;
b0 h000 ð0Þ ¼ 1
(11)
and we let hk be the solution of the matrix difference equation Mhkþ2 þ Chkþ1 þ Khk ¼ 0, h0 ¼ 0;
Mh1 ¼ I.
(12)
bi hðj1iÞ ðxÞh4j .
(13)
We have hðxÞ ¼
j1 4 X X j¼1 i¼0
The calculations run as follows. The characteristic differential Eq. (11) is abhðivÞ ðxÞ þ ðae þ cbÞo2 h00 ðxÞ þ ðeo2 aÞco2 hðxÞ ¼ 0, hð0Þ ¼ h0 ð0Þ ¼ h00 ð0Þ ¼ 0;
abh000 ð0Þ ¼ 1.
(14)
The characteristic polynomial can be conveniently written as PðsÞ ¼ abðs4 þ g2 s2 r4 Þ, where g2 ¼ ðe=b þ c=aÞo2 ;
r4 ¼ co2 ðeo2 þ aÞ=ab.
The roots of the characteristic polynomial PðsÞ are then easily found to be s ¼ ; ; id; id, where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1=2 2g2 þ 2 ðg4 þ 4r4 Þ; d ¼ ðg2 þ 2 Þ. Thus g2 ¼ d2 2 ;
r4 ¼ d2 2 .
By using the Euler basis or the Laplace transform method, we find that the solution of the initial value problem (11) is hðxÞ ¼
d sinhðxÞ sinðdxÞ . abð2 þ d2 Þd
(15)
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The following matrix values are obtained by iterating the initial value problem (12): 2 3 1 0 6a 7 0 0 7, h0 ¼ ; h1 ¼ 6 4 5 1 0 0 0 b 2
6 0 h2 ¼ 6 4 1 b Then
" hðxÞ ¼
3 1 b7 7; 5 0
2 6 6 h3 ¼ 6 4
a2 þ co2 b a2 b
3 0
7 7 7. eo 5 2 b 2
0
bh00 ðxÞ þ ðeo2 aÞhðxÞ
ah0 ðxÞ
ah0 ðxÞ
ah00 ðxÞ þ hðxÞco2
# .
(16)
3.2. Characteristic equation We now proceed to determine the characteristic equation that will give the frequencies for which the modes XðxÞ are to be determined. Since the left end of the beam is fixed, the choice of the matrix basis h; h0 should simplify computations at such end. The substitution of the general solution XðxÞ ¼ hðxÞc1 þ h0 ðxÞc2 ;
0pxpL,
(17)
where c1 , c2 are 2 1 vectors, into the boundary conditions (6) leads to the linear system hð0Þc1 þ h0 ð0Þc2 ¼ 0:c1 þ M1 c2 ¼ 0, ðA21 hðLÞ þ B21 h0 ðLÞÞc1 þ ðA21 h0 ðLÞ þ B21 h00 ðLÞÞc2 ¼ 0. It follows that c2 ¼ 0 and the characteristic equation is detðDðoÞÞ ¼ 0,
(18)
DðoÞ ¼ A21 hðLÞ þ B21 h0 ðLÞ.
(19)
where 0
Here the values hðLÞ, h ðLÞ are to be computed from Eq. (16) and the coefficients A21 , B21 are given in Eq. (6). In particular, by using the Euler basis that leads to Eq. (15), we have 2 3 dA sinhðxÞ B sinðdxÞ coshðxÞ cosðdxÞ 6 7 abð2 þ d2 Þd bð2 þ d2 Þ 6 7 , (20) hðxÞ ¼ 6 coshðxÞ cosðdxÞ dC sinhðxÞ D sinðdxÞ 7 4 5 2 2 bð2 þ d Þ abð2 þ d Þd where A ¼ eo2 þ b2 a, B ¼ eo2 bd2 a, C ¼ co2 þ 2 a, D ¼ c o 2 d2 a and the involved parameters and constants are as defined above. It is clear that for each root of the characteristic Eq. (19), we have the mode Xðx; oÞ ¼ hðx; oÞc1 ,
(21)
ARTICLE IN PRESS J.R. Claeyssen, S.N.J. Costa / Journal of Sound and Vibration 296 (2006) 1053–1058
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where c1 is a non-zero solution of Dc ¼ 0. The dependence of the characteristic equation as well as the modes upon frequency o have been emphasized. When the frequency is a double one, we actually have two modes associated with such frequency. In this situation each column of the matrix h will be then a mode for the coupled Timoshenko model. From Eqs. (16) and (21), we have the displacement mode Uðx; oÞ ¼ aðbh00 ðx; oÞ þ ðeo2 aÞhðx; oÞÞ þ bðah0 ðx; oÞÞ
(22)
Rðx; oÞ ¼ aðah0 ðx; oÞÞ þ bðah00 ðx; oÞ þ co2 hðx; oÞÞ
(23)
and the rotation mode
for scalar constants a and b that both are not simultaneously zero. Here hðx; oÞ is the solution (15) for each root o of the characteristic Eq. (19). We can that verify that Uðx; oÞ and Rðx; oÞ satisfy the characteristic fourth-order differential Eq. (14), that is abyðivÞ ðxÞ þ ðae þ cbÞo2 y00 ðxÞ þ ðeo2 aÞco2 yðxÞ ¼ 0
(24)
or equations that are obtained by differentiating once or twice this later equation. We should observe that this later equation, after dividing by the factor ab, is the fourth-order spatial equation that arises for the amplitude by substitution into the time fourth-order equation that is obtained decoupling system (2), that is q4 w q2 w q 2 y q4 w þ ¼ 0, (25) a2 t2 4 þ b2 ða2 þ t2 Þ 2 qt qx qt2 qx4 where a2 ¼
r ; gkG
b2 ¼
rA ; gEI
t2 ¼
r . gE
(26)
This equation have been also obtained by a variational argument [12] where the frequency spectrum is discussed too. Acknowledgments This work was supported by CNPq and Fapergs. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
S. Clark, Dynamics of Continuous Elements, Prentice-Hall, Englewood Cliffs, NJ, 1972. G.B. Warburton, The Dynamical Behaviour of Structures, Pergamon Press, Oxford, 1976. L. Meirovitch, Principles and Techniques of Vibrations, Prentice-Hall, Englewood Cliffs, NJ, 1997. S.M. Han, H. Benaroya, T. Wei, Dynamics of transversally vibrating beams using four engineering theories, Journal of Sound and Vibration 225 (5) (1999) 935–988. J. Ginsberg, Mechanical and Structural Vibrations, Wiley, New York, 2001. T.C. Huang, The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, Journal of Applied Mechanics 28 (1961) 579–584. M.W.D. White, G.R. Hapler, Vibration modes and frequencies of Timoshenko beams with attached rigid bodies, Journal of Applied Mechanics 62 (1995) 193–199. J.C.R. Claeyssen, On predicting the response of non-conservative linear vibrating systems by using dynamical matrix solutions, Journal of Sound and Vibration 140 (1) (1990) 73–84. J.R. Claeyssen, T. Tsukazan, Dynamical solutions of linear matrix differential equations, Quarterly of Applied Mathematics XLVIII (1) (1990). J.C.R. Claeyssen, G. Canahualpa, C. Jung, Applied Numerical Mathematics 30 (1999) 65–78. J.R. Claeyssen, L.D. Chiwiacowsky, G.S. Suazo, The impulse response in the symbolic computating of modes for beams and plates, Applied Numerical Mathematics 40 (2002) 119–135. V.V. Nestrenko, A theory for transverse vibrations of the Timoshenko beam, Journal of Applied Mathematics and Mechanics 57 (4) (1993) 669–677.
Short Communication
Modes for the coupled Timoshenko model with a restrained end J.R. Claeyssena,, S.N.J. Costab a
Instituto de Matema´tica-Promec, Universidade Federal do Rio Grande do Sul, P.O. Box 10673, 90.001-970 Porto Alegre, RS-Brazil b Instituto de Matema´tica-PPGMAp, Universidade Federal do Rio Grande do Sul, 90.001-970 Porto Alegre, RS-Brazil Received 4 October 2004; received in revised form 23 January 2006; accepted 21 February 2006 Available online 14 June 2006
Abstract The modes of the second-order Timoshenko system for the displacement and rotation of a fixed beam with a restrained end at the left are formulated in terms of a fundamental spatial response. This is done without decoupling the system into fourth-order scalar equations. The restrained end leads to time–space boundary conditions which introduce the frequency as a parameter into the system of equations for determining the modes. These equations involve first-order derivatives and, consequently, the modes are determined by solving a non-conservative differential system. This modal differential equation is discussed in terms of a fundamental matrix response. It is determined by applying a closed formula that was obtained by the first author and involves the characteristic polynomial of the modal differential equation. r 2006 Elsevier Ltd. All rights reserved.
1. Introduction Vibrations of beams according to the Timoshenko theory has been considered by several authors [1–4]. Here, we characterize the modes of the Timoshenko model for the displacement and rotation of a fixed beam with a mass welded to the other end and attached to a translational spring [5]. In the literature, the Timoshenko model is usually decoupled into two fourth-order equations of the same type [6,7]. The frequency equation and mode shapes are then formulated in terms of the classical Euler basis. This later is constructed in terms of the roots of the associated characteristic polynomial of a fourth-order differential equation. The involved constants in the solution modes are then determined by substitution into the original coupled system and boundary conditions. Another approach has been given in Refs. [8–11]. It consists in using a basis generated by a fundamental matrix response which is characterized by given initial conditions. This avoids the need of decoupling the second-order Timoshenko system. It is kept as a system formed by two coupled equations of second order. The eigenanalysis is studied in terms of such fundamental response. It turns out that the amplitude vector of a mode satisfies a non-conservative second-order differential equation. As a basis of solutions, we choose the one that is generated by a fundamental matrix response and its derivative. This response can be computed in closed-form through a formula derived in Ref. [8]. See also Ref. [9] or Ref. [10] for mth-order systems in continuous and discrete time. Corresponding author.
E-mail addresses: [email protected] (J.R. Claeyssen), [email protected] (S.N.J. Costa). 0022-460X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2006.02.025
ARTICLE IN PRESS 1054
J.R. Claeyssen, S.N.J. Costa / Journal of Sound and Vibration 296 (2006) 1053–1058
After determining the modes for the coupled Timoshenko equations, we observe that the components of the modes satisfy the fourth-order equation that is obtained by decoupling the system. 2. Formulation of the problem In Clark [1] or Ginsberg [5], the unforced coupled equations of the Timoshenko model f or the total deflection u and the bending slope c of a uniform beam are derived and given by q2 uðt; xÞ q quðt; xÞ rA cðt; xÞ ¼ 0, kGA qt2 qx qx q2 cðt; xÞ q2 cðt; xÞ quðt; xÞ cðt; xÞ ¼ 0. rI EI kGA qt2 qx2 qx
(1)
Here r is the mass density, A the cross-sectional area, rA the mass per unit length, rI the mass moment of inertia per unit length about the neutral axis which passes through the center, E the modulus of elasticity, G the shear modulus, and k a numerical factor depending on the shape of the cross-section. When a cube of mass m and side dimensions b is welded to the end of a left end clamped beam of length L, and a spring of stiffness k is attached to the cube, the boundary conditions, that include translational and rotatory inertia of the cube as shown in (Fig. 1), are given by uðt; 0Þ ¼ 0, cðt; 0Þ ¼ 0, qu b b€ kGA c ðt; LÞ þ k u þ c þ m u€ þ c ðt; LÞ ¼ 0, qx 2 2 qc b qu 1 € LÞ ¼ 0. EI kGA c þ mb2 cðt; qx 2 qx 6 3. Eigenanalysis In order to determine the eigenfunctions of above system, we substitute the functions u ¼ eiot UðxÞ, c ¼ e CðxÞ into the equations given in Eq. (1). This results in the system iot
kGAU 00 kGAC0 þ rAo2 U ¼ 0, EIC00 þ kGAU 0 þ ðrIo2 kGAÞC ¼ 0.
Fig. 1. A uniform cantilever beam with a restrained end.
(2)
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By substituting the modal functions into the boundary conditions, we have Uð0Þ ¼ 0, Cð0Þ ¼ 0, kb mbo2 0 2 kGA CðLÞ ¼ 0, kGAU ðLÞ þ ðk o mÞUðLÞ þ 2 2 kGAb 0 kGAb mb2 o2 U ðLÞ þ EIC0 ðLÞ þ CðLÞ ¼ 0. 2 2 6
(3)
ð4Þ
In matrix form, it follows that MX00 ðxÞ þ CX0 ðxÞ þ KðoÞXðxÞ ¼ 0,
(5)
where X denotes the vector of the amplitudes U, C for the deflection and the bending slope, respectively, and kGA 0 0 kGA M¼ ; C¼ , 0 EI kGA 0 " # rAo2 0 KðoÞ ¼ . 0 rIo2 kGA We observe that the effect of the shear stress introduces the matrix coefficient C. The frequency appears in the matrix coefficient K as it would be with an Euler–Bernoulli beam. In matrix form, the boundary conditions are written as A11 Xð0Þ þ B11 X0 ð0Þ ¼ 0, A21 XðLÞ þ B21 X0 ðLÞ ¼ 0, where A11 is the 2 2 identity matrix, B11 is the 2 2 zero matrix and 2 3 b 2 2 2 ðk mo m kGA þ Þ k o kGA 6 7 2 6 7 4 kGAb ; B ¼ A21 ¼ 6 7 2 21 b o mb 4 5 kGA 0 2 2 3
(6)
0
3
EI
5.
3.1. Solution basis The general solution of Eq. (5) in terms of initial values can be written as X ðxÞ ¼ h0 ðxÞX ð0Þ þ h1 ðxÞX 0 ð0Þ, h0 ðxÞ ¼ h0 ðxÞM þ hðxÞC, h1 ðxÞ ¼ hðxÞM,
(7)
X ðxÞ ¼ hðxÞc1 þ h0 ðxÞc2
(8)
or for arbitrary vectors c1 and c2 . Here hðxÞ is the solution of the initial value problem Mh00 ðxÞ þ Ch0 ðxÞ þ KðoÞhðxÞ ¼ 0, hð0Þ ¼ 0;
Mh0 ð0Þ ¼ I,
where 0 denotes the 2 2 null matrix and I the 2 2 identity matrix.
(9) (10)
ARTICLE IN PRESS 1056
J.R. Claeyssen, S.N.J. Costa / Journal of Sound and Vibration 296 (2006) 1053–1058
For determining hðxÞ we use the formula derived in Ref. [8]. We first consider the characteristic polynomial PðsÞ ¼ det½s2 M þ sC þ K ¼
4 X
bk s4k .
k¼0
It turns out that b0 ¼ ab;
b1 ¼ 0;
b2 ¼ aeo2 þ co2 b;
b3 ¼ 0;
b4 ¼ co4 e co2 a,
where a ¼ kGA;
b ¼ EI;
c ¼ rA;
e ¼ rI.
Then we find the solution hðxÞ of the initial value problem 4 X
bk hð4kÞ ðxÞ ¼ b0 hðivÞ ðxÞ þ b2 h00 ðxÞ þ b4 hðxÞ ¼ 0,
k¼0
hð0Þ ¼ h0 ð0Þ ¼ h00 ð0Þ ¼ 0;
b0 h000 ð0Þ ¼ 1
(11)
and we let hk be the solution of the matrix difference equation Mhkþ2 þ Chkþ1 þ Khk ¼ 0, h0 ¼ 0;
Mh1 ¼ I.
(12)
bi hðj1iÞ ðxÞh4j .
(13)
We have hðxÞ ¼
j1 4 X X j¼1 i¼0
The calculations run as follows. The characteristic differential Eq. (11) is abhðivÞ ðxÞ þ ðae þ cbÞo2 h00 ðxÞ þ ðeo2 aÞco2 hðxÞ ¼ 0, hð0Þ ¼ h0 ð0Þ ¼ h00 ð0Þ ¼ 0;
abh000 ð0Þ ¼ 1.
(14)
The characteristic polynomial can be conveniently written as PðsÞ ¼ abðs4 þ g2 s2 r4 Þ, where g2 ¼ ðe=b þ c=aÞo2 ;
r4 ¼ co2 ðeo2 þ aÞ=ab.
The roots of the characteristic polynomial PðsÞ are then easily found to be s ¼ ; ; id; id, where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1=2 2g2 þ 2 ðg4 þ 4r4 Þ; d ¼ ðg2 þ 2 Þ. Thus g2 ¼ d2 2 ;
r4 ¼ d2 2 .
By using the Euler basis or the Laplace transform method, we find that the solution of the initial value problem (11) is hðxÞ ¼
d sinhðxÞ sinðdxÞ . abð2 þ d2 Þd
(15)
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The following matrix values are obtained by iterating the initial value problem (12): 2 3 1 0 6a 7 0 0 7, h0 ¼ ; h1 ¼ 6 4 5 1 0 0 0 b 2
6 0 h2 ¼ 6 4 1 b Then
" hðxÞ ¼
3 1 b7 7; 5 0
2 6 6 h3 ¼ 6 4
a2 þ co2 b a2 b
3 0
7 7 7. eo 5 2 b 2
0
bh00 ðxÞ þ ðeo2 aÞhðxÞ
ah0 ðxÞ
ah0 ðxÞ
ah00 ðxÞ þ hðxÞco2
# .
(16)
3.2. Characteristic equation We now proceed to determine the characteristic equation that will give the frequencies for which the modes XðxÞ are to be determined. Since the left end of the beam is fixed, the choice of the matrix basis h; h0 should simplify computations at such end. The substitution of the general solution XðxÞ ¼ hðxÞc1 þ h0 ðxÞc2 ;
0pxpL,
(17)
where c1 , c2 are 2 1 vectors, into the boundary conditions (6) leads to the linear system hð0Þc1 þ h0 ð0Þc2 ¼ 0:c1 þ M1 c2 ¼ 0, ðA21 hðLÞ þ B21 h0 ðLÞÞc1 þ ðA21 h0 ðLÞ þ B21 h00 ðLÞÞc2 ¼ 0. It follows that c2 ¼ 0 and the characteristic equation is detðDðoÞÞ ¼ 0,
(18)
DðoÞ ¼ A21 hðLÞ þ B21 h0 ðLÞ.
(19)
where 0
Here the values hðLÞ, h ðLÞ are to be computed from Eq. (16) and the coefficients A21 , B21 are given in Eq. (6). In particular, by using the Euler basis that leads to Eq. (15), we have 2 3 dA sinhðxÞ B sinðdxÞ coshðxÞ cosðdxÞ 6 7 abð2 þ d2 Þd bð2 þ d2 Þ 6 7 , (20) hðxÞ ¼ 6 coshðxÞ cosðdxÞ dC sinhðxÞ D sinðdxÞ 7 4 5 2 2 bð2 þ d Þ abð2 þ d Þd where A ¼ eo2 þ b2 a, B ¼ eo2 bd2 a, C ¼ co2 þ 2 a, D ¼ c o 2 d2 a and the involved parameters and constants are as defined above. It is clear that for each root of the characteristic Eq. (19), we have the mode Xðx; oÞ ¼ hðx; oÞc1 ,
(21)
ARTICLE IN PRESS J.R. Claeyssen, S.N.J. Costa / Journal of Sound and Vibration 296 (2006) 1053–1058
1058
where c1 is a non-zero solution of Dc ¼ 0. The dependence of the characteristic equation as well as the modes upon frequency o have been emphasized. When the frequency is a double one, we actually have two modes associated with such frequency. In this situation each column of the matrix h will be then a mode for the coupled Timoshenko model. From Eqs. (16) and (21), we have the displacement mode Uðx; oÞ ¼ aðbh00 ðx; oÞ þ ðeo2 aÞhðx; oÞÞ þ bðah0 ðx; oÞÞ
(22)
Rðx; oÞ ¼ aðah0 ðx; oÞÞ þ bðah00 ðx; oÞ þ co2 hðx; oÞÞ
(23)
and the rotation mode
for scalar constants a and b that both are not simultaneously zero. Here hðx; oÞ is the solution (15) for each root o of the characteristic Eq. (19). We can that verify that Uðx; oÞ and Rðx; oÞ satisfy the characteristic fourth-order differential Eq. (14), that is abyðivÞ ðxÞ þ ðae þ cbÞo2 y00 ðxÞ þ ðeo2 aÞco2 yðxÞ ¼ 0
(24)
or equations that are obtained by differentiating once or twice this later equation. We should observe that this later equation, after dividing by the factor ab, is the fourth-order spatial equation that arises for the amplitude by substitution into the time fourth-order equation that is obtained decoupling system (2), that is q4 w q2 w q 2 y q4 w þ ¼ 0, (25) a2 t2 4 þ b2 ða2 þ t2 Þ 2 qt qx qt2 qx4 where a2 ¼
r ; gkG
b2 ¼
rA ; gEI
t2 ¼
r . gE
(26)
This equation have been also obtained by a variational argument [12] where the frequency spectrum is discussed too. Acknowledgments This work was supported by CNPq and Fapergs. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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