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Asymptotic analysis of the vibration spectrum of coupled Timoshenko beams with a dissipative joint
European Journal of Mechanics A/Solids 29 (2010) 629e636
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Asymptotic analysis of the vibration spectrum of coupled Timoshenko beams with a dissipative joint Matthew P. Coleman a, *, Les Schaffer b a b
Department of Mathematics and Computer Science, Fairfield University, Fairfield CT 06824-5195, USA Department of Physics, Fairfield University, Fairfield CT 06824-5195, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 January 2009 Accepted 12 March 2010 Available online 24 March 2010
Complex flexible structures often are composed of simpler objects like beams joined end-to-end. We consider the case of two identical Timoshenko beams, coupled together at a typical energy-dissipating joint. The Wave Propagation Method of Keller and Rubinow is employed to compute the vibration spectrum of the system. It is found that the spectrum consists of a typical double-branched Timoshenko spectrum. In addition, however, for most combinations of energy-conserving end conditions, at least one of these branches decomposes into two sub-branches, one which is damped and one, undamped. 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Timoshenko Coupled beams Dissipative joint Wave propagation Eigenfrequencies Vibration spectrum
1. Introduction The design of large or complex flexible structuresdwhether bridges, buildings, robots, space stations, etc.dentails the coupling of smaller, simpler components, often those that can be modeled as beams, plates or shells. These couplings often include passive or active damping mechanisms. Successful design, of course, depends on a knowledge of the system’s vibration spectrum, i.e., the set of its natural frequencies of vibration. While the asymptotic behavior of the vibration spectrum of coupled EulereBernoulli beams has been well studied (see, e.g. (Chen et al., 1989), (Chen and Zhou, 1990) and (Krantz and Paulsen, 1991)), this seems not to be the case for coupled Rayleigh or Timoshenko beams. Here, then, we investigate the vibration spectrum of two identical Timoshenko beams, coupled by a so-called Type I joint (Chen et al., 1989). We employ the Wave Propagation Method (WPM) of Keller and Rubinow (Chen and Zhou, 1990), (Keller and Rubinow, 1960) in order to derive asymptotic estimates for the spectrum. The use of this Method in the form employed in (Chen and Zhou, 1990) and (Coleman and Wang, 1993) reduces an 8 8 eigenvalue (actually, latent value) problem to a much more tractable 4 4 problem.
* Corresponding author. Tel.: þ1 203 254 4000x2512; fax: þ1 203 254 4163. E-mail address: [email protected]field.edu (M.P. Coleman). 0997-7538/$ e see front matter 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2010.03.004
We choose parameters so that the joint is dissipative, and we consider the problem subject to all ten combinations of energyconserving end conditions. The results show the typical twobranched structure of the Timoshenko eigenvalue problem. However, it is found that, in eight of the ten cases, there exist infinitely many modes which remain undamped. In these cases, at least one of the branches of the spectrum consists of two subbranches, one damped and the other undamped. We then include a comparison of our results, for the special case of no damping, with the exact results for the clampedeclamped single Timoshenko beam. Here, we follow the methodology of (Han et al., 1999), adapted to our dimensionless problem. We find that the asymptotic results match the exact results, even for the lowest frequencies, for small values of the slenderness ratio. As this ratio increases, we must go farther along the spectrum in order to have a good match. We should mention, also, that (Han et al., 1999) actually studies and compares the four standard linear beam models (Euler-Bernoulli, Rayleigh, shear and Timoshenko), and we note that (Challamel, 2006) and (Hodges, 2007) include comparisons of the shear and Timoshenko models. The paper is organized as follows. First, we give a brief derivation of the problem, then follow it with a restatement of the problem in dimensionless form. We then apply WPM to the problem and present the results, followed by a comparison with exact results for the single Timoshenko beam, and ending with observations.
630
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
2. The problem
S : ð10Þ and ð12Þ
(15)
We begin with the dimensional model of the dynamic Timoshenko beam, as given in (Kim and Renardy, 1987) and (Coleman and Wang, 1993). Letting W(x, t) be the lateral displacement; F(x, t) the bending angle; M(x, t) the bending moment, and Q(x, t) the shear force, we have
R : ð11Þ and ð13Þ
(16)
F : ð12Þ and ð13Þ:
(17)
Qx rWtt ¼ 0
(1)
Q þ Mx Ir Ftt ¼ 0
(2)
Wx ¼ F þ
Fx ¼
1 Q K
(3)
1 M: EI
(4)
Here, the constant physical parameters are: r, the linear mass density; E, Young’s modulus; I, the area moment of inertia; K, the shear stiffness (K ¼ k0 GA, where k0 is the shape factor; G, the shear modulus; A, the cross-sectional area); and Ir, the mass moment of inertia (Ir ¼ sI ¼ rI=A, where s is the volume-massdensity). Now, to find the eigenfrequencies ix2, we perform a standard separation of variables:
Wðx; tÞ ¼ eix t wðxÞ; 2 Mðx; tÞ ¼ eix t mðxÞ;
Fðx; tÞ ¼ eix t fðxÞ;
2
2
(18)
Q2 ð0; tÞ ¼ Q1 ð0; tÞ
(19)
F2t ð0; tÞ F1t ð0; tÞ ¼ d1 M1 ð0; tÞ
(20)
W2t ð0; tÞ W1t ð0; tÞ ¼ d2 Q1 ð0; tÞ:
(21)
As we shall show, if d1 0 and d2 0, the system’s energy will satisfy dE/dt 0 and, if d1 > 0 or d2 > 0, we will have dE=dt < 0.
(5) EðtÞ ¼
Solving for f, m and q in terms of w, we have
M2 ð0; tÞ ¼ M1 ð0; tÞ
Proof. The system’s total energy at any time t is given by
2
Q ðx; tÞ ¼ eix t qðxÞ:
Now, we consider two identical such beams, each of length L, with displacements w1(x), L x 0, and w2(x), 0 x L, respectively. Thus, each of w1 and w2 satisfies (9). We subject the left end of the w1 beam, at x ¼ L, to one of (14)e(17); similarly for the right end of the w2 beam, at x ¼ L. We apply the conditions for a so-called Type I joint, as given in (Chen et al., 1989). We generalize the derivations therein (for coupled Euler-Bernoulli beams) to the case of coupled Timoshenko beams, arriving at
rEI 4 Ir 4 EI 000 1 x fðxÞ ¼ w ðxÞ þ 1 þ 2 x w0 ðxÞ K K K
(6)
1 2
Z0 h
i
2 rW1t þ Ir F21t þ KðF1 W1x Þ2 þEI F21x dx
L
þ
ZL h
1 2
i
2 rW2t þ Ir F22t þ KðF2 W2x Þ2 þEI F22x dx:
0
rEI
x4 wðxÞ
(7)
Here, W1t ¼ W1t(x, t), etc. It follows that
rEI 4 0 Ir 4 000 x w ðxÞ: 1 x qðxÞ ¼ EIw ðxÞ Ir þ K K
(8)
E0 ðtÞ ¼
mðxÞ ¼ EIw00 ðxÞ þ
K
r
r Ir 4 00 x w ðxÞ þ x4 wð4Þ ðxÞ þ þ K EI EI
Ir 4 x 1 wðxÞ ¼ 0: K
(9)
Each of the four energy-conserving boundary conditionsdclamped (C) simply-supported or pinned (S), roller-supported or sliding (R) and free (F)dinvolves two of the following:
wðaÞ ¼ 0
(10)
rEI EI 000 fðaÞ ¼ 00 w ðaÞ þ 1 þ 2 x4 w0 ðaÞ ¼ 0 K K
r
000
qðaÞ ¼ 00EIw ðaÞ þ Ir þ
rEI K
(11)
(12)
0
x w ðaÞ;
(13)
at a boundary point x ¼ a. Specifically, we have
C : ð10Þ and ð11Þ
þ EI F1x F1xt dx þ
(14)
ZL
rW2t W2tt þ Ir F2t F2tt
0
þ KðF2 W2x ÞðF2t W2xt Þ þ EI F2x F2xt dx: Upon integrating by parts,
x¼0 E0 ðtÞ ¼ ½EI F1x F1t W1t ðF1 W1x Þx¼L þ
W1t ½rW1tt þ KðF1x W1xx Þ
L
þ F1t Ir F1tt EI F1xx þ KðF1 W1x Þ dx x¼L þ ½EIF2x F2t W2t ðF2 W2x Þx¼0 ZL þ
4
rW1t W1tt þ Ir F1t F1tt þ KðF1 W1x ÞðF1t W1xt Þ
Z0
4 mðaÞ ¼ 00w00 ðaÞ þ x wðaÞ ¼ 0 K
L
The resulting ODE in w is then
Z0
W2t ½rW2tt þ KðF2x W2xx Þ
0
þ F2t Ir F2tt EI F2xx þ KðF2 W2x Þ dx: Now, the clamped boundary condition 0 W1t(L, t) ¼ F1t(L, t) ¼ W2t(L, t) ¼ F2t(L, t) ¼ 0, and both integrals are zero from (1)e (4). Thus,
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
E0 ðtÞ ¼ EI F1x ð0; tÞF1t ð0; tÞ KW1t ð0; tÞ½F1 ð0; tÞ W1x ð0; tÞ EIF2x ð0; tÞF2t ð0; tÞ þ KW2t ð0; tÞ½F2 ð0; tÞ W2x ð0; tÞ ¼ F1t ð0; tÞM1 ð0; tÞ þ W1t ð0; tÞQ1 ð0; tÞ F2t ð0; tÞM2 ðtÞ W2t ð0; tÞQ2 ð0; tÞ and applying (18)e(21) yields
E0 ðtÞ ¼ d1 M12 ð0; tÞ d2 Q12 ð0; tÞ:
QED
The physical realization of a Type I joint can be found in Fig. 1. The joint is represented by the “gap” abcd, where we have a linear-viscous-damper connecting each pair of points aeb, ced, and aed. If each horizontal damper has damping coefficient c1, and the vertical damper has damping constant c2, then we will have
2 d1 ¼ ; c1 h2
r
4 4 w002 ð0Þ þ x w2 ð0Þ ¼ w001 ð0Þ þ x w1 ð0Þ K K
(22)
rEI 4 0 rEI 4 0 000 000 x w2 ð0Þ ¼ EIw1 ð0Þ þ Ir þ x w1 ð0Þ EIw2 ð0Þ þ Ir þ K K (23) h
i rEI 4 000 000 2 EI w2 ð0Þ w1 ð0Þ þ 1 þ 2 x ½w02 ð0Þ w01 ð0Þ ix K K i r 4 Ir 4 h 00 w1 ð0Þ þ x w1 ð0Þ ¼ d1 EI 1 x K K
w2 L ffiffiffiffi q
l2 ¼ EIr L2 x2 b1 ¼ Ir =rL2 2 b2 ¼ EI=KL pffiffiffiffiffiffiffi 0 rEILd1 d1 ¼ pffiffiffiffiffi rEI
d02 ¼
L
(26)
d2 :
Lastly, we will let z2(y)/z2(y), so that each beam’s domain is the interval 1 y 0. We now abuse notation and return to using x, w1 and w2, instead of y, z1 and z2, respectively. The resulting dimensionless ODEs are 4
ð4Þ
w2 ðxÞþðb1 þ b2 Þl w002 ðxÞþ l 4
4
4
b1 b2 l4 1 w1 ðxÞ ¼ 0;
(27)
b1 b2 l4 1 w2 ðxÞ ¼ 0; 1 < x < 0: (28)
The dimensionless versions of the boundary (10)e(13) are, respectively,
wj ð1Þ ¼ 0
(29)
000
b2 wj ð 1Þ þ 1 þ b22 l4 w0j ð 1Þ ¼ 0
(30)
w00j ð 1Þ þ b2 l wj ð 1Þ ¼ 0
(31)
4
000
wj ð 1Þ þ ðb1 þ b2 Þl w0j ð 1Þ ¼ 0; 4
(32)
j ¼ 1, 2. The new joint conditions, then, are
(24)
w002 ð0Þ þ b2 l w2 ð0Þ ¼ w001 ð0Þ þ b2 l w1 ð0Þ 4
4
000
000
(33)
w2 ð0Þ ðb1 þ b2 Þl w02 ð0Þ ¼ w1 ð0Þ þ ðb1 þ b2 Þl w01 ð0Þ
Ir 4 ix 1 x ½w2 ð0Þ w1 ð0Þ K rEI 4 0 000 x w1 ð0Þ : ¼ d2 EIw1 ð0Þ þ Ir þ K 2
z2 ¼
ð4Þ
(see (Chen et al., 1989)). Here h is the vertical thickness of each beam. (Note that the sign of d1 here is the opposite of the corresponding parameter in (Chen et al., 1989). This is only because of the sign convention chosen for M, in (2) and (4)). Upon separating variables and writing formulas (18)e(21) in terms of w1 and w2, we have
y ¼ xL z1 ¼ wL1
w1 ðxÞ þ ðb1 þ b2 Þl w001 ðxÞ þ l
1 d2 ¼ c2
r
631
2
ð25Þ
il
n
4
h
000
000
i
4
(34)
o
b2 w2 ð0Þ þ w1 ð0Þ þ 1 þ b22 l4 ½w02 ð0Þ þ w01 ð0Þ
h i 4 4 w001 ð0Þ þ b2 l w1 ð0Þ ¼ d01 1 b1 b2 l
(35)
4 1 b1 b2 l ½w2 ð0Þ w1 ð0Þ h 000 i 4 ¼ d02 w1 ð0Þ þ ðb1 þ b2 Þl w01 ð0Þ :
(36)
2
3. Dimensionless problem Following (TrailleNash and Collar, 1953) and (Coleman, 2009), we introduce the dimensionless quantities
il
4. Application of the wave propagation method We find that the solutions of the ODEs (27) and (28) are of the form wj(x) ¼ emx, where 2 m ¼ il g1 ;
2 il g2 :
(37)
Here,
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 g1 ¼ tr2 r4 b1 b2 þ 4 ;
l
Fig. 1. Schematic for design of a Type I joint. aeb, ced, and aed are linear-dampers, with damping coefficients c1, c1, and c2 respectively.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 g2 ¼ tr2 þ r4 b1 b2 þ 4 ;
l
(38)
632
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
where r2 ¼ b1 þ b2 =2. We note that these derivations are based on the assumption that we are considering only frequencies above pffiffiffiffiffiffiffiffiffiffiffi 2 the Timoshenko cutoff frequency, that is, l > 1= b1 b2 (as in (Traill-Nash and Collar, 1953) and (Han et al., 1999)). In order to apply WPM, we write the general solutions as 2 2 2 w1 ðxÞ ¼ A1 ðlÞeil g1 ðxþ1Þ þ A2 ðlÞeil g2 ðxþ1Þ þ B1 ðlÞeil g1 ðxþ1Þ
2 2 2 w1 ðxÞ ¼ A3 ðlÞeil g1 x þ A4 ðlÞeil g2 x þ B3 ðlÞeil g1 x 2 þ B4 ðlÞeil g2 x
w2 ðxÞ ¼ A5 ðlÞe
A1 ¼ B1
(52)
A1 ¼ B1
(53)
b2 A1 þ
(39)
2
(51)
qffiffiffiffiffi
2 þ B2 ðlÞeil g2 ðxþ1Þ
il g1 ðxþ1Þ
A1 þ A2 ¼ B1 B2
(40) il g2 ðxþ1Þ
þ A6 ðlÞe
il g1 ðxþ1Þ
2
þ B5 ðlÞe
qffiffiffiffiffi
b1 A2 ¼
qffiffiffiffiffi
b2 B1 þ
qffiffiffiffiffi
b1 B2 :
(54)
Thus, we may write the boundary conditions for w1 at x ¼ 1 in terms of the reflection relation
A1 A2
2
¼
0 a2
a1 c1
B1 B2
¼ N1
B1 ; B2
(55)
and, similarly, for w2 at x ¼ 1, as
þ B6 ðlÞeil g2 ðxþ1Þ 2
A5 A6
(41) w2 ðxÞ ¼ A7 ðlÞeil g1 x þ A8 ðlÞeil g2 x þ B7 ðlÞeil g1 x 2
2
2
2 þ B8 ðlÞeil g2 x :
(42)
(For the sake of simplicity, we write Ai and Bi instead of Ai(l) and Bi(l), respectively, below.) We note here that
½A3 ; A4 ; A7 ; A8 T ¼ J½A1 ; A2 ; A5 ; A6 T
(43)
¼
b1 c2
0 b2
B5 B6
¼ N2
C : a1 ¼ 1; a2 ¼ 1; c1 ¼ 2; b1 ¼ 1; b2 ¼ 1; c2 ¼ 2 S : a1 ¼ a2 ¼ 1; c1 ¼ 0; b1 ¼ b2 ¼ 1; c2 ¼ 0 R : a1 ¼ a2 ¼ 1; c1 ¼ 0; b1 q ¼ ffiffiffiffi b2 ¼ 1; c2 ¼ 0 qffiffiffiffi F : a1 ¼ 1; a2 ¼ 1; c1 ¼ 2 bb2 ; b1 ¼ 1; b2 ¼ 1; c2 ¼ 2 bb2 : 1
1
(57)
½B1 ; B2 ; B5 ; B6 T ¼ J½B3 ; B4 ; B7 ; B8 T ; where
2
eil g1 6 0 J ¼ 6 4 0 0 2
02
0 02
eil g2
0 0 02
eil g1 0
0 0
(44)
3 7 7: 5
(45)
g1 b2 g21
(46)
A1 þ g2 b2 g22
g2 b2 g22
B2 þ
1
b2 l4
g1 b1 þ b2 g21 A1 þ g2 b1 þ b2 g22 A2
g1 b1 þ b2
g21
B1 g2 b1 þ b2
g22
b1 þ O
; 4
1
l
g2 ¼
qffiffiffiffiffi
b2 þ O
1
l
4
ð59Þ
b2 g21 A7 þ b2 g22 A8 þ b2 g21 B7 þ b2 g22 B8
B2 ¼ 0:
ð49Þ
WLOG, we assume that b2 > b1, i.e., that EI=K > Ir =r. (This, in fact, is the case for physically realistic situations.) Then, from (38), we may write
qffiffiffiffiffi
g1 b1 þ b2 g21 B7 g2 b1 þ b2 g22 B8 þ g1 b1 þ b2 g21 A3 þ g2 b1 þ b2 g22 A4 g1 b1 þ b2 g21 B3 g2 b1 þ b2 g22 B4 ¼ 0
ð47Þ
(48)
0 : N2
Now we proceed to apply the joint conditions (33)e(36) to the expressions (40) and (42), resulting in
b2 g21 A1 þ b2 g22 A2 þ b2 g21 B1 þ b2 g22 B2 ¼ 0
N1 0
g1 b1 þ b2 g21 A7 þ g2 b1 þ b2 g22 A8
ð g1 A 1 þ g2 A 2 g1 B 1 g2 B 2 Þ ¼ 0
R1 ¼
A2 g1 b2 g21 B1
(58)
where
eil g2
A1 þ A2 þ B1 þ B2 ¼ 0
The combined reflection relation, then, is given by
½A1 ; A2 ; A5 ; A6 T ¼ R1 ½B1 ; B2 ; B5 ; B6 T ;
Applying the boundary conditions (29)e(32) to the expression for w1 given by (39) gives us, respectively,
g1 ¼
(56)
In each case, the values of the matrix entries, for each boundary condition, are given by
and
B5 : B6
(50)
and, led by our analyses in (Coleman and Wang, 1993) and 4 (Coleman, 2009), we neglect terms of order Oð1=l Þ. In this case, we may write the boundary (46)e(49) as
b2 g21 A3 b2 g22 A4 b2 g21 B3 b2 g22 B4 ¼ 0
ð60Þ
b2 g1 g21 b2 A7 þ b2 g2 g22 b2 A8 b2 g1 g21 b2 B7
b2 g2 g22 b2 B8 þ b2 g1 g21 b2 A3 þ b2 g2 g22 b2 A4 h b2 g1 g21 b2 B3 b2 g2 g22 b2 B4 ¼ d01 b1 b2 b2 g21 A3 i ð61Þ þ b2 g22 A4 þ b2 g21 B3 þ b2 g22 B4
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
h
i
Here, aj, bj and cj, j ¼ 1, 2, are given by (57), x1 and x2 by (68), and we use the shorthand
d02 g1 g21 b1 b2 b1 b2 A3 i h þ d02 g2 g22 b1 b2 b1 b2 A4 i h d02 g1 g21 b1 b2 b1 b2 B3 i h d02 g2 g22 b1 b2 b1 b2 B4
ej ¼ eil
Again, using (50) and neglecting terms of Oð1=l Þ, these simplify to 4
b2 A þ A8 b1 7
b2 B B8 þ b1 7
sffiffiffiffiffi
b2 A þ A4 b1 3
sffiffiffiffiffi
A3 þ B3 A7 B7 ¼ 0 A3 B3 þ A7 B7 ¼
bj ;
j ¼ 1; 2:
a1 ¼ b1 ¼ 1 :
e2il
2
pffiffiffiffi
e2il
a1 ¼ b1 ¼ 1 :
(65)
00
b1 ¼ 1; 2þd1 2d001
a1 ¼ 1; b1 ¼ H1 :
(64) þ B3 Þ
ð71Þ
Thus, for the d1-branch(es) of the spectrum, we have
b2 B B4 ¼ 0 b1 3 (63)
d001 ðA3
pffiffiffi
1 0 ¼ detðR1 JR2 J IÞ ¼ ð2 þ d1 Þð2 þ d2 Þ i h a1 b1 ðd1 2Þe41 þ ða1 þ b1 Þd1 e21 þ d1 þ 2 $ i h a2 b2 ðd2 2Þe42 ða2 þ b2 Þd2 e22 þ d2 þ 2 :
ð62Þ
l
sffiffiffiffiffi
2
Finally
b1 b2 ½A7 þ A8 þ B7 þ B8 1 þ 4 ½A7 þ A8 þ B7 þ B8 A3 A4 B3 B4 ¼ 0: sffiffiffiffiffi
633
e4il pffiffiffiffi 2
2
pffiffiffiffi
00
b1 ¼ d1 þ2 d001 2 00
b1 ¼ 1; d1 þ2 d001 2
or
ðA7 þ A8 þ B7 þ B8 A3 A4 B3 B4 Þ sffiffiffiffiffi sffiffiffiffiffi !
b2 A þ A4 b1 3
¼ d002
b2 B B4 ; b1 3
2
ilm
pffiffiffiffiffiffi pffiffiffiffiffiffi where d001 ¼ d01 b1 and d002 ¼ d02 = b2 . After very much, but straightforward, computation, we solve for B3, B4, B7 and B8, and rewrite (63)e(66) as the reflection relation at x ¼ 0 given by
2
2
3 d1 B3 6 2þd1 6 6 B4 7 x1 6 7 ¼ 6 6 2 4 B7 5 6 4 2þd1 B8 x2
2 2þd1
0 d2 2þd2
x2
0
d1 2þd
2 2þd2
x1
1
ilm ¼ 2
ilm 2
sffiffiffiffiffi
b2 d1 þ d2 þ d1 d2 ; b1 ð2 þ d1 Þð2 þ d2 Þ
x2 ¼ 2
d00 2 1 m pffiffiffiffiffi ln 100 þ pffiffiffiffiffipi; m˛Z; 4 b1 d1 þ 2 2 b1
ð72aÞ
ð72bÞ
n 2 iln ¼ pffiffiffiffiffipi;
a1 ¼ b1 ¼ 1 :
where
x1 ¼ 2
2
(67)
2þd2
sffiffiffiffiffi
iln ¼
a1 ¼ 1; b1 ¼ H1 :
3
2 3 A3 7 2 7 6 7 2þd2 7 6 A4 7 7 4 5; R2 A 7 0 7 5 A8 d2 0
2n þ 1 pffiffiffiffiffi pi; 2 b1 2 d001 1 m ¼ pffiffiffiffiffi ln 00 þ pffiffiffiffiffipi; n; m˛Z; b1 2 b1 2 þ d1
a1 ¼ b1 ¼ 1 :
(66)
b2 d1 d2 ; b1 ð2 þ d1 Þð2 þ d2 Þ
b1
d00 2 1 m ¼ pffiffiffiffiffi ln 100 þ pffiffiffiffiffipi; n; m˛Z: b1 2 b1 d1 þ 2
ð72cÞ
Similarly, the d2-branch(es) are given by
(68)
a2 ¼ b2 ¼ 1 :
and where, for the sake of clarity, we are using d1 and d2 instead of d00 1 and d00 2, respectively. Putting together (43), (44), (58) and (67), we arrive at
2
ilm
3 2 3 2 3 2 3 2 3 A1 B1 B3 A3 A1 6 A2 7 6 7 6 7 6 7 6 7 6 7 ¼ R1 6 B2 7 ¼ R1 J6 B4 7 ¼ R1 JR2 6 A4 7 ¼ R1 JR2 J6 A2 7; 4 A5 5 4 B5 5 4 B7 5 4 A7 5 4 A5 5 A6 B6 B8 A8 A6
n 2 iln ¼ pffiffiffiffiffipi;
b1
d00 2 1 m ¼ pffiffiffiffiffi ln 200 þ pffiffiffiffiffipi; n; m˛Z; b2 2 b2 d2 þ 2
ð73aÞ
2
which will hold for nontrivial [A1, A2, A5, A6]T if and only if
detðR1 JR2 J IÞ ¼ 0:
a2 ¼ 1; b2 ¼ H1 : 2
ilm ¼
2
ilm
6 6 c d1 e2 þ a x e e 2 1 1 2 6 1 2þd1 1 R1 JR2 J I ¼ 6 2 e2 6 b 1 2þd1 1 4 2 e2 þ b x e e c2 2þd 2 2 1 2 1 1
1
2n þ 1 pffiffiffiffiffi pi; 2 b1 2 d002 1 m ¼ pffiffiffiffiffi ln 00 þ pffiffiffiffiffipi; n; m˛Z: b2 2 b2 2 þ d2 2
iln ¼
a2 ¼ b2 ¼ 1 :
(70)
d1 2 e1 1 a1 2þd
3
0
2 e2 a1 2þd 1
0
d2 2 a2 2þd e2 1
2 e2 þ a x e e c1 2þd 2 2 1 2 1
2 e2 a2 2þd 2
0
d1 2 b1 2þd e1 1
2
2 e2 b2 2þd 2 2
ð73bÞ
(69)
Now we have
2
d00 2 1 m pffiffiffiffiffi ln 200 þ pffiffiffiffiffipi; m˛Z; 4 b2 d2 þ 2 2 b2
1
1
d1 2 c2 2þd e 1 1
1
þ b2 x1 e1 e2
2
0 d2 2 b2 2þd e 2 2
1
7 7 7 7: 7 5
ð73cÞ
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Then, for combinations of the boundary conditions (14)e(17), we have CeC: (72a), (73c) CeS: (72b), (73c) CeR: (72a), (73b) CeF: (72b), (73b) SeS: (72c), (73c)
SeR: (72b), (73b) SeF: (72c), (73b) ReR: (72a), (73a) ReF: (72b), (73a) FeF: (72b), (73a).
Lastly, we may put (72) and (73) back into dimensional form:
sffiffiffiffiffi EI 2n þ 1 pi; a1 ¼ b1 ¼ 1 : Ir 2 sffiffiffiffiffi ! pffiffiffiffiffiffiffi 1 EI 1 2 rEI d1 2 pffiffiffiffiffiffiffi þ mpi ; n; m˛Z; ixm ¼ ln L Ir 2 2 þ rEI d1 2 ixn
1 ¼ L
a1 ¼ 1; b1 ¼ H1 : sffiffiffiffiffi ! pffiffiffiffiffiffiffi rEI d1 2 1 EI 1 2 þ mpi ; m˛Z; ixm ¼ ln pffiffiffiffiffiffiffi 2L Ir 2 rEI d1 þ 2 sffiffiffiffiffi 1 EI ¼ npi; a1 ¼ b1 ¼ 1 : L Ir sffiffiffiffiffi ! pffiffiffiffiffiffiffi rEI d 2 1 EI 1 2 þ mpi ; n; m˛Z: ixm ¼ ln pffiffiffiffiffiffiffi 1 L Ir 2 rEI d1 þ 2
ð74aÞ
ð74bÞ
ð74cÞ
2 ixn ¼
sffiffiffiffi 1 K 2n þ 1 pi; ¼ a2 ¼ b2 ¼ 1 : L r 2 sffiffiffiffi ! pffiffiffiffiffiffi 1 K 1 2 K r d2 2 pffiffiffiffiffiffi þ mpi ; n; m˛Z: ixm ¼ ln L r 2 2 þ K r d2
2ð1 þ nÞ E ¼ 0 ; k0 kG 1 I ¼ ; s2 AL2
g2 ¼
~ are the wave numbers, g and g are as given in (38) Here, a and b 1 2 and l2 in (26). We note here that, in terms of these quantities, our dimensionless parameters b1 and b2 become
b1 ¼
and
a2 ¼ 1; b2 ¼ H1 : sffiffiffiffi ! pffiffiffiffiffiffi 1 K 1 Kr d 2 2 þ mpi ; m˛Z; ixm ¼ ln pffiffiffiffiffiffi 2 2L r 2 K r d2 þ 2
If we let d1 ¼ d2 ¼ 0, then our system should behave as a single Timoshenko beam of length 2L. Thus, we may compare our asymptotic results with the solutions resulting from the exact frequency equation(s) given in the literature ((Han et al., 1999), (Karnovsky and Lebed, 2001). It can be shown that the equations given by these two references are equivalent. We restrict our study here to the clampedeclamped case. Following (Han et al., 1999) we derive their frequency equation for our model. We non-dimensionalize (2L / 1), and introduce the following parameters from (Han et al., 1999):
2 ~ ¼ l2 g ; u ¼ l2 : a ¼ l g2 ; b 1
2 ixn
sffiffiffiffi 1 K a2 ¼ b2 ¼ 1 : npi; L r sffiffiffiffi ! pffiffiffiffiffiffi 1 K 1 K r d2 2 2 þ npi ; n; m˛Z; ixm ¼ ln pffiffiffiffiffiffi L r 2 K r d2 þ 2
5. Justification of method e comparison with “exact” solution
ð75aÞ
1 ; b2 ¼ g2 b1 : s2
Following this notation, our asymptotic frequencies from (72b) and (73b) become
Branch 1 :
u np np u ¼ pffiffiffiffiffi or ; ¼ g s b2
n ¼ 1; 2; .
(76)
Branch 2 :
u np u ¼ pffiffiffiffiffi or ¼ np; s b1
n ¼ 1; 2; .
(77)
Meanwhile, our exact frequency equation is
~ 1 cos a cos b
2 2 ~2 g2 a2 ~ 2 a 2 g2 b ~2 þa2 b b
~ 2 2 sin a sin b; 2 2 ~ ~ ~ 2 2 2 2 b g a 2ab a g b
ð75bÞ
(78) which is equivalent to that given in (Han et al., 1999) (p. 966, Table 10, cec). The exact frequencies are
2 ixn
ð75cÞ
We note that we have assumed implicitly that d00 1 s 2 and d00 2 s 2 in the above argument. If, instead, we set d00 1 ¼ 2 in (71), we are led to the following:
pffiffiffiffi 2 b1 ¼ 1 a1 ¼ b1 ¼ 1 : e2il p ffiffiffiffi 2 l b1 ¼ 1 2i a1 ¼ b1 ¼ 1 : e a1 ¼ 1; b1 ¼ H1 : branch disappears: Thus, it seems as though the damping disappears; similarly, for d00 2 ¼ 2. However, the situation here is unstable in that, as d00 1 / 2, the real part of il2 / N, that is, the damping rate appears to become infinite. This interesting situation bears further study; indeed it is similar to the phenomenon of superstability in the single Timoshenko beam with boundary damping ((Shubov, 2002), (Shubov, private communication)).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u a2 þ b ~2 ¼ u ¼ l2 ¼ t b 1 1 þ g2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 ~2 ta þ b : s 1 þ g2
(79)
Again following (Han et al., 1999), we set g ¼ 2.205 and compute u/s from (79), extending their Fig. 18, p. 970 (note that their vertical axis is mislabeled as u*/L*(r*/E*)1/2, when it should be u*L*(r*/E*)1/2 ¼ u/s). As there, we plot u/s vs. 1/s, and we compare with the two asymptotic branches (76) and (77). We note that, if the beam is taken to have circular cross section, then s ¼ 4L/d, where L/d is the beam’s slenderness ratio. Thus, in Fig. 2, we plot the first 13 “exact” frequencies, and compare them to the first 13 asymptotic frequencies. We see a very good match between the exact and asymptotic frequencies, even for the lowest frequencies, for values of 1/s > 0.2, i.e., for L/d < 1.25. For more slender beams corresponding to 1/s ¼ 0.1 (L/d ¼ 2.5), the match is not good for the first few frequencies e and, in particular, we see the crossing and near crossing of the exact frequency curves e but then begins to get better once we
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
635
ω
s
15
10
5
0
0.0
0.1
0.2
0.3
0.4
1 0.5 s
Fig. 2. Clampedeclamped Timoshenko beam, first 13 frequencies. Exact: solid; Asymptotic, Branch 1: dash-dot; Asymptotic, Branch 2: dash.
get to the 10th or 12th frequencies. Further, it seems that the complicated structure of the left-most parts of the frequency curves tends towards the vertical axis as we go farther out in the spectrum. This behavior becomes more pronounced in Fig. 3. Here we plot the frequency curves for the n ¼ 10, 15, 20, 25, 30, 35 and 40th frequencies and, again, compare to the asymptotic frequencies. Above the 15th frequency, we see a good match for values of 1/s > 0.05 (L/d ¼ 5), and we believe that the trend must continue as n increases.
These results agree with the numerical results in (Coleman and Wang, 1993), where it is seen that, for small values of L/d, the match between numerical and asymptotic frequencies is very good even for the lowest frequency, while, as L/d increases, we must go farther “up” the spectrum before there is a good match. Finally, as we allow d1 and d2 to become nonzero, the asymptotic results suggest that the only effect is to change the damping rate, with the above frequencies remaining the same. Although we have no exact values with which to compare, numerical data in (Coleman and Wang, 1993) clearly suggests that this is only an
Fig. 3. Clampedeclamped Timoshenko beam, frequencies 10, 15, 20, 25, 30, 35, 40. Exact: solid; Asymptotic, Branch 1 (n ¼ 7, 11, 14, 21, 28): dash-dot; Asymptotic, Branch 2 (n ¼ 5, 8, 11): dash. The horizontal asymptotes are labeled by height and branch: 7p/g, 14p/g, 21p/g from Branch 1; 5p, 8p and 11p from Branch 2. (We note that it is purely coincidental that frequency 20 converges to a number near 20, frequency 25 to a number near 25, etc.).
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M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
asymptotic behavior. Indeed, the numerical results suggest that the actual damping rates are much smaller than the asymptotic results, at least at the low end of the spectrum; the actual rates approach the asymptotic rates from below. Meanwhile, it is not clear how these damping rates affect the imaginary parts of the frequencies.
Further, we have been unable to find numerical results. We plan on performing such a numerical study for this problem and, eventually, to perform similar analytical and numerical investigations of other types of joints. References
6. Conclusion/observations It is known that, for damping in the middle of the span, there often are modes that remain undamped. While not true for identical coupled EulereBernoulli beams (see (Chen and Zhou, 1990)), it certainly is the case here, except for the combinations CeF and SeR. Indeed, if aj ¼ bj ¼ 1, the corresponding branch of the spectrum consists of two sub-branches, one damped and one undamped. Further, if dj ¼ 0, then the corresponding branch (or union of two sub-branches) will be the same as that for a beam of length 2L which is continuous in the corresponding variable (i.e., from (20) and (21), Ft for d1 ¼ 0 and Wt for d2 ¼ 0). This is to be expected, of course. In particular, if d1 ¼ d2 ¼ 0, then the spectrum will be the same as that for a single beam of length 2L (Coleman and Wang, 1993). The only experimental results that we have been able to find are given in (Chen et al., 1989), for the case when d2 ¼ 0. Unfortunately, those results include only the first twelve frequencies, far fewer than are necessary to give any kind of agreement with asymptotic results (see, e.g., Example 4 in (Coleman and Wang, 1993)). However, the fact that the Euler-Bernoulli results in (Chen et al., 1989) seem to be overestimates certainly is consistent with the pffiffiffiffi fact that jxj w p for the EulereBernoulli beam, while jxjw p for these Timoshenko beams.
Challamel, N., 2006. On the comparison of Timoshenko and shear models in beam dynamics. J. Eng. Mech. 132, 1141e1145. Chen, G., Krantz, S.G., Russell, D.L., Wayne, C.E., West, H.H., Coleman, M.P., 1989. Analysis, designs and behavior of dissipative joints for coupled beams. SIAM J. Appl. Math. 49, 1665e1693. Chen, G., Zhou, J., 1990. The wave propagation method for the analysis of boundary stabilization in vibrating structures. SIAM J. Appl. Math. 50, 1254e1283. Coleman, M.P., 2009. A perturbation method for the accurate estimation of the vibration spectrum for the Timoshenko beam. J. Sound Vibration (preprint). Coleman, M.P., Wang, H.K., 1993. Analysis of the vibration spectrum of a Timoshenko beam. Wave Motion 17, 223e239. Han, S.M., Benaroya, H., Wei, T., 1999. Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vibration 255 (5), 935e988. Hodges, D.H., 2007. An asymptotic derivation of shear beam theory from Timoshenko theory. J. Eng. Mech. 133, 957e961. Karnovsky, I.A., Lebed, O.I., 2001. Formulas for Structural Dynamics: Tables Graphs and Solutions. McGraw-Hill. Keller, S.B., Rubinow, S.I., 1960. Asymptotic solution of eigenvalue problems. Ann. Phys. 9, 24e75. Kim, J.U., Renardy, Y., 1987. Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25, 1417e1429. Krantz, S.G., Paulsen, W.H., 1991. Asymptotic eigenfrequency distributions for the N-segment EulerdBernoulli coupled beam equations with dissipative joints. J. Symbolic Comp. 11, 369e418. Shubov, M.A., 2002. Asymptotic and spectral analysis of the spatially nonhomogenous Timoshenko beam model. Math. Nachr. 241, 125e162. Shubov, M.A., Private communication. Traill-Nash, R.W., Collar, A.R., 1953. The effect of shear flexibility and rotary inertia on the bending vibration of beams. Quart. J. Mech. Appl. Math. 6, 186e222.
Contents lists available at ScienceDirect
European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol
Asymptotic analysis of the vibration spectrum of coupled Timoshenko beams with a dissipative joint Matthew P. Coleman a, *, Les Schaffer b a b
Department of Mathematics and Computer Science, Fairfield University, Fairfield CT 06824-5195, USA Department of Physics, Fairfield University, Fairfield CT 06824-5195, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 January 2009 Accepted 12 March 2010 Available online 24 March 2010
Complex flexible structures often are composed of simpler objects like beams joined end-to-end. We consider the case of two identical Timoshenko beams, coupled together at a typical energy-dissipating joint. The Wave Propagation Method of Keller and Rubinow is employed to compute the vibration spectrum of the system. It is found that the spectrum consists of a typical double-branched Timoshenko spectrum. In addition, however, for most combinations of energy-conserving end conditions, at least one of these branches decomposes into two sub-branches, one which is damped and one, undamped. 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Timoshenko Coupled beams Dissipative joint Wave propagation Eigenfrequencies Vibration spectrum
1. Introduction The design of large or complex flexible structuresdwhether bridges, buildings, robots, space stations, etc.dentails the coupling of smaller, simpler components, often those that can be modeled as beams, plates or shells. These couplings often include passive or active damping mechanisms. Successful design, of course, depends on a knowledge of the system’s vibration spectrum, i.e., the set of its natural frequencies of vibration. While the asymptotic behavior of the vibration spectrum of coupled EulereBernoulli beams has been well studied (see, e.g. (Chen et al., 1989), (Chen and Zhou, 1990) and (Krantz and Paulsen, 1991)), this seems not to be the case for coupled Rayleigh or Timoshenko beams. Here, then, we investigate the vibration spectrum of two identical Timoshenko beams, coupled by a so-called Type I joint (Chen et al., 1989). We employ the Wave Propagation Method (WPM) of Keller and Rubinow (Chen and Zhou, 1990), (Keller and Rubinow, 1960) in order to derive asymptotic estimates for the spectrum. The use of this Method in the form employed in (Chen and Zhou, 1990) and (Coleman and Wang, 1993) reduces an 8 8 eigenvalue (actually, latent value) problem to a much more tractable 4 4 problem.
* Corresponding author. Tel.: þ1 203 254 4000x2512; fax: þ1 203 254 4163. E-mail address: [email protected]field.edu (M.P. Coleman). 0997-7538/$ e see front matter 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2010.03.004
We choose parameters so that the joint is dissipative, and we consider the problem subject to all ten combinations of energyconserving end conditions. The results show the typical twobranched structure of the Timoshenko eigenvalue problem. However, it is found that, in eight of the ten cases, there exist infinitely many modes which remain undamped. In these cases, at least one of the branches of the spectrum consists of two subbranches, one damped and the other undamped. We then include a comparison of our results, for the special case of no damping, with the exact results for the clampedeclamped single Timoshenko beam. Here, we follow the methodology of (Han et al., 1999), adapted to our dimensionless problem. We find that the asymptotic results match the exact results, even for the lowest frequencies, for small values of the slenderness ratio. As this ratio increases, we must go farther along the spectrum in order to have a good match. We should mention, also, that (Han et al., 1999) actually studies and compares the four standard linear beam models (Euler-Bernoulli, Rayleigh, shear and Timoshenko), and we note that (Challamel, 2006) and (Hodges, 2007) include comparisons of the shear and Timoshenko models. The paper is organized as follows. First, we give a brief derivation of the problem, then follow it with a restatement of the problem in dimensionless form. We then apply WPM to the problem and present the results, followed by a comparison with exact results for the single Timoshenko beam, and ending with observations.
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2. The problem
S : ð10Þ and ð12Þ
(15)
We begin with the dimensional model of the dynamic Timoshenko beam, as given in (Kim and Renardy, 1987) and (Coleman and Wang, 1993). Letting W(x, t) be the lateral displacement; F(x, t) the bending angle; M(x, t) the bending moment, and Q(x, t) the shear force, we have
R : ð11Þ and ð13Þ
(16)
F : ð12Þ and ð13Þ:
(17)
Qx rWtt ¼ 0
(1)
Q þ Mx Ir Ftt ¼ 0
(2)
Wx ¼ F þ
Fx ¼
1 Q K
(3)
1 M: EI
(4)
Here, the constant physical parameters are: r, the linear mass density; E, Young’s modulus; I, the area moment of inertia; K, the shear stiffness (K ¼ k0 GA, where k0 is the shape factor; G, the shear modulus; A, the cross-sectional area); and Ir, the mass moment of inertia (Ir ¼ sI ¼ rI=A, where s is the volume-massdensity). Now, to find the eigenfrequencies ix2, we perform a standard separation of variables:
Wðx; tÞ ¼ eix t wðxÞ; 2 Mðx; tÞ ¼ eix t mðxÞ;
Fðx; tÞ ¼ eix t fðxÞ;
2
2
(18)
Q2 ð0; tÞ ¼ Q1 ð0; tÞ
(19)
F2t ð0; tÞ F1t ð0; tÞ ¼ d1 M1 ð0; tÞ
(20)
W2t ð0; tÞ W1t ð0; tÞ ¼ d2 Q1 ð0; tÞ:
(21)
As we shall show, if d1 0 and d2 0, the system’s energy will satisfy dE/dt 0 and, if d1 > 0 or d2 > 0, we will have dE=dt < 0.
(5) EðtÞ ¼
Solving for f, m and q in terms of w, we have
M2 ð0; tÞ ¼ M1 ð0; tÞ
Proof. The system’s total energy at any time t is given by
2
Q ðx; tÞ ¼ eix t qðxÞ:
Now, we consider two identical such beams, each of length L, with displacements w1(x), L x 0, and w2(x), 0 x L, respectively. Thus, each of w1 and w2 satisfies (9). We subject the left end of the w1 beam, at x ¼ L, to one of (14)e(17); similarly for the right end of the w2 beam, at x ¼ L. We apply the conditions for a so-called Type I joint, as given in (Chen et al., 1989). We generalize the derivations therein (for coupled Euler-Bernoulli beams) to the case of coupled Timoshenko beams, arriving at
rEI 4 Ir 4 EI 000 1 x fðxÞ ¼ w ðxÞ þ 1 þ 2 x w0 ðxÞ K K K
(6)
1 2
Z0 h
i
2 rW1t þ Ir F21t þ KðF1 W1x Þ2 þEI F21x dx
L
þ
ZL h
1 2
i
2 rW2t þ Ir F22t þ KðF2 W2x Þ2 þEI F22x dx:
0
rEI
x4 wðxÞ
(7)
Here, W1t ¼ W1t(x, t), etc. It follows that
rEI 4 0 Ir 4 000 x w ðxÞ: 1 x qðxÞ ¼ EIw ðxÞ Ir þ K K
(8)
E0 ðtÞ ¼
mðxÞ ¼ EIw00 ðxÞ þ
K
r
r Ir 4 00 x w ðxÞ þ x4 wð4Þ ðxÞ þ þ K EI EI
Ir 4 x 1 wðxÞ ¼ 0: K
(9)
Each of the four energy-conserving boundary conditionsdclamped (C) simply-supported or pinned (S), roller-supported or sliding (R) and free (F)dinvolves two of the following:
wðaÞ ¼ 0
(10)
rEI EI 000 fðaÞ ¼ 00 w ðaÞ þ 1 þ 2 x4 w0 ðaÞ ¼ 0 K K
r
000
qðaÞ ¼ 00EIw ðaÞ þ Ir þ
rEI K
(11)
(12)
0
x w ðaÞ;
(13)
at a boundary point x ¼ a. Specifically, we have
C : ð10Þ and ð11Þ
þ EI F1x F1xt dx þ
(14)
ZL
rW2t W2tt þ Ir F2t F2tt
0
þ KðF2 W2x ÞðF2t W2xt Þ þ EI F2x F2xt dx: Upon integrating by parts,
x¼0 E0 ðtÞ ¼ ½EI F1x F1t W1t ðF1 W1x Þx¼L þ
W1t ½rW1tt þ KðF1x W1xx Þ
L
þ F1t Ir F1tt EI F1xx þ KðF1 W1x Þ dx x¼L þ ½EIF2x F2t W2t ðF2 W2x Þx¼0 ZL þ
4
rW1t W1tt þ Ir F1t F1tt þ KðF1 W1x ÞðF1t W1xt Þ
Z0
4 mðaÞ ¼ 00w00 ðaÞ þ x wðaÞ ¼ 0 K
L
The resulting ODE in w is then
Z0
W2t ½rW2tt þ KðF2x W2xx Þ
0
þ F2t Ir F2tt EI F2xx þ KðF2 W2x Þ dx: Now, the clamped boundary condition 0 W1t(L, t) ¼ F1t(L, t) ¼ W2t(L, t) ¼ F2t(L, t) ¼ 0, and both integrals are zero from (1)e (4). Thus,
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
E0 ðtÞ ¼ EI F1x ð0; tÞF1t ð0; tÞ KW1t ð0; tÞ½F1 ð0; tÞ W1x ð0; tÞ EIF2x ð0; tÞF2t ð0; tÞ þ KW2t ð0; tÞ½F2 ð0; tÞ W2x ð0; tÞ ¼ F1t ð0; tÞM1 ð0; tÞ þ W1t ð0; tÞQ1 ð0; tÞ F2t ð0; tÞM2 ðtÞ W2t ð0; tÞQ2 ð0; tÞ and applying (18)e(21) yields
E0 ðtÞ ¼ d1 M12 ð0; tÞ d2 Q12 ð0; tÞ:
QED
The physical realization of a Type I joint can be found in Fig. 1. The joint is represented by the “gap” abcd, where we have a linear-viscous-damper connecting each pair of points aeb, ced, and aed. If each horizontal damper has damping coefficient c1, and the vertical damper has damping constant c2, then we will have
2 d1 ¼ ; c1 h2
r
4 4 w002 ð0Þ þ x w2 ð0Þ ¼ w001 ð0Þ þ x w1 ð0Þ K K
(22)
rEI 4 0 rEI 4 0 000 000 x w2 ð0Þ ¼ EIw1 ð0Þ þ Ir þ x w1 ð0Þ EIw2 ð0Þ þ Ir þ K K (23) h
i rEI 4 000 000 2 EI w2 ð0Þ w1 ð0Þ þ 1 þ 2 x ½w02 ð0Þ w01 ð0Þ ix K K i r 4 Ir 4 h 00 w1 ð0Þ þ x w1 ð0Þ ¼ d1 EI 1 x K K
w2 L ffiffiffiffi q
l2 ¼ EIr L2 x2 b1 ¼ Ir =rL2 2 b2 ¼ EI=KL pffiffiffiffiffiffiffi 0 rEILd1 d1 ¼ pffiffiffiffiffi rEI
d02 ¼
L
(26)
d2 :
Lastly, we will let z2(y)/z2(y), so that each beam’s domain is the interval 1 y 0. We now abuse notation and return to using x, w1 and w2, instead of y, z1 and z2, respectively. The resulting dimensionless ODEs are 4
ð4Þ
w2 ðxÞþðb1 þ b2 Þl w002 ðxÞþ l 4
4
4
b1 b2 l4 1 w1 ðxÞ ¼ 0;
(27)
b1 b2 l4 1 w2 ðxÞ ¼ 0; 1 < x < 0: (28)
The dimensionless versions of the boundary (10)e(13) are, respectively,
wj ð1Þ ¼ 0
(29)
000
b2 wj ð 1Þ þ 1 þ b22 l4 w0j ð 1Þ ¼ 0
(30)
w00j ð 1Þ þ b2 l wj ð 1Þ ¼ 0
(31)
4
000
wj ð 1Þ þ ðb1 þ b2 Þl w0j ð 1Þ ¼ 0; 4
(32)
j ¼ 1, 2. The new joint conditions, then, are
(24)
w002 ð0Þ þ b2 l w2 ð0Þ ¼ w001 ð0Þ þ b2 l w1 ð0Þ 4
4
000
000
(33)
w2 ð0Þ ðb1 þ b2 Þl w02 ð0Þ ¼ w1 ð0Þ þ ðb1 þ b2 Þl w01 ð0Þ
Ir 4 ix 1 x ½w2 ð0Þ w1 ð0Þ K rEI 4 0 000 x w1 ð0Þ : ¼ d2 EIw1 ð0Þ þ Ir þ K 2
z2 ¼
ð4Þ
(see (Chen et al., 1989)). Here h is the vertical thickness of each beam. (Note that the sign of d1 here is the opposite of the corresponding parameter in (Chen et al., 1989). This is only because of the sign convention chosen for M, in (2) and (4)). Upon separating variables and writing formulas (18)e(21) in terms of w1 and w2, we have
y ¼ xL z1 ¼ wL1
w1 ðxÞ þ ðb1 þ b2 Þl w001 ðxÞ þ l
1 d2 ¼ c2
r
631
2
ð25Þ
il
n
4
h
000
000
i
4
(34)
o
b2 w2 ð0Þ þ w1 ð0Þ þ 1 þ b22 l4 ½w02 ð0Þ þ w01 ð0Þ
h i 4 4 w001 ð0Þ þ b2 l w1 ð0Þ ¼ d01 1 b1 b2 l
(35)
4 1 b1 b2 l ½w2 ð0Þ w1 ð0Þ h 000 i 4 ¼ d02 w1 ð0Þ þ ðb1 þ b2 Þl w01 ð0Þ :
(36)
2
3. Dimensionless problem Following (TrailleNash and Collar, 1953) and (Coleman, 2009), we introduce the dimensionless quantities
il
4. Application of the wave propagation method We find that the solutions of the ODEs (27) and (28) are of the form wj(x) ¼ emx, where 2 m ¼ il g1 ;
2 il g2 :
(37)
Here,
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 g1 ¼ tr2 r4 b1 b2 þ 4 ;
l
Fig. 1. Schematic for design of a Type I joint. aeb, ced, and aed are linear-dampers, with damping coefficients c1, c1, and c2 respectively.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 1 g2 ¼ tr2 þ r4 b1 b2 þ 4 ;
l
(38)
632
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
where r2 ¼ b1 þ b2 =2. We note that these derivations are based on the assumption that we are considering only frequencies above pffiffiffiffiffiffiffiffiffiffiffi 2 the Timoshenko cutoff frequency, that is, l > 1= b1 b2 (as in (Traill-Nash and Collar, 1953) and (Han et al., 1999)). In order to apply WPM, we write the general solutions as 2 2 2 w1 ðxÞ ¼ A1 ðlÞeil g1 ðxþ1Þ þ A2 ðlÞeil g2 ðxþ1Þ þ B1 ðlÞeil g1 ðxþ1Þ
2 2 2 w1 ðxÞ ¼ A3 ðlÞeil g1 x þ A4 ðlÞeil g2 x þ B3 ðlÞeil g1 x 2 þ B4 ðlÞeil g2 x
w2 ðxÞ ¼ A5 ðlÞe
A1 ¼ B1
(52)
A1 ¼ B1
(53)
b2 A1 þ
(39)
2
(51)
qffiffiffiffiffi
2 þ B2 ðlÞeil g2 ðxþ1Þ
il g1 ðxþ1Þ
A1 þ A2 ¼ B1 B2
(40) il g2 ðxþ1Þ
þ A6 ðlÞe
il g1 ðxþ1Þ
2
þ B5 ðlÞe
qffiffiffiffiffi
b1 A2 ¼
qffiffiffiffiffi
b2 B1 þ
qffiffiffiffiffi
b1 B2 :
(54)
Thus, we may write the boundary conditions for w1 at x ¼ 1 in terms of the reflection relation
A1 A2
2
¼
0 a2
a1 c1
B1 B2
¼ N1
B1 ; B2
(55)
and, similarly, for w2 at x ¼ 1, as
þ B6 ðlÞeil g2 ðxþ1Þ 2
A5 A6
(41) w2 ðxÞ ¼ A7 ðlÞeil g1 x þ A8 ðlÞeil g2 x þ B7 ðlÞeil g1 x 2
2
2
2 þ B8 ðlÞeil g2 x :
(42)
(For the sake of simplicity, we write Ai and Bi instead of Ai(l) and Bi(l), respectively, below.) We note here that
½A3 ; A4 ; A7 ; A8 T ¼ J½A1 ; A2 ; A5 ; A6 T
(43)
¼
b1 c2
0 b2
B5 B6
¼ N2
C : a1 ¼ 1; a2 ¼ 1; c1 ¼ 2; b1 ¼ 1; b2 ¼ 1; c2 ¼ 2 S : a1 ¼ a2 ¼ 1; c1 ¼ 0; b1 ¼ b2 ¼ 1; c2 ¼ 0 R : a1 ¼ a2 ¼ 1; c1 ¼ 0; b1 q ¼ ffiffiffiffi b2 ¼ 1; c2 ¼ 0 qffiffiffiffi F : a1 ¼ 1; a2 ¼ 1; c1 ¼ 2 bb2 ; b1 ¼ 1; b2 ¼ 1; c2 ¼ 2 bb2 : 1
1
(57)
½B1 ; B2 ; B5 ; B6 T ¼ J½B3 ; B4 ; B7 ; B8 T ; where
2
eil g1 6 0 J ¼ 6 4 0 0 2
02
0 02
eil g2
0 0 02
eil g1 0
0 0
(44)
3 7 7: 5
(45)
g1 b2 g21
(46)
A1 þ g2 b2 g22
g2 b2 g22
B2 þ
1
b2 l4
g1 b1 þ b2 g21 A1 þ g2 b1 þ b2 g22 A2
g1 b1 þ b2
g21
B1 g2 b1 þ b2
g22
b1 þ O
; 4
1
l
g2 ¼
qffiffiffiffiffi
b2 þ O
1
l
4
ð59Þ
b2 g21 A7 þ b2 g22 A8 þ b2 g21 B7 þ b2 g22 B8
B2 ¼ 0:
ð49Þ
WLOG, we assume that b2 > b1, i.e., that EI=K > Ir =r. (This, in fact, is the case for physically realistic situations.) Then, from (38), we may write
qffiffiffiffiffi
g1 b1 þ b2 g21 B7 g2 b1 þ b2 g22 B8 þ g1 b1 þ b2 g21 A3 þ g2 b1 þ b2 g22 A4 g1 b1 þ b2 g21 B3 g2 b1 þ b2 g22 B4 ¼ 0
ð47Þ
(48)
0 : N2
Now we proceed to apply the joint conditions (33)e(36) to the expressions (40) and (42), resulting in
b2 g21 A1 þ b2 g22 A2 þ b2 g21 B1 þ b2 g22 B2 ¼ 0
N1 0
g1 b1 þ b2 g21 A7 þ g2 b1 þ b2 g22 A8
ð g1 A 1 þ g2 A 2 g1 B 1 g2 B 2 Þ ¼ 0
R1 ¼
A2 g1 b2 g21 B1
(58)
where
eil g2
A1 þ A2 þ B1 þ B2 ¼ 0
The combined reflection relation, then, is given by
½A1 ; A2 ; A5 ; A6 T ¼ R1 ½B1 ; B2 ; B5 ; B6 T ;
Applying the boundary conditions (29)e(32) to the expression for w1 given by (39) gives us, respectively,
g1 ¼
(56)
In each case, the values of the matrix entries, for each boundary condition, are given by
and
B5 : B6
(50)
and, led by our analyses in (Coleman and Wang, 1993) and 4 (Coleman, 2009), we neglect terms of order Oð1=l Þ. In this case, we may write the boundary (46)e(49) as
b2 g21 A3 b2 g22 A4 b2 g21 B3 b2 g22 B4 ¼ 0
ð60Þ
b2 g1 g21 b2 A7 þ b2 g2 g22 b2 A8 b2 g1 g21 b2 B7
b2 g2 g22 b2 B8 þ b2 g1 g21 b2 A3 þ b2 g2 g22 b2 A4 h b2 g1 g21 b2 B3 b2 g2 g22 b2 B4 ¼ d01 b1 b2 b2 g21 A3 i ð61Þ þ b2 g22 A4 þ b2 g21 B3 þ b2 g22 B4
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
h
i
Here, aj, bj and cj, j ¼ 1, 2, are given by (57), x1 and x2 by (68), and we use the shorthand
d02 g1 g21 b1 b2 b1 b2 A3 i h þ d02 g2 g22 b1 b2 b1 b2 A4 i h d02 g1 g21 b1 b2 b1 b2 B3 i h d02 g2 g22 b1 b2 b1 b2 B4
ej ¼ eil
Again, using (50) and neglecting terms of Oð1=l Þ, these simplify to 4
b2 A þ A8 b1 7
b2 B B8 þ b1 7
sffiffiffiffiffi
b2 A þ A4 b1 3
sffiffiffiffiffi
A3 þ B3 A7 B7 ¼ 0 A3 B3 þ A7 B7 ¼
bj ;
j ¼ 1; 2:
a1 ¼ b1 ¼ 1 :
e2il
2
pffiffiffiffi
e2il
a1 ¼ b1 ¼ 1 :
(65)
00
b1 ¼ 1; 2þd1 2d001
a1 ¼ 1; b1 ¼ H1 :
(64) þ B3 Þ
ð71Þ
Thus, for the d1-branch(es) of the spectrum, we have
b2 B B4 ¼ 0 b1 3 (63)
d001 ðA3
pffiffiffi
1 0 ¼ detðR1 JR2 J IÞ ¼ ð2 þ d1 Þð2 þ d2 Þ i h a1 b1 ðd1 2Þe41 þ ða1 þ b1 Þd1 e21 þ d1 þ 2 $ i h a2 b2 ðd2 2Þe42 ða2 þ b2 Þd2 e22 þ d2 þ 2 :
ð62Þ
l
sffiffiffiffiffi
2
Finally
b1 b2 ½A7 þ A8 þ B7 þ B8 1 þ 4 ½A7 þ A8 þ B7 þ B8 A3 A4 B3 B4 ¼ 0: sffiffiffiffiffi
633
e4il pffiffiffiffi 2
2
pffiffiffiffi
00
b1 ¼ d1 þ2 d001 2 00
b1 ¼ 1; d1 þ2 d001 2
or
ðA7 þ A8 þ B7 þ B8 A3 A4 B3 B4 Þ sffiffiffiffiffi sffiffiffiffiffi !
b2 A þ A4 b1 3
¼ d002
b2 B B4 ; b1 3
2
ilm
pffiffiffiffiffiffi pffiffiffiffiffiffi where d001 ¼ d01 b1 and d002 ¼ d02 = b2 . After very much, but straightforward, computation, we solve for B3, B4, B7 and B8, and rewrite (63)e(66) as the reflection relation at x ¼ 0 given by
2
2
3 d1 B3 6 2þd1 6 6 B4 7 x1 6 7 ¼ 6 6 2 4 B7 5 6 4 2þd1 B8 x2
2 2þd1
0 d2 2þd2
x2
0
d1 2þd
2 2þd2
x1
1
ilm ¼ 2
ilm 2
sffiffiffiffiffi
b2 d1 þ d2 þ d1 d2 ; b1 ð2 þ d1 Þð2 þ d2 Þ
x2 ¼ 2
d00 2 1 m pffiffiffiffiffi ln 100 þ pffiffiffiffiffipi; m˛Z; 4 b1 d1 þ 2 2 b1
ð72aÞ
ð72bÞ
n 2 iln ¼ pffiffiffiffiffipi;
a1 ¼ b1 ¼ 1 :
where
x1 ¼ 2
2
(67)
2þd2
sffiffiffiffiffi
iln ¼
a1 ¼ 1; b1 ¼ H1 :
3
2 3 A3 7 2 7 6 7 2þd2 7 6 A4 7 7 4 5; R2 A 7 0 7 5 A8 d2 0
2n þ 1 pffiffiffiffiffi pi; 2 b1 2 d001 1 m ¼ pffiffiffiffiffi ln 00 þ pffiffiffiffiffipi; n; m˛Z; b1 2 b1 2 þ d1
a1 ¼ b1 ¼ 1 :
(66)
b2 d1 d2 ; b1 ð2 þ d1 Þð2 þ d2 Þ
b1
d00 2 1 m ¼ pffiffiffiffiffi ln 100 þ pffiffiffiffiffipi; n; m˛Z: b1 2 b1 d1 þ 2
ð72cÞ
Similarly, the d2-branch(es) are given by
(68)
a2 ¼ b2 ¼ 1 :
and where, for the sake of clarity, we are using d1 and d2 instead of d00 1 and d00 2, respectively. Putting together (43), (44), (58) and (67), we arrive at
2
ilm
3 2 3 2 3 2 3 2 3 A1 B1 B3 A3 A1 6 A2 7 6 7 6 7 6 7 6 7 6 7 ¼ R1 6 B2 7 ¼ R1 J6 B4 7 ¼ R1 JR2 6 A4 7 ¼ R1 JR2 J6 A2 7; 4 A5 5 4 B5 5 4 B7 5 4 A7 5 4 A5 5 A6 B6 B8 A8 A6
n 2 iln ¼ pffiffiffiffiffipi;
b1
d00 2 1 m ¼ pffiffiffiffiffi ln 200 þ pffiffiffiffiffipi; n; m˛Z; b2 2 b2 d2 þ 2
ð73aÞ
2
which will hold for nontrivial [A1, A2, A5, A6]T if and only if
detðR1 JR2 J IÞ ¼ 0:
a2 ¼ 1; b2 ¼ H1 : 2
ilm ¼
2
ilm
6 6 c d1 e2 þ a x e e 2 1 1 2 6 1 2þd1 1 R1 JR2 J I ¼ 6 2 e2 6 b 1 2þd1 1 4 2 e2 þ b x e e c2 2þd 2 2 1 2 1 1
1
2n þ 1 pffiffiffiffiffi pi; 2 b1 2 d002 1 m ¼ pffiffiffiffiffi ln 00 þ pffiffiffiffiffipi; n; m˛Z: b2 2 b2 2 þ d2 2
iln ¼
a2 ¼ b2 ¼ 1 :
(70)
d1 2 e1 1 a1 2þd
3
0
2 e2 a1 2þd 1
0
d2 2 a2 2þd e2 1
2 e2 þ a x e e c1 2þd 2 2 1 2 1
2 e2 a2 2þd 2
0
d1 2 b1 2þd e1 1
2
2 e2 b2 2þd 2 2
ð73bÞ
(69)
Now we have
2
d00 2 1 m pffiffiffiffiffi ln 200 þ pffiffiffiffiffipi; m˛Z; 4 b2 d2 þ 2 2 b2
1
1
d1 2 c2 2þd e 1 1
1
þ b2 x1 e1 e2
2
0 d2 2 b2 2þd e 2 2
1
7 7 7 7: 7 5
ð73cÞ
634
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
Then, for combinations of the boundary conditions (14)e(17), we have CeC: (72a), (73c) CeS: (72b), (73c) CeR: (72a), (73b) CeF: (72b), (73b) SeS: (72c), (73c)
SeR: (72b), (73b) SeF: (72c), (73b) ReR: (72a), (73a) ReF: (72b), (73a) FeF: (72b), (73a).
Lastly, we may put (72) and (73) back into dimensional form:
sffiffiffiffiffi EI 2n þ 1 pi; a1 ¼ b1 ¼ 1 : Ir 2 sffiffiffiffiffi ! pffiffiffiffiffiffiffi 1 EI 1 2 rEI d1 2 pffiffiffiffiffiffiffi þ mpi ; n; m˛Z; ixm ¼ ln L Ir 2 2 þ rEI d1 2 ixn
1 ¼ L
a1 ¼ 1; b1 ¼ H1 : sffiffiffiffiffi ! pffiffiffiffiffiffiffi rEI d1 2 1 EI 1 2 þ mpi ; m˛Z; ixm ¼ ln pffiffiffiffiffiffiffi 2L Ir 2 rEI d1 þ 2 sffiffiffiffiffi 1 EI ¼ npi; a1 ¼ b1 ¼ 1 : L Ir sffiffiffiffiffi ! pffiffiffiffiffiffiffi rEI d 2 1 EI 1 2 þ mpi ; n; m˛Z: ixm ¼ ln pffiffiffiffiffiffiffi 1 L Ir 2 rEI d1 þ 2
ð74aÞ
ð74bÞ
ð74cÞ
2 ixn ¼
sffiffiffiffi 1 K 2n þ 1 pi; ¼ a2 ¼ b2 ¼ 1 : L r 2 sffiffiffiffi ! pffiffiffiffiffiffi 1 K 1 2 K r d2 2 pffiffiffiffiffiffi þ mpi ; n; m˛Z: ixm ¼ ln L r 2 2 þ K r d2
2ð1 þ nÞ E ¼ 0 ; k0 kG 1 I ¼ ; s2 AL2
g2 ¼
~ are the wave numbers, g and g are as given in (38) Here, a and b 1 2 and l2 in (26). We note here that, in terms of these quantities, our dimensionless parameters b1 and b2 become
b1 ¼
and
a2 ¼ 1; b2 ¼ H1 : sffiffiffiffi ! pffiffiffiffiffiffi 1 K 1 Kr d 2 2 þ mpi ; m˛Z; ixm ¼ ln pffiffiffiffiffiffi 2 2L r 2 K r d2 þ 2
If we let d1 ¼ d2 ¼ 0, then our system should behave as a single Timoshenko beam of length 2L. Thus, we may compare our asymptotic results with the solutions resulting from the exact frequency equation(s) given in the literature ((Han et al., 1999), (Karnovsky and Lebed, 2001). It can be shown that the equations given by these two references are equivalent. We restrict our study here to the clampedeclamped case. Following (Han et al., 1999) we derive their frequency equation for our model. We non-dimensionalize (2L / 1), and introduce the following parameters from (Han et al., 1999):
2 ~ ¼ l2 g ; u ¼ l2 : a ¼ l g2 ; b 1
2 ixn
sffiffiffiffi 1 K a2 ¼ b2 ¼ 1 : npi; L r sffiffiffiffi ! pffiffiffiffiffiffi 1 K 1 K r d2 2 2 þ npi ; n; m˛Z; ixm ¼ ln pffiffiffiffiffiffi L r 2 K r d2 þ 2
5. Justification of method e comparison with “exact” solution
ð75aÞ
1 ; b2 ¼ g2 b1 : s2
Following this notation, our asymptotic frequencies from (72b) and (73b) become
Branch 1 :
u np np u ¼ pffiffiffiffiffi or ; ¼ g s b2
n ¼ 1; 2; .
(76)
Branch 2 :
u np u ¼ pffiffiffiffiffi or ¼ np; s b1
n ¼ 1; 2; .
(77)
Meanwhile, our exact frequency equation is
~ 1 cos a cos b
2 2 ~2 g2 a2 ~ 2 a 2 g2 b ~2 þa2 b b
~ 2 2 sin a sin b; 2 2 ~ ~ ~ 2 2 2 2 b g a 2ab a g b
ð75bÞ
(78) which is equivalent to that given in (Han et al., 1999) (p. 966, Table 10, cec). The exact frequencies are
2 ixn
ð75cÞ
We note that we have assumed implicitly that d00 1 s 2 and d00 2 s 2 in the above argument. If, instead, we set d00 1 ¼ 2 in (71), we are led to the following:
pffiffiffiffi 2 b1 ¼ 1 a1 ¼ b1 ¼ 1 : e2il p ffiffiffiffi 2 l b1 ¼ 1 2i a1 ¼ b1 ¼ 1 : e a1 ¼ 1; b1 ¼ H1 : branch disappears: Thus, it seems as though the damping disappears; similarly, for d00 2 ¼ 2. However, the situation here is unstable in that, as d00 1 / 2, the real part of il2 / N, that is, the damping rate appears to become infinite. This interesting situation bears further study; indeed it is similar to the phenomenon of superstability in the single Timoshenko beam with boundary damping ((Shubov, 2002), (Shubov, private communication)).
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u a2 þ b ~2 ¼ u ¼ l2 ¼ t b 1 1 þ g2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2 ~2 ta þ b : s 1 þ g2
(79)
Again following (Han et al., 1999), we set g ¼ 2.205 and compute u/s from (79), extending their Fig. 18, p. 970 (note that their vertical axis is mislabeled as u*/L*(r*/E*)1/2, when it should be u*L*(r*/E*)1/2 ¼ u/s). As there, we plot u/s vs. 1/s, and we compare with the two asymptotic branches (76) and (77). We note that, if the beam is taken to have circular cross section, then s ¼ 4L/d, where L/d is the beam’s slenderness ratio. Thus, in Fig. 2, we plot the first 13 “exact” frequencies, and compare them to the first 13 asymptotic frequencies. We see a very good match between the exact and asymptotic frequencies, even for the lowest frequencies, for values of 1/s > 0.2, i.e., for L/d < 1.25. For more slender beams corresponding to 1/s ¼ 0.1 (L/d ¼ 2.5), the match is not good for the first few frequencies e and, in particular, we see the crossing and near crossing of the exact frequency curves e but then begins to get better once we
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
635
ω
s
15
10
5
0
0.0
0.1
0.2
0.3
0.4
1 0.5 s
Fig. 2. Clampedeclamped Timoshenko beam, first 13 frequencies. Exact: solid; Asymptotic, Branch 1: dash-dot; Asymptotic, Branch 2: dash.
get to the 10th or 12th frequencies. Further, it seems that the complicated structure of the left-most parts of the frequency curves tends towards the vertical axis as we go farther out in the spectrum. This behavior becomes more pronounced in Fig. 3. Here we plot the frequency curves for the n ¼ 10, 15, 20, 25, 30, 35 and 40th frequencies and, again, compare to the asymptotic frequencies. Above the 15th frequency, we see a good match for values of 1/s > 0.05 (L/d ¼ 5), and we believe that the trend must continue as n increases.
These results agree with the numerical results in (Coleman and Wang, 1993), where it is seen that, for small values of L/d, the match between numerical and asymptotic frequencies is very good even for the lowest frequency, while, as L/d increases, we must go farther “up” the spectrum before there is a good match. Finally, as we allow d1 and d2 to become nonzero, the asymptotic results suggest that the only effect is to change the damping rate, with the above frequencies remaining the same. Although we have no exact values with which to compare, numerical data in (Coleman and Wang, 1993) clearly suggests that this is only an
Fig. 3. Clampedeclamped Timoshenko beam, frequencies 10, 15, 20, 25, 30, 35, 40. Exact: solid; Asymptotic, Branch 1 (n ¼ 7, 11, 14, 21, 28): dash-dot; Asymptotic, Branch 2 (n ¼ 5, 8, 11): dash. The horizontal asymptotes are labeled by height and branch: 7p/g, 14p/g, 21p/g from Branch 1; 5p, 8p and 11p from Branch 2. (We note that it is purely coincidental that frequency 20 converges to a number near 20, frequency 25 to a number near 25, etc.).
636
M.P. Coleman, L. Schaffer / European Journal of Mechanics A/Solids 29 (2010) 629e636
asymptotic behavior. Indeed, the numerical results suggest that the actual damping rates are much smaller than the asymptotic results, at least at the low end of the spectrum; the actual rates approach the asymptotic rates from below. Meanwhile, it is not clear how these damping rates affect the imaginary parts of the frequencies.
Further, we have been unable to find numerical results. We plan on performing such a numerical study for this problem and, eventually, to perform similar analytical and numerical investigations of other types of joints. References
6. Conclusion/observations It is known that, for damping in the middle of the span, there often are modes that remain undamped. While not true for identical coupled EulereBernoulli beams (see (Chen and Zhou, 1990)), it certainly is the case here, except for the combinations CeF and SeR. Indeed, if aj ¼ bj ¼ 1, the corresponding branch of the spectrum consists of two sub-branches, one damped and one undamped. Further, if dj ¼ 0, then the corresponding branch (or union of two sub-branches) will be the same as that for a beam of length 2L which is continuous in the corresponding variable (i.e., from (20) and (21), Ft for d1 ¼ 0 and Wt for d2 ¼ 0). This is to be expected, of course. In particular, if d1 ¼ d2 ¼ 0, then the spectrum will be the same as that for a single beam of length 2L (Coleman and Wang, 1993). The only experimental results that we have been able to find are given in (Chen et al., 1989), for the case when d2 ¼ 0. Unfortunately, those results include only the first twelve frequencies, far fewer than are necessary to give any kind of agreement with asymptotic results (see, e.g., Example 4 in (Coleman and Wang, 1993)). However, the fact that the Euler-Bernoulli results in (Chen et al., 1989) seem to be overestimates certainly is consistent with the pffiffiffiffi fact that jxj w p for the EulereBernoulli beam, while jxjw p for these Timoshenko beams.
Challamel, N., 2006. On the comparison of Timoshenko and shear models in beam dynamics. J. Eng. Mech. 132, 1141e1145. Chen, G., Krantz, S.G., Russell, D.L., Wayne, C.E., West, H.H., Coleman, M.P., 1989. Analysis, designs and behavior of dissipative joints for coupled beams. SIAM J. Appl. Math. 49, 1665e1693. Chen, G., Zhou, J., 1990. The wave propagation method for the analysis of boundary stabilization in vibrating structures. SIAM J. Appl. Math. 50, 1254e1283. Coleman, M.P., 2009. A perturbation method for the accurate estimation of the vibration spectrum for the Timoshenko beam. J. Sound Vibration (preprint). Coleman, M.P., Wang, H.K., 1993. Analysis of the vibration spectrum of a Timoshenko beam. Wave Motion 17, 223e239. Han, S.M., Benaroya, H., Wei, T., 1999. Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vibration 255 (5), 935e988. Hodges, D.H., 2007. An asymptotic derivation of shear beam theory from Timoshenko theory. J. Eng. Mech. 133, 957e961. Karnovsky, I.A., Lebed, O.I., 2001. Formulas for Structural Dynamics: Tables Graphs and Solutions. McGraw-Hill. Keller, S.B., Rubinow, S.I., 1960. Asymptotic solution of eigenvalue problems. Ann. Phys. 9, 24e75. Kim, J.U., Renardy, Y., 1987. Boundary control of the Timoshenko beam. SIAM J. Control Optim. 25, 1417e1429. Krantz, S.G., Paulsen, W.H., 1991. Asymptotic eigenfrequency distributions for the N-segment EulerdBernoulli coupled beam equations with dissipative joints. J. Symbolic Comp. 11, 369e418. Shubov, M.A., 2002. Asymptotic and spectral analysis of the spatially nonhomogenous Timoshenko beam model. Math. Nachr. 241, 125e162. Shubov, M.A., Private communication. Traill-Nash, R.W., Collar, A.R., 1953. The effect of shear flexibility and rotary inertia on the bending vibration of beams. Quart. J. Mech. Appl. Math. 6, 186e222.