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Correct Boundary Conditions for String Model With Boundary Damping

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CORRECT BOUNDARY CONDITIONS FOR STRING MODEL WITH BOUNDARY DAMPING
Igor Andrianov, Denis Byvalin, Marina Matyash
Institute of General Mechanic, RWTH Aachen University Prydniprovskaya State Academy of Civil Engineering and Architecture Dnepropetrovsk National University
1. Abstract
In this paper simplified boundary conditions for string model is obtained starting from the beam model with boundary damping.
2. Introduction
In a number of mechanical systems oscillations can be described by beam equations. Examples of such systems are suspended bridges, tall buildings, power transmission lines, and elevator cables. To suppress the oscillations various types of boundary damping can be applied [1]. Nowadays, various types of passive dampers applied at the boundary have been considered extensively [2-7]. Simplified model which describe these oscillations can be expressed in BVP for wave equation (string model) [2,6,7]. Area of applicability of this approach was studied in [8] for classical BCs. In this paper simplified BCs for string model is obtained starting from BVP for the beam with boundary damping.
3. Governing relations
Governing BVP for stretched beam can be written as follows [9]:
0
222244
=∂∂ρ+∂∂−∂∂
t W F xW T xW EI
; (1)
0
=∂∂=
xW W
for
x
= 0; (2)
t xW xW EI
∂∂∂α−=∂∂
220122
for
x
=
L
; (3)
t W xW T xW EI
∂∂α=∂∂−∂∂
20233
for
x
=
L
. (4)
Here W is the normal displaycement, T is the stretched force, E is the Young modulus, I is the static moment, L is the beam length, x is the spacial coordinate, t is the time; parameters and chracterize demping.
201
α
202
α
From the physical standpoint BVP (1) – (4) describes stretched beam with left end clamped and right one with the boundary damping. BVP (1) – (4) can be reduced to the following form:
0
222220442
=τ∂∂+ξ∂∂−ξ∂∂ε
W W T W
; (5) 0
=ξ∂∂=
W W
for
0
=ξ
; (6)
τ∂ξ∂
∂α−=ξ∂∂
W W
22122
for
1
=ξ
; (7)
τ∂∂α=ξ∂∂ε−ξ∂∂
−
W W T W
2220233
for
1
=ξ
, (8) where
1
22
<<=ε
FL I
;
L x
=ξ
;
ρ=τ
E Lt
;
EF T T
=
20
, ; 1~
0
T
ρα=α
E LI
20121
;
ρα=α
E I L
20222
.
4. Formulation of BC
Further asymptotic technique described in [9] will be used. Only the lower part of the spectrum is under consideration:
W W
~
τ∂∂
. (9) For passing to the string model one must suppose
W W
~
ξ∂∂
, (10) then Eq. (5) can be reduced to the following form
0
20220220
=τ∂∂−ξ∂∂
W W T
. (11) Using series
K
+ε+ε+=
0240120
W W W W
, (12) one obtains the following recurrent system of Eqs. of successive approximations:
4042012201220
ξ∂∂−=τ∂∂−ξ∂∂
W W W T
; (13) . . . . . . . . . . .
Solution of the 2nd order PDE (11) can satisfy only two BCs. For compensation of discrepancy in BCs one must use boundary layer solution
W
b
, satisfied the following estimation:
bb
W W
1
~
−
εξ∂∂
. (14) Solution of boundary layer BVP can be expressed by series:
K
+ε+ε+=
24120
bbbb
W W W W
. (15) The recurrent system of Eqs. for boundary layer is:
0
0202022
=−ξ∂∂ε
bb
W T W
; (16)
∫∫
ξ ξ
ξξτ∂∂−=−ξ∂∂ε
002021202122
d d W W T W
bbb
; (17) . . . . . . . . . . . Now one must obtain BCs for Eqs. (11), (16) from governing BCs (7),(8). Solution of Eq. (16) can be written as follows:
)1(exp)()1(exp)(
01101)1(0
−ξετ+−ξετ=
−−
T C T C W
b
. (18) From decaying condition
0
)1(0
→
b
W
for
0
→ξ
(19) one obtains
0)(
1
=τ
C
. (20) Using Eqs. (18), (20) one can rewrite BCs (7), (8) in the following form, neglecting the small terms: for
ξ
= 1
⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ τ∂∂ε+τ∂ξ∂
∂α−=ε+
ξ∂∂
−−
C T W C T W
010221202202
; (21)
⎟ ⎠ ⎞⎜⎝ ⎛ τ∂∂+τ∂∂α−=ξ∂∂ε
−
C W W T
0220202
. (22) We suppose ,
1~
21
α
222
~
−
εα
22
~
2
α ε
−
and introduce parameter of asymptotic integration (see [8,9,10])
06
~
W C
ε
. (23) For one obtains BCs for Eqs. (11), (16):
2
2
=δ
for
ξ
= 1
τ∂∂α−=ξ∂∂ε
−
0220202
W W T
; (24)
τ∂ξ∂
∂α−ξ∂∂−=ε
−
0221202202
W W C T
. (25)
5. Concluding remarks
Obtained results can be generalized for nonlinear beams [11]. For plates [12] described technique can be used for obtaining BCs for membrane approach.
6. References
[1]
H. Wang, A.R. Elcrat, R.I. Egbert, Modelling and boundary control of conductor galloping,
Journal of
Sound and Vibration
161 (1993) 301-315. [2]
S. Cox, E. Zuazua, The rate at which energy decays in a string damped at one end,
Indiana University Mathematics
Journal
44 (1995), 545-573. [3]
S.P.Chen, K.S. Liu, Z.Y. Liu, Spectrum and stability for elastic systems with global or local Kelvin-Voigt damping,
SIAM Journal on Applied Mathematics
59 (1998) 651-668.
[4]
K.S. Liu, Z.Y. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping,
SIAM Journal on Control and Optimization
36 (1998) 1086-1098.
[5]
J.W. Hijmissen, W.T. van Horssen, On aspect of boundary damping for vertical beams with and without tip-mass,
Journal of
Sound and Vibration
310(3) (2008) 740-754. [6]
Darmawijoyo, W.T. van Horssen, On the weakly damped vibration of a string attached to a spring-mass-dashpot system,
Journal of Vibration and Control
9 (2003) 1231-1248. [7]
Ö. Morgül, B.P. Rao, F. Conrad, On the stabilization of a cable with a tip masses,
IEEE Transactions on Automatic Control
39(10) (1994) 2140-2145. [8]
I.V. Andrianov, W.T. van Horssen, On the transversal vibrations of a conveyor belt: Applicability of simplified models,
Journal of
Sound and Vibration
313 (2008) 822-829. [9]
G. Chen, S.G. Krantz, D.W. Ma, C.E Wayne, H.H. West, The Euler-Bernoulli beam equation with boundary energy dissipation,
Operator Methods for Optimal Control Problems, S.J. Lee, ed., Lecture Notes in Pure and Applied Mathematics
108 (1987) 67-96. [10]
J. Awrejcewicz, I.V. Andrianov, L.I. Manevitch,
Asymptotic Approaches in the Nonlinear Dynamics: New Trends and Applications,
Springer, Berlin, 1998. [11]
Darmawijoyo, W.T. van Horssen, On boundary damping vibration for a weakly nonlinear wave equation,
Nonlinear Dynamics
30 (2002) 179-191. [12]
M.A. Zarubinskaya, W.T. van Horssen, On aspect of boundary damping for a rectangular plate,
Journal of
Sound and Vibration
292 (2006) 844-853.
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