a review - Acta Technica Corviniensis

The wire movement is reported to the fixed system
Oxz. Also, a moving system attaches against the force
O 1 x 1 z with
x = Vt + x 1
(1)
The wire motion equation reported to the fixed
referential is
2
2
∂ w ∂w
∂ w
m + a + kw − T = P cosωtδ(
x −Vt)
, (2)
2
2
∂t
∂t
∂x
where k is the elastic constant, and a – the damping
constant of the continuous elastic support.
Also, the boundary conditions have to be considered
lim w = 0 . (3)
x−Vt
→∞
Actually, the steady state behaviour is interesting
and due to that, the change of variable (1) is
recommended, where the motion is reported to the
moving referential. Practically, the following
relations will be applied
n n n
n
∂ ∂
→ ,
∂ ⎛ ∂ ∂ ⎞
n n
→ V
n
∂x ∂x1
t
⎜ −
t x
⎟ . (4)
∂ ⎝ ∂ ∂ 1 ⎠
The equation of motion (2) and the boundary
conditions (3) become
2
2
2 ∂ w ∂w
∂ w
( mV − T ) − aV − 2mV
+
2
∂x
∂x
1
1 ∂x1∂t
2
∂w
∂ w
+ kw + a + m = P cosωtδ(
x1
) (5)
2
∂t
∂t
lim w = 0 . (6)
x1
→∞
Considering the steady state harmonic behaviour, the
complex variables
iωt
i t
t
w( x1 , t)
= w(
x1
) e , P t Pe
ω i0 iω
( ) = = Pe e (7)
have to verify the following equation
2
2 d w
dw
( mV − T ) − V ( a + 2ωmi)
+
2
dx
dx
1
2
+ k − ω m + ωai)
w = Pδ(
x )
(8)
( 1
and the boundary conditions
lim w = 0 . (9)
x1
→∞
To solve the problem defined by the equation (8) and
the boundary conditions (9), the Green’s functions
method may be applied [11]. In fact, the solution is
given as
∞
w ( x1 ) =
∫
G(
x1,
ξ)
P δ(
ξ)dξ
= P G(
x1,0)
, (10)
− ∞
where G(x 1 , ξ) is the Green’s function. This function
represents the wire response in the x 1 section of the
moving reference frame, due to a unit harmonic
force applied in the section of the same moving
reference frame. It has to be observed the fact that
the wire response is defined by the Green’s function
and this function is the receptance.
The Green’s function can be built as a linear
combination of the eigenfunctions of the differential
operator of the equation (8). To find this function,
the starting point is the homogenous equation
1
ACTA TECHNICA CORVINIENSIS – Bulletin of Engineering
2
2 d w
dw
( mV −T
) −V
( a + 2ωmi)
+
2
dx
dx
1
1
2
+ ( k − ω m + ωai)
w = 0
(11)
and its solution
λx1
w ( x 1 ) = Ae . (12)
Then, the characteristic equation is obtained
2 2
2
( mV − T ) λ − V ( a + 2ωmi)
λ + k − ω m + ωai
= 0 (13)
After some calculations, this equation takes the
following form
2 2 2
2 2
( V − c ) λ − 2V
( ζω0
+ ωi)
λ + ω0
− ω + 2ζω0ωi
= 0 (14)
where
a
2 k T
ζ = , ω = , c =
(15)
0
2 mk m m
The solutions of the characteristic equation
represent the eigenvalues
ω
0
λ1,2
= ⋅μ1,2
(16)
c
2 2 2
α(
ζ + Ωi)
± 1− Ω − α (1 − ζ ) + 2ζΩi
where μ 1,2 =
(17)
2
α −1
V ω
with α = , Ω = . (18)
c
ω 0
There are two cases, the so-called sub-critical and
overcritical cases.
1. The sub-critical case (α < 1) – the force velocity is
smaller than the velocity of the elastic wave in the
contact wire; the critical velocity has value of c. In
this case, the eigenvalues real parts have opposite
signs
Re λ 1 < 0, Re λ 2 > 0.
(19)
In fact, the Green’s function G(x 1 , ξ) has two forms
satisfying the boundary conditions
−
− λ 2 x1
G ( x 1 , ξ ) = A e for − ∞ < x 1 < ξ (20)
+
+ λ1
x1
G ( x 1 , ξ ) = A e for ξ < x 1 < ∞ ,
−
where A and A + depend on the ξ variable. These
functions will be calculated using both continuity and
jump conditions.
The Green’s function has to be continuous in x 1 = ξ
− λ 2 ξ + λ1
ξ
A e = A e . (21)
Its derivation in respect to x 1 has a jump in x 1 = ξ
+
−
∂G
( ξ + 0, ξ)
∂G
( ξ − 0, ξ)
1 1
−
=
(22)
2
∂x1 ∂x1
T α −1
respectively
+ λ
1 1
1 ξ − λ2
ξ
λ1
A e − λ2
A e = . (23)
2
T α −1
Upon solving the equations (21) and (23), it is
obtained
k
− μ 2ξ
k
− μ ξ
1
T
− 1 e
T
A =
, + 1 e
2
A =
(24)
2
kT ( μ1
− μ2)(
α −1)
kT ( μ1
− μ2)(
α −1)
and then, the Green’s function
T
− 1 e
G ( x1,
ξ)
=
for − ∞ < x
2
1 < ξ (25)
kT ( μ − μ )( α −1)
1
k
μ 2 ( x1
−ξ)
2
36
2013. Fascicule 2 [April–June]