Ramona PinŃoi
Through the correlation of the compacting process with the vibration parameters, the compaction of the fresh concrete is optimized. The employed dynamic model is a linear viscous one. The dissipative energy in the vibrated concrete is given by the vibration parameters and the hysteresis loop is determined analyzing the dissipative energy in corre lation with the degree of concrete compaction.
: hysteresis loop, vibration parameters, concrete compaction
The compaction of fresh concrete can be optimized by effectively correlating the degree of compaction with the vibration parameters (amplitude, frequency). In this case compliance with the required viscosity, which determines the internal en ergy dissipation, must ensure the viscous damping coefficient c for the fresh con crete put into practice and subjected to an adequate vibration regime.
In the context of coherence, relations were established between the amplitude of the vibrating support excited by the force
F
=
m
0r
ω
2sin
ω
t
, in which2 0 0
m
r
ω
F
=
, and the mass of the fresh concrete from the mold (pattern).The dynamic model is shown in Figure 1 and is characterized by the linear, viscous connection, with a damping coefficient c, chosen between the concrete mass and the vibrating platform, meaning that the viscous connection force is of the type
F
(
x
)
=
cv
=
c
(
x
1−
x
2)
.ANALELE UNIVERSITĂłII
“EFTIMIE MURGU” RE IłA
! Dynamic Model
In Figure 1, the vibrating platform with mass m1 and instantaneous displace ment x1 = x1(t) is excited by an inertial vibrator with eccentric masses having the static moment mor and the angular velocityω.
Mass m2 of the fresh concrete is considered as a whole and has the instanta neous displacement x2= x2(t) and the coefficient of viscosity of the fresh concrete c. The entire assembly rests elastically on a system of springs with the elastic con stant k.
Deformation equations in the complex domain are of the form [1]:
0
)
(
)
(
1 2 2 2
0 1 2 1 1 1
=
−
+
=
+
−
+
z
z
c
z
m
e
F
kz
z
z
c
z
m
jωt(1) where: j( t i)
i i
A
e
z
=
ω −ϕ , with i=1,2 andj
=
−
1
. Thus:t j
e
F
t
F
(
)
=
0 ω with ReF
(
t
)
=
F
0cos
ω
t
)
cos(
)
cos(
2 2
2 2
1 1
1 1
ϕ
ω
ϕ
ω
−
=
=
−
=
=
t
A
z
R
x
t
A
z
R
x
e e
. (1’)
c
R
F
A
m
c
R
F
A
)
(
)
(
0 2
2 2 2 2 0 1
ω
ω
ω
=
+
=
(2)
where F0=m0rω2 and represents the amplitude of the perturbing force and
0 2 2 4 4 6 6
)
(
r
r
r
r
R
ω
=
ω
+
ω
+
ω
+
is the transfer function of attenuation of the dynamic force, applied in amplitudes of the instantaneous displacements.The ratio of the amplitudes
2 1
A
A
=
λ
is obtained in the form of:c
m
c
A
A
22 22
2
1
ω
λ
=
=
+
(3)or
2
1
ν
λ
=
+
(3’)where
c
m
ω
ν
ω
ν
(
)
=
=
2 is the damping ratio."
The dissipative energy in the vibrated concrete is given by [2]:
2 2
A
c
W
d=
π
ω
(4)where
2 1 1
2
1
ν
λ
=
+
=
A
A
A
.Thus:
2 2 1
1
ν
ω
π
+
=
c
A
W
d . (4’)If one replaces
c
m
ω
ω
ν
ν
=
(
)
=
2 in (4’), it results:2 2 2 2
3 2
1
m
c
c
A
W
dω
ω
π
+
=
(5)For the continuous variation of ω, the maximum dissipative energy
W
dmax can0
)
(
22 22 2 2 2 2 2 2 1 3
=
+
−
=
m
c
m
c
A
c
d
dW
dω
ω
π
ω
(6) from which 2m
c
d
=
±
ω
, with the only physical solution2
m
c
d
=
ω
.The curve has two points of extension, namely a point of minimum
−
−
* 2,
W
dm
c
Min
and a point of maximum
+
+
* 2,
W
dm
c
Max
, and the curvature sense (concave, convex) is given by 2
0
2
=
ω
d
W
d
, that is [3]:
0
)
3
)(
(
2
22 22 2 2 2 2 2 2 2 2
=
−
+
−
=
ω
ω
ω
ω
m
m
c
c
m
d
W
d
(7)
The roots are:
3
2 1
m
c
−
=
ω
,ω
2=
0
,3
2 2
m
c
+
=
ω
and have the sign of0
2 2ω
d
W
d
for
+
∪
−
∞
−
∈
,
3
0
,
3
2 2
m
c
m
c
ω
and0
2 2ω
d
W
d
for
+∞
+
∪
−
∈
3
,
0
3
,
2 2
m
c
m
c
ω
.The curve representation
W
d=
W
d(
ω
)
is shown in Figure 2 with the significant points:
−
−
* 2,
W
dm
c
Min
,
+
+
* 2,
W
dm
c
Max
, inflection
−
−
W
im
c
I
3
,
2
1 ,
+
+
W
im
c
I
3
,
2
1 ,
and in which
W
d*=
W
dmax represents the maximum energy.The maximum energy
W
dmax=
W
d*, results from the condition that2
m
c
d=
ω
: 2 2 2 1 * max2
)
(
m
c
A
W
W
d d dπ
ω
=
The corresponding energy of the inflection point I,
3
2
m
c
I
=
ω
, is:2 2 2 1 2
2 2 2 2 2
3 2 2
1
3
4
3
3
)
(
m
c
A
m
m
c
c
c
m
c
A
W
Iω
Iπ
=
π
+
=
(9)or
*
2
3
)
(
I dI
W
W
ω
=
(9’)! Representation of Wd
The maximum value of the dissipated energy,
W
dmax, results from the condition
ω
=
=
ω
2
m
c
d , in which
m
Ns
m
c
c
*=
=
2ω
=
60
*
100
π
=
6
π
*
10
3 andπ
100
=
Ω
, : representing the pulsation of the disturbing force so that1
* 2
*
=
=
c
m
ω
ν
.=
+
=
where
2
1 2
A
A
=
, and the energy isW
(
ω
)
=
π
c
*ω
dA
22 or2 1 2 2 *
2
)
(
A
m
c
W
ω
=
π
(10)For
A
1=
0
,
5
*
10
−3m
,m
2=
60
kg
,m
Ns
c
*=
6
π
*
10
3 andπ
ω
ω
50
2
1
1
=
=
,ω
2=
ω
d=
ω
=
100
π
,ω
ω
150
π
2
3
3
=
=
,π
ω
ω
ω
4=
I=
3
=
3
*
100
,ω
5=
2
ω
=
200
π
,the following values for the dissipated energy given by formula (5) are ob tained:
j
W
W
1
,
85
2
1
)
(
1 11
=
=
ω
ω
( )
j
W
W
2(
ω
)
=
d*ω
d=
2
,
35
j
W
2
,
17
2
3
3
=
ω
j
W
W
4(
3
ω
)
=
I(
ω
I)
=
1
,
98
j
W
5(
2
ω
)
=
1
,
857
and which satisfy the condition
*
2
,
35
1
,
98
2
2
2
3
*=
=
=
dI
W
W
.The function 2
2 2 2
3 2 1
)
(
m
c
c
A
W
dω
ω
π
ω
+
=
can be represented numerically with the parameter c, the current variable ω=0...500 rad/s considering a step of 0,1rad/s.# $ %
The transmitted force is
F
T=
Q
(
t
)
=
−
c
(
x
1+
x
2)
, wherex
1(
t
)
=
A
1cos
ω
t
withx
1(
t
)
=
−
ω
A
1sin
ω
t
(
)
[
A
A
t
A
t
]
c
t
Q
(
)
=
−
ω
−
1+
2cos
ϕ
sin
ω
+
2sin
ϕ
cos
ω
(11)or
(
)
[
A
A
t
A
t
]
c
t
Q
(
)
=
ω
1+
2cos
ϕ
sin
ω
−
2sin
ϕ
cos
ω
(11’) The time variable, t, is eliminated from equation (11) or (11’), taking into ac count thatx
1=
A
1cos
ω
t
so that 1 1
cos
A
x
t
=
ω
and 21 2 1
1
sin
A
x
t
=
±
−
ω
.One finally obtains:
(
)
−
−
+
±
=
ω
cos
ϕ
1
sin
ϕ
)
(
1 1 2 2 1 2 1 2 1 1A
x
A
A
x
A
A
c
x
Q
(12) Where: 2 2 2 2 2sin
ω
ω
ϕ
m
c
m
+
=
, 2 2 2 2cos
ω
ϕ
m
c
c
+
=
and in which
c
m
tg
ϕ
=
2ω
was taken into consideration.Replacing in (12) the functions sinφ and cosφ it results:
+
−
−
+
+
±
=
2 2 2 2 2 1 1 2 1 2 1 2 1 2 2 2 2 1 2 11
)
1
(
ω
ω
ω
ω
m
c
m
x
A
A
A
x
A
m
c
c
A
A
A
c
x
Q
(13) where, introducing 2 2 2 2 1 2ω
m
c
c
A
A
+
=
, one has1 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2 2 2 3 1
2
)
(
x
m
c
m
c
x
A
m
c
c
m
c
x
Q
ω
ω
ω
ω
ω
+
−
−
+
+
±
=
(14)&
Based on the schematization of the visco linear model for the compacting of fresh concrete under vibration, the following can be determined:
a) the variation law of the dissipated energy in steady state vibration regime for the compaction process;
b) the maximum value of the dissipated energy through the correlation of the excitation pulsation with the concrete mass and the viscous constant of the dissipation energy;
c) determination of the hysteresis loops using the vibration parameters A1,ω and the mass m2and damping constant c of the fresh concrete.
'
[1] Bratu P.,VibraŃiile sistemelor elastice, Ed. Tehnică, BucureGti, 2000. [2] Hu C., Rheologie des betons fluids, These de doctorat, L'ecole Natio
nale des Ponts dt Chaussees, Paris, 1995.
[3] Darabi B., Dissipation of Vibration Energy Using Viscoelastic Granular Materials,PhD Thesis, University of Sheffield, U.K., 2013.
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