CORRECTION VIBRATION ANALYSIS OF A TRIMORPH PLATE

CORRECTION VIBRATION ANALYSIS OF A TRIMORPH PLATE

JOURNAL OF THEORETICAL
AND APPLIED MECHANICS
49, 3, pp. 641-664, Warsaw 2011
CORRECTION
VIBRATION ANALYSIS OF A TRIMORPH PLATE FOR
OPTIMISED DAMAGE MITIGATION
Akuro Big-Alabo
University of Port Harcourt, Department of Mechanical Engineering, Rivers State, Nigeria
e-mail: [email protected]
Matthew P. Cartmell
University of Glasgow, School of Engineering, Systems, Power and Energy Research Division,
Glasgow, Scotland, UK; e-mail: [email protected]
Page 645 – Eq. (2.7)1
∂3u
∂3u
∂3v
∂3v
+
(B
+
2B
)
+
(B
+
2B
)
+
B
12
66
12
66
11
∂x3
∂x∂y 2
∂x2 ∂y
∂y 3
4
4
4
i
h
∂ w
∂ w
∂ w
− D11 4 + 2(D12 + 2D66 ) 2 2 + D11 4
∂x
∂x ∂y
∂y
2
2
∂ w
∂w
∂ w
∂ 2 w ∂u
= ρh 2 + c
− q(x, y, t) + A11 2 + A12 2
∂t
∂t
∂x
∂y ∂x
∂u
∂2w
∂ 2 w ∂v
∂v ∂ 2 w
+ A12 2 + A11 2
+ 2A66
+
∂x
∂y ∂y
∂y
∂x ∂x∂y
h ∂ 2 w 2 ∂ 2 w 2 i
2
2
∂ 2 w 2
∂ w∂ w
−B11
+
−
2B
−
4B
12
66
∂y 2
∂x2
∂x2 ∂y 2
∂x∂y
B11
Page 645 – Eq. (2.7)3
∂2u
∂2u
∂2v
∂3w
+
A
+
(A
+
A
)
−
B
66
12
66
11
∂x2
∂y 2
∂x∂y
∂x3
3
2
∂ w
∂ u
−(B12 + 2B66 )
= ρh 2
2
∂x∂y
∂t
A11
2
A. Big-Alabo, M.P. Cartmell
Page 646 – Eq. (2.9)
B12 + 2B66 πx
B12 + 2B66 πy
3 B11
u(t)
cot
+
v(t) cot
−
π
2
3
2
ab
a
b
a b
b
2(D12 + 2D66 ) D11 4 D11
−π
+
+ 4 w(t) = ρhẅ + cẇ − q(t)
a4
a2 b2
b
A
A
πx
πy
11
12
+ 2 u(t)w(t) cos
sin
−π 3
3
a
ab
a
b
A
A
πx
πy
11
12
−π 3 3 + 2 v(t)w(t) sin
cos
b
a b
a
b
1
1
πx πy
πy
πx
3
+
v(t)
cot
cos
+2π A66
u(t)
cot
w(t) cos
2
2
ab
b
a b
a
a
b
1
h
1
1
−π 4 B11 4 + 4 + 2B12 2 2
b
a
a b
i
1
πx
πy
2 πx
2 πy
+4B66 2 2 cot
sin
cot
sin
[w(t)]2
a b
a
b
a
b
(2.9)
πy
π
πx
A11 2 A66
(A12 + A66 )u(t) cot
cot
−π
+ 2 v(t)
ab
a
b
a2
b
B
B
+
2B
πy
11
12
66
+π 3 3 +
w(t) cot
= ρhv̈
b
a2 b
b
A
π
πx
πy
A66 11
(A12 + A66 )v(t) cot
cot
− π2
+
u(t)
ab
a
b
a2
b2
B
B12 + 2B66 πx
11
+π 3 3 +
w(t) cot
= ρhü
2
b
ab
a
−π 3
B
11
a3
+
Page 647 – Eq. (2.11) and expression for the B coefficient
ẅ + C ẇ + Dw(t) − B[w(t)]2 = Q sin(ωt)
B=
π 4 B11 2B12 B11 + 2 2 + 4
ρh a4
a b
b
Page 648 – Objective functions (1)
(1) ẅ + C ẇ + Dw(t) − B[w(t)]2 = Q sin(ωt)
(2.11)
3
Vibration analysis of a trimorph plate...
Page 650 – Title for Fig. 3
The title for Fig. 3 ends with ξ = 0.016 and not ξ = 0.01 as it appears
currently.
Page 652 – Table 3
The highlighted cells are to be corrected with the values shown in the table
below.
S/N
Table 3. Nonlinear analysis of the trimorph plate response for the different
layer arrangement
Layer
arrangem.
1-2-3
respectively
Plate
Plate
Plate
Opera- DamSteady- Steadyden- stiffness stiffness
ting
ping
-state
-state
sity
(B)
(BE)
freq.
const.
response set-in
[kg/m3 ] [N/mkg] [N/m2 kg] [rad/s] [Ns/mkg]
[m]
time [s]
1 Al/PVDF/PZT 3109.50 2.229·107 2.102·107 4721.08
1.700
2.004·10−4
6
2 Al/PZT/PVDF 3109.50 2.228·107 1.706·107 4719.77
1.699
2.003·10−4
6
3 PVDF/Al/PZT 3109.50 2.439·105 3.506·107 493.83
15.803
2.081·10−4
3
3
4 PVDF/PZT/Al 3109.50 2.458·10
5
3.558·10
7
495.74
15.864
2.065·10
−4
5 PZT/PVDF/Al 3109.50 1.846·10
6
3.231·10
7
1358.64
5.978
1.981·10
−4
3
6 PZT/Al/PVDF 3109.50 1.811·10 1.773·10 1345.61 5.921 2.013·10
B denotes bending stiffness while BE denotes bending-extension stiffness
−4
3
6
7
Page 654 – Fig. 10
Fig. 10. Frequency-domain response of PVDF/Al/PZT trimorph plate
configuration
4
A. Big-Alabo, M.P. Cartmell
Page 655 – Fig. 11
Fig. 11. Frequency-domain response of PVDF/PZT/Al trimorph plate
configuration
Page 656 – Line 4
... linear analysis is lower ... instead of ... linear analysis is higher ...
Page 656 – Second paragraph, line 6
... are respectively higher and lower than that of ... instead of ... are lower
than that of ...