9.7 Size-Dependent Euler-Bernoulli Beams with Topologically Optimized Microstructure 383
k 2
u , x +
1
2
w , x
2
− (k 3 + k 4 ) w , xx − M T = ¯
M
x=0, L
(9.98)
or δ w , x
x=0, L
= 0.
We introduce the following non-dimensional variables and parameters
¯
x =
x
L
, ¯
u =
u
h
,
¯
w =
w
h
, λ =
L
h
, l 0 =
l
h
,
¯
k 1 , ¯
k 2 , ¯
k 3
=
k 1
K 0
,
k 2
K 0 h
,
k 3
K 0 h 2
, ¯
k 4 =
k 4
K 0 l 2 ,
¯
b 0 , ¯
b 1 , ¯
b 2
=
b 0
I 0
,
b 1
I 0 h
,
b 2
I 0 h 2
, ¯
t =
t
L
K 0
I 0
,
¯
ω =
L
2 I 0
K 0
ω
2
,
¯
f =
f Lλ
2
K 0
, ¯
q =
q Lλ
K 0
,
¯
C =
Cλ
K 0
,
¯
θ = α 0 θ,
¯
N T =
N T λ
2
K 0
,
¯
M T =
M T λ
K 0 L
,
(9.99)
where K 0 = AE 0 and I 0 = Aρ 0 ; the quantities k 1 and b 0 correspond to homogeneous
microbeams with Young’s modulus E 0 ; material density is denoted by ρ 0 and α 0
stands for the linear temperature extension coefficient.
The following counterpart non-dimensional equations of beam motion can be
derived (bars over non-dimensional quantities are omitted)
k 1
u , x +
1
2
w , x
2
, x
−
k 2 w , xx
, x
− N T, x + f = b 0 u , tt − b 1 w , xtt ,
(9.100)
1
λ 2
k 1
u , x +
1
2
w , x
2
w , x
, x
−
1
λ 2
k 2 w , xx w , x
, x
−
1
λ 2
N T w , x
, x
+
+
1
λ 2
k 2
u , x +
1
2
w , x
2
, xx
−
1
λ 2
k 3 + l
2
0 k 4
w , xx
, xx
−
− M T, xx + q + C , x = b 0 w , tt +
1
λ 2 b 1 u , xtt −
1
λ 2 b 2 w , xxtt .
(9.101)
The corresponding non-dimensional boundary conditions have the following
form:
(i) clamping-clamping
u = w = w , x
x=0, 1
= 0,
(9.102)