382
9 Topologic Optimization of Vibrations of Size-Dependent Beams
N , x + f = b 0 u , tt − b 1 w , xtt ,
(9.89)
Nw , x
, x
+ (M , xx + Y , xx ) + q + C , x = b 0 w , tt + b 1 u , xtt − b 2 w , xxtt , (9.90)
and the corresponding beam boundary conditions are as follows:
N = ¯
N
x=0, L
= 0 or δ u| x=0, L = 0,
(9.91)
Nw , x + M , x + Y , x + C − b 1 u , tt + b 2 w , xtt = ¯
Q
x=0, L
(9.92)
or δ w| x=0, L = 0,
M + Y = ¯
M
x=0, L
or δ w , x
x=0, L
= 0.
(9.93)
Equations (9.91)–(9.93) imply that there are six boundary conditions for the sizedependent (non-classical) non-homogeneous Bernoulli-Euler beam, on the contrary
to the classical beam associated with four boundary conditions.
Equations (9.84), (9.89) and (9.90) allow one to derive the equations of motion
in terms of displacements:
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 ,
(9.94)
k 1
u , x +
1
2
w , x
2
w , x
, x
−
k 2 w , xx + N T
w , x
, x
+
+
k 2
u , x +
1
2
w , x
2
, xx
−
(k 3 + k 4 ) w , xx
, xx
− M T, xx + q + C , x = b 0 w , tt + b 1 u , xtt − b 2 w , xxtt .
(9.95)
The corresponding boundary conditions can be defined in the following way:
k 1
u , x +
1
2
w , x
2
− k 2 w , xx − N T = ¯
N
x=0, L
= 0 or δ u| x=0, L = 0,
(9.96)
k 1
u , x +
1
2
w , x
2
− k 2 w , xx − N T
w , x +
+
k 2
u , x +
1
2
w , x
2
− k 3 w , xx − M T
, x
−
k 4 w , xx
, x
+ C − b 1 u , tt + b 2 w , xtt = ¯
Q
x=0, L
= ¯
Q , or δ w| x=0, L = 0.
(9.97)