Page 50 - 《振动工程学报》2026年第2期

366 振 动 工 程 学 报 第 39 卷

w w b
0
( ) [ (L) (L) ]
= ρ
n p
M 3,7+N m p n p 2h φ (x L 1 )φ (y L 1 ) dy L 1 dx L 1
m p
−a −b
w w b [ ]
0
( ) (L i ) (L i )
= ρ 2h φ (x L i )φ (y L i ) dy L i dx L i
M 3,7+N+i m p n p
−a −b m p n p
w w b
0
( ) [ (L i ) (L i ) ]
= ρ
n p
m p
M 3,6+2N m p n p 2h φ (x L i )φ (y L i ) dy L i dx L i
−a −b
w w w w b 2 ∂φ (R i ) )
a b [ ] a n p (y R 1
3
(M 47 ) = ρ (R) (R) +ρ h φ (R i ) )
m p n p 2h y R i φ (x R 1 )φ (y R 1 ) dy R 1 dx R 1 m p (x R i dy R 1 dx R 1
n p
m p
0 −b 0 −b 3
∂y R 1
w w b w w b 2 ∂φ (R i ) )
a
a
( ) [ (R i ) (R i ) ] 3 (R i ) n p (y R i
= ρ φ )φ +ρ h φ )
M 4,7+i m p n p 2h y R i m p (x R i n p (y R i ) dy R i dx R i m p (x R i dy R i dx R i
0 −b 0 −b 3
∂y R i
w w b w w b 2 ∂φ (R N ) )
a
a
( ) [ (R N ) (R N ) ] 3 (R N ) n p (y R N
= ρ φ )φ +ρ h φ )
M 4,6+N m p n p 2h y R N m p (x R i n p (y R N ) dy R N dx R N m p (x R N dy R N dx R N
0 −b 0 −b 3
∂y R N
w w b [ ] w w b 2 ∂φ (L 1 ) )
0
a
( ) (L 1 ) (L 1 ) 3 (L 1 ) n p (y L 1
= ρ 2h y L 1 φ (x L 1 )φ (y L 1 ) dy L 1 dx L 1 +ρ h φ (x L 1 ) dy L 1 dx L 1
M 4,7+N m p n p
−a −b m p n p 0 −b 3 m p
∂y L 1
w w b w w b 2 (L i ) )
0
a
( ) [ (L i ) (L i ) ] 3 (L i ) ∂φ (y L i
n p
= ρ +ρ )
m p
n p
m p
M 4,7+N+i m p n p 2h y L i φ (x L i )φ (y L i ) dy L i dx L i h φ (x L i dy L i dx L i
−a −b 0 −b 3
∂y L i
w w b w w b 2 ∂φ (L N ) )
0
a
( ) [ (L N ) (L N ) ] 3 (L N ) n p (y L N
= ρ 2h y L N φ (x L N )φ (y L N ) dy L N dx L N +ρ h φ (x L N ) dy L N dx L N
M 4,6+2N m p n p
−a −b m p n p 0 −b 3 m p
∂y L N
 (R 1 ) 
w w b [ ] w w b 2  ∂φ (x R 1 ) 
a
a
3 
(M 57 ) = ρ +r 0 )φ (R 1 ) )φ (R 1 ) +ρ h − m p φ (R 1 ) )  

m p n p 2h −(x R 1 m p (x R 1 n p (y R 1 ) dy R 1 dx R 1  n p (y R 1  dy R 1 dx R 1
0 −b 0 −b 3
∂x R 1

w w b { } w w b 2  ∂φ (R i ) ) 
a
a
( ) (R i ) (R i ) 3  m p (x R i (R i )  
= ρ +r 0 +a(i−1)]φ )φ +ρ h − φ ) 

2h −[x R i
M 5,7+i m p n p
m p (x R i n p (y R i ) dy R i dx R i  n p (y R i  dy R i dx R i
0 −b 0 −b 3
∂x R i

w w b { } w w b 2  ∂φ (R N ) ) 
a
a
( ) (R N ) (R N ) 3  m p (x R N (R N )  
=ρ +r 0 +a(N −1)]φ )φ +ρ h − φ 

)
M 5,6+N m p n p 2h −[x R N m p (x R N n p (y R N ) dy R N dx R N  n p (y R N  dy R N dx R N
0 −b 0 −b 3
∂x R N

w w b [ ] w w b 2  ∂φ (L 1 ) ) 
0
0
( ) (L 1 ) (L 1 ) 3  m p (x L 1 (L 1 )  
= ρ −r 0 )φ )φ +ρ h − φ 

)
M 5,7+N m p n p 2h −(x L 1 m p (x L 1 n p (y L 1 ) dy L 1 dx L 1  n p (y L 1  dy L 1 dx L 1
−a −b −a −b 3
∂x L 1
 (L i ) 
w w b w w b 2 )
0
0
( ) { (L i ) (L i ) } 3   ∂φ (x L i (L i )  
m p
= ρ +ρ h − 

)
M 5,7+N+i m p n p 2h −[x L i −r 0 −a(i−1)]φ (x L i )φ (y L i ) dy L i dx L i  φ (y L i  dy L i dx L i
n p
m p
n p
−a −b −a −b 3
∂x L i
w w b
0
( ) { (L N ) (L N ) }
=ρ −r 0 −a(N −1)]φ )φ +
M 5,6+2N m p n p 2h −[x L N m p (x L N n p (y L N ) dy L N dx L N
−a −b

w w b 2  ∂φ (L N ) ) 
0 m p (x L N 
3 
ρ h − φ (L N ) (y L N  dy L N dx L N



)
−a −b 3  n p
∂x L N
M 73 、 M 74 、 M 75 、 M 7+i,3 、 M 7+i,4 、 M 7+i,5 、 M 6+N,3 、 M 6+N,4 、 M 6+N,5 、 M 7+N,3 、 M 7+N,4 、 M 7+N,5 、 M 7+N+i,3 、 M 7+N+i,4 、
M 7+N+i,5 、 M 6+2N,3 、 M 6+2N,4 和 M 6+2N,5 是 m t n t ×1 维的列向量，可表示为：
T
T
M 73 = M ，M 74 = M ，M 75 = M T
37 47 57
T
M 7+i,3 = M T 3,7+i ，M 7+i,4 = M T 4,7+i ，M 7+i,5 = M 5,7+i
T
M 6+N,3 = M T 3,6+N ，M 6+N,4 = M T 4,6+N ，M 6+N,5 = M 5,6+N
T
M 7+N,3 = M T 3,7+N ，M 7+N,4 = M T 4,7+N ，M 7+N,5 = M 5,7+N
M 7+N+i,3 = M T ，M 7+N+i,4 = M T ，M 7+N+i,5 = M T
3,7+N+i 4,7+N+i 5,7+N+i
M 6+2N,3 = M T ，M 6+2N,4 = M T ，M 6+2N,5 = M T
3,6+2N 4,6+2N 5,6+2N
M 77 、 M 7+i,7+i 、 M 6+N,6+N 、 M 7+N,7+N 、 M 7+N+i,7+N+i 和 M 6+2N,6+2N 是 M矩阵 m t n t ×m t n t 维的分块矩阵：
[
w w b 2 ∂φ (R 1 ) (x R 1 ) ∂φ (R 1 ) (x R 1 ) 2 ∂φ (R 1 ) (y R 1 )
a
3
(M 77 ) =ρ h 3 m p φ (R 1 ) ) m q φ (R 1 ) )+ h φ (R 1 ) ) n p +
m p n p ,m q n q n p (y R 1 n q (y R 1 m p (x R 1
0 −b 3 ∂x R 1 ∂x R 1 3 ∂y R 1
∂φ (R 1 ) (y R 1 ) ]
φ (R 1 ) ) n q +2hφ (R 1 ) )φ (R 1 ) )φ (R 1 ) )φ (R 1 )
m q (x R 1 m p (x R 1 n p (y R 1 m q (x R 1 n q (y R 1 ) dy R 1 dx R 1
∂y R 1
[
w w b 2 ∂φ (R i ) ) ∂φ (R i ) ) ∂φ (R i ) )
a
( ) 3 m p (x R i (R i ) m q (x R i (R i ) 2 3 (R i ) n p (y R i
=ρ h φ ) φ )+ h φ ) +
M 7+i,7+i m p n p ,m q n q n p (y R i n q (y R i m p (x R i
0 −b 3 3
∂x R i ∂x R i ∂y R i
∂φ (R i ) (y R i ) ]
φ (R i ) ) n q +2hφ (R i ) )φ (R i ) )φ (R i ) )φ (R i )
m q (x R i m p (x R i n p (y R i m q (x R i n q (y R i ) dy R i dx R i
∂y R i