Page 51 - 《振动工程学报》2026年第2期

第 2 期 朱光楠，等：卫星大型柔性天线全局模态法力学特性分析 367

[
w w b 2 ∂φ (R N ) ) ∂φ (R N ) ) ∂φ (R N ) )
a
( ) 3 m p (x R N (R N ) m q (x R N (R N ) 2 3 (R N ) n p (y R N
=ρ h φ (y R N ) φ (y R N )+ h φ (x R N ) +
M 6+N,6+N m p n p ,m q n q
0 −b 3 n p n q 3 m p
∂x R N ∂x R N ∂y R N
∂φ (R N ) (y R N ) ]
φ (R N ) ) n q +2hφ (R N ) )φ (R N ) )φ (R N ) )φ (R N )
m q (x R N m p (x R N n p (y R N m q (x R N n q (y R N ) dy R N dx R N
∂y R N
[
w w b 2 ∂φ (L 1 ) ) ∂φ (L 1 ) ) ∂φ (L 1 ) )
a
( ) 3 m p (x L 1 (L 1 ) m q (x L 1 (L 1 ) 2 3 (L 1 ) n p (y L 1
=ρ h φ (y L 1 ) φ (y L 1 )+ h φ (x L 1 ) +
M 7+N,7+N m p n p ,m q n q
0 −b 3 n p n q 3 m p
∂x L 1 ∂x L 1 ∂y L 1
∂φ (L 1 ) (y L 1 ) ]
φ (L 1 ) ) n q +2hφ (L 1 ) )φ (L 1 ) )φ (L 1 ) )φ (L 1 )
m q (x L 1 m p (x L 1 n p (y L 1 m q (x L 1 n q (y L 1 ) dy L 1 dx L 1
∂y L 1
[
w w b 2 (L i ) ) (L i ) ) (L i ) )
a
m q
n p
m p
( ) 3 ∂φ (x L i (L i ) ∂φ (x L i (L i ) 2 3 (L i ) ∂φ (y L i
=ρ h ) ) +
m p
M 7+N+i,7+N+i m p n p ,m q n q φ (y L i φ (y L i )+ h φ (x L i
n p
n q
0 −b 3 3
∂x L i ∂x L i ∂y L i
(L i ) ]
∂φ (y L i )
(L i ) n q (L i ) (L i ) (L i ) (L i )
φ (x L i ) +2hφ (x L i )φ (y L i )φ (x L i )φ (y L i ) dy L i dx L i
m q m p n p m q n q
∂y L i
[ (L N ) (L N ) (L N )
w w b 2 ∂φ ) ∂φ ) ∂φ )
a
( ) 3 m p (x L N (L N ) m q (x L N (L N ) 2 3 (L N ) n p (y L N
=ρ h φ ) φ )+ h φ ) +
M 6+2N,6+2N m p n p ,m q n q n p (y L N n q (y L N m p (x L N
0 −b 3 3
∂x L N ∂x L N ∂y L N
∂φ (L N ) (y L N ) ]
φ (L N ) ) n q +2hφ (L N ) )φ (L N ) )φ (L N ) )φ (L N )
m q (x L N m p (x L N n p (y L N m q (x L N n q (y L N ) dy L N dx L N
∂y L N
式中， m p ,m q = 1,2,··· ,m t n p ,n q = 1,2,··· ,n t 。
；
K是 (6+2Nm t n t +4N)×(6+2Nm t n t +4N)维的矩阵：
 
 0 6×6 0 6×Nm t n t 0 6×Nm t n t 0 6×4N 
 
 
 
 
 0 Nm t n t ×6 K R 0 Nm t n t ×Nm t n t 0 Nm t n t ×4N  

K =    
 
 0 Nm t n t ×6 0 Nm t n t ×Nm t n t K L 0 Nm t n t ×4N  

 
 
0 4N×6 0 4N×Nm t n t 0 4N×Nm t n t 0 4N×4N