准静态铁木辛柯梁
编辑
铁木辛柯梁的变形。
θ
x
=
φ
(
x
)
{\displaystyle \theta _{x}=\varphi (x)}
不等于
d
w
/
d
x
{\displaystyle dw/dx}
。
在静力学中铁木辛柯梁理论没有轴向影响，假定梁的位移服从于
u
x
(
x
,
y
,
z
)
=
−
z
φ
(
x
)
;
u
y
(
x
,
y
,
z
)
=
0
;
u
z
(
x
,
y
)
=
w
(
x
)
{\displaystyle u_{x}(x,y,z)=-z~\varphi (x)~;~~u_{y}(x,y,z)=0~;~~u_{z}(x,y)=w(x)}
式中
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
是梁上一点的坐标，
u
x
,
u
y
,
u
z
{\displaystyle u_{x},u_{y},u_{z}}
是位移矢量的三维坐标分量，
φ
{\displaystyle \varphi }
是对于梁的中性面的法向转角，
w
{\displaystyle w}
是中性面的在
z
{\displaystyle z}
方向的位移。
控制方程是以下常微分方程的解耦系统：
d
2
d
x
2
(
E
I
d
φ
d
x
)
=
q
(
x
,
t
)
d
w
d
x
=
φ
−
1
κ
A
G
d
d
x
(
E
I
d
φ
d
x
)
.
{\displaystyle {\begin{aligned}&{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\left(EI{\frac {\mathrm {d} \varphi }{\mathrm {d} x}}\right)=q(x,t)\\&{\frac {\mathrm {d} w}{\mathrm {d} x}}=\varphi -{\frac {1}{\kappa AG}}{\frac {\mathrm {d} }{\mathrm {d} x}}\left(EI{\frac {\mathrm {d} \varphi }{\mathrm {d} x}}\right).\end{aligned}}}
静态条件下的铁木辛柯梁理论，若在以下條件成立時，等同于欧拉-伯努利梁理论
E
I
κ
L
2
A
G
≪
1
{\displaystyle {\frac {EI}{\kappa L^{2}AG}}\ll 1}
此時，可忽略上面控制方程的最后一项，得到有效的近似，式中
L
{\displaystyle L}
是梁的长度。
对于等截面均匀梁，合并以上两个方程，
E
I
d
4
w
d
x
4
=
q
(
x
)
−
E
I
κ
A
G
d
2
q
d
x
2
{\displaystyle EI~{\cfrac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}=q(x)-{\cfrac {EI}{\kappa AG}}~{\cfrac {\mathrm {d} ^{2}q}{\mathrm {d} x^{2}}}}
动态铁木辛柯梁
编辑
在铁木辛柯梁理论中若不考虑轴向影响，则给出梁的位移
u
x
(
x
,
y
,
z
,
t
)
=
−
z
φ
(
x
,
t
)
;
u
y
(
x
,
y
,
z
,
t
)
=
0
;
u
z
(
x
,
y
,
z
,
t
)
=
w
(
x
,
t
)
{\displaystyle u_{x}(x,y,z,t)=-z~\varphi (x,t)~;~~u_{y}(x,y,z,t)=0~;~~u_{z}(x,y,z,t)=w(x,t)}
式中
(
x
,
y
,
z
)
{\displaystyle (x,y,z)}
是梁内一点的坐标，
u
x
,
u
y
,
u
z
{\displaystyle u_{x},u_{y},u_{z}}
是位移矢量的三维坐标分量，
φ
{\displaystyle \varphi }
是对于梁的中性面的法向转角，
w
{\displaystyle w}
是中性面
z
{\displaystyle z}
方向的位移.
从以上假设，铁木辛柯梁，考虑到振动，要用线性耦合偏微分方程描述：[3]
ρ
A
∂
2
w
∂
t
2
−
q
(
x
,
t
)
=
∂
∂
x
[
κ
A
G
(
∂
w
∂
x
−
φ
)
]
{\displaystyle \rho A{\frac {\partial ^{2}w}{\partial t^{2}}}-q(x,t)={\frac {\partial }{\partial x}}\left[\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\right]}
ρ
I
∂
2
φ
∂
t
2
=
∂
∂
x
(
E
I
∂
φ
∂
x
)
+
κ
A
G
(
∂
w
∂
x
−
φ
)
{\displaystyle \rho I{\frac {\partial ^{2}\varphi }{\partial t^{2}}}={\frac {\partial }{\partial x}}\left(EI{\frac {\partial \varphi }{\partial x}}\right)+\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)}
其中因变量是梁的平移位移
w
(
x
,
t
)
{\displaystyle w(x,t)}
和转角位移
φ
(
x
,
t
)
{\displaystyle \varphi (x,t)}
。注意不同于欧拉-伯努利梁理论，转角位移是另一个变量而非挠度斜率的近似。此外，
ρ
{\displaystyle \rho }
是梁材料的密度（而非线密度）；
A
{\displaystyle A}
是截面面积；
E
{\displaystyle E}
是弹性模量；
G
{\displaystyle G}
是剪切模量；
I
{\displaystyle I}
是轴惯性矩；
κ
{\displaystyle \kappa }
，称作铁木辛柯剪切系数，由形状确定，通常矩形截面
κ
=
5
/
6
{\displaystyle \kappa =5/6}
；
q
(
x
,
t
)
{\displaystyle q(x,t)}
是载荷分布（单位长度上的力）；
m
:=
ρ
A
{\displaystyle m:=\rho A}
J
:=
ρ
I
{\displaystyle J:=\rho I}
这些参数不一定是常数。
对于各向同性的线弹性均匀等截面梁，以上两个方程可合并成[4][5]
E
I
∂
4
w
∂
x
4
+
m
∂
2
w
∂
t
2
−
(
J
+
E
I
m
k
A
G
)
∂
4
w
∂
x
2
∂
t
2
+
m
J
k
A
G
∂
4
w
∂
t
4
=
q
(
x
,
t
)
+
J
k
A
G
∂
2
q
∂
t
2
−
E
I
k
A
G
∂
2
q
∂
x
2
{\displaystyle EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+m~{\cfrac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\cfrac {EIm}{kAG}}\right){\cfrac {\partial ^{4}w}{\partial x^{2}~\partial t^{2}}}+{\cfrac {mJ}{kAG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}=q(x,t)+{\cfrac {J}{kAG}}~{\cfrac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {EI}{kAG}}~{\cfrac {\partial ^{2}q}{\partial x^{2}}}}
轴向影响
编辑
如果梁的位移由下式给出
u
x
(
x
,
y
,
z
,
t
)
=
u
0
(
x
,
t
)
−
z
φ
(
x
,
t
)
;
u
y
(
x
,
y
,
z
,
t
)
=
0
;
u
z
(
x
,
y
,
z
)
=
w
(
x
,
t
)
{\displaystyle u_{x}(x,y,z,t)=u_{0}(x,t)-z~\varphi (x,t)~;~~u_{y}(x,y,z,t)=0~;~~u_{z}(x,y,z)=w(x,t)}
其中
u
0
{\displaystyle u_{0}}
是
x
{\displaystyle x}
方向的附加位移，则铁木辛柯梁的控制方程成为
m
∂
2
w
∂
t
2
=
∂
∂
x
[
κ
A
G
(
∂
w
∂
x
−
φ
)
]
+
q
(
x
,
t
)
J
∂
2
φ
∂
t
2
=
N
(
x
,
t
)
∂
w
∂
x
+
∂
∂
x
(
E
I
∂
φ
∂
x
)
+
κ
A
G
(
∂
w
∂
x
−
φ
)
{\displaystyle {\begin{aligned}m{\frac {\partial ^{2}w}{\partial t^{2}}}&={\frac {\partial }{\partial x}}\left[\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\right]+q(x,t)\\J{\frac {\partial ^{2}\varphi }{\partial t^{2}}}&=N(x,t)~{\frac {\partial w}{\partial x}}+{\frac {\partial }{\partial x}}\left(EI{\frac {\partial \varphi }{\partial x}}\right)+\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\end{aligned}}}
其中
J
=
ρ
I
{\displaystyle J=\rho I}
，
N
(
x
,
t
)
{\displaystyle N(x,t)}
是外加轴向力。任意外部轴向力的平衡依靠应力
N
x
x
(
x
,
t
)
=
∫
−
h
h
σ
x
x
d
z
{\displaystyle N_{xx}(x,t)=\int _{-h}^{h}\sigma _{xx}~dz}
式中
σ
x
x
{\displaystyle \sigma _{xx}}
是轴向应力，梁的厚度设为
2
h
{\displaystyle 2h}
。
包含轴向力的梁方程合并为
E
I
∂
4
w
∂
x
4
+
N
∂
2
w
∂
x
2
+
m
∂
2
w
∂
t
2
−
(
J
+
m
E
I
κ
A
G
)
∂
4
w
∂
x
2
∂
t
2
+
m
J
κ
A
G
∂
4
w
∂
t
4
=
q
+
J
κ
A
G
∂
2
q
∂
t
2
−
E
I
κ
A
G
∂
2
q
∂
x
2
{\displaystyle EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+N~{\cfrac {\partial ^{2}w}{\partial x^{2}}}+m~{\frac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\cfrac {mEI}{\kappa AG}}\right)~{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}+{\cfrac {mJ}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}=q+{\cfrac {J}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {EI}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}}
阻尼
编辑
如果，除轴向力外，我们考虑与速度成正比的阻尼力，形如
η
(
x
)
∂
w
∂
t
{\displaystyle \eta (x)~{\cfrac {\partial w}{\partial t}}}
铁木辛柯梁的耦合控制方程成为
m
∂
2
w
∂
t
2
+
η
(
x
)
∂
w
∂
t
=
∂
∂
x
[
κ
A
G
(
∂
w
∂
x
−
φ
)
]
+
q
(
x
,
t
)
{\displaystyle m{\frac {\partial ^{2}w}{\partial t^{2}}}+\eta (x)~{\cfrac {\partial w}{\partial t}}={\frac {\partial }{\partial x}}\left[\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\right]+q(x,t)}
J
∂
2
φ
∂
t
2
=
N
∂
w
∂
x
+
∂
∂
x
(
E
I
∂
φ
∂
x
)
+
κ
A
G
(
∂
w
∂
x
−
φ
)
{\displaystyle J{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=N{\frac {\partial w}{\partial x}}+{\frac {\partial }{\partial x}}\left(EI{\frac {\partial \varphi }{\partial x}}\right)+\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)}
合并方程为
E
I
∂
4
w
∂
x
4
+
N
∂
2
w
∂
x
2
+
m
∂
2
w
∂
t
2
−
(
J
+
m
E
I
κ
A
G
)
∂
4
w
∂
x
2
∂
t
2
+
m
J
κ
A
G
∂
4
w
∂
t
4
+
J
η
(
x
)
κ
A
G
∂
3
w
∂
t
3
−
E
I
κ
A
G
∂
2
∂
x
2
(
η
(
x
)
∂
w
∂
t
)
+
η
(
x
)
∂
w
∂
t
=
q
+
J
κ
A
G
∂
2
q
∂
t
2
−
E
I
κ
A
G
∂
2
q
∂
x
2
{\displaystyle {\begin{aligned}EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}&+N~{\cfrac {\partial ^{2}w}{\partial x^{2}}}+m~{\frac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\cfrac {mEI}{\kappa AG}}\right)~{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}+{\cfrac {mJ}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}+{\cfrac {J\eta (x)}{\kappa AG}}~{\cfrac {\partial ^{3}w}{\partial t^{3}}}\\&-{\cfrac {EI}{\kappa AG}}~{\cfrac {\partial ^{2}}{\partial x^{2}}}\left(\eta (x){\cfrac {\partial w}{\partial t}}\right)+\eta (x){\cfrac {\partial w}{\partial t}}=q+{\cfrac {J}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {EI}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}\end{aligned}}}