第六章 管柱振动

n =1 ∞
d 2Tin (t ) dTin (t ) 2 2 + ci + λin ai Tin (t ) = 0 2 dt dt 2 d X in (t ) 2 + λin X in (l ) = 0 dl 2
• 因为钻柱的振动是周期性的,可令
Tin (t ) = e jnωt λin = −α in + jβ in
b.牛顿定律 牛顿定律
∂ 2v ∂ 2v ρAdx 2 = AG 2 dx ∂t ∂x
∂ 2v ∂ 2v ρ 2 =G 2 ∂t ∂x
横波（剪切波，S波）
6.3 钻柱纵向振动
• 运动方程
2 ∂ 2ui ∂u i 2 ∂ ui = ai − ci 2 2 ∂t ∂t ∂x
ci =
2πµλ D Ai ρ i ln 2 Roi
• 边界条件
– 地面边界
θ1 ( 0, t ) = Θ ( t )
– 钻头边界条件：井底不平，使钻头上下振动是 导致钻柱振动的直接原因
• 激励位移法钻头边界条件
θb ( Lm, t ) = Θ b( t )
• 激励力法钻头边界条件
∂θ m (l , t ) Gm J m = T (t ) ∂l l = Lm
2
α in − β in
2
1 − j2α in β in = 2 (n 2ω 2 − jci nω ) ai
2 2
n ω α in − β in = 2 ai c i nω 2α in β in = 2 ai
2 2
α in = α in =
nω 2ai nω 2ai
ci 1+ 1+ nω 2 ci −1+ 1+ nω
(n = 1,2,⋯, ∞ )
• 分别得出i 段顶端位移函数和负荷函数
n =1 ∞ Fi (0, t ) = E i Ai ∑ [(α in k in + β in µ in ) cos(nωt ) n =1 + (−α in µ in + β in k in ) sin(nωt )] y in = υ in γ in = δ in α inσ in + β inτ in k in = 2 2 (α in + β in ) Ei Ai β inσ in − α inτ in µ in = 2 2 (α in + β in ) Ei Ai u i (0, t ) = ∑ [ y in cos(nωt ) + γ in sin(nωt )]
0
u i −1 ( Li −1 , t ) sin(nωt )dt
2E A σ in = i −1 i −1 Tp
∫
Tp
0
∂u i −1 (l , t ) cos(nωt )dt ∂l l = Li −1
2 Ei −1 Ai −1 τ in = Tp
∫
Tp
0
∂u i −1 (l , t ) sin(nωt )dt ∂l l = Li −1
• 分离变量解 ui (l,t) = Xi (l)Ti (t)
d 2Ti (t ) d 2 X i (l ) dTi (t ) dt 2 + c dt = dl 2 i 2 2 X i (l ) ai Ti (t ) ai Ti (t )
u i (l , t ) = ∑ Tin (t ) X in (l )
∂θ i −1 (l ,Βιβλιοθήκη Baidut ) Gi −1 J i −1 ∂l l = Li −1 ∂θ i (l , t ) = Gi J i ∂l l =0
• 周期性条件
θ i (l , t ) = θ i (l , t + Tp )
∂θ i (l , t ) ∂θ i (l , t + Tp ) = ∂t ∂t
Oin (l ) = sin(α in l )[k in ch ( β in l ) + γ in sh ( β in l )]
+ cos(α in l )[µ in sh ( β in l ) + y in ch ( β in l )]
Pin (l ) = cos(α in l )[k in sh ( β in l ) + γ in ch ( β in l )]
M max =
σ max
ρRω 2 l 2
8g
ρy max (ω + ω r ) 2 l 2 + π 2g
M max y max M max = = I Wx
6.5.2 钻柱内钻井液的影响
• 钻柱内钻井液对钻柱的作用力
∂ 2u ∂ 2u ρAdx 2 = AE 2 dx ∂t ∂x
∂ 2u ∂ 2u ρ 2 =E 2 ∂t ∂x
纵波（P波）
2 杆的横向振动
振动方程 a.设v(x,t)为杆 处的横向位移，微元体受横向合力为 设 为杆x处的横向位移 为杆 处的横向位移，
∂v ∂ 2 v ∂v ∂ 2v AG + 2 dx − AG = AG 2 dx ∂x ∂x ∂x ∂x
• 将i 段顶端的位移和负荷展开为傅立叶级数 形式
u i (0, t ) = ∑ [υ in (l ) cos(nωt ) + δ in (l ) sin(nωt )] n =1 ∞ Fi (0, t ) = ∑ [σ in (l ) cos(nωt ) + τ in (l ) sin( nωt )] n =1 2 Tp 2 Tp υ in = ∫ U (t ) cos(nωt )dt δ in = ∫ U (t ) sin(nωt )dt 0 Tp Tp 0
6.5 钻柱的横向振动
• 微分方程
∂2 y ∂2 ρA 2 + 2 ∂t ∂x ∂2 y EI 2 − Sy = 0 ∂x
• 分离变量法可以获得方程的解
y ( x, t ) = ∑ nπ f n (t ) sin x l
2 nπ 4 S nπ 2 f n (t ) = C sin a + t l ρA l 2 nπ 4 S nπ 2 + D cos a + t l ρA l
− sin(α in l )[µ in ch ( β in l ) + y in sh ( β in l )]
e jnωt = cos(nωt ) + j sin( nωt ) • 利用
u i (l , t ) = ∑ {Oin (l ) cos(nωt ) + Pin (l ) sin(nωt ) + j[Oin (l ) sin(nωt ) − Pin (l ) cos(nωt )]}
2. 杆的纵向振动
a.设u(x,t)为杆 处的纵向位移，微元体受纵向合力为 设 为杆x处的纵向位移 为杆 处的纵向位移，
∂u ∂ 2u ∂ 2u ∂u AE + 2 dx − AE = AE 2 dx ∂x ∂x ∂x ∂x
b.牛顿定律 牛顿定律
• 固有频率
nπ ωn = l
2
EI S 1+ 2 ρA nπ EI l
• 轴向激振载荷存在的情况下钻柱横向振动
y ( x, t ) = ∑
4
F0 sin ω 0 t nπρA(ω n
2
nπ sin x 2 l − ω0 )
2 4
y max
ρARω r l = π (nπ ) 4 EI − ρA(ω + ω r ) 2 l 4 + (nπ ) 2 Sl 2
∂ 2u ∂ 2u ρdx 2 = T 2 dx ∂t ∂x ∂ 2u ∂ 2u ρ 2 =T 2 ∂t ∂x
2 ∂ 2u 2 ∂ u =c 2 ∂t ∂x 2
无限弦的解 达朗贝尔解
1 u ( x, t ) = [u0 ( x − ct ) + u0 ( x + ct ) 2 1 x + ct + ∫ v0 (ξ )dξ c x −ct
• t 时刻 x 的位移 • 与 x 相距 ct 的两点的初始位移的和 • 与 x 相距 ct 以内的初始速度的积分
c. 两端固定弦的振动解
∂ 2u ∂ 2u = c2 2 ∂t 2 ∂x
u (0, t ) = u (l , t ) = 0
ɺ Tɺ(t ) X " ( x) = = −λ 2 c T (t ) X ( x)
n =1 ∞
• 对于实际问题, 虚振动没有意义, 舍掉
u i (l , t ) = ∑ [Oin (l ) cos( nωt ) + Pin (l ) sin( nωt )]
n =1 ∞
∂ui (l , t ) = Fi (l , t ) = Ei Ai ∂l ∞ ∂Pin (l ) ∂Oin (l ) Ei Ai ∑ cos(nωt ) + sin(nωt ) ∂l ∂l n =1
∞
6.4 钻柱扭转振动
• 运动方程
2 ∂ 2θ i ∂θ i 2 ∂ θi = ai − ci 2 2 ∂t ∂x ∂t
(i = 1,2,3⋯ , m)(0 ≤ l ≤ Li )
di =
µ 2πDw 2 Roi 2 λ ρ i J i ( Dw 2 − 4 Roi 2 )
• 连续条件
θ i −1 ( Li −1 , t ) = θ i (0, t )
• 连续条件
u i −1 ( Li −1 , t ) = u i (0, t ) ∂u i −1 (l , t ) Ei −1 Ai −1 ∂l l = Li −1 ∂u i (l , t ) = Ei Ai ∂l l =0
• 周期性条件
u i (l , t ) = u i (l , t + Tp ) ∂u i (l , t ) ∂u i (l , t + Tp ) = ∂t ∂t
• 边界条件
– 地面边界
u1 ( 0, t ) = U ( t )
– 钻头边界条件：井底不平，使钻头上下振动是 导致钻柱振动的直接原因
• 激励位移法钻头边界条件
ub ( Lm, t ) = U b( t )
• 激励力法钻头边界条件
∂u m (l , t ) E m Am = B(t ) ∂l l = Lm
2
• 第二式的通解为
X in (l ) = Φ in sin(λin l ) + Θ in cos(λin l )
• 令
Φ in Θ in = y in − jγ in
shl = − j sin( jl ) chl = cos( jl ) = −k in − jµ in
X in (l ) = Oin (l ) − jPin (l )
2 σ in = Tp
∞
∫
Tp
0
F (t ) cos(nωt )dt
2 τ in = Tp
∫
Tp
0
F (t ) sin(nωt )dt
(n = 1,2,⋯, ∞ )
• 当i≠1时
2 υ in = Tp
∫
Tp
0
u i −1 ( Li −1 , t ) cos(nωt )dt
2 δ in = Tp
∫
Tp
设
u ( x, t ) = X ( x)T (t )
ɺ X ( x)Tɺ(t ) = c X " ( x)T (t )
2
X " ( x ) + λX ( x ) = 0 nπ X ( x) = sin x l
∞
nπc T (t ) = sin t l
nπc nπc nπ u ( x, t ) = ∑ an cos t + bn sin t sin x l l l n =1
第六章 管柱振动
6.1 管柱振动的形式
• 钻柱在井眼内表现为四种振动形式
– 纵向振动 – 横向振动 – 扭转振动 – 涡动
• 按危害程度由大到小排列应为：反转运动 (涡动)、横振、扭振、纵振
6.2 连续系统振动的一般概念
1. 弦的振动
弦振动的方程
微元段dx的 微元段 的y方向的合力为
∂θ T tan(θ + dx) − T tan(θ ) ∂x 很小时， 当θ 很小时， ∂u ∂θ ∂u ∂ 2u tan(θ ) = tan(θ + dx) = + 2 dx ∂x ∂x ∂x ∂x