懸垂線 (1回目)

2024/7/1(月)
懸垂線 (1回目)
 
(catenary)
懸垂線(カテナリー、紐を垂らしたときの曲線)
 
■ 導出
▼ 定義

 
T:紐の張力
θ:水平方向からの張力方向への角度
ds:x～x+dx間(点A,B間)の紐の長さ
ρ:紐の線密度
g:重力加速度
f(x):紐の高さ
 
▼ 微分方程式の導出
y = y(x)  … 紐の形のグラフ
y'(x) = dy/dx = tanθ(x)  … ①
 
点A:θ(x), T(x)
点B:θ(x+dx), T(x+dx)
点AB間の紐の長さds
ds = √(dx² + dy²) = dx√{1 + (dy/dx)²}
 
水平方向
T(x)cosθ(x) = T(x+dx)cosθ(x+dx)
どのxでも同じ値なので
T(x)cosθ(x) = const.
H = T(x)cosθ(x)  … ② と置く(水平張力)
 
垂直方向
dy/dx = y'(x)
T(x)sinθ(x) + dsρg = T(x+dx)sinθ(x+dx)
ρgdx√{1 + (dy/dx)²} = T(x+dx)sinθ(x+dx) - T(x)sinθ(x)
{T(x+dx)sinθ(x+dx)-T(x)sinθ(x)}/dx = ρg√{1 + (dy/dx)²}
 
(d/dx){T(x)sinθ(x)} = ρg√{1 + (dy/dx)²}  … ③
 
①,②より
T(x)sinθ(x) = T(x)cosθ(x)sinθ(x)/cosθ(x)
= T(x)cosθ(x)tanθ(x) = T(x)cosθ(x)(dy/dx)
= H(dy/dx)
を③に代入
(d/dx){H(dy/dx)} = ρg√{1 + (dy/dx)²}
 
(d/dx)²y = (ρg/H)√{1 + (dy/dx)²}
 
λ = ρg/H, f(x) = dy/dxと置く
(d/dx)f = (d/dx)²y = λ√{1 + f²}  … ④
 
▼ 積分
∫{1/√(1+x²)}dxを解く
 
x = tanθと置く
dx/dθ = (d/dθ)tanθ = (d/dθ){sinθ/cosθ}
= {(d/dθ)sinθ}/cosθ + sinθ{(d/dθ)(1/cosθ)}
= 1 + sin²θ(1/cos²θ) = 1 + tan²θ
= 1 + (1-cos²θ)/cos²θ = 1/cos²θ
dx = (1/cos²θ)dθ
1 + tan²θ = 1/cos²θ
 
∫{1/√(1+x²)}dx
= ∫{1/√(1+tan²θ)}(1/cos²θ)dθ
= ∫{1/√(1/cos²θ)}(1/cos²θ)dθ
= ∫(cosθ/cos²θ)dθ
= ∫{cosθ/(1-sin²θ)}dθ
t = sinθと置く
dt/dθ = cosθ
dθ = (1/cosθ)dt
∫{cosθ/(1-sin²θ)}dθ
= ∫{cosθ/(1-t²)}(1/cosθ)dt
= ∫{1/(1-t²)}dt
 
1/(t+1) - 1/(t-1)
= 1/(1+t) + 1/(1-t)
= {(1-t)+(1+t)}/{(1+t)(1-t)}
= 2/{(1+t)(1-t)} = 2/(1-t²)
 
∫{1/(1-t²)}dt = (1/2)∫{2/(1-t²)}dt
(1/2)∫{1/(t+1) - 1/(t-1)}dt
= (1/2)log|t+1| - log|t-1| + C
= (1/2)log(|t+1|/|t-1|) + C
= (1/2)log(|sinθ+1|/|sinθ-1|) + C
= (1/2)log(|1+sinθ|/|1-sinθ|) + C
= (1/2)log{(1+sinθ)²/(1²-sin²θ)} + C
= (1/2)log{(1+sinθ)²/cos²θ} + C
= log|(1+sinθ)/cosθ| + C
= log|tanθ + √(1/cos²θ)| + C
= log|tanθ + √{(cos²θ+sin²θ)/cos²θ}| + C
= log|tanθ + √(1+tan²θ)| + C
= log|x + √(1+x²)| + C
 
∫{1/√(1 + x²)}dx = log|x + √(1 + x²)| + C
 
▼ 微分方程式を解く
④式
(d/dx)f = (d/dx)²y = λ√{1 + f²}
df / √{1 + f²} = λdx
 
∫{1/√(1 + f²)}df = λ∫dx
log|f + √(1+f²)| = λx + A
f + √(1+f²) = exp(λx + A)
√(1+f²) = exp(λx + A) - f
1 + f² = {exp(λx + A) - f}² 
1 + f² = exp(λx + A)² + f² - 2fexp(λx + A)
2fexp(λx + A) = exp(λx + A)² - 1
f = (1/2){exp(λx + A) - exp(-λx - A)}
y = ∫(dy/dx)dx = ∫fdx =
= (1/2)∫{exp(λx + A) - exp(-λx - A)}dx
= {1/(2λ)}{exp(λx + A) + exp(-λx - A)} + B
= (1/λ){exp(λx + A) + exp(-λx - A)}/2 + B
 
λ = ρg/H, H = T(x)cosθ(x) = const.
y(x) = (1/λ){exp(λx + A) + exp(-λx - A)}/2 + B
y'(x) = (1/2){exp(λx + A) - exp(-λx - A)}
 
▼ ハイパボリック関数(双曲線関数)
cosh(t) = {exp(t) + exp(-t)}/2
sinh(t) = {exp(t) - exp(-t)}/2
と定義される
 
y(x) = (1/λ){exp(λx + A) + exp(-λx - A)}/2 + B
= (1/λ)cosh(λx + A) + B
y'(x) = (1/2){exp(λx + A) - exp(-λx - A)}
= sinh(λx + A)
 
▼ 紐の微小長さds
y(x) = (1/λ)cosh(λx + A) + B
y'(x) = dy/dx = sinh(λx + A)
 
cosh²(t) - sinh²(t)
= [{exp(t)+exp(-t)}² - {exp(t)-exp(-t)}²]/4
= {2exp(t)exp(-t) + 2exp(t)exp(-t)}/4 = 1
cosh²(t) - sinh²(t) = 1
 
1 + (dy/dx)² = 1 + sinh²(λx + A) = cosh²(λx + A)
 
ds = dx√{1 + (dy/dx)²} = cosh(λx + A)dx
 
 
■ 結果
▼ 定義
g:重力加速度(m/s²)
ρ:紐の密度(kg/m)
H:水平張力(N)
y(x):懸垂線(カテナリー)
y'(x):懸垂線の傾き
ds:x～x+dx間の紐の長さ
 
▼ 懸垂線y(x)
A, B:積分定数
λ = ρg/H, H = T(x)cosθ(x) = const.
y(x) = (1/λ)cosh(λx + A) + B
y'(x) = sinh(λx + A)
ds = cosh(λx + A)dx