Capitolo 17 – Le Vibrazioni Meccaniche

LE VIBRAZIONI MECCANICHE
cosω tcosϕ − senω tsenϕ
=
2 2 2 2
( cos∆ωtcosω1t sen ∆ωtsenω1t)
⋅( cos∆ϕcosϕ1 − sen ∆ϕsenϕ1)
+
− ( sen ∆ωtcosω1t + cos∆ωtsenω1t) ⋅
( sen ∆ϕcosϕ cos∆ϕsenϕ )
= − ⋅
⋅ + =
1 1
( ∆ω ∆ϕ ∆ω ∆ϕ )
( ∆ω ∆ϕ ∆ω ∆ϕ )
( ∆ω ∆ϕ ∆ω ∆ϕ )
( ∆ω ∆ϕ ∆ω ∆ϕ )
= cos tcos cosϕ − cos tsen senϕ cosω
t +
1 1 1
− sen tcos cosϕ − sen tsen senϕ senω
t +
1 1 1
− sen tsen cosϕ + sen tcos senϕ cosω
t +
1 1 1
− cos tsen cosϕ + cos tcos senϕ senω
t =
=
⎡
−⎢
⎣
1 1 1
⎡(
cos∆ωtcos∆ϕ ⎢
⎣ − ( ∆ωt sen ∆ωtsen ∆ϕ)
cos
∆ϕ + ∆ωt ∆ϕ)
( sen ∆ωtcos∆ϕ cos∆ωtsen ∆ϕ)
cosϕ
( t t )
− ϕ + ⎤
1
⎥ cosω1t
+
cos sen sen cos senϕ
⎦
+ + ⎤
1
⎥ senω1t
=
− sen ∆ω sen ∆ϕ + cos∆ω cos∆ϕ senϕ
⎦
[ ( ∆ω ∆ϕ) ( ∆ω ∆ϕ)
]
− ( ∆ω + ∆ϕ) − ( ∆ω + ∆ϕ)
= cos t + cosϕ − sen t + senϕ cosω
t +
1 1 1
[ ]
cos t senϕ sen t cosϕ senω
t
1 1 1
Sostituendo si ottiene:
⎧⎪
[ X + X cos( t + ) ] cos + ⎫
1 2 ∆ω ∆ϕ ϕ1
⎪
xt () = ⎨
⎬cosω1t
+
⎩⎪ − X sen( t )
2 ∆ω + ∆ϕ senϕ1⎭⎪
⎧⎪
[ X + X cos( t + ) ] sen + ⎫
1 2 ∆ω ∆ϕ ϕ1
⎪
− ⎨
⎬senω1t
⎩⎪ + X sen( ∆ωt + ∆ϕ)
cosϕ
⎭⎪
Poniamo:
2 1
[ 1 2 ( ) ] ϕ1
− X ( ∆ωt + ∆ϕ)
[ 1 2 ( ) ] ϕ1
X ( t )
X cosΦ= X + X cos ∆ωt + ∆ϕ cos +
sen senϕ
2 1
− X senΦ= X + X cos ∆ωt + ∆ϕ sen +
+ sen ∆ω + ∆ϕ cosϕ
2 1
1
1
707