Vibration of continuous systems
Beams
16/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
Calculate the natural frequencies and mode shapes for the transverse vibration of a
beam of length l that is fixed at one end and pinned at the other end.
16/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
Calculate the natural frequencies and mode shapes for the transverse vibration of a
beam of length l that is fixed at one end and pinned at the other end.
16/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
Calculate the natural frequencies and mode shapes for the transverse vibration of a
beam of length l that is fixed at one end and pinned at the other end.
16/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
Calculate the natural frequencies and mode shapes for the transverse vibration of a
beam of length l that is fixed at one end and pinned at the other end.
16/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
17/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
17/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
18/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
the first 3 mode shapes of the clamped–pinned beam (arbitrarily normalized to unity)
18/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
19/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
19/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
20/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Beams
20/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
21/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
a rectangular membrane
τ ∇2 w(x, y, t) = ρw(x, y, t)
21/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
a rectangular membrane
τ ∇2 w(x, y, t) = ρw(x, y, t)
∇2 =
∂2
∂2
+
(in rectangular coordinates)
∂x2
∂y 2
21/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
a rectangular membrane
τ ∇2 w(x, y, t) = ρw(x, y, t)
∇2 =
∂2
∂2
+
(in rectangular coordinates)
∂x2
∂y 2
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
2
2
∂x
∂y
c
∂t2
c=
p
τ /ρ
21/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
a rectangular membrane
τ ∇2 w(x, y, t) = ρw(x, y, t)
∇2 =
∂2
∂2
+
(in rectangular coordinates)
∂x2
∂y 2
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
2
2
∂x
∂y
c
∂t2
c=
p
τ /ρ
21/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
a rectangular membrane
τ ∇2 w(x, y, t) = ρw(x, y, t)
∇2 =
∂2
∂2
+
(in rectangular coordinates)
∂x2
∂y 2
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
2
2
∂x
∂y
c
∂t2
c=
p
τ /ρ
21/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
Sol’n:
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
Sol’n:
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
∂x2
∂y 2
c
∂t2
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
Sol’n:
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
∂x2
∂y 2
c
∂t2
Assume separation of variables sol’n:
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
Sol’n:
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
∂x2
∂y 2
c
∂t2
Assume separation of variables sol’n: w(x, y, t) = X(x)Y (y)T (t)
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
Sol’n:
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
∂x2
∂y 2
c
∂t2
Assume separation of variables sol’n: w(x, y, t) = X(x)Y (y)T (t)
1 T̈
X 00
Y 00
=
+
2
c T
X
Y
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
Sol’n:
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
∂x2
∂y 2
c
∂t2
Assume separation of variables sol’n: w(x, y, t) = X(x)Y (y)T (t)
1 T̈
X 00
Y 00
=
+
2
c T
X
Y
T̈
= −ω 2 (ω is a cst)
T c2
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
Sol’n:
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
∂x2
∂y 2
c
∂t2
Assume separation of variables sol’n: w(x, y, t) = X(x)Y (y)T (t)
1 T̈
X 00
Y 00
=
+
2
c T
X
Y
T̈
= −ω 2 (ω is a cst)
T c2
X 00
Y 00
= −ω 2 −
X
Y
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
Consider the vibration of a square membrane clamped at all the edges
Compute the natural frequencies and mode shapes for the case a = b = 1
Sol’n:
∂ 2 w(x, y, t)
∂ 2 w(x, y, t)
1 ∂ 2 w(x, y, t)
+
= 2
∂x2
∂y 2
c
∂t2
Assume separation of variables sol’n: w(x, y, t) = X(x)Y (y)T (t)
1 T̈
X 00
Y 00
=
+
2
c T
X
Y
X 00
Y 00
= −ω 2 −
X
Y
T̈
= −ω 2 (ω is a cst)
T c2
X 00
= −α2
X
Y 00
= −γ 2
Y
(α, γ are csts)
22/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
23/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
23/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion.
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion. Hence, A3 = A4 = 0.
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion. Hence, A3 = A4 = 0. Then,
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion. Hence, A3 = A4 = 0. Then,
X(x)Y (y) = A1 sin αx sin γy + A2 sin αx sin γy
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion. Hence, A3 = A4 = 0. Then,
X(x)Y (y) = A1 sin αx sin γy + A2 sin αx sin γy
BC: w = 0 along the line x = 1 gives
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion. Hence, A3 = A4 = 0. Then,
X(x)Y (y) = A1 sin αx sin γy + A2 sin αx sin γy
BC: w = 0 along the line x = 1 gives
A1 sin α sin γy + A2 sin α cos γy = 0
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion. Hence, A3 = A4 = 0. Then,
X(x)Y (y) = A1 sin αx sin γy + A2 sin αx sin γy
BC: w = 0 along the line x = 1 gives
A1 sin α sin γy + A2 sin α cos γy = 0 =⇒ sin α (A1 sin γy + A2 cos γy) = 0
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion. Hence, A3 = A4 = 0. Then,
X(x)Y (y) = A1 sin αx sin γy + A2 sin αx sin γy
BC: w = 0 along the line x = 1 gives
A1 sin α sin γy + A2 sin α cos γy = 0 =⇒ sin α (A1 sin γy + A2 cos γy) = 0
√
α = nπ
γ = mπ
ωmn = π n2 + m2
m, n = 1, 2 . . . , ∞
24/24
OKTAV
Mechanical Vibrations
Vibration of continuous systems
Membranes
For the clamp BC along x = 0
T (t)X(x)Y (y) = T (t)(A3 sin γy + A4 cos γy) = 0 =⇒ A3 sin γy + A4 cos γy = 0
must hold for any value of y
γ 6= 0 otherwise, the system has a rigid-body motion. Hence, A3 = A4 = 0. Then,
X(x)Y (y) = A1 sin αx sin γy + A2 sin αx sin γy
BC: w = 0 along the line x = 1 gives
A1 sin α sin γy + A2 sin α cos γy = 0 =⇒ sin α (A1 sin γy + A2 cos γy) = 0
√
α = nπ
γ = mπ
ωmn = π n2 + m2
m, n = 1, 2 . . . , ∞
w(x, y, t) =
∞ X
∞
p
p
X
(sin mπx sin nπy) Amn sin n2 + m2 cπt + Bmn cos n2 + m2 cπt
m=1 n=1
24/24
OKTAV
Mechanical Vibrations