beam vibrationsmpatil/courses/beamvibrations.pdf · euler-bernoulli beam theory • plane sections...
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Beam Vibrations
Chapter 8
f(x, t)
y(x, t)
m(x), EI(x)
f(x, t)dx
m(x)dxdx
Q(x, t)
M(x, t) Q(x, t) + ∂Q(x,t)∂x dx
M(x, t) + ∂M(x,t)∂x dx
Forget BCsfor now
f(x, t)dx
M(x, t) Q(x, t) + ∂Q(x,t)∂x dx
M(x, t) + ∂M(x,t)∂x dx
Q(x, t)m(x)dxdx∙
Q(x, t) +∂Q(x, t)
∂xdx
¸−Q(x, t) + f(x, t)dx
=m(x)dx∂2y(x, t)
∂t2
∂Q(x, t)
∂x+ f(x, t) = m(x)
∂2y(x, t)
∂t2
f(x, t)dx
M(x, t) Q(x, t) + ∂Q(x,t)∂x dx
M(x, t) + ∂M(x,t)∂x dx
Q(x, t)m(x)dxdx∙
M(x, t) +∂M(x, t)
∂xdx
¸−M(x, t)
+
∙Q(x, t) +
∂Q(x, t)
∂xdx
¸dx+ f(x, t)dx
dx
2= 0
Rotary Inertia Neglected
∂M(x, t)
∂x+Q(x, t) = 0
Euler-Bernoulli Beam Theory
• Plane sections remain plane and perpendicular to the beam axis– No shear deformation
• Timoshenko Beam Theory: Shear deformation is included
M(x, t) = EI(x)∂2y(x, t)
∂x2
xz
y
ρ
M
yθ
θ
θ =dy
dx
dx = ρθ
1
ρ=dθ
dx
² =(ρ− η)dθ − ρdθ
ρdθ
= −ηρ
dxη
ρ
dθ
θ
² = −η dθdx
I =
ZZA
η2dA
M = −ZZ
A
σηdA
= −ZZ
A
E²ηdA (σ = E²)
=
ZZA
Eθ0η2dA
M = EIθ0
M = EIy00
∂Q(x, t)
∂x+ f(x, t) = m(x)
∂2y(x, t)
∂t2
∂M(x, t)
∂x+Q(x, t) = 0
M(x, t) = EI(x)∂2y(x, t)
∂x2
− ∂2
∂x2
∙EI(x)
∂2y(x, t)
∂x2
¸+ f(x, t) = m(x)
∂2y(x, t)
∂t2
Simple Boundary Conditions
freefixed
Q =/ 0 y = 0
M =/ 0 θ = 0
Q = 0 y =/ 0
M = 0 θ =/ 0
mixedpinnedQ = 0 y =/ 0
M =/ 0 θ = 0
Q =/ 0 y = 0
M = 0 θ =/ 0
Cantilevered Beam
freefixed
Simply-supported Beam
pinned pinned
Free-free Beam
free free
Cantilevered Beam
− ∂2
∂x2
∙EI(x)
∂2y(x, t)
∂x2
¸+ f(x, t) =m(x)
∂2y(x, t)
∂t2
y(0, t) = 0
θ(0, t) =∂y(x, t)
∂x
¯x=0
= 0
M(L, t) = EI(x)∂2y(x, t)
∂x2
¯x=L
= 0
Q(L, t) = − ∂
∂x
∙EI(x)
∂2y(x, t)
∂x2
¸¯x=L
= 0
Free Vibration
− ∂2
∂x2
∙EI(x)
∂2y(x, t)
∂x2
¸= m(x)
∂2y(x, t)
∂t2
• Separation of Variables
y(x, t) = Y (x)F (t)
− d2
dx2
∙EI(x)
d2Y (x)
dx2
¸F (t) =m(x)Y (x)
d2F (t)
dt2
Y (0) = 0
dY (x)
dx
¯x=0
= 0
d2Y (x)
dx2
¯x=L
= 0
d
dx
∙EI(x)
d2Y (x)
dx2
¸¯x=L
= 0
− 1
m(x)Y (x)
d2
dx2
∙EI(x)
d2Y (x)
dx2
¸=
1
F (t)
d2F (t)
dt2= −ω2
d2F (t)
dt2+ ω2F (t) = 0
⇒ F (t) = A sinωt+B cosωt = C cos(ωt+ φ)
d2
dx2
∙EI(x)
d2Y (x)
dx2
¸= ω2m(x)Y (x)
Y (0) = 0 Y 0(0) = 0
Y 00(L) = 0 [EIY 00]0(L) = 0
• For prismatic beam with constant mass distribution and cross-sectional bending stiffness
• Solution:
d4Y (x)
dx4− ω2m
EIY (x) = 0
d4Y (x)
dx4− β4Y (x) = 0 β4 =
ω2m
EIY (0) = 0 Y 0(0) = 0
Y 00(L) = 0 Y 000(L) = 0
Y (x) = A sinβx+B cosβx+C sinhβx+D coshβx
Boundary Conditions
Y (x) = A sinβx+B cosβx+ C sinhβx+D coshβx
Y 0(x) = Aβ cosβx−Bβ sinβx+Cβ coshβx+Dβ sinhβxY 00(x) = −Aβ2 sinβx−Bβ2 cosβx+Cβ2 sinhβx+Dβ2 coshβxY 000(x) = −Aβ3 cosβx+Bβ3 sinβx+Cβ3 coshβx+Dβ3 sinhβx
⎡⎢⎢⎣0 1 0 11 0 1 0
− sinβL − cosβL sinhβL coshβL− cosβL sinβL coshβL sinhβL
⎤⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩ABCD
⎫⎪⎪⎬⎪⎪⎭ = 0
Non-Trivial Solutions
det
⎡⎢⎢⎣0 1 0 11 0 1 0
− sinβL − cosβL sinhβL coshβL− cosβL sinβL coshβL sinhβL
⎤⎥⎥⎦ = 0
⇒ 1 + cos(βL) cosh(βL) = 0
ω = β2rEI
m
Solutions: Eigenvalues
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩β1Lβ2Lβ3Lβ4Lβ5L
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩1.87514.69417.854810.99614.137
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭ ⇒
ω1 = 1.87512q
EImL4
ω2 = 4.69412q
EImL4
ω3 = 7.85482q
EImL4
ω4 = 10.9962q
EImL4
ω5 = 14.1372q
EImL4
Eigenfunctions/ModeshapesYr(x) = Ar sinβrx+Br cosβrx+Cr sinhβrx+Dr coshβrx
Dr = −BrCr = −Ar(− sinβrL− sinhβrL)Ar + (− cosβrL− coshβrL)Br = 0
⇒Br = −sinβrL+ sinhβrL
cosβrL+ coshβrLAr
Yr(x) = Ar [sinβrx− sinhβrx
− sinβrL+ sinhβrL
cosβrL+ coshβrL(cosβrx− coshβrx)
¸
Orthogonality
d2
dx2
∙EI(x)
d2Yr(x)
dx2
¸= ω2rm(x)Yr(x)
Ys(x)d2
dx2
∙EI(x)
d2Yr(x)
dx2
¸= ω2rm(x)Ys(x)Yr(x)
Z L
0
Ys(x)d2
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx = ω2r
Z L
0
m(x)Ys(x)Yr(x)dx
Z L
0
Ys(x)d2
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx
= Ys(x)d
dxEI(x)
d2Yr(x)
dx2
¯L0
−Z L
0
dYs(x)
dx
d
dx
∙EI(x)
d2Yr(x)
dx2
¸dx
= Ys(x)d
dx
∙EI(x)
d2Yr(x)
dx2
¸¯L0
− dYs(x)dx
∙EI(x)
d2Yr(x)
dx2
¸¯L0
+
Z L
0
d2Ys(x)
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx
For simple BCs onlyZ L
0
EI(x)d2Ys(x)
dx2d2Yr(x)
dx2dx = ω2r
Z L
0
m(x)Ys(x)Yr(x)dx
Z L
0
EI(x)d2Ys(x)
dx2d2Yr(x)
dx2dx = ω2r
Z L
0
m(x)Ys(x)Yr(x)dx
Z L
0
EI(x)d2Yr(x)
dx2d2Ys(x)
dx2dx = ω2s
Z L
0
m(x)Yr(x)Ys(x)dx
0 = (ω2r − ω2s)Z L
0
m(x)Yr(x)Ys(x)dx
0 =
Z L
0
m(x)Yr(x)Ys(x)dx
0 =
Z L
0
EI(x)d2Yr(x)
dx2d2Ys(x)
dx2dx
NormalizationZ L
0
EI(x)
∙d2Yr(x)
dx2
¸2dx = ω2r
Z L
0
m(x) [Yr(x)]2 dx
SelectYr(x) such that :
Z L
0
m(x) [Yr(x)]2 dx = 1
⇒Z L
0
EI(x)
∙d2Yr(x)
dx2
¸2dx = ω2r
Initial (unforced) Response
y(x, t) =∞Xr=1
Yr(x)ηr(t)Yr(x) are mass normalizedmode shapes
ηs(t) + ω2sηs(t) = 0
ηs(t) = ηs(0) cosωst+ηs(0)
ωssinωst
ηs(0) =
Z L
0
m(x)Ys(x)y0(x)dx
ηs(0) =
Z L
0
m(x)Ys(x)v0(x)dx
Forced Response
y(x, t) =∞Xr=1
Yr(x)ηr(t)Yr(x) are mass normalizedmode shapes
ηs(t) + ω2sηs(t) = Ns(t)
Ns(t) =
Z L
0
Ys(x)f(x, t)dx
Interesting BCs: spring
k
Q(L, t) = − ∂
∂x
∙EI(x)
∂2y(x, t)
∂x2
¸¯x=L
= −ky(L, t)
EId3Y (x)
dx3
¯x=L
− kY (L) = 0
For constant EI and separation of variables
⎡⎢⎢⎣0 1 0 11 0 1 0
− sinβL − cosβL sinhβL coshβL−(βL)3 cosβL− k sinβL (βL)3 sinβL− k cosβL (βL)3 coshβL− k sinhβL (βL)3 sinhβL− k coshβL
⎤⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩ABCD
⎫⎪⎪⎬⎪⎪⎭ = 0
k =kL3
EI
det
⎡⎢⎢⎣0 1 0 11 0 1 0
− sin βL − cosβL sinhβL cosh βL
−(βL)3 cosβL − k sin βL (βL)3 sin βL − k cos βL (βL)3 cosh βL − k sinh βL (βL)3 sinh βL− k cosh βL
⎤⎥⎥⎦ = 0
0 0.2 0.4 0.6 0.8 10
2
4
6
8
kbar/(kbar+3)
β L
Z L
0
Ys(x)d2
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx
= Ys(x)d
dx
∙EI(x)
d2Yr(x)
dx2
¸¯L0
− dYs(x)dx
∙EI(x)
d2Yr(x)
dx2
¸¯L0
+
Z L
0
d2Ys(x)
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx
= kYs(L)Yr(L) +
Z L
0
d2Ys(x)
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx
Z L
0
EI(x)d2Ys(x)
dx2d2Yr(x)
dx2dx+ kYs(L)Yr(L) = ω2r
Z L
0
m(x)Ys(x)Yr(x)dx
0 =
Z L
0
m(x)Yr(x)Ys(x)dx
0 =
Z L
0
EI(x)d2Yr(x)
dx2d2Ys(x)
dx2dx+ kYs(L)Yr(L)
SelectYr(x) such that :
Z L
0
m(x) [Yr(x)]2 dx = 1
⇒Z L
0
EI(x)
∙d2Yr(x)
dx2
¸2dx+ k [Yr(L)]
2 = ω2r
Modal Equations
y(x, t) =∞Xr=1
Yr(x)ηr(t)Yr(x) are mass normalizedmode shapes
ηs(t) + ω2sηs(t) = Ns(t)
ηs(0) =
Z L
0
m(x)Ys(x)y0(x)dx
ηs(0) =
Z L
0
m(x)Ys(x)v0(x)dx
Ns(t) =
Z L
0
Ys(x)f(x, t)dx
Interesting BCs: mass
M
Q(L, t) = − ∂
∂x
∙EI(x)
∂2y(x, t)
∂x2
¸¯x=L
= −M∙∂2y(x, t)
∂t2
¸x=L
For constant EI, separation of variables and frequency response
EId3Y (x)
dx3
¯x=L
+ ω2MY (L) = 0
⎡⎢⎢⎣0 1 0 11 0 1 0
− sinβL − cosβL sinhβL coshβL− cosβL+MβL sinβL sinβL+MβL cosβL coshβL+MβL sinhβL sinhβL+MβL coshβL
⎤⎥⎥⎦⎧⎪⎪⎨⎪⎪⎩ABCD
⎫⎪⎪⎬⎪⎪⎭ = 0
ω = β2rEI
mM =
M
mL
det
⎡⎢⎢⎣0 1 0 11 0 1 0
− sin βL − cosβL sinhβL cosh βL− cos βL+MβL sinβL sinβL+MβL cos βL coshβL+MβL sinhβL sinhβL+MβL coshβL
⎤⎥⎥⎦ = 0
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
Mbar/(Mbar+1)
β L
SDOF Massless Beam
Z L
0
Ys(x)d2
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx
= Ys(x)d
dx
∙EI(x)
d2Yr(x)
dx2
¸¯L0
− dYs(x)dx
∙EI(x)
d2Yr(x)
dx2
¸¯L0
+
Z L
0
d2Ys(x)
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx
= −ω2rMYs(L)Yr(L) +Z L
0
d2Ys(x)
dx2
∙EI(x)
d2Yr(x)
dx2
¸dx
Z L
0
EI(x)d2Ys(x)
dx2d2Yr(x)
dx2dx−ω2rMYs(L)Yr(L) = ω2r
Z L
0
m(x)Ys(x)Yr(x)dx
0 =
Z L
0
m(x)Yr(x)Ys(x)dx+MYs(L)Yr(L)
0 =
Z L
0
EI(x)d2Yr(x)
dx2d2Ys(x)
dx2dx
SelectYr(x) such that :Z L
0
m(x) [Yr(x)]2 dx+M [Yr(L)]
2 = 1
⇒Z L
0
EI(x)
∙d2Yr(x)
dx2
¸2dx = ω2r
Modal Equations
y(x, t) =∞Xr=1
Yr(x)ηr(t)Yr(x) are mass normalizedmode shapes
ηs(t) + ω2sηs(t) = Ns(t)
Ns(t) =
Z L
0
Ys(x)f(x, t)dx
ηs(0) =
Z L
0
m(x)Ys(x)y0(x)dx+MYs(L)y0(L)
ηs(0) =
Z L
0
m(x)Ys(x)v0(x)dx+MYs(L)v0(L)
Variational Approach
• Extended Hamilton’s Principle
t2
t1
(δT − δV + δW )dt = 0
Kinetic Energy
T =L
0
1
2m(x)
∂y(x, t)
∂t
2
dx
=1
2
L
0
m(x) [y(x, t)]2dx
δT =L
0
m(x)y(x, t)δy(x, t)dx
δT =
Z L
0
m(x)y(x, t)δy(x, t)dxZ t2
t1
δTdt =
Z t2
t1
Z L
0
m(x)y(x, t)δy(x, t)dxdt
=
Z L
0
Z t2
t1
m(x)y(x, t)δy(x, t)dtdx
=
Z L
0
[m(x)y(x, t)δy(x, t)]|t2t1 dx
−Z L
0
Z t2
t1
m(x)y(x, t)δy(x, t)dtdx
Potential (Strain) Energy
V =
Z L
0
ZZA
∙1
2E²(x, η, t)2
¸dAdx
=1
2
Z L
0
ZZA
"E
µ−η ∂
2y(x, t)
∂x2
¶2#dAdx
=1
2
Z L
0
E
µ∂2y(x, t)
∂x2
¶2 ZZA
η2dAdx
=1
2
Z L
0
EI(x) [y00(x, t)]2dx
δV =
Z L
0
EI(x)y00(x, t)δy00(x, t)dx
= [EI(x)y00(x, t)] δy0(x, t)|L0
−Z L
0
[EI(x)y00(x, t)]0δy0(x, t)dx
= [EI(x)y00(x, t)] δy0(x, t)|L0− [EI(x)y00(x, t)]0 δy(x, t)
¯L0
+
Z L
0
[EI(x)y00(x, t)]00δy(x, t)dx
Virtual Work
δW =L
0
[f(x, t)δy] dx
Z t2
t1
Z L
0
n−m(x)y(x, t)− [EI(x)y00(x, t)]00 + f(x, t)
oδy(x, t)dxdt
−Z t2
t1
[EI(x)y00(x, t)] δy0(x, t)|L0 dt
+
Z t2
t1
[EI(x)y00(x, t)]0δy(x, t)
¯L0dt
+
Z L
0
[m(x)y(x, t)δy(x, t)]|t2t1 dx = 0
m(x)y(x, t) + [EI(x)y00(x, t)]00= f(x, t)
Boundary Conditions : x = 0 & x = L
either [EI(x)y00(x, t)] = 0 or δy0(x, t) = 0
either [EI(x)y00(x, t)]0= 0 or δy(x, t) = 0
Simple BCs
Interesting BCM
k
T =1
2
Z L
0
m(x) [y(x, t)]2 dx+1
2M [y(x, t)]2
¯x=L
V =1
2
Z L
0
EI(x) [y00(x, t)]2dx+
1
2ky(x, t)2
¯x=0
δW =
Z L
0
[f(x, t)δy] dx
Z t2
t1
Z L
0
n−m(x)y(x, t)− [EI(x)y00(x, t)]00 + f(x, t)
oδy(x, t)dxdt
−Z t2
t1
[EI(x)y00(x, t)] δy0(x, t)|L0 dt
+
Z t2
t1
½[EI(x)y00(x, t)]
0δy(x, t)
¯L0− ky(x, t)δy(x, t)|x=0 − My(x, t)δy(x, t)|x=L
¾dt
+
Z L
0
[m(x)y(x, t)δy(x, t)]|t2t1 dx+ My(x, t)δy(x, t)|x=L|t2t1= 0
m(x)y(x, t) + [EI(x)y00(x, t)]00= f(x, t)
Boundary Conditions : x = 0 & x = L
either [EI(x)y00(x, t)] = 0 or δy0(x, t) = 0
x = 0 : either [EI(x)y00(x, t)]0+ ky(x, t) = 0 or δy(x, t) = 0
x = L : either [EI(x)y00(x, t)]0 −My(x, t) = 0 or δy(x, t) = 0