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