FEM for continuum dynamics

FEM for continuum dynamics

Piotr Pluciński e-mail: pplucin@L5.pk.edu.pl

Jerzy Pamin

e-mail: jpamin@L5.pk.edu.pl

Lecture contents

1 Dynamic equilibrium state

2 Free vibrations Beam element

3 Example

Dynamic equilibrium state

Y Z

X

S

n

t

t ^d ρb(x, y, z)

ρb ^d (x, y, z)

ρ¨ u(x, y, z) P

V

Forces

ρb – mass force density vector [N/m ³ ]

ρb ^d – volume damping force density vector [N/m ³ ] ρ¨ u – inertia force density vector [N/m ³ ]

t – traction force density vector [N/m ² ]

t ^d – surface damping force

density vector [N/m ² ]

Dynamic equilibrium state

Y Z

X

S

n

t

t ^d ρb(x, y, z)

ρb ^d (x, y, z)

ρ¨ u(x, y, z) P

V

Forces

ρb – mass force density vector [N/m ³ ]

ρb ^d – volume damping force density vector [N/m ³ ] ρ¨ u – inertia force density vector [N/m ³ ]

t – traction force density vector [N/m ² ]

t ^d – surface damping force

density vector [N/m ² ]

Dynamic equilibrium state

Y Z

X S

n t

t ^d ρb(x, y, z)

ρb ^d (x, y, z)

ρ¨ u(x, y, z) P

V

Body equilibrium equations

Z

S

t − t ^d
dS + Z

V

ρ b − b ^d − ¨ u
dV = 0

Static boundary conditions t − t ^d = σn where σ – stress tensor Using Green–Gauss–Ostrogradski theorem

Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

Dynamic equilibrium state

Y Z

X S

n t

t ^d ρb(x, y, z)

ρb ^d (x, y, z)

ρ¨ u(x, y, z) P

V

Body equilibrium equations

Z

S

t − t ^d
dS + Z

V

ρ b − b ^d − ¨ u
dV = 0

Static boundary conditions t − t ^d = σn where σ – stress tensor

Using Green–Gauss–Ostrogradski theorem Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

Dynamic equilibrium state

Y Z

X S

n t

t ^d ρb(x, y, z)

ρb ^d (x, y, z)

ρ¨ u(x, y, z) P

V

Body equilibrium equations

Z

S

t − t ^d
dS + Z

V

ρ b − b ^d − ¨ u
dV = 0

Static boundary conditions t − t ^d = σn where σ – stress tensor Using Green–Gauss–Ostrogradski theorem

Z

S

σndS = Z

V

L ^T σdV where L – differential operator matrix

Equilibrium equations

Navier equations Z

V



L ^T σ + ρ 

b − b ^d − ¨ u 

dV = 0 ⇐⇒L ^T σ + ρ 

b − b ^d − ¨ u 

= 0 ∀P ∈ V

σ ij,j + ρ 

b i − b ^d i − ¨ u i



= 0

Weak formulation – weighting function w ∼ = δu – kinematically admissible displacement variation

Z

V

(δu) ^T L ^T σ + ρ b − b ^d − ¨ u

dV = 0 ∀δu

Equilibrium equations

Navier equations Z

V



L ^T σ + ρ 

b − b ^d − ¨ u 

dV = 0 ⇐⇒L ^T σ + ρ 

b − b ^d − ¨ u 

= 0 ∀P ∈ V σ ij,j + ρ 

b i − b ^d i − ¨ u i



= 0

Weak formulation – weighting function w ∼ = δu – kinematically admissible displacement variation

Z

V

(δu) ^T L ^T σ + ρ b − b ^d − ¨ u

dV = 0 ∀δu

Equilibrium equations

Navier equations Z

V



L ^T σ + ρ 

b − b ^d − ¨ u 

dV = 0 ⇐⇒L ^T σ + ρ 

b − b ^d − ¨ u 

= 0 ∀P ∈ V σ ij,j + ρ 

b i − b ^d i − ¨ u i



= 0

Weak formulation – weighting function w ∼ = δu – kinematically admissible displacement variation

Z

V

(δu) ^T L ^T σ + ρ b − b ^d − ¨ u

dV = 0 ∀δu

Equilibrium equations

Navier equations Z

V



L ^T σ + ρ 

b − b ^d − ¨ u 

dV = 0 ⇐⇒L ^T σ + ρ 

b − b ^d − ¨ u 

= 0 ∀P ∈ V σ ij,j + ρ 

b i − b ^d i − ¨ u i



= 0

Weak formulation – weighting function w ∼ = δu – kinematically admissible displacement variation (complying with kinematic b.cs)

Z

V

(δu) ^T L ^T σ + ρ b − b ^d − ¨ u

dV = 0 ∀δu

− Z

V

(Lδu) ^T σdV + Z

S

(δu) ^T σndS + Z

V

(δu) ^T ρ b − b ^d − ¨ u
dV = 0

Equilibrium equations

Navier equations Z

V



L ^T σ + ρ 

b − b ^d − ¨ u 

dV = 0 ⇐⇒L ^T σ + ρ 

b − b ^d − ¨ u 

= 0 ∀P ∈ V σ ij,j + ρ 

b i − b ^d i − ¨ u i



= 0

Weak formulation – weighting function w ∼ = δu – kinematically admissible displacement variation (complying with kinematic b.cs)

Z

V

(δu) ^T L ^T σ + ρ b − b ^d − ¨ u

dV = 0 ∀δu

− Z

V

(Lδu) ^T σdV + Z

S

(δu) ^T σn t − t ^d

dS + Z

V

(δu) ^T ρ b − b ^d − ¨ u
dV = 0

Equilibrium equations

Navier equations Z

V



L ^T σ + ρ 

b − b ^d − ¨ u 

dV = 0 ⇐⇒L ^T σ + ρ 

b − b ^d − ¨ u 

= 0 ∀P ∈ V σ ij,j + ρ 

b i − b ^d i − ¨ u i



= 0

Weak formulation – weighting function w ∼ = δu – kinematically admissible displacement variation (complying with kinematic b.cs)

Z

V

(δu) ^T L ^T σ + ρ b − b ^d − ¨ u

dV = 0 ∀δu

− Z

V

(Lδu) ^T σdV + Z

S

(δu) ^T t − t ^d
dS+

Z

V

(δu) ^T ρ b − b ^d − ¨ u
dV = 0

Equilibrium equations

Navier equations Z

V



L ^T σ + ρ 

b − b ^d − ¨ u 

dV = 0 ⇐⇒L ^T σ + ρ 

b − b ^d − ¨ u 

= 0 ∀P ∈ V σ ij,j + ρ 

b i − b ^d i − ¨ u i



= 0

Weak formulation – weighting function w ∼ = δu – kinematically admissible displacement variation – it is virtual work principle

Z

V

(δu) ^T L ^T σ + ρ b − b ^d − ¨ u

dV = 0 ∀δu Z

V

(Lδu) ^T σdV = Z

S

(δu) ^T t − t ^d
dS + Z

V

(δu) ^T ρ b − b ^d − ¨ u
dV

Equilibrium equations

Navier equations Z

V



L ^T σ + ρ 

b − b ^d − ¨ u 

dV = 0 ⇐⇒L ^T σ + ρ 

b − b ^d − ¨ u 

= 0 ∀P ∈ V σ ij,j + ρ 

b i − b ^d i − ¨ u i



= 0

Weak formulation – weighting function w ∼ = δu – kinematically admissible displacement variation – it is virtual work principle

Z

V

(δu) ^T L ^T σ + ρ b − b ^d − ¨ u

dV = 0 ∀δu

Z

V

(Lδu) ^T σdV = Z

S

(δu) ^T t − t ^d
dS + Z

V

(δu) ^T ρ b − b ^d − ¨ u
dV

internal virtual work external virtual work

Equilibrium equation of discretized structure

Equilibrium equation (FEM approximation: u ^eh (x, t) = N ^e (x)q ^e (t))

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation (FEM approximation: u ^eh (x, t) = N ^e (x)q ^e (t))

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(L ^e N ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(L ^e N ^e B ^e

δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

(δq ^e ) ^T

Z

V ^e

B ^eT σ ^e dV ^e − Z

S ^e

N ^eT 

t ^e − t ^de  dS ^e

− Z

N ^eT ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

( δq ^e I T ^e δQ

) ^T

Z

V ^e

B ^eT σ ^e dV ^e − Z

S ^e

N ^eT 

t ^e − t ^de  dS ^e

− Z

V ^e

N ^eT ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

( I T ^e δQ) ^T

Z

V ^e

B ^eT σ ^e dV ^e − Z

S ^e

N ^eT 

t ^e − t ^de  dS ^e

− Z

N ^eT ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0 (δQ) ^T

E

X

e=1

I T ^eT

Z

V ^e

B ^eT σ ^e dV ^e − Z

S ^e

N ^eT 

t ^e − t ^de  dS ^e

− Z

V ^e

N ^eT ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0 (δQ) ^T

∀δQ

E

X

e=1

I T ^eT

Z

V ^e

B ^eT σ ^e dV ^e − Z

S ^e

N ^eT 

t ^e − t ^de  dS ^e

− Z

N ^eT ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

Z

V ^e

(L ^e δu ^e ) ^T σ ^e dV ^e − Z

S ^e

(δu ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(δu ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

Z

V ^e

(B ^e δq ^e ) ^T σ ^e dV ^e − Z

S ^e

(N ^e δq ^e ) ^T 

t ^e − t ^de  dS ^e

− Z

V ^e

(N ^e δq ^e ) ^T ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

E

X

e=1

I T ^eT

Z

V ^e

B ^eT σ ^e dV ^e − Z

S ^e

N ^eT 

t ^e − t ^de  dS ^e

− Z

V ^e

N ^eT ρ 

b ^e − b ^de − ¨ u ^e  dV ^e



= 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T ^eT

Z

V ^e

B ^eT σ ^e dV ^e

 +

E

X

e=1

I T ^eT

Z

V ^e

N ^eT ρ¨ u ^e dV ^e

 +

+

E

X

e=1

I T ^eT

Z

S ^e

N ^eT t ^de dS ^e + Z

V ^e

N ^eT ρb ^de dV ^e



=

=

E

X

e=1

I T ^eT

Z

S ^e

N ^eT t ^e dS ^e + Z

V ^e

N ^eT ρb ^e dV ^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε, linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e q ^e = D ^e B ^e T I ^e Q, u ¨ ^e = N ^e ¨ q ^e = N ^e T I ^e Q ¨ viscous damping:

t ^de = µ _d u ˙ ^e = µ _d N ^e ˙q ^e = µ _d N ^e T I ^e Q, ˙ ρb ^de = µ _b u ˙ ^e = µ _b N ^e ˙q ^e = µ _b N ^e T I ^e Q ˙

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T ^eT

Z

V ^e

B ^eT σ ^e dV ^e

 +

E

X

e=1

I T ^eT

Z

V ^e

N ^eT ρ¨ u ^e dV ^e

 +

+

E

X

e=1

I T ^eT

Z

S ^e

N ^eT t ^de dS ^e + Z

V ^e

N ^eT ρb ^de dV ^e



=

=

E

X

e=1

I T ^eT

Z

S ^e

N ^eT t ^e dS ^e + Z

V ^e

N ^eT ρb ^e dV ^e



Consideration of kinamatic and constitutive equations linear elasticity: σ = Dε, linear kinematic relation: ε = Lu

σ ^e = D ^e L ^e u ^e = D ^e L ^e N ^e q ^e = D ^e B ^e T I ^e Q, u ¨ ^e = N ^e ¨ q ^e = N ^e T I ^e Q ¨ viscous damping:

t ^de = µ _d u ˙ ^e = µ _d N ^e ˙q ^e = µ _d N ^e T I ^e Q, ˙ ρb ^de = µ _b u ˙ ^e = µ _b N ^e ˙q ^e = µ _b N ^e T I ^e Q ˙

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T ^eT

Z

V ^e

B ^eT D ^e B ^e dV ^e

 I T ^e Q+

+

E

X

e=1

I T ^eT

Z

V ^e

ρN ^eT N ^e dV ^e

 I T ^e Q+ ¨

+

E

X

e=1

I T ^eT

Z

S ^e

µ d N ^eT N ^e dS ^e + Z

V ^e

µ b N ^eT N ^e dV ^e

 I T ^e Q = ˙

=

E

X

e=1

I T ^eT

Z

S ^e

N ^eT t ^e dS ^e + Z

V ^e

N ^eT ρb ^e dV ^e



Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T ^eT

( Z

V ^e

B ^eT D ^e B ^e dV ^e k ^e

) I T ^e Q+

+

E

X

e=1

I T ^eT

( Z

V ^e

ρN ^eT N ^e dV ^e m ^e

) I T ^e Q+ ¨

+

E

X

e=1

I T ^eT

( Z

S ^e

µ d N ^eT N ^e dS ^e + Z

V ^e

µ b N ^eT N ^e dV ^e c ^e

) I T ^e Q = ˙

=

E

X

e=1

I T ^eT

( Z

S ^e

N ^eT t ^e dS ^e + Z

V ^e

N ^eT ρb ^e dV ^e f ^e

)

Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I

T ^eT k ^e T I ^e Q +

E

X

e=1

I

T ^eT m ^e T I ^e Q + ¨

E

X

e=1

I

T ^eT c ^e T I ^e Q = ˙

E

X

e=1

I T ^eT f ^e

Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

I T ^eT k ^e T I ^e

K ^e

Q +

E

X

e=1

I

T ^eT m ^e T I ^e M ^e

Q + ¨

E

X

e=1

I

T ^eT c ^e T I ^e C ^e

Q = ˙

E

X

e=1

I T ^eT f ^e

F ^e

Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

K ^e Q +

E

X

e=1

M ^e Q + ¨

E

X

e=1

C ^e Q = ˙

E

X

e=1

F ^e

Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Equilibrium equation of discretized structure

Equilibrium equation

E

X

e=1

K ^e K

Q +

E

X

e=1

M ^e M

Q + ¨

E

X

e=1

C ^e C

Q = ˙

E

X

e=1

F ^e F

Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Equilibrium equation of discretized structure

Equilibrium equation

KQ + M ¨ Q + C ˙ Q = F

Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Equilibrium equation of discretized structure

Equilibrium equation – forced vibrations with damping M ¨ Q(t) + C ˙ Q(t) + KQ(t) = F(t)

M – inertia (mass) matrix C – damping matrix K – stiffness matrix

F – external nodal force vector

Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Equilibrium equation of discretized structure

Equilibrium equation – forced vibrations with damping M ¨ Q(t) + C ˙ Q(t) + KQ(t) = F(t)

M – inertia (mass) matrix C – damping matrix K – stiffness matrix

F – external nodal force vector

Equilibrium equations – free vibrations w/o damping

M ¨ Q(t) + KQ(t) = 0

Eigenvibrations

Approximation

u ^e (x, t) = N ^e (x)q ^e (t) = N ^e (x)q ^e _A sin(ωt + ϕ)

N ^e – shape function matrix

q ^e _A – eigenvibration amplitude vector ω – angular frequency

ϕ – phase shift ω = 2πf = ^2π _T

where f, T – frequency and period, respectively

Free vibrations

Displacement depending on time (vibration)

q ^e (t) = q ^e _A sin(ωt + ϕ)

q ¨ ^e (t) = −ω ² q ^e _A sin(ωt + ϕ) after assembly and substitution into equilibrium equation

K − ω ² M
Q A = 0 equation is satisfied when

det K − ω ² M
= 0

Free vibrations

Displacement depending on time (vibration)

q ^e (t) = q ^e _A sin(ωt + ϕ) q ¨ ^e (t) = −ω ² q ^e _A sin(ωt + ϕ)

after assembly and substitution into equilibrium equation

K − ω ² M
Q A = 0 equation is satisfied when

det K − ω ² M
= 0

Free vibrations

equilibrium equation =⇒ eigen problem

q ^e (t) = q ^e _A sin(ωt + ϕ) q ¨ ^e (t) = −ω ² q ^e _A sin(ωt + ϕ) after assembly and substitution into equilibrium equation

K − ω ² M
Q A sin(ωt + ϕ) = 0 ∀t

equation is satisfied when

det K − ω ² M
= 0

Free vibrations

equilibrium equation =⇒ eigen problem

q ^e (t) = q ^e _A sin(ωt + ϕ) q ¨ ^e (t) = −ω ² q ^e _A sin(ωt + ϕ) after assembly and substitution into equilibrium equation

K − ω ² M
Q A sin(ωt + ϕ) = 0 ∀t

equation is satisfied when

det K − ω ² M
= 0

Free vibrations

equilibrium equation =⇒ eigen problem

q ^e (t) = q ^e _A sin(ωt + ϕ) q ¨ ^e (t) = −ω ² q ^e _A sin(ωt + ϕ) after assembly and substitution into equilibrium equation

K − ω ² M
Q A = 0

equation is satisfied when

det K − ω ² M
= 0

Free vibrations

equilibrium equation =⇒ eigen problem

q ^e (t) = q ^e _A sin(ωt + ϕ) q ¨ ^e (t) = −ω ² q ^e _A sin(ωt + ϕ) after assembly and substitution into equilibrium equation

K − ω ² M
Q A = 0 equation is satisfied when

det K − ω ² M
= 0 or Q A = 0

Free vibrations

equilibrium equation =⇒ eigen problem

q ^e (t) = q ^e _A sin(ωt + ϕ) q ¨ ^e (t) = −ω ² q ^e _A sin(ωt + ϕ) after assembly and substitution into equilibrium equation

K − ω ² M
Q A = 0 equation is satisfied when

det K − ω ² M
= 0

Free vibrations

Beam element

A, I, ρ

x ^e z ^e

0 l ^e

q ₁ ^e

q ₂ ^e

q ₃ ^e

q ₄ ^e x ^e z ^e

i j

Shape functions N ^e = [N ₁ ^e N ₂ ^e N ₃ ^e N ₄ ^e ]

0 l ^e

1

x ^e N ₁ ^e (x ^e ) = 1 − 3

 x ^e l ^e

 2

+ 2

 x ^e l ^e

 3

0 l ^e

1

x ^e N ₃ ^e (x ^e ) = 3

 x ^e l ^e

 2

− 2 

x ^e l ^e

 3

0 l ^e x ^e

N ₂ ^e (x ^e ) = x ^e h

1 −

 x ^e l ^e

i 2

0 l ^e x ^e

N ₄ ^e (x ^e ) = x ^e

 

x ^e l ^e

 2

− 

x ^e l ^e

 

Free vibrations

Stiffness matrix for beam element

k ^e = Z l ^e

0

B ^eT D ^e B ^e dx ^e

B ^e = LN ^e , L =

"

− d ² dx ^e2

#

, D ^e = [E ^e I ^e ]

k ^e = E ^e I ^e l ^e3









12 6l ^e −12 6l ^e 6l ^e 4l ^e2 −6l ^e 2l ^e2

−12 −6l ^e 12 −6l ^e 6l ^e 2l ^e2 −6l ^e 4l ^e2









Free vibrations

Mass (inertia) matrix for beam element

m ^e = Z l ^e

0

ρA ^e N ^eT N ^e dx ^e

µ ^e = ρA ^e – mass density per unit length [kg/m]

m ^e = µ ^e l ^e 420









156 22l ^e 54 −13l ^e 22l ^e 4l ^e2 13l ^e −3l ^e2 54 13l ^e 156 −22l ^e

−13l ^e −3l ^e2 −22l ^e 4l ^e2









Example

Cantilever

E, I, µ, l Q 1 Q 3

Z

K − ω ² M
Q A = 0

 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0

⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1

Q 2

Q 3

Q 4 X Z

K − ω ² M
Q A = 0







 EI

l ³









12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− ω ² µl 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 Q _A1 Q _A2 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0

⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1 Q 3

Z

K − ω ² M
Q A = 0

















12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− ω ² µl ⁴ EI

1 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 Q _A1 Q _A2 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0

⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1

Q 2

Q 3

Q 4 X Z

K − ω ² M
Q A = 0

















12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− ω ² µl ⁴ EI

λ 1 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 Q _A1 Q _A2 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0

⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1 Q 3

Z

K − ω ² M
Q A = 0

















12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− λ 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 Q _A1 Q _A2 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0

⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1

Q 2

Q 3

Q 4 X Z

K − ω ² M
Q A = 0

















12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− λ 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 0 0 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0

⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1 Q 3

Z

K − ω ² M
Q A = 0

















12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− λ 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 0 0 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0

⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1

Q 2

Q 3

Q 4 X Z

K − ω ² M
Q A = 0

















12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− λ 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 0 0 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0

⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1 Q 3

Z

K − ω ² M
Q A = 0

















12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− λ 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 0 0 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0 ⇒ λ 1 = 12.48 λ 2 = 1211.52

⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

E, I, µ, l Q 1

Q 2

Q 3

Q 4 X Z

K − ω ² M
Q A = 0

















12 6l -12 6l 6l 4l ² -6l 2l ² -12 -6l 12 -6l 6l 2l ² -6l 4l ²









− λ 420









156 22l 54 -13l 22l 4l ² 13l -3l ² 54 13l 156 -22l -13l -3l ² -22l 4l ²























 0 0 Q _A3 Q _A4









=







 0 0 0 0









 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

  Q _A3 Q _A4



=  0 0



 12 -6l -6l 4l ²



− λ 420

 156 -22l -22l 4l ²

   

= 0 ⇒ λ 1 = 12.48 λ 2 = 1211.52 ⇒

ω 1 = 3.53 l ²

s EI

µ ω 2 = 34.81

l ² s

EI

µ

Example

Cantilever

Eigen vibration modes

Free vibration modes are determined from one of two linearly dependent equations upon substitution of relevant eigenvalue.

0 l ^e

Q A3 = 0.728l·Q A4

Q _A4

x ^e for ω ₁

0 l ^e x ^e

for ω ₂

Q _A3 = 0.131l·Q _A4

Q _A4