02b_ modal decomposition and superposition

8/10/2019 02b_ Modal Decomposition and Superposition

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d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis 2.3 modal decomposition

equation of motion including Rayleighdamping

M u + D u + K u = r D = 1 M + 2 K

ansatz using modal matrix

u = u u = u, u = u

inserting ansatz and multiplication by T

decoupled system of differential equations

T

M | {z }Mdiagu +

TD | {z }Ddiag

u +T

K | {z }Kdiag u =

Tr|{z}r

with diagonal matrices Mdiag, Ddiag and K diag

Mdiag =

2

6664M1 0 0

0 M2 0... ... . . . ...

0 0 MNEQ

3

7775 Kdiag =2

6664K1 0 0

0 K2 0... ... . . . ...

0 0 KNEQ

3

7775NEQ decoupled differential equations

Mi ui+ Di ui+ Ki ui = ri i= 1, , NEQ

eigenvalue analysis - modal transformation 20
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d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis 2.4 single dof dynamics

displacement u

normalized time dt

displacem

entu

velocity u and acceleration u

normalized time dt

u,

u

reformulation of homogenious equation of motion

u + D

Mu +

K

Mu= 0 u + 2 u + 2 u= 0

undamped natural frequency , damping rate

=

rK

M =

D

2 M

damped natural frequency d, period T

d = p1 2 T =2

dexponential ansatz and initial conditions

u(t) = et

u0 cos dt +

u0+ u0

dsin dt

alternativeu(t) = C e

t

| {z }boundscos[d t + ]

| {z }oscillationwithC=

vuutu20 + [ u0 +u0]2

2d

= arctanu0 +u0

d u0

elastic pendulum - general analytical solution 24
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d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis 2.4 single dof dynamics

displacement u

normalized time dt

displacem

entu

velocity u and acceleration u

normalized time dt

u,

u

displacement

u(t) = et u0 cos d t

+u0+ u0

dsin d t

velocity

u(t) = et u0 cos d t

u0+ 2

u0

dsin d t

acceleration

u(t) = et 2 [221] u0+ 3 u0

dsin d t

[2 u0+ 2

u0] cos d t

elastic pendulum - general analytical solution 25
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d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis 2.5 modal symbiosis

given: solutions in modal space

ui, ui, ui u, u, u

modal synthesis of coupled solutions

u = u u = u, u = u

result: analytic solution of coupled system of equations

modal reduction

modal analysis for first few eigenvalues modal synthesis

characteristic main dynamics of system

modal sythesis 30
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d.kuhl, institute of mechanics and dynamics, university of kassel 2 eigenvalue analysis 2.6 summary

semidiscrete initial value problem

M u + D u + K u = r(t), u(t0) = u0, u(t0) = u0, u(t0) = u0

coupled system ofNEQsecond order differential equations system matrices M, D, K and load vector r

initial conditions u0, u0, u0

eigenvalue analysis, generalized eigenvalue problem

hK

2M

i = 0

transformation into a standard eigenvalue problemhK

2I

i = 0, K= L

1K L

T, = L

T

calculation of eigenvalues 2i fori [1, NEQ] using standard eigenvalue problem

calculation of eigenvectors of standard eigenvalue problem

hK 2i Ii i = 0 transformation eigenvectors of standard eigenvalue problem into eigenvectors of generalized

eigenvalue problem

i = LT

i

analytical solution of coupled dynamical systems 39
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modal decomposition of equation of motion

generation of modal matrix containing all eigenvectors i

=h1 2 NEQ

i transformation of equation of motion into a diagonal (decoupled) system

M u + D u + K u = r

transformation of system matrices

Mdiag= T

M , Ddiag = T

D , Kdiag = T

K , r = Tr

modal decomposed equation of motion, set ofNEQscalar valued second order differential

equations

Mdiag i

ui+ D

diagi

ui+ K

diag iu

i= r

i

transformation of initial conditions, components

u = 1

u, u = 1

u, u = 1

u, ui0, ui0, ui0


d k hl i i f h i d d i i i f k l 2 i l l i 2 6
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analytical solution

solution of individual scalar valued differential equations for modal displacements

i = sKdiag i

Mdiag i

, i = Ddiag i

2 i

Mdiagi

, di = i q1 2i

displacements for general initial conditions ui0 and ui0

ui(t) = eiit

ui0cos dit +

ui0+ iiui0

disin dit

velocities and accelerations

ui(t)=e

iit" ui0 cos dit iiui0+

2iui0

di sin dit#ui(t)=e

iit

"

2i [2

2i 1] ui0+ i

3iui0

disin di t[2ii ui0+

2iui0]cos dit

#

modal synthesis of the modal solution to the physical solution

collect in modal vectors of displacements, velocities and accelerations

u, u, u transformation of modal displacements into physical displacements

u = u, u = u, u = u

final analytical solution of original semidiscrete initial value problem

ui, ui, ui

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02b_ modal decomposition and superposition

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