computational methods in transient dynamics
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8/7/2019 Computational Methods in Transient Dynamics
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MECALOG Computational Methods in Transient Dynamic
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Computational Methods in
Transient Dynamics
H. Shakourzadeh
Training Manager
September 2003
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MECALOG Computational Methods in Transient Dynamic
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Overview
Mechanics
Theoretical
Applied
Computational
Solids & Structures
Fluids
Multiphysics
Nano & Micromechanics
Statics
Dynamics
Linear
Nonlinear
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Dynamic Terminology
Periodic motion :is a steady-state oscillatory motion of sustained amplitude and constant period
For linear systems :
• The principal of superposition holds,
• The mathematical techniques are available and well-developed.
For nonlinear systems :
•All systems tend to become nonlinear with increasing amplitude of oscillation.
• Techniques are less well known and difficult to apply
Free vibration :takes place when a system oscillates under the action of forces inherent in the systemitself.
Forced vibration :takes place under the excitation of external forces
Harmonic excitation:Common sources of harmonic excitation are unbalance in rotating machines.
Transient vibration /dynamic :A dynamical system is excited by a suddenly applied nonperiodic excitation F(t).
A transient motion is a nonsteady-state oscillatory motion of diminishing amplitude.
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Free Vibration of a Spring-mass System
Equation of motion for a spring without damping :
Equilibrium of forces leads to :
General solution :
Evaluating the constants from initial conditions and
Special case with and :
m
0 kxxm
tBtAx nn cossin m
kn
k
with
0x 0x
tV
x nn
sin0
00 x 00 V x
txt
xx nn
n
cos0sin0
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Damping effect on the Oscillation
Free vibration of a mass-spring system
Free vibration without damping (C = 0.)
Free vibration with critical damping ( C = Ccr )
Free vibration with underdamping ( C = 0.2 Ccr)
Free vibration with overdamping ( C = 5Ccr)
Time
Displacement
m
K
C
KmC cr 2
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Impulse ExcitationImpulse is the time integral of the force :
The impulse of a very large magnitude force acting for a very short time is
finite.
Unit impulse when Fimp = 1 :
The response to the unit impulse is of
importance to the transient problemsand designed by h(t).
dttFFimp )(
ttF )( is delta function
0 t tall for
t tfor 10
dttFu
imp
t
F imp
F
t
tt
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Impulse Excitation
mF mdvFdt t
t
t
t
mdvFdt V mFimp
txt
xx nn
n
cos0sin0
x(0) = 0
m
Fx
imp)0(
tm
Ftx n
n
imp
sin)(
m
kn with
1impF tm
th nn
sin1
)(
Newton law :
The impulse Fimp acting on the mass will result in a sudden change in its velocity.
Using the general equation of a mass-spring system :
Response for a unit impulse :
Having the response h(t) to a unit impulse excitation, it is possible to establish
the response of the system excited by an arbitrary force f(t).
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Arbitrary ExcitationAn arbitrary force is considered
as series of impulses.
Each impulse contributes to the
response at time t =
where h is the unit impulse function
For a linear system the principle of superposition holds :
Another form of the last equation :
t
t
tF
t t
ttFF imp
)( thttFx
dtthtFx
0 )(
d dtthentLetting )(
d hFx
0
)(
Arbitrary impulse equation
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Impulse ExampleResponse of a single degree of freedom
system to the following step excitation
Damping is neglected
Solution :
Response for a unit impulse :
Substituting into the arbitrary impulse equation
F0
0 t
F(t)
tm
th nn
sin1)(
d tm
Ftx n
t
n
)(sin0
0
)cos1(0 tm
Ftx n
n
t
kx/F0
2
0
1
The pick response to the step excitation of
magnitude F0 is equal to twice the statical deflection
m
k
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Example : Dropping Body
Physical problem :
How far a body can be dropped without incurring damage?
Applications :
Landing of airplanes
Transportation of packaged articles
Analytical solution :Consider a spring-mass system dropped through a height h :
The differential equation of motion as long as the spring remains in contact with the floor :
General solution of the differential equation of motion :
Initial conditions to satisfied :
m
k
h
x
ghx 2
mgkxxm
C tCosBtSinAx
At t=0
gx
0x
ghx 2
tCosg
tSingh
x
12
2
m
kavec
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Maximum displacement and acceleration :
Static solution :
Maximum dynamic acceleration :
Remark :
The deceleration is always greater than g and grows nonlinearly with respect
to the height h.
Example : Dropping Body
mgK st 2
gst
0xWhen 12
st
hgx
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Numerical Integration in Time
When the differential equation cannot be integrated in closed form :
Dynamic force cannot be expressed by simple analytic functions
The system is nonlinear
….
Finite difference method :
The continuous variable t is replaced by the discrete variable t i
The differential equation is solved progressively in time increments t
The solution is approximate :
Sufficiently small time increment results good accuracy.
Central difference method :
Any differential equation of motion for a dynamic system relates acceleration,
velocity and displacement with time variable t.
Based on the Taylor expansion of xi+1 and xi-1 about the pivotal point i :
0,,, txxxf
)(2
111
iii xx
tx )2(
1112
iiii xxx
tx;
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Industrial Problems
High numbers of degrees of freedom
Highly nonlinear behavior due to :
Large displacements and deformation
Large rotations in space
Nonlinear material :
Plasticity, viscosity, damage, rupture, etc.
Dynamic nonlinear solicitation in time
Contact between several parts
Time-dependent response
Each unknown variable varying in timeand in space
Need of numericalcomputations
Computer methods
Need of discretization intime and in space
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Computational Mechanics
Computational Mechanics
Linear
Nonlinear
Finite Element Method
Finite Difference Method
Boundary Element Method
Finite Volume Method
Spectral Method
Mesh-Free Method
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Nonlinear Softwares
Implicit Softwares
SAP by Ed Wilson
(Berkeley 1960)
NONSAP
(Nonlinear)ADINA
Brown UniversityMARC
(1969)
ABAQUS
(1972)
Westinghouse(Nuclear)
ANSYS(1969)
Hydro-codes HEMP
Wilkins(1964)
Costantino
(1967)
SAMSON (1969)
US Air Force
WRECKER
(1972)WHAMS
(1975)
SADCAT(1975)
HONDO
(1975) PRONTO
DYNA(1976)
PAMCRASH
(1980)
LS-DYNA(1989)
RADIOSS
(1987)
Explicit Softwares
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Application Fields
Structural Mechanics
Fluid-Structure interaction
Material characterization
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A Physical Problem
Crushing of thin-walled members
Impact of a tubular thin-walled beam on a rigid wall
Thin-walled beam
Initial velocity
Rigid wall
M
V0
M’ , V0
Contributory mass
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Crushing of a tubular beam
Initial velocity field
High pressure region
V0
Wave front
Wave front traveling
backward at material
wave speed
High pressure region
V0 Crimpling in the
high pressure region
V1
Variable
contact
Plastic
hinge
Global buckling phase
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Modeling of a Physical Problem
1. Geometry (Physical model)
1D, 2D or 3D ? Beam, Shell or Solid ?
2. Physical laws (conservation)
Physical laws (conservation)
Mass conservation
Energy conservation
Momentum conservation (equilibrium)
3. Space Discretisation:
Finite Difference (FD)
Finite Volume (FV)
Finite Element (FE)
Contact : use of specific elements (spring)
4. Time Discretisation:
Newmark scheme
5. Formulation:Choice of time and space discretisations
Explicit formulation
Implicit formulation
simplifications
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FormulationsHow to combine time and space discretisations?
1. Lagrangian Formulation
– The mesh points coincide with the material points
– Elements are deformed with material
– Element deformation = Material deformation
Higher the mesh deformation, lower the quality of results2. Eulerian Formulation
– Nodes fixed in space, Material goes through the mesh
– Fixed nodes No degradation of mesh in large deformation problems
– Problems in solid mechanics:
Boundary conditions application Constitutive laws
3. Arbitrary Lagrangian Eulerian Formulation (ALE)– Between two previous formulations
– Internal nodes move to minimize element distortion
– Boundary nodes remain on the boundary of domain
More complicated formulation
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How to describe the kinematic of geometrically nonlinear structures?
What tracking process ? With respect to which reference state?
Total Lagrangian Description : The FEM equations are formulated with respect
to a fixed reference (initial) configuration.
Updated Lagrangian Description : The reference configuration is the last
accepted solution (not fixed) => Used in implicit codes
Corotational Description : The motion is splitted into two parts :
Rigid body motion
Deformational motion
Lagrangian Descriptions
x0
z0
Initial undeformed configuration C0 Current configuration Cn Following configuration Cn+1
TL and UL descriptions
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Lagrangian Formulations
Corotaional formulation :
CR configuration is obtained as a rigid body motion of the base configuration
A CR reference for each element
The coordinate system is Cartesian
The element deformations are measured with respect to the corotated configuration.
x0
z0
Initial undeformed configuration C0
Corotational configuration (CR)
Current
Corotational formulation
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Finite Element Formulation
Equilibrium equation for a continuum : Differential equation
with boundary conditions for applied forces and velocities.
Variational form by Galerkin method : Integral form
Discrete form by Finite Element Method :
t
vb
x
ii
j
ij
b : Body force
: Material density
v : velocity
0
d t
vb
xv i
ij
iji
: Virtual nodal velocitiesiv
0extbodyint
F F F t vM
t v
Or : bodyintext F F F t
vM
extF
intF
bodyF
: External applied load vector
: Internal force vector
: Body force vector
M : Mass matrix
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General Discrete Equations
Overall equation of motion for translational velocities with anti-hourglass
and contact forces :
Overall equation of motion for rotational velocities with anti-hourglass
moments :
conthgrbodyintext F F F F F t
vM
hgrF : Anti-hourglass resistant forces
contF : Contact forces
hgrintext M M M t
I
extM
intM
hgrM
: Externally applied moments
: Internal moments
: Anti-hourglass resistant forces
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Explicit Method
Express the equilibrium equation at time step « n »:
Only [M] has to be inverted.
[M] is diagonal with lumped mass approach
Easy to invert [M] (not expensive)
Every D.O.F. is treated separately
)( next nn t F X K X M
)()( int1
nnext n t F t F M X
dvBX K t F nT nnn )(intwith
m4
1
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Example 1 : Time integration
System mass-spring without damping
1 d.o.f.
Equilibrium without damping:
k is a function of x (nonlinear spring)
Determine :
Explicit scheme with Central Difference Method
f ext(t)
m
kx
)(t f kxxm ext
?)(
?)(
?)(
t x
t x
t x
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Example 1 : Solution
Explicit time discretisation:
Express the equilibrium at moment tn
ttn-1 tn
tn+1
nx1nx
x2
1n
2
1nx
1nx
knownunknown
)( next nn t f kxxm
m
kxt f x nnext n
)( m
t f t f x nnext n
)()( int
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Example 1 : (continued)
Centeral Differences:
Constant derivative
ttn-1 tn tn+1
nx1nx
x2
1n
2
1nx
1nx
nx
t xxx nnn
21
21
t xxxnnn 211
xn+1 is obtained with a precision 2t
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Example 2 :Shocked bar
Propagation of a shockwave in a bar
Small strain assumption
(elastic behavior)
x dx
Discretized Model :f ext(t)
Ni-1 Ni Ni+1
xx
: Equilibrium
E : Hypo-Elastic Material
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Example 2 : Solution
1. Loop over nodes : ( Central Difference Calculation)
2. Loop over elements : ( Stress Calculation)
With one integration point on element
Hypo-elasticity
m
t f t f x nnext n
)()( int t xxx nnn
21
21
t xxxnnn 211
x
v
11
12
1221
21
21
nn
nn
nxx
xx
N1 N2
I.P.
E 21
21
nnE with
t E nnn 211
t
nnn
1
21
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Example 2 : Continued
3. Nodal Forces :
4. Resultant Forces at nodes :
5. Loop over nodes
111
nnAf
112
nnAf
f 1 f 2
N1 N2
j
njni t f t f )()( 11
Ni
f j f j+1
m
t f t f x nnext n
)()( 1int11
Go to the first step
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Remarks
1. Explicit method Element by Element Approach
No need of global resolution
2. Direct calculation of , and for linear or nonlinear stiffness
3. The expression of x is quadratic in terms of
xxx
m
t f t f x nnext n
)()( int
t xxx nnn
21
21
t xxx nnn 211
kxt f )(intwith
linear or nonlinear
t
MECALOG C i l h d i i i
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MECALOG Computational Methods in Transient Dynamic
Explicit Flow Chart
Time integration
t t t
extF •Loop over elements
i
j
j
iij
x
v
x
v
2
1
)( ijij f
t t t t ijijij
•Assemble hrgF F ,int cont F
iii mF v
intF
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