Flexible Multibody Dynamics

cuckootrainMechanics

Oct 31, 2013 (3 years and 7 months ago)

75 views

Vermelding onderdeel organisatie

1

SPAÇAR: A Finite Element Approach in

Flexible Multibody Dynamics

UIC Seminar

September 27, 2004

University of Illinois at Chicago


Laboratory for Engineering Mechanics

Faculty of Mechanical Engineering

Arend L. Schwab


Google: Arend Schwab [I’m Feeling Lucky]

September 27, 2004

2

Acknowledgement

TUdelft:

Hans Besseling

Klaas Van der Werff

Helmut Rankers

Ton Klein Breteler

Jaap Meijaard

… MSc students



UTwente:

Ben Jonker

Ronald Aarts

… MSc students





September 27, 2004

3

Contents


Roots


Modelling


Some Finite Elements


Eqn’s of Motion


Examples


Discussion

September 27, 2004

4

Engineering Mechanics at Delft

From Analytical Mechanics in 50’s:

Warner T. Koiter


On the Stability of Elastic Equilibrium
, 1945

To Numercial Methods in Applied Mechanics in 70’s:

Hans Besseling


The complete analogy between the matrix equations and the continuous
field equations of structural analysis
, 1963

September 27, 2004

5

Mechanism and Machine Theory

Application of Numerical
Methods to:



Kinematic Analysis


Type and Dimension
Synthesis


Dynamic Analysis



CADOM

project:
C
omputer
A
ided
D
esign of
M
echanisms, 1972

Rankers, Van der Werff, Klein Breteler, Schwab,
et al.


September 27, 2004

6

Mechanism and Machine Theory,
Kinematics

Denavit & Hartenberg, 1955



Rigid Bodies


Relative Coordinates
(few)


Kinematic Constraints
(few)



September 27, 2004

7

Mechanism and Machine Theory,
Kinematics

Klaas Van der Werff, 1975


Finite Element Approach



Flexible Bodies


Absolute Coordinates and
Large Rotations
(many)


Kinematic Constraints =
Rigidity of Bodies
(many)



Note: Decoupling of the positional nodes and the orientational nodes.


September 27, 2004

8

Multibody System Dynamics

Finite Element Approach

Key Idea:

Specification of Independent Deformation Modes of the Finite Elements


Coordinates:

(
x
p
,

p
,
x
q
,

q
) total 6


Deformation Modes:

total 6
-
3=3


September 27, 2004

9

Multibody System Dynamics

Finite Element Approach

Pro’s:


Easy FEM assembly of the system equations


Easy mix of partly Rigid and/or Flexible elements


Small set of elements for Large class of Multibody Systems


Absolute Coordinates and Large Rotations


Gen. Deformation can act as Relative Coordinates

Con’s:


Many coordinates, many constraints


Non
-
Constant Mass Matrix

September 27, 2004

10

Multibody System Dynamics

Finite Element Approach

Generalized Deformation can act as Relative
Coordinates

Ex. Hydraulic Cylinder

September 27, 2004

11

Multibody System Dynamics

Compare to Rigid Bodies with Constraints

Milton Chace & Nicky Orlandea, DRAM, ADAMS, 1970

Constraints are at the Joints

Constraints are in the Bodies

FEM approach

Rigid Bodies with Constraints

September 27, 2004

12

3D Beam Element

Coordinates:

(
x
p
,

p
,
x
q
,

q
) total 14


Deformation Modes:

total 14


6 = 8




Cartesian Coordinates
x
p

= (x, y, z)
p


and Euler Parameters

p

=(

0
,

1
,

2
,

3
)
p



= 0

= 0




2 = 6

September 27, 2004

13

3D Hinge Element

Coordinates:

(

p
,

q
) total 8


Deformation Modes:

total 8


3 = 5




= 0

= 0




2 = 3

September 27, 2004

14

3D Truss Element

Coordinates:

(
x
p
,
x
q
) total 6


Deformation Modes:

total 6


5 = 1




September 27, 2004

15

3D Wheel Element

Coordinates:

(
x
w
,

p,

x
c
) total 10


Some Counting:

Pure rolling Rigid body has

3 degrees of freedom (velocities).


We need 10
-
3=7 Constraints on the
Velocities.


Pure rolling is 2 Velocity Constraints,
Lateral and Longitudinal.


Leaves 7
-
2=5 Deformation Modes



September 27, 2004

16

3D Wheel Element

Coordinates:

(
x
w
,

p,

x
c
) total 10


Deformation Modes:

Generalized Slips:

September 27, 2004

17

Ex. Universal or Cardan Joint

Physical Model

Two FEModels: (a) with 4 Rigid Hinges, and (b) with 2 Flexible Hinges

September 27, 2004

18

Ex. Universal or Cardan Joint

Two FEModels: (a) with 4 Rigid Hinges, and (b) with 2 Flexible Hinges

September 27, 2004

19

Dynamic Analysis

In the spirit of
d’Alembert

and
Lagrange

we transform the DAE

in terms of generalized independent coordinates q
j

with x
i
=F
i
(q
j
) resulting in

From which we solve

and Numerically Integrate as an ODE.


Note: the Elastic Forces are according to



and


September 27, 2004

20

Ex. ILTIS Road Vehicle Benchmark

The ILTIS Vehicle

Suspension

FEM Model

85 Elements

239 Gen. Def.

70 Nodes

226 Gen. Coord.

10 DOF’s

-

Rigid Cabin

-

4 Independently
Suspended Wheels

-

CALSAP Tire Model

September 27, 2004

21

Ex. ILTIS Road Vehicle Benchmark

Static Equilibrium Results

September 27, 2004

22

Ex. ILTIS Road Vehicle Benchmark

Handling Performance Test:

Ramp
-
to
-
Step Steer Manoeuvre at v = 30 m/s.

CALSPAN tire model



Zero Lateral Slip

September 27, 2004

23

Ex. Slider
-
Crank Mechanism

Slider
-
Crank Mechanism from Song & Haug, 1980

Rigid Crank,Flexible Connecting Rod


=150 rad/s, 2% damping


Transient Solution


Periodic Solution


First Eigenfrequency of pinned joint
connecting rod

0
= 832 rad/s


September 27, 2004

24

Linearized Equations of Motion

Equations of Motion can be Analytically Linearized at a Nominal Motion

Even for Systems having Non
-
Holonomic Constraints!






with:

M: reduced Mass Matrix


C: Tangent Velocity dependant Matrix


K: Tangent Stiffness Matrix





q
k
: Kinematic Coordinates variations


A: Non
-
Holonomic Constraints


B: Tangent Reonomic Constraints Matrix






September 27, 2004

25

Ex. Slider
-
Crank Mechanism

Nominal Periodic Motion and small Vibrations described by the
Linearized Equations of Motion

Slider
-
Crank Mechanism from Song & Haug, 1980

Rigid Crank,Flexible Connecting Rod


=150 rad/s, 2% damping


Transient Solution


Periodic Solution



September 27, 2004

26

Linearized Equations of Motion at
Nominal Periodic Motion

Periodic Solutions for small Vibrations superimposed on a
Nominal Periodic Motion

Linearized Equations of Motion at Nominal Motion:

The Coefficients in the Matrices are Periodic with Period T=2

/


Transform these Matrices into Fourier Series:

and assume a periodic solution of the form:

September 27, 2004

27

Linearized Equations of Motion at
Nominal Periodic Motion

Periodic Solutions for small Vibrations superimposed on a
Nominal Periodic Motion

Substitution into the Linearized Equations of motion and balance of every
individual Harmonic leads to:

These are (2k+1)*dof linear equations from which we can solve the

2k+1 harmonics:

Which form the solution of the small Vibration problem:

September 27, 2004

28

Ex. Slider
-
Crank Mechanism

Slider
-
Crank Mechanism from Song & Haug, 1980

FEModel: 2 Beam Elements for

the Flexible Connecting Rod

Rigid Crank,Flexible Connecting Rod


=150 rad/s, 2% damping

Transient Solution

Periodic Solution



September 27, 2004

29

Ex. Slider
-
Crank Mechanism


Damping 1% and 2%

First Eigenfrequency of pinned joint
connecting rod

0
= 832 rad/s

Resonace at 1/5, 1/4, and 1/3 of

0



Linearized Results



Full Non
-
Linear Results

Maximal Midpoint Deflection/l for a range of

’s

September 27, 2004

30

Ex. Slider
-
Crank Mechanism


First Eigenfrequency
of pinned joint
connecting rod

0
= 832 rad/s

Resonace at 1/5, 1/4,
and 1/3 of

0


Quasi Static Solution


Individual Harmonics of the Midpoint Deflection/l for a range of

’s

September 27, 2004

31

Ex. Dynamics of an Uncontrolled Bicycle

Cornell University, Ithaca, NY, 1987:
Yellow Bike in the Car Park

September 27, 2004

32

Ex. Dynamics of an Uncontrolled Bicycle

Cornell University, Ithaca, NY, 1987:
Yellow Bike in the Car Park

September 27, 2004

33

Ex. Dynamics of an Uncontrolled Bicycle

Modelling Assumptions:

-

rigid bodies

-

fixed rigid rider

-

hands
-
free

-

symmetric about vertical plane

-

point contact, no side slip

-

flat level road

-

no friction or propulsion




Note: This model is energy conservative

September 27, 2004

34

Ex. Dynamics of an Uncontrolled Bicycle

3 Degrees of Freedom:

4 Kinematic Coordinates:

FEModel: 2 Wheels, 2 Beams, 6 Hinges

September 27, 2004

35

Ex. Dynamics of an Uncontrolled Bicycle

Forward Full Non
-
Linear Dynamic Analysis with an initial small side
-
kick

Forward Speed:

v = 3.5 m/s

v = 4.5 m/s

September 27, 2004

36

Ex. Dynamics of an Uncontrolled Bicycle

Investigate the Stability of the Steady Forward Upright Motion by
means of the Linearized Equations of Motion at this Steady Motion

Linearized Equations of Motion for Systems having Non
-
Holonomic Constraints
in State
-
Space form:

Assume an exponential motion for the small variations:

September 27, 2004

37

Ex. Dynamics of an Uncontrolled Bicycle

Rootloci of


from the Linearized Equations of Motion with as a Parameter the
Forward Speed v

Asymptotically Stable in the
Speed Range:

4.1 < v < 5.7 m/s

September 27, 2004

38

Ex. Dynamics of an Uncontrolled Bicycle

What happens for v>5.7 m/s?

Forward Speed:

v = 6.3 m/s

September 27, 2004

39

Conclusions

-
SPAÇAR is a versatile FEM based Dynamic Modeling System for Flexible
and/or Rigid Multibody Systems .

-
The System is capable of modeling idealized Rolling Contact (Non
-
Holonomic Constraints).

-
The System uses a set of minimal independent state variables, which
avoid the use of differential
-
algebraic equations.

-
The Equations of Motion can be Linearized Analytically at any given
state.