Dynamics of Multi -body Systems [608254]

Dynamics of Multi -body Systems

Prof.dr.ing . Csaba Antonya
[anonimizat]

Lecture2
Multi-Body Dynamics DATR
Lecture 2 / Kinematic analysis

2
The rigid body
Generalized coordinates
Coordinate transformation
Structure of MBS
DOF
Position Analysis
Velocity Analysis
Acceleration Analysis
 …. Newton -Raphson (next lecture)

Lecture2
Multi-Body Dynamics DATR
Simulation of a mechanical system
3 Physical reality
Structural
and
geometrical
modeling Mathematical Model
Numerical model
Computer implementation
(programming)
Simulation software Post-processing, graphical and animation

Lecture2
Multi-Body Dynamics DATR
Kinematic Analysis Stages
Position Analysis Stage
Challenging

Velocity Analysis Stage
Simple

Acceleration Analysis Stage
OK
ONE step is critical:
Write down the constraint equations associated with
the joints present in the system

Lecture2
Multi-Body Dynamics DATR
Kinematic analysis
Kinematics is the study of motion, i.e., the
study of displacement, velocity, and acceleration, regardless (independent) of the forces that produce the motion.
Typically, the time history of one element in the system is prescribed
We are interested in how the rest of the element in the system move
Requires the solution linear and nonlinear systems of equations
The role of kinematics is to ensure the functionality of the multi-body system
5

Lecture2
Multi-Body Dynamics DATR
Why is Kinematics Important?
It can be an end in itself…
Kinematic Analysis – Interested how
components of a certain MBS move when
motion[s] are applied
Kinematic Synthesis – Interested in finding how
to design a MBS to perform a certain operation in a certain way

It is also an essential ingredient when
formulating the Kinetic problem

People are more interested in the Dynamic Analysis rather than in the Kinematic Analysis

Lecture2
Multi-Body Dynamics DATR
Nomenclature
Rigid body:
In the MB model the technical system is
transformed into a set of simplified
entities – rigid bodies
The MB model should include at least the
effects under consideration ( ie. position,
velocities, accelerations, forces), but not
more
The MB model should be
as complex as necessary,
but as simple as possible.
7

Lecture2
Multi-Body Dynamics DATR
Nomenclature
Body -fixed Reference Frame (also called Local
Reference Frame, LRF)

Cartesian generalized coordinates of body i

Generalized coordinates of MB system:

Lecture2
Multi-Body Dynamics DATR
Step back … structural analysis of mechanical systems
Structural analysis is the study of the
nature of connection among the members of a MBS and its mobility.
It is concerned primarily with the fundamental relationships among the degrees of freedom, the number of links, the number of joints, and the type of joints used in a mechanism.
It should be noted that structural analysis only deals with the general functional characteristics of a MBS and not with the physical dimensions of the links.
9

Lecture2
Multi-Body Dynamics DATR
Structural analysis of MBS
BODIES (or LINKS)
The individual rigid bodies
making up a machine or mechanism are called members,
bodies or links .
A component forming a part of a chain; generally rigid with provision at each end for connection to two other links
Bodies are rigid : the distance
between any two points fixed to them remains constant

10

Lecture2
Multi-Body Dynamics DATR
Representation of a rigid body
Aim of the kinematic and dynamic
simulation is the computation of position (velocity and acceleration) as a function of time.

The position of an element (rigid body) can is given by a position vector and an orientation vector.
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Lecture2
Multi-Body Dynamics DATR
Rigid Body
Made of a rigid solid material that
constitutes the material out of which mechanism or machine links are imagined to be made.
They don't expand, contract or distort in any other way that affects real material.
This idea allows us to study pure motion, without interference of reality in the form of elasticity, thermal expansion, viscosity or other phenomena that detract from an ideal situation.
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Lecture2
Multi-Body Dynamics DATR
Rigid Body (cont.)
The distance between any two points, so
as to move with it, fixed to the body remains constant, i.e., does not change.
Reflections, such as the turning over of the planar triangle, are not allowed –
regardless of distances between points being preserved.
13

Lecture2
Multi-Body Dynamics DATR
Position of a body in a plain

14 xi
OiOi
xy
yi
ϕ
i
[]T
i O O ii iyxq ϕ,,=

Lecture2
Multi-Body Dynamics DATR
Description of position
Vector
15 OP

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Multi-Body Dynamics DATR
Reference frames
Two type of reference frames
Fix
Mobile

16 xy
O

Lecture2
Multi-Body Dynamics DATR
Description of position in a reference frame
A vector a can then be resolved into
components and , along the axes x
and y :

Cartesian components of the vector
17 xy
O
aa x
a y
in the own reference frame
xyaa   a ij 

Lecture2
Multi-Body Dynamics DATR
Properties

18

Lecture2
Multi-Body Dynamics DATR
Representation
19 x
xy
yaaaa  aij a
10
01     j i 


Lecture2
Multi-Body Dynamics DATR
Rotation

20 OOiP
xi
xy
yi
ϕ
i
ii
P ii
P P y x x ϕ ϕ sin cos− =
ii
P ii
P P y x y ϕ ϕ cos sin+ =



⋅=


⋅
−=

i
Pi
P
i i
Pi
P
i ii i
PP
yxA
yx
yx
ϕϕϕϕ
cos sinsin cos

Lecture2
Multi-Body Dynamics DATR
Transfer matrix
21
Expressing a given vector in one
reference frame (local) in a different reference frame (global)
Also called a change of base.

Lecture2
Multi-Body Dynamics DATR
Translation and rotation

22

Lecture2
Multi-Body Dynamics DATR
point P in two different reference frames:
a local reference frame (LRF) and a global reference frame (GRF)

Local reference frame is typically fixed (rigidly
attached) to a body that is moving in space

Global reference frame is the “world” reference frame: it’s not moving, and serve as the universal reference frame

23

Lecture2
Multi-Body Dynamics DATR
O’
XY
O'Psx’
y’
φ rP
rP
24
Coordinate transformation

Lecture2
Multi-Body Dynamics DATR
Velocity of the point

25

Lecture2
Multi-Body Dynamics DATR
Acceleration of the point
26

Lecture2
Multi-Body Dynamics DATR
Generalized Coordinates
Generalized coordinates:
A set of quantities (variables) that allow you to uniquely
determine the state of the mechanical system
You need to know the location of each body
You need to know the orientation of each body

The quantities (variables) are bound to change in time since our system moves
In other words, the generalized coordinates are functions of time

The rate at each the generalized coordinates change is capture by the set of generalized velocities
Most often, obtained as the straight time derivative of the generalized
coordinates

There are multiple ways of choose the set of
generalized coordinates that describe the state of a
system

Lecture2
Multi-Body Dynamics DATR
Relative coordinates

28 BO
2
5
%2
5
%2
5
%2
5
%Y
X
m1gθ1
L
2
5
%2
5
%25
%2
5
%L
m2g2L
O’2E
θ12O’1

Lecture2
Multi-Body Dynamics DATR
Absolute coordinated

29 BO
2
5
%2
5
%2
5
%2
5
%y
x
y1’
x1’
m1gθ1
L
2
5
%2
5
%2
5
%25
%y2’
x2’θ2L
m2g2L
O’2O’1
E

Lecture2
Multi-Body Dynamics DATR
Relative vs. Absolute Generalized Coordinates
Relative coordinates:
Angle θ1 uniquely specified both position and
orientation of body 1
Angle θ12 uniquely specified the position and
orientation of body 2 with respect to body 1
few GCs
Absolute (and Cartesian) generalized coordinates:

x1, y1, θ1 position and orient body 1 wrt GRF (global
RF)
x2, y2, θ2 position and orient body 2 wrt GRF (global
RF)
many generalized coordinates
3 for each body in the system (six for this example)
easy to express locations but many GCs

30

Lecture2
Multi-Body Dynamics DATR
Degrees of Freedom (DOF) – free element
Degrees of Freedom – number of independent
coordinates required to completely specify
the position of the body
An object in space has six degrees of freedom .
Translation –
position along X, Y,
and Z axis (three
degrees of freedom)
 Rotation –
rotation angle about
X, Y, and Z axis
(three degrees of freedom)
31

Lecture2
Multi-Body Dynamics DATR
Degrees of Freedom (DOF) in PLANE
Three
independent
coordinates needed to specify the location of the link AB, x
A, yA, and
angle θ
An unconstrained body in the plane
has three degrees of freedom
MBS with L bodies has 3L degrees of
freedom
32

Lecture2
Multi-Body Dynamics DATR
Kinematic pairs (joints)
The bodies are connected in order to transmit
motion from the driver (input link) to the
follower (output link)
The connections (joints between the bodies)
are called kinematic pairs (or joints)
The joint consists of a pair of mating surfaces – one for each of the joined links, formed by
the contact between two bodies and allows relative motion between them.
Guarantees the contact between two
members and constrains their relative motion
Combination of two links kept in permanent
contact permitting particular kind(s) of relative motion(s) between them

33

Lecture2
Multi-Body Dynamics DATR
Degrees of Freedom (DOF) – joints: f
34
Kinematic constraints are constraints
between bodies that result in the decrease
of the degrees of freedom
The number of independent parameters that is required to determine the relative position of one link (rigid body) with respect to the other link (body) connected by the kinematic pair
f = 1 … 5

Lecture2
Multi-Body Dynamics DATR
Joints
DOF=1
35
DOF=1
DOF=1
DOF=2

Lecture2
Multi-Body Dynamics DATR
Joints
36
DOF=2
DOF=3
DOF=3

Lecture2
Multi-Body Dynamics DATR
Joints
37
DOF=4
DOF=5
DOF=4

Lecture2
Multi-Body Dynamics DATR
The Six Lower Pairs Joints
Name (Symbol) DOF Contains
Revolute (R)
1 (Δθ) R
Translating (Prismatic) (P)
1(Δx) P
Helical(H)
1(Δθ ) RP
Cylindric (C)
2(Δθ ), (Δx) RP
Spherical (S)
3 (Δθ), (Δφ ), (Δψ ) RRR
Planar (F)
3(Δθ ), (Δx), (Δ y)
RPP

Lecture2
Multi-Body Dynamics DATR
Remember ….

39 BODY
KINEMATIC PAIR / JOINT
KINEMATIC CHAIN
MBS
MACHINE

Lecture2
Multi-Body Dynamics DATR
Degrees of Freedom (DOF)
Number of coordinate values required to
completely describe the position of all
bodies in a MBS
Total DOF ≡Mobility:
Number of inputs required to determine the
position of all links of a mechanism
Pairing elements (e.g. joints) in a chain remove DOF (i.e. reduce mobility) by
constraining the position of two or more links at once

40

Lecture2
Multi-Body Dynamics DATR
Mobility analysis
Consider a single body in the plane

41

Lecture2
Multi-Body Dynamics DATR
Mobility analysis
Adding another free body adds another 3
DOF

42

Lecture2
Multi-Body Dynamics DATR
Mobility analysis
But joining the two bodies with a revolute
joint reduces the total DOF by 2:

43

Lecture2
Multi-Body Dynamics DATR
Mobility analysis
Consider DOF contributions in a planar
chain of n -bodies:
DOF of free links -> 3n
Fixed base link – > -3(base link’s DOF are
removed)
Each 1 DOF joint -> -2 (cf. revolute joint
example)
Each 2 DOF joint -> -1
Let the number of 1 DOF joints = f1
Let the number of 2 DOF joints = f2
44

Lecture2
Multi-Body Dynamics DATR
Kutzbach’s (modified Groubler) equation
DOF = degree of freedom or mobility
L = number of links, including ground link
J1 = number of 1 DOF joints (full joints)
J2 = number of 2 DOF joints (half joints)
DOF ≤ 0 structure
mechanism DOF > 0 DOF = 3( L – 1) – 2J1 – J2
(Known also as: Chebychev -Grubler -Kutzbach Relation <CGK>)
Degrees of Freedom (DOF)

45

Lecture2
Multi-Body Dynamics DATR
Implications of mobility DOF
DOF = 0 motion is impossible and the MBS
forms a structure
DOF = 1 The MBS can be driven by a
single input motion
DOF = 2 MBS requires two separate input
motions to produce constrained
(definite) motion
DOF = 3 etc.
DOF < 0 MBS has redundant constraints
– it is over-constrained (preloaded
structure) and is called a statically
indeterminate structure (the forces in
every link cannot be determined)
11 2
A B 11 2 3
A B
C