Lesson 3
Definitions
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Your textbook provides you with a short section on some of the fundamental definitions in
Kinematics. Review this section and familiarize yourself with the following definitions:
Rigid bodies
Mechanisms
Kinematic analysis
Kinematic synthesis
Kinematic chains
Kinematic joints
A summary of some of these topics is provided in the following sections.
Reference Frames
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In a multibody system, we need to define:
One non-moving (also called global, absolute, or inertial) frame: x-y (in 2D); x-y-z (in 3D)
One body-fixed frame (also called local) per body: (in 2D); (in 3D);
A body-fixed frame carries the corresponding body index (subscript).
z
Array (Vector) of Coordinates
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Array of coordinates for a particle i (point) is defined as:
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2D:
3D:
z
x i
r = y i z i ri
i
x ri = i y i
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ri
x i , yi and z i are the Cartesian coordinates of the particle
For n particles: i = 1, ..., n
r1 r= r n •
Array of coordinates for a rigid body can be defined as:
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2D:
3D:
xi qi = yi i
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xi y i
zi qi = i i i i
i
z
Oi
i
Oi
i
i
i
r
ri
i
xi , yi and zi are the Cartesian coordinates of the origin of the body-fixed frame
Rotational coordinates in 3D, such as i , i and i will be discussed later in this course
Array of coordinates for a multibody system can be defined in different ways. For example
for this planar four-bar we can define
B
1 A
q = 2 3
3
y
2
Or, we may define
1
q1 q = q 2 x
q 3
where qi ; i = 1, 2, 3 are either 3-vectors (2D) or 6-vectors (3D)
Degrees of Freedom
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An important characteristics of any mechanical system is its number of degrees-of-freedom
(DoF)
In most planar systems we can determine the number of DoF intuitively, however, in spatial
systems we may have a hard time to do the same. In most cases, we could apply an analytical
formula to find the number of DoF!
A triple pendulum has 3 DoF
A four-bar mechanism has 1 DoF
Constraints
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The dependency between the coordinates of a mechanism is called a constraint. For the 4-bar
mechanism if the array of coordinates is defined as
3
1 2
q = 2 2
3
3
Then we can write the following position constraints
1
1 cos 1 + 2 cos 2 + 3 cos 3 a = 0
1
b
1 sin 1 + 2 sin 2 + 3 sin 3 b = 0
a
Note:
No. of DoF = No. of coordinates – No. of independent constraints
Kinematic Constraints
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Assume a multibody system having k degrees-of-freedom:
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An array of n coordinates is defined as q
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The Jacobian matrix is defined as an m x n matrix
D
q –
Velocity constraints are expressed as m equations
D q = 0
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Acceleration constraints are expressed as m equations
q = 0
D q
+ D
Position constraints are expressed as m equations, where k = n - m
(q) = 0
Driver Constraint
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For kinematic analysis of a multibody system with k degrees-of-freedom we must define k
driver constraints. Most mechanisms have only one DoF!
Driver constraints at position, velocity, and acceleration levels are of the form
(d )
( q) f (t) = 0
(d ) (d ) D q f (t) = 0
(d )
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(d )
q + (d ) D
Dq
f (t) = 0
Driver constraints normally have very simple forms
A driver constrain is normally a function of a single coordinate
Example: Slider-Crank Mechanism
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A 2D example; Driver constraint and the derivatives
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Case 1: crank (i) is the driver
(d,1)
i 0i 0i t 12 t 2 = 0
(d,1) i 0i t = 0
i = 0
(d,1)
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Case 2: slider (j) is the driver
(d,1)
x j a cos t = 0
x j + a sin t = 0
(d,1) x j + a 2 cos t = 0
(d,1)
Kinematic Analysis
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At t = i t solve the following equations:
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Position analysis: Solve the following m + k nonlinear algebraic equations for q
(q) = 0
(q) f (t) = 0
(d )
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Velocity analysis: Solve the following m + k linear algebraic equations for q
D q = 0
(d )
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D q f (t) = 0
Acceleration analysis: Solve the following m + k linear algebraic equations for q
q = 0
D q
+ D
(d )
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(d )
(d )
q + (d ) D
Dq
f (t) = 0
The time variable t is incremented from an initial time to a final time at reasonable
increments (the size of the increments t may vary from one problem to another). The
position, velocity, and acceleration constraints are solved at every time-step t.