Mechanisms
Prof.dr.ing. Csaba Antonya
antonya@unitbv.ro
DATR
Lecture 4
Mobility, spatial mechanisms
Structural synthesis – in 2-3 weeks
Today:
Kinematic analysis of mechanisms
Graphical kinematics.
Four bar mechanisms.
Slider-crank mechanism
MECHANISMS
2 Lecture 4
DATR
Chapter 2. Kinematic analysis of mechanisms
Kinematics is the study of motion, i.e., the
study of displacement, velocity, and
acceleration, regardless of the forces that
produce the motion.
Typically, the time history of one (or
more) element in the system is prescribed
We are interested in how the rest of the
element in the system move
MECHANISMS
Displacement, velocity and acceleration in
the form of:
s(t), v(t), a(t)
s(s1), v(v1), a(a1)
3 Lecture 4
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Displacement Analysis
2 main methods:
Analytical
Graphical
Analytical kinematics is a systematic
process that is most suitable for developing
into a computer program. However, for very
simple systems, analytical kinematics can
be performed by hand calculation
In graphical kinematics, instead of
analytically solving sets of linear or non-
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linear equations, the unknowns are found
graphically. Direct measurements from a
graph provide the values for the unknowns.
4 Lecture 4
DATR
Graphical kinematics of mechanisms
By sketching the mechanisms in different
positions and measuring linear
displacements and angular displacements
The analysis is performed by systematically
enforcing the condition of links being rigid
bodies as well as the geometric constraints
imposed by the joints.
Graphical method is inexact, inconvenient
for repeated analysis
Many times graphical displacement analysis
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are developed for the purpose of computer
animation
5 Lecture 4
DATR
Graphical kinematics of mechanisms
Step 1: Identify the links (bodies) and joints.
Draw a schematic of the mechanism at known
scale by replacing all rigid bodies in the
mechanism with appropriate and
representative lines.
Step 2: Determine the number of DOF and
input actuators (can be performed via the
Mobility).
Step 3: Represent the input links in several
distinct positions (usually equidistant
positions) according to the input actuator’s law
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of motion and draw the entire mechanism
Step 4: Measure displacements, orientations,
coordinates of the links
6 Lecture 4
DATR
Graphical kinematics of planar mechanisms
Position analysis of planar linkage
mechanisms relies on essentially three
analytical tools.
The same tools, expressed in terms of
algebraic equations, have their
counterparts in all analytical approaches
as well.
These are computation of intersections
between
A pair of lines
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A line and a circle
Two circles
7 Lecture 4
DATR
Graphical kinematics of planar mechanisms
Any single-loop, planar mechanism will
have two unknowns (dependent
variables)
Need a method to solve for these
variables in terms of known mechanism
parameters
MECHANISMS
8 Lecture 4
DATR
Graphical kinematics of the Four Bar Linkage
Defined by the length of the links : l1 l2 l3 l4
B l3
3
C B
l C
2 2
4
l4
A D
1
A l1 D Structural analysis DOF=1
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9 Lecture 4
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Nomenclature
CRANK: Link that makes a complete
revolution and is pivoted to ground.
ROCKER: Link that has oscillatory (back
and forth) rotation and is pivoted to
ground.
COUPLER (or connecting rod): Link that
has complex motion and is not pivoted to
ground.
GROUND: defined as any link or links that
are fixed (nonmoving) with respect to the
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reference frame
10 Lecture 4
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Four bar mechanisms - applications
Gripping
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11 Lecture 4
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Four bar mechanisms - applications
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12 Lecture 4
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Four bar mechanisms - applications
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13 Lecture 4
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Four bar mechanisms - applications
MECHANISMS
14 Lecture 4
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Four bar mechanisms - applications
Bycicle
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15 Lecture 4
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Four bar mechanisms - applications
Boeing 787 Flaps mechanisms
MECHANISMS
16 Lecture 4
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Four bar mechanisms - applications
MECHANISMS
17 Lecture 4
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Grashof’s law
In a planar four bar revolute pair kinematic
chain if the sum of the lengths of the
shortest and the longest links is less than or
equal to the sum of the lengths of the other
two intermediate links at least one link will
have full rotation.
Mechanisms obtained from the kinematic
chain satisfying these conditions are known
as Grashofian Mechanisms.
Mechanisms obtained from the kinematic
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chain which are not obeying these
conditions are known as Non-Grashofian
Mechanisms.
18 Lecture 4
DATR
Grashof’s classification of Four Bar Mechanism
Double Crank Mechanism
Crank – Rocker Mechanisms
Double – Rocker Mechanism
l – lengths of the longest link
s – lengths of the shortest link
l+s<p+q l+s=p+q l+s>p+q
MECHANISMS
Shortest bar is the Short bar is the side Shortest bar is the
frame (s) link (s) coupler (s) Parallelogram, can DOUBLE –
DOUBLE CRANK – DOUBLE be flattened ROCKER
CRANK ROCKER ROCKER
19 Lecture 4
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Double Crank
MECHANISMS
20 Lecture 4
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Crank – Rocker
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21 Lecture 4
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DOUBLE ROCKER
CRANK – ROCKER
MECHANISMS
DOUBLE CRANK
l+s<p+q
22 Lecture 4
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parallelogram
l+s=p+q
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23 Lecture 4
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l+s>p+q
MECHANISMS
DOUBLE ROCKER
24 Lecture 4
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Extreme position of the output (Crank Rocker)
The input link and the coupler are extended
as a straight line
MECHANISMS
25 Lecture 4
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Extreme position of the output (Crank Rocker)
the input link and coupler are folded as a
straight line.
MECHANISMS
26 Lecture 4
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The toggle positions are determined by
the co-linearity of two of the moving links
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27 Lecture 4
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Graphical kinematics, the input link
Input link in revolute motion
The position (orientation) of the input link
measured with the angle φ2
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28 Lecture 4
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One independent parameter (DOF=1)–the angle φ2
The angle φ2=0o, 30o, 60o, 90o, …, 180o …,
330o, 360o (=0o).
MECHANISMS
29 Lecture 4
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Obtaining the position of C
At the intersection of circle with radius l3
(with center in B) and l4 (with center in D)
– the sketch is at scale
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30 Lecture 4
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The mechanism with input φ2=0o (scale
drawing). The angle φ4 is measured.
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31 Lecture 4
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The mechanism with input φ2=60o (scale
drawing). The angle φ4 is measured.
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32 Lecture 4
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Kinematic curves
Graphic representation of two variables
depending on each other.
The variables represented are usually
position, velocity and acceleration of a
point or a link in a mechanism related to
the parameter of its input link.
Four bar mechanism the angle φ4 related
to the input angle φ2
MECHANISMS
33 Lecture 4
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Graphical kinematic of the four bar mechanisms
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34 Lecture 4
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Graphical kinematic of the four bar mechanisms
Known lengths
Known input
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35 Lecture 4
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Slider-crank mechanism (crank - slider)
Defined by the length of the links : l1 l2
and e
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36 Lecture 4
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Slider-crank mechanism
Most mechanisms are driven by motors,
and slider-cranks are often used to
transform rotary motion into linear
motion.
Case slider as the input link and the crank
as the output link: the mechanism
transfers translational (linear) motion into
rotary motion. The pistons and crank in an
internal combustion engine
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37 Lecture 4
DATR
Internal Combustion Engine
In an Internal Combustion Engine, link 1
is the engine block. Link 2 is the
crankshaft, link 3 is the connecting rod
and link 4 is the piston.
The pin joint between 2 and 1 is named
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the main bearing. The pin joint between
the 2 and 3 is named the rod bearing. The
pin joint between 3 and 4 is named the
wristpin.
38 Lecture 4
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Nomenclature
The sliding contact between 4 and 1 is the
cylinder bore in the engine block and the
displacement of 4 is called the stroke
In the case of an engine, the line of action of
the wristpin, shown by the dashed line,
passes through the center of the main
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bearing and the offset d is zero. When d=0,
the crank-slider mechanism is called an “in-
line crank-slider”.
The crank-slider shown in figure is an “offset
crank-slider” 39 Lecture 4
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Inversions of a mechanism
Inversions change which of the
mechanism’s links is fixed
An n-link mechanism has n inversions
MECHANISMS
40 Lecture 4
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Inversions of the slider-crank
Standard slider-crank
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41 Lecture 4
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Inversions of the slider-crank
Freeing the slide, and fixing link 2
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42 Lecture 4
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Inversions of the slider-crank
Fixing link 3 (note: 4 is still free to rotate)
MECHANISMS
43 Lecture 4
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Inversions of the slider-crank
Fixing 4 (1 slides, but does not rotate,
w.r.t. 4)
MECHANISMS
44 Lecture 4
MECHANISMS DATR
45
Lecture 4
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Slider-crank mechanisms - applications
MECHANISMS
46 Lecture 4
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Slider-crank mechanisms - applications
MECHANISMS
47 Lecture 4
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Slider-crank mechanisms - applications
MECHANISMS
48 Lecture 4
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Slider-crank mechanisms - applications
MECHANISMS
49 Lecture 4
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Slider-crank mechanisms - applications
Airbus 320 Flaps mechanism
MECHANISMS
50 Lecture 4
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Slider-crank mechanisms – extreme positions
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51 Lecture 4
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Graphical kinematics, the input link
Input link in revolute motion
The position (orientation) of the input link
measured with the angle φ2
MECHANISMS
52 Lecture 4
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One independent parameter (DOF=1)–the angle φ2
The angle φ2=0o, 30o, 60o, 90o, …, 180o …,
330o, 360o (=0o).
MECHANISMS
53 Lecture 4
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Obtaining the position of C
At the intersection of circle with radius l3
(with center in B) and the line of the
translational joint D) – the sketch is at scale
MECHANISMS
54 Lecture 4
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The mechanism with input φ2=0o (scale
drawing). The displacement s4 is measured.
MECHANISMS
55 Lecture 4
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Graphical kinematic of the crank-slider mechanisms
MECHANISMS
56 Lecture 4
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Graphical kinematic of the crank-slider mechanisms
MECHANISMS
57 Lecture 4
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Graphical kinematic of the crank-slider mechanisms
MECHANISMS
58 Lecture 4
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Graphical kinematic of the McPherson suspension
MECHANISMS
59 Lecture 4
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Planar 1 DOF with P (prismatic) and R (revolute) joints
MECHANISMS
60 Lecture 4
