Dynamics of Second-Class Planar Mechanisms
by Means of a Computer Aided Modular
Approach.
Sebastián Durango.
Universidad Autónoma de Manizales.
Antigua Estación del Ferrocarril, Manizales, Colombia.
Email: [email protected]
Abstract—A computer-aided method for the dynamic
analysis of planar mechanisms is presented. It can be
classified as a general program (GP) based on chains. The
method is developed using a modular approach by means of
Assur’s groups, in which each structural group may be
analyzed and then codified as a computer function,
regarding the kinematical, kinetostatic and dynamic
aspects. A Graphical User Interface (GUI) is proposed
using SIMULINK. Examples of analysis for simple and
complex mechanisms are presented.
Keywords — Kinematical chains, kinematics, dynamics,
modular approach, Assur groups, planar mechanisms,
kinematical units.
spatial problem. Typical examples of joint based
programs are Working Model, Visual Nastran and
Cosmos.
Libraries developed using the kinematic unit approach
contain a number of kinematic chains with a special
characteristic that can be assembled in some way to form
a mechanism. The most important feature of this kind of
programs is that it comprises the advantages of both SP
programs and joint based GP programs, Reference [10]
lists them as follows: computational efficiency and
flexibility.
References about academic and commercial kinematic
unit based programs are limited, finding [13] and some
cited by [10], table 2.
I. INTRODUCTION.
II. STRUCTURAL CLASSIFICATION.
Computer aided analysis methods for mechanical
systems are divided by [16] into two categories:
1) Special Purpose (SP).
2) General Purpose (GP).
Special purpose methods are typically limited to a
specific mechanism, for example the programs Fourbar,
Slider, and Dynacam designed by [17] for the kinematics
and dynamic analysis of four bar mechanisms, crankslider mechanisms and cam mechanisms.
General-purpose methods are codified as libraries
without any specific mechanism, but including the
necessary elements to virtually assembly a mechanism.
Libraries could be developed over the kinematic joint
concept or over the kinematic unit concept.
For joint based programs the library contains a
number of kinematic joints. The principal advantage of
this approach is the effectiveness to define a complex
In 1919 Russian scientist L. V. Assur proposed a new
method to classify mechanisms based on the concept of
structural groups that corresponded to the kinematic unit
concept.
A structural group is a kinematic chain with no
degrees of freedom relative to the links with its free
elements form pairs, and that cannot be divided in
simpler kinematic chains with the same characteristic.
This concept is the keystone of the computational
efficiency of the modular approach because it’s possible
to find independent analytical solutions for the kinematic
and kinetostatic aspects of each group. The number of
degrees of freedom (DOF) for a group is then:
(1)
If we consider only mechanisms with v class pairs,
then (1) becomes:
(2)
Where n is the number of links and pV is the number
of pairs with one DOF. If a mechanism contains pairs of
two DOF those can be transformed into pairs of one DOF
in an instantaneous equivalent mechanism.
The primary link and the fixed link forming a V class
pair are named conventionally a first class mechanism.
The formation of any planar mechanism can be
understood as the serial union of groups that satisfies (1)
and one or more first class mechanisms. Consider for
example the mechanism in Fig. 1, the structural law of
formation is as follows: the mechanism has one degree of
freedom with a first class rotational mechanism formed
by the fixed link and link 1, then is added a second class
group formed by links 2 and 3, link 1 and 2 form a pair in
B and the fixed link and link 3 form a pair in D. Finally a
second class group formed by links 4 and 5 is added
between links 2 and 3. The law of formation can be
expressed as:
When an analytical approach is used, the solution of
each group is developed and then codified as a library
module. This module could be used at any time for the
solution of a specific mechanism that contains it in its
structural classification. In any case the solution requires
knowing the kinematic of every connection pair on the
group.
The kinematics of the connection pairs is determined
by means of the problem conditions, or using the solution
of the previously connected groups. The task is then to
follow the structural law of formation of the mechanism,
see Fig. 2, and solve sequentially every group that forms
the mechanism.
Fig. 2. Aspects of dynamic analysis by structural classification.
No specific solutions for structural groups are
presented here, for additional information on kinematics
analysis using structural classification see [8], [4], [13]
and [10].
IV. KINETOSTATICS
Fig. 1. Second class mechanism and its structural classification.
A mechanism with N degrees of freedom contains N
first class mechanisms; each of them has an independent
coordinate as input parameter. The set of input
parameters for a specific mechanism is known as
generalized coordinates.
For more information on structural classification see
[12], [2], [13], [4], [8], and [10].
III. KINEMATICS
A general and systematic kinematic analysis could be
developed by means of structural classification. There
are several graphic and analytical methods available, for
example, reference [5] has developed a parametric
methodology for the computer aided graphic solution.
Reference [6] proposes two tasks for force analysis:
a) The study of the action of external forces, link’s
weight, friction forces and inertial forces over the links
and over the kinematical pairs and supports; and the
determination of the means to reduce dynamic loads that
appear during the mechanism movement.
b) The study of the movement law of the mechanism
under the action of given forces and the determination of
the means to keep the movement law requested for the
mechanism.
The first task is known as force analysis; the second is
known as dynamics of mechanisms.
The force analysis or kinetostatics requires knowing
of the movement law and applied forces for all links in
the mechanism. A standard method in the mechanism
theory consists to solve this task by means of the
D’Alembert principle.
If a modular approach is used, the force analysis for
each module can be developed independently because
each structural group is statically determined by virtue of
it not having degrees of freedom regarding the links that
forms pairs with it. Consider for example the two links
Assur’s Group with three rotational joints presented in
Fig. 3. To solve the kinetostatic problem is necessary to
know the kinematics of all links, then using the
D’alembert principle we can formulate:
(3)
(4)
(5)
Where
, is the inertial force over link i.
, is the inertial moment over link i.
, is the sum of all external forces over link i.
, is the sum of all externals moments over
link i.
, is the sum of the moment of all external
forces about the mass center i.
The set of expressions (3), (4) and (5) is linear to solve
the vector variables FA and FB. To solve the FC reaction
it’s possible to develop the sum of forces over link 2 or 3.
The modular approach demands to solve the system
analytically, then the expressions obtained to FA, FB and
FC can be codified as an independent module.
as functions of time, as well as joint forces, if desired,
also as a functions of time’.
The problem to estimate the acceleration of the input
links could be developed in several ways. If the
mechanism has only one input the Zhukovski’s theorem
about a rigid handle could bee used. If the mechanism
has more than one input a numeric strategy like the one
presented by [10] could be developed.
To obtain the movement of the mechanism is
necessary to know the initial conditions and solve the
second-order differential equations of movement in the
independent coordinates; a numeric ODE solver typically
does this.
For the purpose of this work the determination of the
acceleration in the input mechanism was developed using
the Zhukovski’s theorem that is described briefly,
adapted from [2]:
‘All forces and moments could be reduced to the input
link, called reduction element. If over any link actuates a
force P, another one applied in some point of the
reduction element could substitute this. Such substitution
has sense in that case, when the given force P and the
one that substituted it, are equivalents in certain
aspect…In particular its possible to use as a condition
that the elemental work did by the given force P and the
substitution one are equals’. The equivalent or reduction
force could be determined by the Zhukovski’s theorem
procedure.
For the dynamic analysis of a mechanism generally
the forces that perform over the mechanism are estimated
separately, and then the reduction could be determined
for the resistance production forces, friction forces, and
any force performing on the system. This kind of
determination allows analyzing the influence of each
force over the movement.
VI. APPLICATION PROBLEMS
Fig. 3. Diagram of dynamic equilibrium. Two links Assur’s group
with three rotational joints.
To solve the kinetostatic problem for a specific
mechanism the procedure requires first, solving the last
group in the structural law, then this information is used
to solve the previous group in the law until the primary
mechanisms are solved, see Fig. 2. Finally, for the
primary mechanisms the moment or force necessary to
achieve the required movement is solved.
V. DYNAMICS
Concerning to the dynamic of mechanisms task [9]
propose: ‘In time response analysis, we are given a
mechanism of known geometry, mass and inertia that is
subject to known external loads and known driving
forces or torques (such a torque-speed relationship).
Time response analysis produces kinematic information
about the linkage positions, velocities and accelerations
Possible application of the modular approach includes
the kinematics, kinetostatics and time response for planar
mechanisms. The kinds of mechanisms could be solved
depending of the modules codified in the library. For this
work, only first class mechanisms and second-class
groups were codified in a toolbox (library) in two ways,
MatLab functions and SIMULINK blocks. The initial
development was done in MatLab, therefore the library
included more modules. The idea of using SIMULINK
modules is to develop a Graphical User Interface (GUI)
that simplifies the analysis of particular mechanisms.
Not only closed chains could be analyzed, for open
chains with planar movement the toolbox is applicable,
see for example [7] where a kinematics and kinetostatic
analysis over a manipulator in planar movement is
developed. The characteristics of the library, called Assur
Toolbox are presented in table 1.
TABLE I.
CHARACTERISTICS OF ASSUR TOOLBOX.
MatLab User Defined Functions
Group
Kinetost..
with
friction
Time
resp.
SIMULINK
(GUI)
Kinematics
Kinem.
Kinetost.




















The results of the analysis are presented in Fig. 6 and
Fig. 7.
The mechanism develops the rectilinear motion using
57% (206o) of its crank input motion.

Fig. 4. Chebyschev’s four bar mechanism for rectilinear motion.
A. Four Bar Mechanism.
Fig. 4 shows a Chebyschev’s four bar mechanism for
rectilinear motion designed for a rectilinear trajectory
L = 435 mm and maximum deviation concerning to the
main line of  5 mm. The dimensions are r = 99 mm,
l = 245 mm and a = 196 mm. The purpose of the
analysis is to verify the approximate rectilinear motion
designed for the point M. The analysis requires the
structural classification of the mechanism and a known
law of movement for the input link. In this case the
structural classification is IR0,1IIRRR2,3 and the law of
movement for  is assumed as 0 ≤  ≤ 360o.
The kinematic analysis was developed using the
SIMULINK Assur Toolbox library; Fig. 5 presents the
SIMULINK model. In the model the blocks represent the
rotational primary mechanism (IG 0,1), second class RRR
Assur group (II GGG 2,3), fixed rotational pairs (O, C),
particular point (M) and input movement (vcte).
Each structural block accepts a three-dimensional
input vector with position, velocity and acceleration
parameters. In the same way the output of each block is a
three-dimensional vector with position, velocity and
acceleration solutions for the characteristic kinematic
dimensions and characteristic points of the links.
Fig. 5. Kinematic analysis for the Chebyschev’s rectilinear
mechanism using Assur Toolbox GUI in SIMULINK.
Fig. 6. Chebyschev’s rectilinear mechanism: M point trajectory.
Fig. 7. Chebyschev’s rectilinear mechanism: M point trajectory.
Detail.
B. Mechanical Finger.
Fig. 8 shows a complex mechanism for a mechanical
finger presented by [18]. The mechanism required a
kinematic and kinetostatic analysis for the determination
of its movement characteristics. The structural
classification
for
the
mechanism
is
IP0,1IIRRR2,3IIRRR4,5IIRRR6,7IIRRR8,9IIRRR10,11.
For the analysis a constant input speed of 5 mm/s 
was assumed. The kinematic and kinetostatic analysis
was developed using the modular approach see Fig. 2.
The analysis was developed using MatLab’s modules for
Assur Toolbox. A flow diagram for the kinematics
analysis is proposed in Fig. 9.
The kinetostatic analysis was developed considering
only the inertial effects.
Fig. 9. Flow diagram for mechanical finger kinematic analysis.
Only five basic modules were necessary for the finger
analysis, three for kinematic aspects –prismatic primary
mechanism, IIRRR Assur group and particular point
module– and two for kinetostatic aspects –prismatic
primary mechanism and IIRRR Assur group–.
The results of the analysis are presented in Fig. 10,
Fig. 11, Fig. 12, Fig. 13.
Fig. 8. Mechanical finger.
Fig. 10.
Mechanical finger: Point P trajectory.
Fig. 11.
Mechanical finger: Mass center accelerations.
Fig. 14.
Mechanical finger: computer aided position analysis.
VII. CONCLUSION.
Fig. 12.
Fig. 13.
Mechanical finger: Pair reactions.
The Assur Toolbox library is a general-purpose
program for the dynamic analysis of planar mechanisms
with a modular approach designed over the kinematic
unit concept.
The library has codified the solutions of several
kinematic units as MatLab’s functions and partially as
SIMULINK’s blocks, see table 1. At this time the library
includes the kinematics and kinetostatic aspects for
primary mechanisms and second-class Assur groups and
comprises approximations to the kinetostatic with
friction problem and dynamic problem for the rotational
primary mechanism and first kind second-class Assur
group.
Mechanical finger: input force.
An alternative to validate the kinematic analysis is to
develop a computer aided graphic analysis, as [5]
proposes. Fig. 14 presents a position’s graphic analysis
using the SAM51 software. The maximum relative error
comparing SAM51 and Assur Toolbox position analysis
was 0,1% in the Y coordinate and 0,01% in the X
coordinate, see [18].
Fig. 15.
An approximation to Assur Toolbox library in
SIMULINK.
There are some specific requirements for the use of
modules in Matlab; especially the language syntax could
be an obstacle to the user. The idea to develop a GUI by
means of SIMULINK blocks is promising. The analysis
of a Chebyschev’s four-bar mechanism presented in
section VI. A) prove to be easier and intuitive as regards
to equivalent MatLab analysis.
Table 2 presents a comparison between the library and
other general purpose programs based on chains, adapted
from [10].
Fig. 15 presents an approximation to the Assur
Toolbox GUI in SIMULINK.
The library has proved its versatility on the analysis of
several engineering problems. The mechanical finger
presented in [18] and the manipulator model presented in
[7] are two relevant examples.
TABLE II.
Program
[REF]
2links
4links
Analysis
Kinemat
[5]
[6]
[7]
[8]
[9]
PROGRAMS FOR MODULAR ANALYSIS AND ITS CHARACTERSTICS.
Kinematic
Unit
[4]
Kinet
Kinetost
Dyna
with
[10]
[11]
[12]
[13]
friction
SIMPLA


[19]
CRANMEC


[14]

[15]
[16]
[15]
N. N.

[11]
KIDYIAN




[3]
CADME




[10]
ASSUR




[17]

Partially
Partial
[19]
TOLBOX
Adapted from [10]
AUTHOR INFO
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 Sebastián Durango Idárraga. I. M. M.Sc.
 Assistant Professor. Mechanics Department,
Universidad Autónoma de Manizales
 Antigua Estación del Ferrocarril, Manizales,
Caldas, Colombia.
 Tel. +057(6) 881 1306
 [email protected]