A.
G.
ERDMAN
Assistant
Professor of
Mechanical Engineering,
University of
Minnesota, Minneapolis,
Minn.,
Formerly,
Graduate Research Assistant,
Rensselaer
Polytechnic
Institute. Assoc.
Mem. ASME
G.
N.
SANDOR
ALCOA
Foundation Professor of
Mechanical Design, Chairman, Division
of Machines and
Structures. Fellow
ASME
R.
G.
OAKBERG
Assistant
Professor of Civil
Engineering. Rensselaer
Polytechnic
Institute, Troy,
N. Y.
A General Method
for
KinetoElastodynamic Analysis and Synthesis
of
Mechanisms
Kinetoelastodynamics is the study of the motion of mechanisms consisting of elements which may deflect due to external loads or internal body forces. This paper describes the initial phases in the development of a general method of kinetoelastodynamic analysis and synthesis based on the flexibility approach of structural analysis, ãwhich may be applied to any planar or spatial mechanism. Dynamic error is investigated due to flexural, longitudinal, and torsional element strain, and system inertia fluctuations; the treatment of Coulomb and viscous friction is indicated. KinetoElastodynamic Stretch Rotation Operators are derived which will rotate and stretch both planar and spatial link vectors reflecting rigid body motion plus elastic deformations of the
link.
A numerical example is presented to demonstrate the elastodynamic analysis technique.
Introduction
L
K
past ten years have seen tremendous strides in the field of kinematics. Kinematic synthesis has become a powerful design tool for the engineer, surpassing in certain respects previous kinematic design techniques, which may now be classified as classical kinematics. There is, however, one major shortcoming of present kinematic synthesis techniques. This is the rigidity assumption which prevails throughout the literature with few notable exceptions. . Mechanisms consisting of links, gears, sliders, etc., are not rigid in actuality—they are elastic and deflect when subject to high static or dynamic forces. In lowspeed motion, if static forces are not high, the designer usually will not need to concern himself with the inherent elasticity of a mechanism system, but in highspeed applications he may find the same mechanism inoperable due to high fluctuating inertia forces. The motivation behind the search for new synthesis procedures recognizing the elastic properties of mechanisms has srcinated from just such occurrences. For example, a gripper mechanism in a highspeed printing press may be synthesized kinematically with all links regarded as rigid. In the actual performance, however, such a linkage may accomplish its task only up to onehalf design speed; above that speed, the gripper may miss the target. The literature has recognized the need for dynamic analysis and
1
Based on the initial phase of a dissertation by the first author toward partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering), Machines and Structures Division, School of Engineering, Rensselaer Polytechnic Institute. Contributed by the Mechanisms Committee and presented at the Winter Annual Meeting, Washington, D. C, November 28December
2, 1971, of THE
AMERICAN SOCIETY
OF
MECHANICAL
ENGINEEBS.
Manuscript received at ASME Headquarters, May 30, 1970. Paper
No.
71WA/DE6. synthesis techniques. A publication on mechanism dynamics, however, may fall under many subcategories such as kinematic derivatives
[17]
2
,
harmonic analysis [8], balancing
[917],
vibrations [1822, 40], elastic analysis [2324], structural analysis [3538], stability [3941], time response [1517, 4247] or backlash and impact [4852]. (Benedict [43] cites a fairly complete

set of references in dynamics which will not be repeated here.) The great majority of work in dynamics of mechanisms falls under the kinematic derivatives category (sometimes obtained by way of the Lagrangian approach) which assumes rigid mechanism elements. Some authors [8, 1822, 2734, 39, 40] have dealt with elastic complex systems, i.e., systems with mixed elastic and nonelastic members. Because of the complexity of the solutions, usually only one element is considered elastic, and then only with one degree of freedom of de^ formation: torsion, extensibility, or lateral bending. Lagrangian mechanics or energy methods are often employed to derive equations of motion, but many simplifying assumptions, which must be made in order to solve these equations, unfortunately make the model and solution often impractical. Since designers deal with completely elastic systems, these methods do not lead to meaningful solutions except in very simple cases. The authors of the paper believe that structural dynamics techniques based on the finite element method constitute a more desirable approach to analyzing and synthesizing a completely elastic system. Recently, Winfrey [26] has performed a kinematic analysis combined with the structural dynamics stiffness technique to yield the rigid body plus elastic motion. Boronkay and Mei [24] have analyzed a multiple input flexible link mechanism using the finite element method where the revolute joints are replaced by flexible joints. The
kinetoelastodynamic
analysis method presented in this paper, based on the flexibility method of structural analysis, is adaptable for the treatment of the effects of viscous Numbers in brackets designate References at end of paper.
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and coulomb friction, inertia fluctuations seen by the input element, and power input and output of the system. A few authors have published articles concerning dynamic synthesis, but in a limited sense. Most of these contributors have synthesized for prescribed kinematic derivatives assuming a rigid system. This is termed point dynamic synthesis [2]. Sherwood examines the fluctuation of the input angular velocity of the rigid mechanism due to varying torques and resynthesizes by redistribution of mass [16, 17] or adjusting one link length
[15].
Skreiner [47] resynthesizes a rigid four bar path generator in order to minimize shaking and pin forces by addition of mass and springs to the system. Burns and Crossley [3233] perform a
kinetoelastostatic
synthesis on a four bar function generator with a flexible coupler. They describe a graphical solution for a constant torque output. Shoupe [30] has synthesized function generators which would be rigid except for one highly flexible member. The general method of
kinetoelastodynamic
synthesis proposed in this paper will, for the first time, it is believed, include all kinematic and dynamic effects influencing the motion of elastic mechanisms. Since there are some inconsistencies in the literature in the nomenclature involving mechanism dynamics, the authors would like to propose definitions for the following general expressions. Kinematic Analysis. Examination of the displacements, velocity ratios, acceleration ratios, etc., of a mechanism with all its members regarded as rigid. The reference variable is a position parameter. Dynamic Analysis. Determination of the displacements, velocities, accelerations, etc., of a mechanism, including derivations of inertia forces of a mechanism made up of rigid members. The reference variable is time. Elastic Analysis. Examination of the stresses and deflections of an elastic system due to static load in order to determine system flexibilities or stiffness. Elastodynamic Analysis. Examination of displacements, velocities, accelerations, stresses, strains, etc., of a moving elastic mechanism. Inertia forces are calculated by assuming all of the members rigid. KinetoElastodynamic Analysis. Examination of the displacements, velocities, accelerations, stresses, strains, etc., of a moving elastic mechanism. Effects of elastic deformation upon the inertia forces are included in the analysis.
A,E,I '«2
A E I
3 ã»
(b)
A,E.I
TA
~~M~
Fig.
1 Models for elastic analysis: (a) cantilever beam, (fe) two force member, and (c) simply supported beam with end moment
Kinematic Synthesis. Creations of a mechanism which satisfies various combinations of prescribed positions, velocity ratios acceleration ratios, etc., assuming all members as rigid and massless. The reference variable is a position parameter. Dynamic Synthesis. Creation of a mechanism which satisfies various combinations of prescribed positions, velocities, accelerations, etc., considering members as rigid and as having concentrated or distributed masses. The reference variable is time. Dynamic Balancing. Same as dynamic synthesis, but including minimization of shaking forces and/or moments within a mechanism and those transmitted to its supports. KinetoElastostatic Synthesis. Creation of a mechanism which satisfies various combinations of prescribed positions, velocity ratios, acceleration ratios, force and torque transmissions, etc. The reference variable is a position parameter. Mechanism members are assumed to be elastic. KinetoElastodynamic Synthesis. Creation of a mechanism which satisfies various combinations of positions, velocities, accelerations, force and torque transmissions, stresses, strains, etc., at predetermined running speeds. Mechanism membei's are assumed to be elastic and have concentrated or distributed masses. KinetoElastodynamics. The study of the motions of mechanisms consisting of elements which may deflect due to external loads or internal body forces.
Analysis of Elastic Systems
Any mechanism may be considered a structure if its rigidbodykinematic degrees of freedom are removed. The structure is a system of several elements which may have internal elastic degrees of freedom. A fourbar linkage may be converted into a structure by modeling the input link as a cantilever or fixedfree beam, as in Fig. 1(a). (Winfrey [26] concludes that a rotating elastic rod vibrates as a rotating elastic cantilever beam.) The beam has length
I,
crosssectional area
A,
modulus of elasticity
E,
and crosssectional moment of inertia
I
about its
Z
axis, normal to the plane of the mechanism. The element forces /i,
fi,
and element moment /
3
cause end deflections
d
1:
dt,
and angular deflection
d
3
which are expressed as:
pl~l
di
_d
3
_
=
[F]
r/i 
/ã
L/.J
where
[F]
is the element flexibility matrix:
m =
l/AE
0 0 0
l
3
/3EI P/2EI
0
l*/2EI l/EI
(1)
(2)
A twoforce member representing a link with two pin joints, shown in Fig. 1(6), can only transmit longitudinal force. Thus, its flexibility matrix has only one term:
[F] = [l/AE]
[dd =
[l/Am\fi]
(3) (4)
A link subject to torsion, Fig. 2(a), (such as the shaft connecting the motor to the input link) will have a flexibility matrix as follows: . so that where
[F]
=
[l/QJ]
[dd =
[i/ojm,]:
(5) (6) di = relative torsional (angular) deformation in length
l G
= modulus of elasticity in shear / = polar moment of inertia of the (circular) crosi oss section
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Spatial linkages may also be modeled according to their type of connection. A spatial link with two spherical joints
(SS)
has only a longitudinal degree of freedom, and in essence is a twoforce member. An RS (revolutespherical) link, Fig. 2(6), will possess two elastic degrees of freedom. The flexibility matrix for this case is
m

[
/AE
0
so that
L«4j
[F]
0
l»/3EI_
O
(8) (9)
Fig.
2 Models for elastic analysis: (a) torsional member and (b) spatial RS member
fa
= torque transmitted through length / A link with a slider connected by way of a revolute joint acts as a freely supported beam at the slider end. In some cases a mechanism link is not just a simple straight beam, but a flexibility matrix may be pieced together with a basic knowledge of beam theory. For example, in Fig. 3 the coupler link is composed of two elements separated by a fixed angle
a.
Element 1 may be treated as a simple cantilever beam with three elastic degrees of freedom, while element 2 is treated as a simply supported beam with a moment
fa
on the left end (due to element 1) and a longitudinal force
fa
as in Fig. 1(c). Thus, the flexibility matrix for element 2 will be:
[F] = 'l/AE
0
0
1
EIJ
/ZEI.
(7)
Gears may also be modeled to determine their elastic deformations. Gear teeth are investigated [59] by considering the tooth as a cantilever beam. The total deformation of the tooth consisting of the result of direct compression at the point of contact between teeth and of beam deflection and shear may be calculated.
Flexibility Approach
A mechanism is composed of various combinations of elements, each of which can be represented by a known structural model as was demonstrated in the preceding section. The deflections of the entire mechanism system may be derived by performing an elastic analysis via the flexibility approach. The mechanism will have system, or generalized external forces acting upon it which will be represented by the column matrix [P
3
]
j
= 1, ..
.TO,
where
m
is the number of system forces. The number of elastic degrees of freedom of the system is the sum of the elastic degrees of freedom of its elements; each degree of freedom being represented by an element coordinate
x
it
i
= 1, . .
,n,
where
n
is the number of element coordinates. In order to transfer the system forces into element or internal forces [/;],
i =
1, . . . .
n,
each acting in the respective element coordinate direction, an
n
X
m
force transfer matrix
[B]
is derived by the methods of static analysis. This matrix is dependent upon the configuration of the system and, therefore, is a function of the independent variable—say the input angle of the singledegreeoffreedom rigid bodykinematic system. A matrix of element flexibilities,
[F]
, an
n
X
n
matrix which is
independent of the input angle,
is composed of the element flexibility matrices along its diagonal. Premultiplying the element forces by the matrix of element flexibilities will yield the element deformations
[di], i =
1 . . .
n.
Finally, the element deformations are transformed into system, or generalized deflections
[5j], j =
1, . . .,
in,
by premultiplying by the transpose of the force transfer matrix,
[B]
',anmX» matrix, as follows: where and
[5]
=
IB]'[F][B]IP},
[/I =
[B][P],
Id]
=
[F]
If],
(10) (11)
(12)
and where the matrix product
[B]'[F] [B]
represents the system flexibility matrix [ff], an m X TO matrix which depends on the input angle.
Fig.
3 Case I. Fourbar path generator with disk mass located at path point P. Elements 1, 2, 3, and 4 are elastic
FourBar Path or Motion Generator
In order to illustrate the application of the flexibility method of mechanism analysis, an elastodynamic analysis will be performed on a planar fourbar path or motion generator, Fig. 3, in order to determine the elastic displacement of the path point
P
through its cycle of motion, where element 3 is the input member. The coupler link consists of two elements separated by a fixed angle
a.
Three cases will be described in increasing levels of accuracy, each of which include the results of the previous level and build upon it, taking additional effects into account to come closer to reality. Case 1. As a first model, a completely elastic moving system made up of elements 1, 2, 3, and 4 (Fig. 4), will be analyzed. The links are assumed massless compared to an inertial mass
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Fig.
4 Element diagram for Case 1 showing the element coordinates and system forces
in a desired system configuration, characterized by the input angle
4>
=
</>
(the
jth
position of the mechanism), is examined. Elements 1 and 3 are modelled as cantilever or fixedfree beams, element 2 is a simply supported beam, and element 4 is a twoforce member. Notice that element 3 has no moment acting at its free end because the beam is massless and no moment may be transmitted to it through a pin joint. Notice, also, that element 2 has a moment applied to its left end which is present due to the fixed angle between elements 1 and 2. There are eight element coordinates
(n
= 8) representing the eight elastic degrees of freedom of this model. The force transfer matrix
[B]
and the matrix of element flexibilities
[F]
for case 1 are derived in the Appendix. Case 2. The second model is similar to the first, except that each element has a concentrated or a disk mass located at each joint (as shown in Fig. 7) resulting in eight system forces
(m
= 8) instead of only the three system forces at the tracer point. The system forces, directed in the eight elementcoordinate directions, represent the inertia forces of each element due to each equivalent mass. Equation (10)is still valid,except for the following changes in matrix dimensions: [Py] for
j
= 1, . . ., 8, (8 X 1 instead of 3 X 1);
[B'\
a new 8X8 matrix, replaces the 8 X 3
[B]
matrix;
([B
l
]
remains the same, since only the deformations at the path point are of interest). The new force transfer matrix is, in partitioned form:
Fig.
5 Force diagram of coupler for Case 1
[B'\
[B]
0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
C(f>\
8<j>\
0 1 0 0 0 0 0 0 0 0 1
(13) Where
[B]
is the previous 8X3 matrix (equation (52)), and, in symbolic form,
C<pX
=
cos
<j>
—
\) and
S<j>\
= sin
j>
—
X). Thus: (14) Notice that any number of additional inertia forces may be added to the above system, but the matrices will increase in size. For instance, if there are seventeen system forces and ten element coordinates, then the ten element deflections will be expressed as:
8i
s
3
_
B
(3 X 8)
F
(8 X 8)
B
(8 X 8) Pi
_Ps.
Fig.
6 Force diagram of element 2 for Case T
located at the path point P. Thus the external or generalized forces acting on the system are the horizontal and vertical inertial forces Pi and
Pi
plus an mertial torque
Ps,
all located at the path point
(m =
3). In order to perform on elastodynamic analysis on the mechanism, the departure from the rigidbody position
Fig.
7 Element diagram for Case 2 showing system forces Pi through
Pt
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