Reduction of Input Torque and Joint Reactions in High-Speed
Mechanical Systems with Reciprocating Motion
VIGEN ARAKELIAN1,2
1LS2N-ECN UMR 6004, 1 rue de la Noë,
BP 92101, F-44321 Nantes,
FRANCE
2MECAPROCE / INSA-Rennes,
20 av. des Buttes de Coesmes,
CS 70839, F-35708 Rennes,
FRANCE
Abstract: - In high-speed machinery, the variable inertia forces generated by reciprocating masses often
introduce undesirable effects, such as a significant increase in the required input torque and joint forces. This
paper addresses the challenge of reducing input torque and joint reaction forces in such mechanisms by
employing two compression linear springs positioned between the slider and the frame. These springs
counterbalance the slider's inertia force, thereby diminishing both the input torque and joint reactions. It is
important to note that the elastic forces exerted by these springs remain internal to the mechanical system,
preserving the balance of shaking forces and moments of the mechanism on the frame. The analytical
framework developed in this study focuses on minimizing the root mean square and maximum values of the
inertia force effects. A significant scientific achievement is attaining a given goal through an analytical
solution. Notably, this is the first instance where this problem has been formulated and solved using explicit
expressions. The effectiveness of the proposed technique is also demonstrated through CAD simulations,
showing a substantial reduction in input torque and joint reactions.
Key-Words: - fast-moving machinery, input torque, joint reactions, inertia forces, slider-crank mechanism, root-
mean-square approximation.
Received: July 13, 2023. Revised: May 4, 2024. Accepted: June 13, 2024. Published: July 24, 2024.
1 Introduction
In the domain of industrial machinery, operating at
high speeds has become imperative, with inertial
forces taking precedence. The inertia inherent in
moving links often results in significant fluctuations
in a mechanism's input torque throughout its cycle.
Given that motors are designed for peak operating
conditions, mitigating these torque requirements
holds economic appeal. Such mitigation not only
allows for the use of less powerful motors but also
contributes to reduced noise levels and enhanced
longevity for select components, [1].
Devices for compensating inertia forces on input
torque can be categorized into two main groups
based on their installation location within the
machine: i) Compensating cam systems installed at
the machine's input element and unloading all
transmissions from the motor to the input link; ii)
Cyclic mechanism compensators with elastic
connections directly engage the moving mass,
minimizing the fluctuations of the input torque and
reactions on mechanism links. With the inclusion of
a compensation device in the cyclic mechanism,
surplus energy is stored and subsequently
reintroduced into the mechanical system.
Compensation devices utilized in this context
include elements capable of accumulating and
releasing potential or kinetic energy with minimal
losses per cycle, such as springs, torsion rods,
pneumatic and hydraulic devices, inertial systems,
etc. Let us explore some studies dedicated to
addressing this issue.
The reduction of input torque can be achieved
through the optimal distribution of moving masses,
[2], [3], [4], [5], [6], [7], [8], [9], [10]. Previous
studies have focused on optimizing the mass
parameters of moving links to achieve torque
decrease. Additionally, researchers have explored
another approach: the incorporation of springs into
the mechanism for input torque compensation.
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The synthesis of spring parameters proposed in
[11] and [12] allows the minimization of input
torque in planar mechanisms. Various methods for
input torque compensation, such as incorporating
cam sub-systems as discovered in [13], [14], [15],
[16], [17], [18], have been investigated. This
approach involves altering the input inertial
parameters through prescribed cam profiles.
Additionally, alternative solutions involving
articulated dyads and linkages, proposed in [19],
[20], [21], [22] have been examined to minimize
torque fluctuations. In these cases, optimal
redistribution of moving masses to ensure torque
compensation is achieved by attaching additional
structural sub-systems to the original mechanism.
Flywheels driven by noncircular gears, as
discussed in [23], [24], [25], [26] allow a means to
completely balance the input torque. Nonlinear
programming techniques have also been used to
develop linkages with optimal dynamic properties,
allowing the minimization of torque fluctuations
[27], [28], [29].
Among the more efficient methods for input
torque balancing is the creation of cam-spring
mechanical systems, as detailed in [30], [31], [32],
[33], [34], [35], [36], [37]. In such systems, the
spring absorbs energy when torque demand is low
and releases energy when demand is higher,
providing precise compensation for load variations
attributable to periodic torque. The synthesis and
design of cam-spring mechanisms generally ensure
complete compensation for periodic torque-induced
load variations. Redundant drives have developed as
a method for input torque compensation.
In [38], a servomotor was employed to achieve
torque balancing in the linkage by varying the input
speed function. Similarly, a redundant servomotor
was utilized to address a similar challenge of
simultaneous shaking moment and input torque
balancing in four-bar linkages, as discussed in [39].
Furthermore, the combination of redundant drivers
and gear trains has led to the proposal of various
balancers for torque compensation in cyclic
mechanisms, as presented in [40], and [41].
The study [42] addresses the problem of input
torque compensation with the optimal connection of
two identical slider-crank mechanisms (Figure 1).
In mechanical design, the challenge of balancing
inertia forces on the frame and compensating for
input torque are typically addressed as separate
tasks. Traditionally, a mechanism can be balanced
using well-known methods [10], and its input torque
can then be compensated for using a complementary
device.
Fig. 1: Input torque compensation with the optimal
connection of two identical slider-crank
mechanisms, [42]
Fig. 2: Simultaneous inertia force balancing and
torque compensation in slider-crank mechanisms,
[10]
However, in the study [10], a novel design
approach has been introduced, which advocates for
simultaneous inertia force balancing and torque
compensation in slider-crank mechanisms (Figure
2).
This paper deals with the inertia force
compensation in high-speed mechanisms with
reciprocating motion. It is carried out by providing
two springs mounted between the slider and the
frame, which compensate for the inertia force of the
slider and, as a result, reduce the input torque and
joint reactions of the mechanism.
2 Statement of the Problem
Figure 3 shows a slider-crank mechanism. The
inertia force resulting from reciprocating motion can
be expressed as a series:
(1)
with
…
where, is the length of the crank, m is the
mass associated with reciprocating motion,
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is the length of the coupler link, is the
rotation angle of the crank and cont its angular
velocity.
Fig. 3: Slider-crank mechanism with added springs
This series comprises an infinite number of
terms, with each term denoting a simple harmonic
motion characterized by a known frequency and
amplitude. Higher-frequency amplitudes tend to be
negligible, rendering only a small number of lower-
frequency amplitudes significant. Considering the
fourth or higher harmonics is seldom necessary.
Thus, the inertia force of the reciprocating
motion can be expressed as:
(2)
taking into account that .
In high-speed slider-crank mechanisms, this
force is substantial, leading to increased input torque
requirements and joint forces. The objective of this
study is to introduce a solution aimed at minimizing
the input torque and joint reactions. To achieve this
goal, the slider-crank mechanism is equipped with
two compression springs featuring linear
characteristics, mounted between the slider and the
frame. It is important to note that the stroke length is
s=2r, and the additional springs generate the
following extra forces:
and (3)
where, and are the stiffness coefficients of the
springs.
In order to minimize the input torque and joint
reactions due to the reciprocating inertia force, it is
necessary to minimize:
int
(4)
Two solutions are considered below: on the base
of the root mean square and maximum values
minimization of function (4).
3 Minimization by Root Mean
Square Value
For minimization of the root mean square (RMS)
value
(5)
it is necessary to minimize integral:
(6)
where,
(7)
(8)
(9)
Hence, upon integration, the function that
necessitates minimization is as follows:
(10)
To determine the minimum of the function , we
impose the following conditions:
(11)
from which we obtain:
(12)
where,
(13)
(14)
(15)
(16)
Therefore, the stiffness coefficients of the
springs are determined from equation (12):
(17)
and
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(18)
4 Minimization by Chebichev’s
Approximation
The quadratic approximation generally results in
minor deviations on average from the given
function. However, across various segments, the
deviation can occasionally escalate to values
significantly divergent from the average. The best
function approximation is free from these flaws by
introducing the minimum possible value of the
maximum deviation from the given function:
int .
According to Chebyshev's theorem on the best
function approximation, it is necessary to find such
polynomial coefficients (in the present study the
stiffness coefficients of the springs and ) for
which the maximum value of function (4) will be
minimum, i.e.
int
(19)
In order to achieve such a minimization it is
necessary and enough that the force F determined
from (4) no less then ways reaches its limit
values consecutively changing its sign in the
interval , i.e.
int
(20)
Fig. 4: Best function approximation
Given that for , the difference should not
exceed its limit value (Figure 4), its derivative at
these points is reduced to zero, i.e. for half rotation
of the input crank is sufficient.
(21)
Considering the extreme points, we obtain two
additional equations, totaling. Please note that
the function (4) is symmetrical and the minimization
for half rotation of the input crank is
sufficient.
Thus, we obtain:
(22)
and we determine:
(23)
(24)
However, taking into account that
, we obtain andconsequently
.
By substituting the values of
into equation (20), we obtain:
(25)
where,
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
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Therefore, the stiffness coefficients of the
springs are as follows:
(38)
and
(39)
Considering that the angles already been
determined and that the inertial force in the axial
crank-slider mechanism are symmetric function, it is
possible to obtain their more precise values of
and by including their exact values in the
equation (25).
Therefore, we derive the following equation:
int
int
int
(40)
and determine
(41)
(42)
Let us illustrate the proposed method for
unloading the input torque and reactions in joints
from variable inertia forces through a numerical
example.
5 Illustrative Example with
Simulation Results
Let us consider a slider-crank mechanism with
following parameters: ,
, m.
Firstly, root mean square minimization is
considered.
The exact value of the reciprocating inertial force
can be represented as follows Σφάλμα! Το αρχείο
προέλευσης της αναφοράς δεν βρέθηκε.:
(43)
where,
Then, from (13)-(16) we determine
m2, ,
, and obtain the
following values of the springs’ stiffness
coefficients: and .
The root mean square minimization leads to the
resulting force, which varies in the interval [-
137.9N; 146.9N].
Let us consider now the minimization by
Chebichev’s approximation. In this case, the
expressions (41) and (42) have the following
numerical values:
(44)
(45)
and the springs’ stiffness coefficients:
and with
. Thus, the variation of reciprocating inertia
force after minimization by Chebichev’s approach
becomes uniform, which varies in the interval [-
143N; 143N].
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Figure 5 shows the variations of the
reciprocating inertia force before compensation
, the compensation force developed by springs
and the resulting force after compensation
. It should be noted that the values of the
resulting forces compensated by root mean square
and maximum values are very close (the difference
is no more than 5N) and they are shown by one
graph in Figure 4. Obtained results show that the
suggested compensation technique allows to reduce
the maximum value of the reciprocating inertia
force until 85%.
Fig. 5: Variations of reciprocating inertia force
before compensation , springs’ elastic force
and the resulting force after compensation
Fig. 6: Variations of the input torque
The examined mechanism has been simulated
with the software ADAMS. Figure 6 presents the
variations of the input torque before (dotted line)
and after (full line) compensation. The numerical
simulation showed that the redaction of the
maximum values of the input torque is 71%.
Figure 7 presents the variations of the reaction in
prismatic pair before (dotted line) and after (full
line) compensation. In this case, the redaction of the
maximum value of the joint reaction is 64%.
We also wish to highlight the versatility of the
proposed compensation technique, which remains
effective even in scenarios where external forces are
exerted on the slider. These forces can be
seamlessly incorporated into a function (4) through
analytical representation. It is important to note that
the efficacy of compensation, as well as the optimal
values for the stiffness coefficients of the springs,
are contingent upon the specific characteristics of
these external forces.
Fig. 7: Variations of the reaction in prismatic pair
However, it should be also noted that adding a
spring between the slider and the frame in a crank-
slider mechanism can have a significant impact on
the system's resonance. This alters the stiffness and
mass characteristics of the mechanical system,
potentially shifting the system's natural frequency. If
this new natural frequency approaches the frequency
of external excitation, it can lead to amplification of
oscillations and increase the risk of resonance.
When external forces are applied to the mechanism,
resonance can occur if the frequency of these
external excitations corresponds to the system's
natural frequency. This can result in significant
oscillations, potentially leading to decreased
performance. In such cases, to minimize the effects
of resonance, it is essential to optimize the
parameters of the springs calculated from inertia
force compensation, based on the dynamic
characteristics of the system and the anticipated
operating conditions. This may require a thorough
analysis of the system's dynamic behavior and
experimental testing to validate the selected spring
choices.
6 Conclusions
When a machine element having a large mass is
given a reciprocating movement, the periodical
variations in speed bring the variable dynamic loads,
which on the one hand have several undesirable
effects on the frame, as vibrations, on the other
hand, they increase the joint reactions and require
that great driving force must be applied.
In this paper, an arrangement for the
compensation of inertia forces in the mechanisms
with reciprocating moving links is proposed. It is
shown that by a simple system containing two linear
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compression springs, it is possible to minimize
simultaneously the input torque and the joint
reactions. It is important to note that the added
elastic forces, which compensate for the alternative
inertia forces are internal relative to the mechanical
system, i.e. they do not perturb the shaking force
and shaking moment balance of a mechanism. On
the basis of an analytical approach, the conditions
for the compensation are formulated by the
minimization of the root-mean-square and
maximum values of the inertia force of the
reciprocating moving mass. The efficiency of the
suggested technique is illustrated by the numerical
example in which 71% of the input torque and 64%
of the reaction in the prismatic pair are achieved.
It should be noted that previous works aimed at
achieving a similar goal employed more complex
design solutions, using additional mechanisms with
cams and other similar elements. The advantages of
this study include the simplicity of the
compensating device's design and the fact that the
solution is obtained purely analytically, significantly
increasing the clarity of the solution and its ease of
application in various engineering projects.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare.
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