Simple Approximate Solutions for Dynamic Response of Suspension
System
JACOB NAGLER
NIRC, Haifa, Givat Downes
ISRAEL
Abstract: - An approximate simplified analytic solution is proposed for the one DOF (degree of freedom) static
and dynamic displacements alongside the stiffness (dynamic and static) and damping coefficients (minimum
and maximum/critical values) of a parallel spring-damper suspension system connected to a solid mass-body
gaining its energy by falling from height h. The analytic solution for the prescribed system is based on energy
conservation equilibrium, considering the impact by a special G parameter. The formulation is based on the
works performed by Timoshenko (1928), Mindlin (1945), and the U. S. army-engineering handbook (1975,
1982). A comparison between the prescribed studies formulations and current development has led to
qualitative agreement. Moreover, quantitative agreement was found between the current prescribed suspension
properties approximate value - results and the traditionally time dependent (transient, frequency) parameter
properties. Also, coupling models that concerns the linkage between different work and energy terms, e.g., the
damping energy, friction work, spring potential energy and gravitational energy model was performed.
Moreover, approximate analytic solution was proposed for both cases (friction and coupling case), whereas the
uncoupling and the coupling cases were found to agree qualitatively with the literature studies. Both coupling
and uncoupling solutions were found to complete each other, explaining different literature attitudes and
assumptions. In addition, some design points were clarified about the wire mounting isolators stiffness
properties dependent on their physical behavior (compression, shear tension), based on Cavoflex catalog.
Finally, the current study aims to continue and contribute the suspension, package cushioning and containers
studies by using an initial simple pre – design analytic evaluation of falling mass- body (like cushion,
containers, etc.).
Key-Words: - suspension, spring, damper, energy conservation, friction, displacement, stiffness, coupling.
Received: May 21, 2021. Revised: December 7, 2021. Accepted: December 27, 2021. Published: January 9, 2022.
1 Introduction
Understanding the impact phenomenon of free fall
body from height h has been studied by many
researchers (all enlisted references). The problem
has many applications in the packaging and
automotive suspension industries.
The first to model vibrations by using advantage
formulations of energy conservation was
Timoshenko [1] in 1928 (pp. 74-76). He has
suggested one DOF (degree of freedom) model
(vibrating mass hanging on a spring and moves
along the vertical direction only) of energy
conservation between the kinetic energy (mv2/2, m –
mass weight, v - mass velocity) and the spring
potential energy (kx2/2 - kxst2/2, k-spring stiffness, xst
- static displacement) in order to evaluate the system
frequency. About
two decades later in 1945, Mindling [2] has
proposed his own one DOF model based on energy
conservation for the free falling mass-body attached
to a spring. During his derivation development, he
considered an energetic balance between the impact
force combined with the kinetic force and the
gravitational potential energy. He has derived the
expressions for the maximum impact force (P),
maximum dynamic displacement (xd) and the
maximum acceleration factor (G).
Three decades later, during the years 1971-1975, the
U.S. army, has published several studies [3] – [4]
related to the cushioning and containers design,
respectively; e.g. involved with analytic
approximations for the acceleration, height and time
during the shock damage. On the one hand, Ref. [3]
deals with simple analytic approximation of impact
acceleration dependent on time duration and height,
Ref. [4] made a generalized model that predicts g-
level response in terms of drop height, static stress,
thickness of the cushion and temperature. Note that
Ref. [5] is being an updated version of Ref. [3]
published in 1982.
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Based on the shock velocity and acceleration
shapes (triangular saw, semi-sinusoidal acceleration
shapes, velocity- time step and simple gravitational
falling impact), both Cavoflex [6] and Endine [7]
companies have developed energy model that links
to the natural frequency to determine the wire rope
isolators' suspension coefficients as dependent on
the static and dynamic states. Note that Cavoflex
company model has similar characteristics as
current model; however, the stiffness is calculated
by using the frequency parameter value.
Alternative methods to compute the shock
dynamic response are mainly based on the full
analytic and numerical solutions of the excitation
equilibrium (motion) to determine the amplitudes
and shock excitations that the suspension physical
parameters are dependent on.
Many researchers, as will be mentioned here
solved dynamic simple harmonic motion. Full
analytic solution for the dynamic motion (mass,
damper and spring) influenced by the sinusoidal
excitations and gravitational effect in relative to the
static state was given by Kaper [8]. Some examples
dealing with falling objects have been investigated
by French & Kirk (free falling mass-body on a
located spring was measured experimentally: spring
force and position vs. analytic method) [9], Wong
(Board- falling by using one-dimensional spring
model) [10], Nagurka & Huang [11] (falling
bouncing ball modelled by mass-spring-damper
transient model which includes contact time and
impact force analyses).
Frequency-time dependent solution divided
between static and dynamic states of different
suspension products was performed and investigated
by Tse et al. [12], Schwanen [13] in the context of
wire spring modeling, Zhang [14] in the context of
rubber isolators dynamic properties and Jazar [15]
(different mechanical vibration elements).
Alternative solutions that concerns soft products
(cushion and rubber vibration elements) modeling
using stress-strain relations attached to energy or
frequency expressions have been performed recently
by Polukoshko et al. [16] and Ge [17], respectively.
In addition, some methods based on lumped system
or Lagrange energy solutions have been provided by
Rajasekaran [18] and Bahreyni [19] in the context of
earthquake and mass sensor micro-device modeling,
respectively.
Enclosing the brief review, it should be
mentioned that full frequent-time dependent
solution of the well-known mass-spring-damper
differential equation has been solved analytically
and numerically by many researchers, for instance,
Jacobsen and Ayre [20] (full analytic derivation for
transient response to step and pulse reaction
functions), Thomson [21] (continue the work of
Mindlin [2] in the context of full frequent spring
solution with damper), Constantinou et al. [22]
(have developed relations for the suspensions
physical properties in terms of energy expressions
dependent on the frequency parameter in the context
of earthquake absorbing response structures design),
Ray et al. [23] (have developed approximate
asymptotic expressions for the visco-elastic
suspension system dependent on the energetic
terms), Coleman [24] (analytic solution for step
external input), Cruz-Duarte et al. [25] (have
performed similar falling mass model as proposed in
the current study although it was solved
numerically). Moreover, analytic solution that
assumes initial displacement appears in the opening
of Escalante-Martínez et al. study [26] (p. 2) who
also developed fractional developed differential
equation of viscoelastic fluid in MR
(magnetorheological) damper (solved numerically).
Giraud [27] has developed advanced frequency
lamped parameter – dependent solution for
piezoelectric transducer. Of course, it should be
mentioned there are many text books that deals with
the dynamic vibrations solving the full motion
frequency-dependent differential equation [28-33].
The current energy conservation approach is
based upon the previous studies method [1]-[7].
Moreover, the current study considers the damping
coefficient formulating an energy balance based on
the force equilibrium states (static and dynamic) that
accompanied with the G-acceleration dynamic
factor to evaluate the suspension properties
(displacements, spring stiffness, damping
coefficient) during the mass-body free falling.
2 Analytic Problem Formulation
Suppose we have mass-body that is connected
to a suspension system composed of spring with
linear elasticity and a damper with neglected
self-mass as shown in Fig. 1. Now, the system
(mass, spring and damper) is rest falling from
height h such as the energy conservation energy
will be expressed as follows:
2 2 2
Gravitational energy Kinetic energy
Spring potential energy
1 1 1
()
2 2 2
p d st d d st st k
E mg h x x k x k x mv E
(1)
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Fig. 1: Free falling of mass attached to a suspension
system illustration for dynamic and static
suspension states.
Fig. 2: Free falling of mass attached to a suspension
system illustration with friction force influence (Fr).
The initial gravitational energy (
p
E
) term due to the
falling mass body is
mgh
subtracted from the
spring initial static displacement (
st
x
) and was
added with the compressed dynamic displacement
(
d
x
) to generate the expression
()
d st
mg h x x
.
The potential energy equals to the kinetic energy
(with velocity
22
st
v g h x gh
since
st
hx
) that in reversal/return equals to the
potential energy difference of spring with linear
elasticity (
22
11
22
p spring d d st st
E k x k x
alongside the dynamic
d
k
and static
st
k
stiffness
coefficients, respectively).
Next step, we will define by using energy terms
the following equality between the damping energy
expression and the kinetic energy, by:
2
min 1
2
p dumper d st k
E c v x x mv E
(2)
where
min
c
is the minimum value of the damping
coefficient, whereas the maximum or critical size
will be defined by the relation
max 2
critical st
m
c Gg c k m
v
. The reason we
have two kinds of distinct energy equalities for
damper and spring (even though their mutual
assembly configuration) is because each component
act separately in relative to the kinetic and
gravitational energies due their different functioning
(spring is aided to absorb the gravitational energy
while the damper should decelerate the kinetic
energy) [5, p.5-8]. Although one should remember
that in case of damping the dynamic oscillation, the
frequency is dependent on the damping coefficient
such as
2
/ / 2
dk m c m
, whereas the
critical damping coefficient
2
critical
c km
and the
natural frequency equals to
/
nkm
(see Yu
and Wu [37]). The connection between the
components is expressed by the distance
difference
d st
xx
. Moreover, the number of
equations (5) should be compatible with the number
of the unknowns' quantity (
, , , ,
d st st d
k k x x c
).
Next step is to develop the shock (maximum
dynamic force) and the static forces as follows:
d d d
st st st
F k x Gmg
F k x mg
(3)
where
G
is the loading coefficient (ratio between
dynamic and absorption weight) and the link
between the spring dynamic and static stiffness
coefficients will be assumed to behave linearly, by:
,
d st
k qk
)4)
while the ratio
q
is constant. Substituting Eq. (3) and
(4) into (1), yields the algebraic relations for the
dynamic and static spring stiffness coefficients and
displacements, respectively:
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22
22
min max
2
min
22
,
22
22
,
22
1, 2 ,
2
1
/24
d st
d st
critical
d st
critical
d st d st st
hh
x G x q
G G q G G q
G G q G G q mg
k mg k
h h q
mv m
c c Gg c km
x x v
v v m
cc Gg x x x x k
(5)
In case we have a number of equal parallel springs
(n) combination, relations (4) become:
22
22
min max
2
min
22
,
22
22
,
22
, 2 ,
2
1
/24
d st
d st
critical st
d st
critical
d st d st st
hh
x G x q
G G q G G q
G G q mg G G q mg
kk
h n h qn
m v m Gg m
c c c k
n x x n v n
v v m
cc Gg x x x x nk
(6)
where the dynamic and static forces
fulfil
//
d st
F Gmg n F G
. As one might observe
the solution is not time dependent since we calculate
the maximum value of the first energy step (peak)
by the approximation method.
For the purpose of the current research extension,
alternative solution that links between the energy
absorption of both spring and damper might yield
the following energy equation:
22
11
22
d st d st d d st st
mg h x x cv x x k x k x
(7)
In turn, by assuming that
2s
c k m
alongside
the previous assumptions and relations (3)-(4),
becomes (
,0
st
xq
):
22
10
42
st st
v G G G q h
xx
qq
g
(8)
where
st
x
should be solved numerically while all
the other properties that are dependent on the static
displacement should be derived using relations (3)-
(4). In case
mg
cv
is substituted into Eq. (7)
alongside relations (3)-(4), the following linear
relations in the explicit form will be resulted:
2
2
2
2
min max
2
min
2, 1 2
1
2, 2
, 2 ,
2
1
/24
st st
dd
critical st
d st
critical
d st d st st
h G mg
xk
Gqh
q
hG mg
x k G q
G q h
m v mg
c c G c k m
x x v
v v m
cc Gg x x x x k
(9)
The obtained coupled expressions (9) will be
compared continually in the next section with
literature reference calculations.
Another possible development for future study
that will be mentioned here is the case of friction
force participation (as illustrated in Fig. 2), the
calculation should include the work of the friction
force which is expressed by
r d n d s
F Q x x
,
where
n
Q
is the wall acting perpendicular normal
force (gaining the maximum value of
mg
while
being proportional to
, constantmg
) and
d
is the dynamic friction (whereas all friction
parameters are assumed to be constants),
respectively [34]. Adding the latter friction term
into the dynamic equation (7) by replacing
cv
with
d
cv mg
results with another non-linear
algebraic equation for
st
x
that should be solved
numerically in similar way to the previous
equilibrium (8) (with the friction energy extension
additive term
1
2
dG
q
):
221
1 1 0
4 2 2
d
st st
v G G G G h
xx
q q q
g
(10)
In case
mg
cv
is substituted into Eq. (9) alongside
relations (3)-(4), the following linear relations in the
explicit form are resulted:
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2
2
2
2
min max
2
min
2, 1 1 2
11
2, 2
, 2 ,
2
1
/24
st st d
d
d d d
d
critical st
d st
critical
d st d st st
h G G mg
xk
q q h
GG
qq
hG mg
x k G q G q
G q G q h
m v mg
c c G c k m
x x v
v v m
cc Gg x x x x k
(11)
Note that full accurate dynamic solution is presented
by [34, 36 - 37].
Next step, a connection between the solved
impact relations (5) and the following original well-
known ODE (ordinary differential equation)
dynamic equation accompanied with the boundary
conditions will be presented:
0
0
B.C.: (0) 0 ; (0) 2
mx cx kx mg
x x v gh
(12)
where
nת
are the natural frequency and the
damping ratio, respectively. x coordinate is
measured in relative to the initial location of the
mass suspension upper surface (cushion [4]) which
considered to be positive upward [11].
Hence, the dynamic stiffness and the damping
coefficient will be:
2
/ 2 / ;
2/
nd
k m k f m n
c km n
(13)
where
f
is the natural frequency [rounds/sec]
and represents the number of isolators,
respectively. Note that in the case of perpendicular
wall friction (Fig. 2 illustration), one should add the
term
mg
to Eq. (12), whereas the friction
coefficient is a function that might be varying in the
range
sd
such as the accurate differential
equation form as brought in [34, 36 - 37] is:
0
0
Dynamic motion: sgn 0;
Static state ( and ):
;
k
r s slide
r
mx cx kx mg f x
f mg
F mg x v
cx kx mg mx F
(14)
Note that a code in Python that might solve the
problem is attached as a link in the Appendix
section.
Although the full solution of the under damping
system ( ) (6) as a response to step
function without friction is [11, 24, 26]:
(15)
where is the initial velocity, is the
initial state displacement, is the natural system
resonance frequency, is the
dynamic frequency, and is the damping
factor. Note that if , then the resulted static
initial state is . Remark that a numerical
code for solving Eq. (12) alongside main cases of
dynamic behavior between two masses relationship
are presented in the Appendix section. For the
purpose of the forthcoming comparative discussion
concerning the maximum acceleration value (to
determine the G parameter), the second order
analytic differentiation manipulation of Eq. (15)
might lead to the following acceleration expression:
(16)
where and
. Now, using [4]
(trigonometric identity) and [11] ( )
relations/assumptions on Eq. (9), the maximum
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value of the evaluated impact force and acceleration
becomes [4]:
(17)
which is obtained for the critical natural frequency
( ) value, dependent on the stiffness to mass
square root ratio ( ).
In case a friction term is involved, Eq. (15) becomes
approximately:
(18)
The current literature cited equations that will be
used to compare with the current formulations are:
2
2
2
[2,3,5]: (linear system) or 0.09906 (non-linear system);
2
; ;
2 / 2 / 2 / 2 / 2 ;
;
d
st d st
st d
d impact d impact d d
impact impact impact
hh
xGG
WW
x k k
kk
k F x ma x hW x W G h
v
F ma a
2
22
0
2
;
2
12
/ / [ ];
22
2 / 2 [ sec/ ];
11
[4]: 1 2 / 1 , , 1;
2 2 2 2 2
[6,38]: / ; 2 / ; / ;
d
n s s
critical s n s
nn
st d d n st st
x
GG
f g x k m Hz
h
c k k m N m
p cg cg c
G w w h g p v
g k k km
k k q k f m n x mg k
2
2
/ ; - dimensionless
[7]: / [ / ];
/ [ ];
/ / [ ];
= 2 [ / ];
dd
d
st st
dd
st n
x mgG k q
k m Gg v N m
x mg k m
x v k m m
k f m N m
- dimensionless. G
(19)
where
W mg
.
3 Results and Discussion: Wire
Mounting Damper Application
To exemplify the use of relations (5) and (9)
versus relations (19) we will use the example
appearing in the Cavoflex company catalogue [6] (p.
38). Suppose we have a container with sensitive
equipment that should withstand falling from
on a hard surface
while
8G
and the total mass equals to
. The hit velocity is equal
to
2 1.4[ / sec]v gh m
. In addition, the natural
frequency is limited of not exceeding 12Hz.
Table 1. Suspension system parameters values comparison for example.
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Table 1 results alongside relations (5) and (9)
might lead to the following comprehensions:
Numerically, the obtained results are in the
same order as the literature results.
However, current calculation values of Eqs.
(5) are quantitatively agree with Refs. [2, 3,
5] and partially with static calculation [4]
(data is not sufficient available on the
dynamic stiffness coefficient calculation)
and [7]. The reason for numerical
differences are derived due to the direct
dependency on the natural frequency
oscillation that increases the dynamic
stiffness and reduces the dynamic distance
[6, 7] (reduced reliance only on basic
energy conservation, but considering full
frequency-time energetic behaviour) while
the current calculation concentrating on
basic energy conservation. Accordingly, the
isolator stiffness dynamic and static
properties values (expressions (5) and (9))
have different design requirements
(displacements are smaller for larger
stiffness values and reversal) than current
uncoupling results based on Refs. [6, 7]
(although the numeric values are close
quantitatively to the coupling solution (9)).
Also, an acceptable agreement was found
between Ref. [4] and Refs. [2, 3, 5].
Moreover, the coupling approximate
solution (Eq. (9)) values are quantitatively
close to the non-coupling solution for the
parameter values
1q
and
2q
,
respectively. The distinction between
solutions (5) and (9) is derived due to the
participation of the damping energy
potential term in the energy balance
equation, which in turn causes to the static
and dynamic displacement reduction in
relative to the uncoupling case (5).
One might observe in relations (5) that the
height of the falling (h) has a great influence
alongside other coefficients (q and G impact
intensity), in determining the static and
dynamic displacement (xst, xd), damping (c)
and stiffness (kst, kd). The falling height can
be expressed in the automotive industry
applications (falling into a pit or bump
obstacle) [40 - 41] or in aspects of storing a
package falling from the height of a heavy
truck or a level [2, 5, 10, 40 - 41].
The prominent advantageous of expression
solutions (5), (6), (9) ,(11), (16), and (18)
for different type of suspension state is that
it enables to make initial simple pre - design
evaluation of falling mass- body (like
cushion, containers, etc.) and to compare it
with advanced experimental and numerical -
simulation tools.
Note that G value should be a summation of
two (displacement) states: dynamic and
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static, such as the total G value is equal to
the dynamic state value added to the static
state; e.g. G = Gd + 1.
In cases where the loading magnitude reach
to its critical value (over 1 ton), then the
dynamic stiffness value has different value
than the static one ( , maximum
compression ratio: ) [6, 39].
Accordingly, the compression stiffness
always greater than the dynamic stiffness
( ) due its
mechanical mechanism enables the spring
shrinkage/compression, resulting the spring
stiffness diminishing until extreme state of
suspension buckling is likely to occur
(maximum coil loading durability).
In the case of tension loading, the opposite
case of compression will be fulfilled
( , but mostly with
smaller stiffness
ratio ) [39].
In case of shear configuration, the
suspension stiffness behaves as the tension
case ( ) until it
reaches to its critical loading. Afterwards,
the state flips to the compressive stiffness
case, so it becomes ( );
for the maximum loading configuration the
stiffness values are close such as
.
Finally, examining the Cavoflex catalog tables
[6], it can be observed that most of the differences
between the tables are expressed in the dynamic and
static displacements and spring stiffness
coefficients:
A. On the one hand, the static displacement
numerical value is substantially different
(ratio of 2 times). On the other hand, the static
displacement is greater than dynamic state in
the case of .
B. In the context of dynamic and static spring
stiffness values, dependent on the loading
parameter (q) parameter. The stiffness
coefficients in the case were found to
be significantly higher than the case .
4 Conclusion
In this study we present a general approximate
framework for analytically calculating the
suspension impact parameters (stiffness,
displacement and damping coefficients - (minimum
and maximum/critical values)) in the dynamic and
static states, based on simple energy conservation
and force equalities. Next, comparison with
literature has revealed that the obtained results are in
the same order. Moreover, current calculation values
were found to agree numerically with some of the
references, which have common assumptions or
conditions.
In addition, some design points were clarified
about the wire mounting isolators stiffness
properties dependent on their physical behaviour
(compression, shear tension), based on Cavoflex
company catalogue. We also mentioned briefly the
wall friction phenomena, analysing it both
numerically and analytically. Moreover, a coupling
energy equation was derived dependent on the
damping energy, friction work (with and without
cases), spring potential energy and gravitational
energy and was found to agree qualitatively with the
uncoupled case and some literature studies. Both
solutions, coupling and uncoupling, have been
revealed to complete each other and might explain
different literature attitudes and assumptions.
Finally, appendix section concerning the
numerical modelling of simple suspension systems
variations of two mass bodies is brought for the
reader use alongside free code.
I strongly believe that the prescribed approximations
could be beneficial to understand the impact
phenomena of cushion, containers and suspension
systems design to achieve simple and immediate
results on the complete (overall) suspension system
properties for the static and dynamic states
(displacements, stiffness, damping and frequency
parameters).
Appendix: A Simple suspension systems
variations of two mass bodies – mass loading vs.
Carrier
Simple suspension systems variations of
two mass bodies; mass loading vs. Carrier
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models are presented and demonstrated using
Matlab/Octave code.
Table of possible cases variations to
evaluate dynamic response of suspension
systems vibrations of two mass bodies: Mass
loading vs. Carrier is presented.
The target of this document is to evaluate in
each case of masses relation the corresponding
values of k (stiffness, spring coefficient) and b
(damping, damper coefficient) by using dynamic
numerical analysis.
Applications example: Carrier vs. its cargo
in different situations (loading, unloading,
travelling, etc.)
A computer program code in Python
language that includes the friction influence
could be found in the following link:
stackoverflow.com/questions/56754421/solving-
a-system-of-mass-spring-damper-and-coulomb-
friction
Table 2. Dynamic case for two masses relation and their mathematical representation
Matlab/Octave code
clc
clear all
close all
flag=1;
zeta=??;
g=-9.81; %[m/s^2]
k=??; %[N/m]
M=??; %[kg]
m=??; %[kg]
c=2*zeta*sqrt(k*(m+M)); %[Ns/m]
w_n=sqrt(k/(m+M)) %[rad/sec]
t=0:??:??; %[sec] t = [time t_0:
increment : t_final] %starting time for
impact detection
%%%%
function F=F_sin(A,w,D,t)
F=A*sin(w*t)+D;
endfunction
%%%
if flag == 1 %Unloading case - both
masses are moving together horizontally -
under horizontal impact force
x0=??; %[m]
v0=0; %[m/sec]
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delta=??; %[N]
D=delta; %[N]
A=0; %[N]
elseif flag == 2 %Loading case - both
masses are moving together after free fall of
mass M on the carrier m
x0=0; %[m]
v0=(M./(M+m))*sqrt(-2*g*h0);%[m/sec]
D=(m+M)*g; %[N]
A=0; %[N]
elseif flag == 3 %Both masses are free
falling together on the suspension system
x0=??; %[m]
v0=sqrt(-2*g*h0); %[m/sec]
D=(m+M)*g; %[N]
A=0; %[N]
elseif flag == 4 %Carrier m loaded with
mass M is vibrated along a given way under
sinusoidal force
c=2*zeta*sqrt(k*(m+M)); %[Ns/m]
w_n=sqrt(k/(m+M)) %[rad/sec]
v0=0; %[m/sec]
h0=??; %[m]
A=-??*(m+M)*g; %[N]
w=2*pi*??; %[rad/s]
D=(m+M)*g; %[N]
endif
x0=[h0 v0]; %x0 = [Initial
Position [m], Initial Velocity[m/sec]]
sistema=@(t,x)[x(2);(-c*x(2)-
k*x(1)+F_sin(A,w,D,t))/(m+M)];
[t,x] = ode45(sistema,t,x0);
accel=(-c*x(:,2)-
k*x(:,1)+F_sin(A,w,D,t))/m;
%%%Plot%%%
figure(1)
plot(t,x(:,1),'b');xlabel('time[sec]');ylabel('Po
sition [m]');grid on;
figure(2)
plot(t,3.6*x(:,2),'r');xlabel('time[sec])');ylab
el('Velocity [Km/sec]');grid on;
figure(3)
plot(t,-
accel./g,'g');xlabel('time[sec]');ylabel('G -
units');grid on;
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Data Availability
All data, models, and code generated or used during
the study appear in the submitted article.
Conflict of Interest
The corresponding author (Jacob Nagler) states that
there is no conflict of interest.
Sources of Funding for Research Presented
in a Scientific Article or Scientific Article
Itself
No funding is involved.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/de
ed.en_US
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DOI: 10.37394/23202.2022.21.2
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