Energy Distribution in Spring Pendulums
M. C. de Sousa,1, a) F. A. Marcus,2, b) I. L. Caldas,1 and R. L. Viana2
1)
Institute of Physics, University of São Paulo, 05508-090, São Paulo, São Paulo,
Brazil
2)
Department of Physics, Federal University of Paraná, C.P. 19044, 81531-990, Curitiba, Paraná,
Brazil
arXiv:1704.04532v1 [nlin.CD] 14 Apr 2017
(Dated: 18 April 2017)
Spring pendulums are intrinsically nonlinear coupled systems that present spring-mass and pendulum like
motions. We analyze the dynamics of spring pendulums considering that the total energy is distributed among
the spring-mass and pendular motions, as well as the coupling between them. Namely, we write the total
energy as a sum of three terms: spring, pendulum and coupling, and we determine how the average energy of
each term varies as a function of the system parameters. Integrating numerically the equations of motion for
a large set of initial conditions covering the entire phase space, we identify regions in the parameter space that
correspond to strong and weak coupling. We also determine the values of parameters for which the average
spring, pendulum or coupling energy term vanishes, and the scaling laws that govern this feature.
The spring pendulum is a typical example of nonlinear system with complex behavior, such as
chaos and resonances. Its equations of motion
are used to model many physical systems, ranging from mechanical systems to plasma physics.
In a spring pendulum, the spring-mass and pendulum motions are coupled and, therefore, they
exchange energy. We find in the literature studies
about the energy exchange in individual trajectories. In this paper, we consider a great number of
trajectories and we analyze the energy exchange
in a more general way. We divide the total energy of the system in three terms that correspond
to the spring and pendulum motions, and the
coupling between them. We average the energy
terms in time and space, and we investigate how
the energy distribution changes according to the
parameters of the system. We analyze the coupling energy term to understand how it mediates
the energy exchange between the spring-mass and
pendular motions. The average coupling energy
term also allows us to identify the values of the
parameters for which the coupling in the system
is strong or weak. When the coupling is strong,
the spring and pendulum motions exchange energy constantly and it is difficult to distinguish
the two kinds of movement.
I.
INTRODUCTION
The spring pendulum, also known as elastic or extensible pendulum, is a paradigm for low-dimensional Hamiltonian systems with nonlinear characteristics. Its main
a) [email protected]
b) [email protected];
Current affiliation: Department of
Physics, Technological Institute of Aeronautics, 12228-900, São
José dos Campos, São Paulo, Brazil
features described in the literature are the energy exchange between the pendulum and the spring-mass movements in the parametric resonance1–3 condition, and the
order-chaos-order transition the system undergoes as its
parameters increase.
One of the first studies about a parametric resonance
for the spring pendulum and other coupled systems with
two degrees of freedom is the work by Vitt and Gorelik
in 19334 . From the 70’s onward, the spring pendulum
has been studied considering approximate analytical solutions for the parametric resonance5–9 , Poincaré sections
of numerical solutions to show the effects of the nonlinear
coupling10–12 , the Lyapunov exponents of chaotic trajectories, the power spectra of regular and chaotic trajectories, the applicability of the KAM theorem11 , and the
order-chaos-order transition the system presents13 .
Besides its dynamical features, the spring pendulum is
also relevant due to its qualitative representation of many
systems of great physical interest. Among these, some
examples are the orbits of celestial bodies14,15 , such as
satellites (both natural and artificial) and asteroids16,17 ,
the classical analogue for the vibrational modes of triatomic molecules producing the Fermi resonance in the
infrared and Raman spectra18–20 , the interaction between
light waves in a nonlinear medium21 , and wave coupling
in plasma physics22 .
Most of the publications about the spring pendulum
consider the behavior of individual trajectories and describe the system using analytical approximations for
small angles and low total energy that lead to the parametric resonance condition. However, many other features can be found by numerically integrating the nonlinear system of equations for a wide range of energy13
and initial conditions, as we discuss in this paper.
The spring pendulum dynamics depends on its total
energy and a control parameter that represents the ratio of the simple pendulum and spring-mass frequencies.
In this context, we numerically integrate the nonlinear
equations of motion for values of energy and control parameter that make the system not tractable analytically.
Moreover, we do not focus on the study of individual
2
trajectories. We analyze the dynamics of the system as a
whole, including all kinds of trajectories it may present:
periodic and quasi-periodic orbits, chaotic orbits, and
resonant islands.
The spring pendulum is an intrinsically coupled system, i.e. the coupling arises from the physical configuration of the system. In this paper, we analyze the intrinsic
coupling in spring pendulums and how it mediates the
energy exchange between the spring-mass and pendular
motions. To do so, we describe the system using coordinates that relate directly to the spring and pendulum
motions, and we write the Hamiltonian as a sum of three
terms, spring-mass, pendulum and coupling, representing, respectively, the energy associated to the spring and
pendulum motions and their coupling. Within this analysis, we find how the energy is distributed among the
considered three energy terms, and we construct scaling
laws for the energy components in terms of the total energy and the control parameter.
Considering this distribution of energy among three
terms (spring, pendulum and coupling), we identify a
transition from strong to weak coupling as we increase
the total energy of the system. When the coupling in
the system is strong, the spring-mass and the pendulum
move as a unique system and, most of the time, it is difficult to distinguish the two kinds of movement. For weak
coupling, the spring-mass and the pendulum slightly interact with each other. In this case, we can identify the
spring and pendulum individual motions for certain periods of time.
The mathematical description of the spring pendulum
is presented in Section II, where we introduce the coordinates that relate directly to the spring and pendulum
motions. In Section III, we distribute the total energy
of the system among three energy terms, spring, pendulum and coupling, and we justify our definitions for each
energy term. In Section IV, we analyze the spring pendulum dynamics as a whole, considering a great number
of trajectories that represent all the possible behaviors
of the system: invariant tori, resonant islands and chaos.
We calculate the average spring, pendulum and coupling
energy terms and we analyze how the energy distribution varies according to the total energy and the control
parameter. We obtain the scaling laws that govern the
energy distribution for this nonlinearly coupled system,
and we identify the regions of strong and weak coupling
in the parameter space. Finally, we draw our conclusions
in Section V.
II.
THE SPRING PENDULUM
The spring pendulum is composed by a mass m attached to the free extremity of a massless spring with
stiffness constant k and relaxed length (length in the absence of forces) l0 . The other extremity of the spring is
fixed at the center of the Cartesian coordinate system
(x, y) = (0, 0) as shown in Figure 1. The system moves
y
x
l0
l
q2
0
k
l
m
q1
FIG. 1. Schematic representation of a spring pendulum.
only in the vertical plane, and its stable equilibrium position corresponds to (x, y) = (0, −l), where l = l0 + mg/k
and g is the acceleration of gravity.
We write the Hamiltonian of the spring pendulum in
Cartesian coordinates as
ET = H =
p2x + p2y
k p
+ mgy + ( x2 + y 2 − l0 )2 ,
2m
2
(1)
where ET is the total energy of the system.
It is also common to consider the Cartesian coordinate
system centered at the spring pendulum stable equilibrium position8,10 . This can be done by using the dimensionless coordinates q1 = x/l and q2 = (y + l)/l shown
in Figure
1. In addition, we consider H = H/kl2 and
p
t = k/m t in order to obtain the dimensionless Hamiltonian:
p21 + p22
+ f (q2 − 1)
2
q
1
+ ( q12 + (q2 − 1)2 + f − 1)2 ,
2
ET = H =
(2)
where the parameter f is defined as f = mg/kl. From
the stable equilibrium condition l = l0 + mg/k, we have
l0 /l = 1 − f . It implies that f must be in the interval
]0, 1[, since l0 /l is a positive quantity.
To better understand how the total energy is distributed in the spring pendulum, one should describe
the system using coordinates that relate directly to the
spring-mass and pendulum motions, such as the canonical coordinates
q
ρ = f − 1 + q12 + (q2 − 1)2 ,
(3)
q1
θ = arctan
.
1 − q2
In the polar coordinate system defined by expressions
(3), ρ represents the spring extension or compression
from l0 , and θ is the angle formed between the mass m
and the vertical axis pointing down, as shown in Figure
3
FIG. 2. Time evolution of individual trajectories for f = 0.25 and (a) ET = −0.200, (b) ET = 0.275. The coordinates (ρ, θ)
are represented in the upper panels, whereas the lower panels show the time evolution of the momenta (pρ , pθ ).
1. Using these new coordinates, we rewrite Hamiltonian
(2) as
p2θ
1 2
ρ2
pρ +
ET = H =
+
2
2
(ρ + 1 − f )
2
− (ρ + 1 − f )f cos θ .
(4)
The total energy is the only constant of motion of
the non-integrable system with two degrees of freedom.
Hamiltonian (4) written in the polar coordinates (ρ, θ)
gives us a better view about the system and the two
types of motion it presents: spring-mass and pendulum.
However, its equations of motion are not very tractable
for direct integration. To perform our analysis, we use
a fourth-order symplectic integrator23 to solve the equations of motion of Hamiltonian (2). The results in the
(ρ, θ) coordinates are obtained through the canonical
transformation (3).
In the spring pendulum, the spring-mass and pendular motions are coupled by the products of ρ and cos θ,
and pθ by (ρ + 1 − f ), as can be seen from Hamiltonian (4) and the corresponding equations of motion. It
is important to notice that this coupling is intrinsic i.e.,
the coupling arises from the configuration of the physical
system. By replacing the fixed length rod of a simple pendulum with a spring, we create an intrinsically coupled
system and, thus, the spring-mass and pendular motions
exchange energy through the coupling. A system with
such proprieties is known as autoparametric system1 .
In the literature, there are some papers that study the
energy exchange for individual trajectories4–9 . In Figure
2, we show the time evolution of individual trajectories
for two different values of total energy ET . Figures 2.(a)
are characteristic of the parametric resonance in coupled
systems. For this kind of trajectory, we observe that the
spring-mass energy is transfered gradually to the swinging motion and back. When the spring transfers its energy to the pendulum, it remains almost still and only
the pendulum motion is appreciable. The opposite occurs
when the pendulum transfers energy back to the spring.
The pendulum moves just slightly, whereas the spring is
compressed and stretched with great energy. The behavior described is known as autoparametric resonance3 and
it occurs when the ratio of spring-mass and pendulum
normal modes is 2:1, which in our mathematical description of the system corresponds to f = 0.25.
Although the autoparametric resonance depicted in
Figure 2.(a) is largely studied in the literature4–9 , it occurs only for very specific configurations of the system:
low energy, small angles and f = 0.25. For general configurations of the system, the scenario represented in Figures 2.(b) is much more common. In these figures, both
the spring and the pendulum are in constant motion and
they exchange energy regularly.
Figure 2 shows the behavior of individual trajectories
for the spring pendulum. However, to analyze the dynamics of the system more generally, one should observe
its Poincaré sections. A Poincaré section shows the intersections of the 4-dimensional trajectories of the spring
pendulum with a plane. We represent several trajectories in the Poincaré section, exhibiting the different kinds
4
FIG. 3. (Color online) Poincaré sections showing the different kinds of trajectories presented by the system for f = 0.25 and
(a) ET = −0.200, (b) ET = 0.275. The trajectories in green (light grey) correspond to those depicted in Figure 2.
of behavior the system may present for a fixed value of
total energy and f . We represent one coordinate and
its associated momentum (p1 × q1 ) in the Poincaré section, whereas the other coordinate assumes a fixed value
(q2 = 0), and the points are plotted whenever its associated momentum is positive (p2 > 0).
Figure 3 shows the Poincaré sections of the spring pendulum for the same energies used in Figure 2. The green
(light grey) trajectories in Figure 3 correspond to those
depicted in Figure 2. For the same value of total energy
and f , the trajectories in black show us that the system
may present regular or chaotic behavior according to the
initial conditions. For ET = −0.200 and f = 0.25 as in
Figure 3.(a), the Poincaré section of the system is regular. Increasing the value of ET , the Poincaré section
presents regular islands immersed in a chaotic sea as can
be seen in Figure 3.(b).
The area covered by regular or chaotic trajectories in
the Poincaré sections of the system depends on the total
energy ET and the parameter f . The literature shows us
that the spring pendulum presents an order-chaos-order
transition as we increase one of these parameters10–13 .
For low values of ET or f , the system is regular. Increasing the total energy or the parameter f , the system
starts to present chaotic trajectories. The area covered
by chaos expands with ET and f , but at some point it
begins to diminish. For sufficiently large values of ET or
f , the system becomes regular once again.
III. ENERGY DISTRIBUTION: SPRING, PENDULUM
AND COUPLING TERMS
In a simple pendulum, the length of the rod is fixed.
In the spring pendulum, the fixed length rod is replaced
by a spring whose length varies in time, coupling the
spring and the pendulum motions. Since the spring-mass
and the pendulum motions are nonlinearly coupled, we
can regard the total energy terms in (4) as those due
to a simple pendulum, a spring-mass and the coupling
between them.
Following this idea, we consider that the total energy ET of the spring pendulum is distributed among
three distinct terms: spring-mass, pendulum and coupling. The spring term ES is the energy associated with
a spring-mass system moving vertically under the action
of gravity. The spring term represents the kinetic energy
of the spring-mass, as well as its elastic and gravitational
potential energy. It is a function of (ρ, pρ ) only, and we
write the spring term ES as
ES =
p2ρ + ρ2
− (ρ + 1 − f )f.
2
(5)
The pendulum term EP is a function of (θ, pθ ). It
corresponds to the energy stored in a simple pendulum,
in which the mass m is suspended by a rod of fixed length
l:
EP =
p2θ
− f cos θ.
2
(6)
We choose l as the fixed length of our simple pendulum
model because this is the length of the extended spring
in the stable equilibrium position of the spring pendulum
described by Hamiltonian (4).
In the spring pendulum, the spring-mass and pendulum motions are nonlinearly coupled as can be seen in
the second and fourth terms of Hamiltonian (4). It is important to notice that this intrinsic coupling is different
from the usually considered coupling between two distinct oscillators. The coupling in the spring pendulum is
associated to the energy exchange between the two kinds
of movement described by the energy terms (5) and (6).
We define the coupling energy term EC for the spring
pendulum as the amount of energy that arises from this
5
FIG. 4. Time evolution of the spring, pendulum and coupling energy terms for the trajectories depicted in Figure 2. For both
trajectories, the total energy of the system remains constant, as can be seen in the lowest panels.
nonlinear coupling:
1
p2θ
− 1 − (ρ − f )f cos θ
EC =
2 (ρ + 1 − f )2
+ (ρ + 1 − f )f.
(7)
As one would expect, the coupling energy term is a function of both the spring and pendulum coordinates: ρ, θ,
and pθ . Furthermore, using our definitions for the energy
terms (5)-(7), the total energy (4) of the spring pendulum
is given by
ET = ES + EP + EC .
(8)
The coupling energy term defined by (7) is very suitable for the limit cases the system may present. Supposing that only the spring-mass system moves in time,
whereas the pendular motion is suppressed, we have
θ = 0, pθ = 0, and EC = f (constant). On the other
hand, if the spring-mass holds still under the action of
gravity in the vertical position, it can be viewed as a rod
of fixed length l. In this case, only the pendulum moves
in time with ρ = f , pρ = 0, and EC = f (constant).
For these limit cases, where only the spring-mass or the
pendulum moves, the coupling energy term EC remains
constant and equals f . We point out that the value of
this constant depends on the referential chosen for the
potential energy due to gravity. In our definitions, we
chose Vg = 0 for y = 0, and thus we have EC = f . If
one chooses Vg = 0 for y = −l, which corresponds to the
stable equilibrium position of the system, then EC = 0
for the limit cases described above.
Although EC = 0 for the limit cases, when Vg = 0
for y = −l, the position of the referential Vg = 0 varies
with l, and consequently with the parameter f . For this
reason, the referential Vg = 0 for y = −l is not suitable
for the analysis we carry out in the following section.
Throughout this paper, we work with Vg = 0 for y = 0,
which is a fixed referential that does not vary with any
parameter.
Figure 4 depicts the time evolution of the energy terms
(5)-(7) for the individual trajectories represented in Figure 2. In Figure 2.(a), the system is close to the limit
case we have described in which either the spring or the
pendulum moves at a time. When only the spring-mass
or the pendulum moves, all the energy terms (5)-(7) remain constant, as can be seen in Figure 4.(a). When the
spring-mass and the pendulum motions exchange energy,
the coupling energy term EC oscillates, causing small oscillations in the spring and pendulum energy terms depicted in Figure 4.(a).
In Figure 2.(b), the spring-mass and the pendulum
move constantly. In this case, all the energy terms in
Figure 4.(b) oscillate regularly as the spring and the pendulum motions exchange energy. In both panels of Figure 4, the total energy of the system remains constant,
whereas the energy terms ES , EP and EC vary in time.
The energy distribution we propose, including a term
due to the nonlinear coupling, reveals new aspects of the
spring pendulum dynamics, as we discuss in the following section. It allows us to analyze how the energy is
transferred between the spring-mass and the pendulum
6
FIG. 5. (Color online) Normalized average energy as a function of the parameter f for positive values of the total energy ET .
The red (dot dashed) curves correspond to the normalized average energy of the spring term, the blue (dashed) curves represent
the pendulum energy term, and the green (solid) curves represent the coupling energy term. The vertical dotted lines in black
indicate the position of minimum and maximum values.
like motions, and how the two kinds of movement are coupled. The energy distribution introduces a new approach
to the study of spring pendulums and other systems with
nonlinear coupling.
IV. SCALING LAWS FOR THE ENERGY
DISTRIBUTION
In the literature, we find some studies about energy exchange for individual trajectories in spring pendulums4–9 .
These studies focus especially in the parametric resonance shown in Figure 2.(a). However, the energy exchange observed for the parametric resonance occurs only
when the system moves with low energy, small angles and
f = 0.25. In general, the energy exchange in spring pendulums is very different from the one observed for the
parametric resonance. For individual trajectories, it may
be similar to Figure 2.(b) or even chaotic.
In this paper, we do not study individual trajectories
as in Figure 2. We analyze the energy exchange in spring
pendulums in a more general way. We consider a great
number of trajectories to reproduce all the possible behaviors of the system, such as in Figure 3. We calculate
the average energy terms for the system and we obtain
scaling laws showing how the energy distribution varies
with the parameters.
The spring pendulum described by Hamiltonian (4)
presents just two parameters: the total energy ET and
the parameter f that accounts for the physical characteristics of the system. For each value of ET and f , we
choose around 20000 initial conditions evenly distributed
throughout the phase space. These initial conditions correspond to all kinds of trajectories, such as, invariant tori,
resonant islands and chaos. Thus, we build a statistic for
the system that considers the entire phase space and all
the dynamical properties it presents.
For each initial condition in phase space, we integrate
the Hamilton’s equations to obtain the time evolution of
the trajectory and we evaluate the average energy terms
in the time interval t = [0, 500]. Considering all the initial
conditions, we calculate the average energy terms E i for
the whole phase space, obtaining numerical results that
are both spatial and temporal averages. We then normalize the average energies for each component (spring,
pendulum and coupling) to the interval [0, 1].
Figure 5 shows the normalized average energy terms for
positive values of the total energy ET . From this figure,
we observe that the average energy terms follow a regular
pattern as we increase the value of ET . For low values
7
FIG. 6. Values of ET and f for which the normalized average energy of the (a) spring, (b) pendulum, and (c) coupling terms
vanish. Numerical data may be fitted by second and third order polynomials as shown by the solid and dashed back curves in
the graphics.
of total energy as in panels (a)-(e), the coupling energy
term is generally higher than the spring and pendulum
energy terms. It indicates a strong coupling in the system
for such values of ET . When the coupling is strong, the
spring and pendulum motions exchange a great amount
of energy and it is difficult to distinguish the two types
of motion.
In Figures 5.(a)-(e), we notice that for specific values
of the parameter f , the average spring and/or pendulum energy term vanishes when the total energy lies in
the interval ET =]0.00, 0.77]. The values of f for which
|E S |normalized = 0 or |E P |normalized = 0 are represented
by the vertical dotted lines in black in Figure 5. These
vertical lines show us that when the average spring or
pendulum energy term vanishes, the slope of the curve
representing the coupling energy term changes.
For ET =]0.00, 0.18[, the value of f for which
|E S |normalized = 0 also corresponds to the maximum
value of the coupling energy term. On the other hand,
for ET = [0.18, 0.77], the maximum value of the coupling energy term corresponds to |E P |normalized = 0.
When ET = 0.129 or 0.560 (Figures 5.(b) and 5.(e)),
|E S |normalized = 0 and |E P |normalized = 0 for the same
value of f and, thus, |E C |normalized = 1. In this situation, all the energy of the system, in average, is concentrated in the coupling energy term, whereas the average
spring and pendulum energy terms vanish.
Figures 6.(a) and 6.(b) show the position of
|E S |normalized = 0 and |E P |normalized = 0 as a function of the total energy ET and the parameter f . The
null values of the spring and pendulum energy terms vary
regularly according to these parameters, and they can be
fitted by second and third order polynomials represented
by the solid and dashed back curves in the graphics.
In Figure 6, the curves for the spring and pendulum
energy terms present two distinct regions, and we use
a different polynomial for each region. The regions in
the spring energy term are ET =]0.0, 0.19[ and ET =
[0.19, 0.56]. The curve for the pendulum energy term
is divided in two regions ET = [0.02, 0.15] and ET =
]0.15, 0.77]. The transition from one region to the other
corresponds to changes in the behavior of the normalized
average energy terms |E i |normalized .
For intermediate values of the total energy as in Figure
5.(f), the average coupling energy term is generally lower
than the average spring and pendulum energy terms,
whereas the average pendulum energy term is the highest one. For intermediate and high values of ET , there
are no longer values of f for which |E C |normalized = 1,
indicating a weaker coupling in the system.
For ET = [0.15, 2.14], the average coupling energy term
vanishes for specific values of the parameter f . When
|E C |normalized = 0, the average spring and pendulum
energy terms are maximum, as can be seen by the dotted
vertical lines in Figures 5.(e) and 5.(f). It means that
the spring and pendulum energy terms concentrate, in
average, all the energy of the system, whereas the average
coupling energy term vanishes.
Figure 6.(c) shows the position of |E C |normalized = 0
as a function of the total energy ET and the parameter
f . The position of |E C |normalized = 0 varies regularly
with the parameters, and it can be fitted by a second
order polynomial. Unlike the curves for the spring and
pendulum energy terms in panels (a) and (b), the curve
representing the coupling energy term is not divided in
different regions, and a single polynomial describes the
behavior of the entire curve.
Figures 5 and 6 show the behavior of the normalized
average energy terms for positive values of the total energy of the system. For negative values of the total energy, the |E i |normalized terms present a very different behavior, as can be seen in Figure 7.(a). When ET < 0,
the average coupling energy term is always greater than
the average spring and pendulum energy terms, whereas
the average spring energy term is the smallest one. From
Figure 7.(a), we also notice that for ET < 0, some values
of parameter f are not allowed because they do not comply with the condition of minimum energy in the system
8
FIG. 7. (Color online) Normalized average energy as a function of the parameter f for negative and null values of the
total energy ET .
ET > f 2 /2 − f .
When the total energy of the system is null, the average coupling energy term presents a very interesting
behavior as can be seen in Figure 7.(b). For ET = 0,
|E C |normalized is constant and equal to 0.5 for all values
of f . It means that, when ET = 0, the coupling energy
term concentrates, in average, exactly half of the system
energy and it does not depend on the parameter f .
Figure 8 shows the normalized average coupling energy term as a function of the total energy ET and the
parameter f . For negative and low positive values of the
total energy, the coupling in the system is stronger, as
indicated by the orange (light grey) color in Figure 8.
The average coupling energy term reaches its maximum
values (|E C |normalized ' 1) in the white region of the
picture. For intermediate values of the total energy, the
coupling in the system becomes weaker and reaches its
minimum values (|E C |normalized ' 0) in the black region of Figure 8. This back region is similar to the curve
|E C |normalized = 0 displayed in Figure 6.(c).
After the back region for which |E C |normalized ' 0
in Figure 8, the average coupling energy term starts to
increase again. The purple (intermediate grey) color in
the picture indicates that the coupling in the system is
moderate when the total energy ET is high. For these
values of ET , the pendular motion dominates the system,
whereas the average spring energy term is the smallest
one.
V.
CONCLUSIONS
We analyzed a spring pendulum and how its energy is
distributed in the system. The spring pendulum is an intrinsically nonlinear coupled system that can behave as a
pendulum, a vertical spring or a combination of both. In
this sense, we considered the total energy of the spring
pendulum distributed among three terms: spring, pen-
FIG. 8. (Color online) Normalized average coupling energy
term as a function of ET and f . The hatched area is not
allowed because these values of ET and f do not comply with
the condition of minimum energy ET > f 2 /2 − f .
dulum and coupling. The spring energy term represents
a spring moving vertically under the action of gravity.
The pendulum term is the energy stored in a simple pendulum. Since the spring and pendulum motions are coupled, the system also presents a coupling energy term
that mediates the energy exchanges between the spring
and the pendulum motions.
For single trajectories, we showed that our definition
for the energy terms agrees with the behavior of the system. When only the spring moves, whereas the pendulum remains still, the spring energy term is equal to the
total energy of the system. The situation is analogous
when only the pendulum moves. On the other hand,
when both the spring and the pendulum are in motion,
they exchange energy through the coupling energy term,
which oscillates in time according to the rate and the
direction of the energy exchange.
We also used our definition for the energy terms to
study the general behavior of the spring pendulum as
a function of its total energy and the parameter that
accounts for the physical characteristics of the system.
We evaluated the energy terms for a great number of
trajectories, obtaining results that are both temporal and
spatial averaged.
From the average energy terms, we were able to identify regions of strong and weak coupling in the parameter
space of the system. For negative and low positive values of total energy, we showed that the coupling energy
term is greater, in average, than the spring and the pendulum energy terms. It means that the spring pendulum
presents a strong coupling, with the spring and the pendulum exchanging a great amount of energy. The two
subsystems behave as a unique new system and, most of
the time, it is difficult to identify the individual springmass and pendular motions.
For intermediate values of total energy, we found that
the coupling in the system is weak. The coupling energy
9
term reaches its minimum values, and it may be null
for specific regions in the parameter space. When the
coupling is weak, the spring and the pendulum slightly
interfere in each other motion and it is easy to identify
the two different kinds of movement in certain periods of
time.
When the total energy is high, we observed a moderate
coupling. In this case, the pendulum energy term dominates the dynamics of the system, whereas the spring
energy term is the lowest one.
The study we presented in this paper allowed us to observe new features of the spring pendulum dynamics, and
to determine, on average, how the coupling mediates the
energy exchange between the different kinds of movement
the system may present. We point out that the methods
we developed are not restricted to the analysis of spring
pendulums. They may also be applied to other nonlinearly coupled systems, providing new perspectives and
contributing for a better understanding about their dynamics, and how the energy distribution regulates the
behavior of nonlinear systems.
SUPPLEMENTARY MATERIAL
See Supplementary Material for the parameter space
representing the normalized average spring and pendulum energy terms. We also present a video that shows
how the normalized average energy terms vary according
to the parameters of the system.
ACKNOWLEDGMENTS
We thank Dr. Kai Ullmann for the discussions that
contributed to the present work. We acknowledge financial support from the Brazilian scientific agencies: São
Paulo Research Foundation (FAPESP) under Grants No.
2015/05186-0 and No. 2011/19296-1, Conselho Nacional
de Desenvolvimento Cientı́fico e Tecnológico (CNPq) under Grants No. 457030/2014-3 and No. 157317/2015-3,
and Coordenação de Aperfeiçoamento de Pessoal de Nı́vel
Superior (Capes).
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