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Math 225 001
Harmonic Motion
Simple harmonic motion is the periodic, oscillatory motion of a mass in an undamped,
unforced system [6]. Mathematically, such a system is described by a simple sine or cosine wave
as:
,
…(1)
where A is the amplitude of the motion, φ is the phase angle, and ω is the angular frequency. One
application of simple harmonic motion is the mass-spring oscillator, which consists of a mass
attached to one end of a spring. The equation that describes this system is governed by Newton’s
second law which states that:
,
where m is the inertial mass, and
…(2)
is the acceleration. When the mass is displaced from its
equilibrium position, the spring exerts a force that resists the displacement and is given by
Hooke’s Law:
,
…(3)
where k is the stiffness of the spring, and y is the displacement of the mass. Since acceleration is
the second derivative of position, simple harmonic motion is governed by the following second
order ordinary differential equation:
…(4)
In addition, due to their mechanical nature, most systems experience damping; this is a force that
can either oppose or amplify the oscillatory motion of the spring and is written as:
…(5)
where b is the damping coefficient, and
is the velocity of the mass. Finally, the system can
experience forcing when other external forces, such as gravity, electricity, or magnetism act upon
the system. Using Newton’s second law, these individual force equations can be combined into a
single second order ordinary differential equation that describes the motion of the mass-spring
system and is given by:
,
where
…(6)
are the external forces applied to the system. For such a system, the term
resonance describes the state at which the system oscillates at its natural frequency of vibration,
also known as its resonance frequency, and absorbs a maximum amount of energy [8]. Different
variations of equation (6), ranging from simple to highly complex, are used in a variety of fields
including quantum physics [7], biology [3], oceanography [4], and environmental science [1].
One application of harmonic motion is the mathematical model of mood variation in
bipolar patients which is used to study mental disorders in the fields of psychology and medicine
[2]. Bipolar II disorder is psychological disorder that is characterized by alternating hypomania
and depressive episodes. By making the assumption that these mood swings are periodic and
intensify without medical treatment, a negatively damped harmonic oscillator can be used to
model their variation. The equation that governs the oscillatory mood variations of an untreated
patient with bipolar II disorder is generally presented in the form:
α > 0,
where x is the patient’s emotional state,
…(7)
is the rate of the mood changes between hypomania
and major depression, ω is the natural frequency of the oscillator, and α is the damping
coefficient. This mathematical approach to studying bipolar disorder is a valuable technique, as it
offers an alternative method for examining the dynamic characteristics of psychological
illnesses. However, the unbounded oscillations which result from the negative damping
coefficient make the model problematic and unrealistic; such unlimited oscillations imply that
untreated individuals will experience mood swings that become infinitely severe over time.
Nevertheless, this equation is useful as it allows researchers to make predictions regarding
various medical treatments and their potential for controlling the mood variations of patients
with the disorder. Thus, when applied to the study of mental illnesses, harmonic motion
equations prove to be valuable tools for understanding and analyzing the oscillatory emotional
behaviour of individuals suffering from bipolar II disorder.
Simple harmonic motion is also used in the subject areas of environmental science and
agricultural economics when studying the vibrations and dynamic movement of trees in response
to mechanical harvesting techniques [1]. Trees are dynamic structures that respond to forces with
complex movements that are directly related to their mass distribution and organ stiffness. When
mechanically harvesting fruit, a trunk shaker applies vibrations to the tree’s stalk. These
vibrations are transferred to the branches of the tree, causing the fruit to detach. Trees use a
variety of different damping techniques to dissipate energy; some sources of damping include
friction generated in its stem and roots and aerodynamic drag produced by its canopy area. The
forced vibrations applied by the machine when combined with the natural damping mechanisms
of the tree, cause the tree to behave as a damping harmonic oscillator. Using Newton’s second
law, the equation for the motion of the tree can be written as:
,
…(8)
where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, and
,
,
, and f(t) are, respectively, the acceleration, velocity, position, and applied force
vectors as functions of time, t. Resonance modes are observed when the force applied by the
mechanical shakers produces a frequency that is close to that of any of the natural frequencies of
the tree or its organs. Resonance results in large displacements in tree motion which, in turn,
increase the risk of damage to the tree. Thus, resonance modes are important pieces of
information that can be used to design and employ harvesting practices that lead to sustainable
crops. Because the characteristics of trees widely vary, with each tree having its own unique
shape, size, and branching pattern, it often becomes challenging to take into account all the
different variables and accurately model the system. Nevertheless, harmonic oscillator equations
are important tools for studying the affects of mechanical harvesting and for increasing the
profitability of crops.
Harmonic motion describes the oscillatory behaviour of a system that is governed by a
second order ordinary differential equation [6]. Since these models can be valuable resources in
many fields of study, an understanding of harmonic oscillators and their basic equation form is
extremely important. Harmonic motion models have a diverse range of application including
physics where they describe the dissipation of energy in a two-capacitor system [5],
oceanography where they describe the variation in water depth of the shallow open sea as a result
of wind currents [4], and biology where they describe the propagation of electric field signals
throughout the interior of a cell [3]. These models can vary in their level of difficulty from
simplistic systems with undamped, free oscillations, to highly complex systems influenced by
damping and forcing [6]. Thus, due to their accuracy and broad relevance, harmonic motion
models are fundamental tools for understanding, studying, and making predictions about the
world in which we live.
References
[1]
S. Castro-García, G. L. Blanco-Roldán, and J. A. Gil-Ribes et al. “Dynamic Analysis of
Olive Trees in Intensive Orchards Under Forced Vibration.” Trees – Structure and
Function 22.6 (2008): 795-802.
[2]
D. Daugherty, T. Urrea-Roque, and J. Troyer et al. “Mathematical Models of Bipolar
Disorder.” Communications in Nonlinear Science and Numerical Simulation 14.7 (2009):
2897-2908.
[3]
F. X. Hart. “The Mechanical Transduction of Physiological Strength Electric Fields.”
Bioelectromagnetics 29.6 (2008): 447-455.
[4]
K. T. Jung, H. W. Kang, and H. J. Lee et al. “Ekman Motion in Shallow Open Sea in the
Presence of Time-Harmonic Variation of Water Depth.” Continental Shelf Research 27.9
(2007): 1287-1302.
[5]
K. Lee. “The Two-Capacitor Problem Revisited: A Mechanical Harmonic Oscillator
Model Approach.” European Journal of Physics 30.1 (2009): 69-84.
[6]
K. Nagle, E. B. Saff, and A. D. Snider. Fundamentals of Differential Equations and
Boundary Value Problems. Pearson Education, Boston, fifth edition, 2008. Sections 4.1,
4.9, 4.10.
[7]
J. R. Taylor, C. D. Zafiratos, and M. A. Dubson. Modern Physics for Scientists and
Engineers. Pearson Education, New Jersey, second edition, 2004. Section 7.5.
[8]
“Mechanical Resonance.” Wikipedia, The Free Encyclopedia. 28 Feb. 2009.
<http://en.wikipedia.org/wiki/Resonance_(acoustics_and_mechanics)>.