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Brian Covello's project examines the mass and spring relationship through differential equations. Information below is taken from wikipedia.org A differential equation is a mathematical equation that relates some function of one or more variables with its derivatives. Differential equations arise whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. Because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions — the set of functions that satisfy the equation. Only the simplest differential equations are solvable by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. For example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball's velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity. The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering. All of these disciplines are concerned with the properties of differential equations of various types. Pure mathematics focuses on the existence and uniqueness of solutions, while applied mathematics emphasizes the rigorous justification of the methods for approximating solutions.
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Brian Covello
Project 2: Mass-‐Spring System With Rubber Band
Background: Second order differential equations are frequently used to model the physical world.
This project aims to model the behavior of the model below:
For ease of calculations and conversion to maple format, the second-‐order differential
equations were reduced to a series of first order differential equations.
1) !"!"= 𝑥, !"
!"= 10 − 𝑘!𝑦
2) !"!"= 𝑥, !"
!"= 10 − 𝑘!𝑦 − 𝑏𝑥
3) !"!"= 𝑥, !"
!"= 10 − 𝑘!𝑦 − 𝑘!ℎ 𝑦 𝑓𝑜𝑟 𝑦 > 0 ; !"
!"= 𝑥, !"
!"= 10 − 𝑘!𝑦 − 0 𝑓𝑜𝑟 , 𝑦 < 0
4) !"!"= 𝑥, !"
!"= 10 − 𝑘!𝑦 − 𝑘!ℎ 𝑦 − 𝑏𝑥
For representation of the piecewise function in (3), the heaviside function was utilized.
This function can accurately describe model the restoring force of a rubber band when it is
stretched and the lack of force by the rubber band when it is compressed.
To this end, this project begins with an elementary
analysis of simple harmonic oscillator motion as defined
by gravity. From there, the effect of a dampener will be
examined with respect to the oscillatory motion. The
dampener will then be taken away, and a rubber band
will be added to the system. Finally, the combined effects
of a rubber band and a dampened system are analyzed.
We assume m=1 and gravitational force =10.
1) Simple harmonic oscillator 2) Harmonic oscillator with damping
3) Oscillator and rubber band, no
damping
4) Oscillator and rubber band with damping
Brian Covello
Calculations and Data:
Ideal Mass Spring System:
We begin by modeling an ideal mass-‐spring system with no rubber band given by the
equation: !"!"= 𝑥, !"
!"= 10 − 𝑘!𝑦 , where k1=12.5. In this model, the restoring force is supplied by
the spring. The phase portrait is represented on the left, and the time plots are represented on the
right with x as red and y as blue.
In the above graphs, y(0)=2 and x(0)=0. Thus, initial velocity is zero, and the
displacement from equilibrium is +2. Notice that when the system is displaced from
equilibrium 2 units in the downward direction, the velocity oscillates between -‐4 and 4.
The oscillations remain constant towards infinity in time. Below is a representation of the
system with no initial displacement from equilibrium and no initial velocity. This system
represents the effects of gravity upon the system starting at the equilibrium position.
Notice that gravity alone is insufficient to compress the spring.
Brian Covello
These simple harmonic oscillators have sinusoidal solutions as expected. The lack of
dampening is manifest in the constant amplitude of the time plot solutions. Lastly,
frequency of simple harmonic oscillators is represented by 𝜔 = 𝑘! and 𝑓 =!!! ;𝑇 = !
!.
Thus period is a constant 1.77 any initial values. This system provides a good basis for
continuing our exploration, but it fails to provide a realistic representation of the system
under realistic conditions. Real systems are dampened, thus we introduce the damping
coefficient b and a new model.
Damped Mass Spring System:
This system can be modeled by the equation !"!"= 𝑥, !"
!"= 10− 𝑘!𝑦 − 𝑏𝑥. Where k1
remains the 12.5 value from the previous model and b is the damping coefficient. The
spring mass system given by: 𝑚 !!!!!!
+ 𝑏 !"!"+ 𝑘!𝑦 = 0 leads to three different cases of
solutions. This reformation is possible due to static equilibrium conditions.
Case 1 (Over-‐damped): b2-‐4mk>0
Case 2 (Critically Damped): b2-‐4mk=0
Case 3 (Under damped): b2-‐4mk<0
It is highly unlikely that a natural system will be critically damped, however, case 2
provides a basis for understanding the differences in the following behaviors when b is
small compared to when b is large. Specifically, one expects a bifurcation value to occur at
𝑏 = 4𝑚𝑘 = 4 ∗ 12.5. We begin by analyzing the under damped case where b=1 at an
initial velocity of zero, and a displacement of 4.
Brian Covello
Here the phase plot spirals in to zero and the frequency is lower. Specifically, the frequency
is now reduced, having the value 𝜔! =|!!!!!"|!!
= 3.5, and T=1.795s. There is an
exponentially decaying amplitude and solutions with sine and cosine characteristics. Even
with different initial conditions these characteristics remain.
We now turn our attention to the over-‐damped case where b=10. Notice the time it takes
for the system to decay is substantially lower.
Here we notice no oscillatory of the system. At different initial conditions some initial
increase in displacement followed by continuous decaying amplitude.
Brian Covello
Mass-‐Spring and Rubber Band without Damping:
The third situation may be modeled by the piece wise equations !"!"= 𝑥, !"
!"= 10 −
𝑘!𝑦 − 𝑘!ℎ 𝑦 𝑓𝑜𝑟 𝑦 > 0 ; 𝑎𝑛𝑑 !"!"= 𝑥, !"
!"= 10 − 𝑘!𝑦 − 0 𝑓𝑜𝑟 , 𝑦 < 0. This was modeled using the
Heaviside function in maple. This function allows for modeling of the force exerted by the rubber
band as it is stretched AND the lack of force by the rubber band as it is compressed. We chose
k2=4.7, y(0)=4 and x(0)=0.
These graphs indicate a return to the ordinary oscillatory behavior of the ideal system. We
notice the period is once again 1.77.Below is initial conditions x(0)=0, y(0)=0. Notice there
is consistent behavior even with different initial conditions. With a lack of conservative
forces, the system will continue to oscillate as t approaches infinity.
Brian Covello
In these models, K2 represents the elasticity constant of the rubber band.
Mass-‐Spring with Rubber Band and Damping
This final model brings all additional models to focus. In this case the Heaviside
function remains, and the system is damped. This model is represented by the equation !"!"= 𝑥, !"
!"= 10 − 𝑘!𝑦 − 𝑘!ℎ 𝑦 − 𝑏𝑥 , where b is the damping coefficient, K2 is the elasticity
constant of the rubber band and k1 is the spring constant. We will let K1=12.5 and K2=4.7, varying
the damping coefficient. In the graphs below, y(0)=0, x(0)=0, and b=1.
Below are the graphs for initial conditions x(0), y(0)=4, and b=1. Here the conservative
forces act to decay exponentially the amplitude for larger values of t. The system is tending
towards its equilibrium position.
Brian Covello
Note the similarities between these graphs and the under damped case as portrayed
in the damped mass spring system above. Likewise, one may also notice the over damped
behavior when b=10 as depicted below:
One interesting case occurs as the damping coefficient becomes small. When b becomes
smaller, the system approaches resonance frequency. Below y(0)=10, x(0)=0, and
b=0.0001.
Brian Covello
Notice that near resonance frequencies, the system is continuously in motion as time
approaches infinity with varying amplitudes giving rise to their own frequency ranges.
Resonance frequencies have long been associated with collapse of bridges and occur quite
often in nature.
One expects the same type of consistent behavior to have a bifurcation value near that
of the damped mass spring system, portraying critical dampness around 7.07. At starting
conditions y(0)=4, x(0)=0, with b= 50, k1=12.5, k2=4.7 the following graphs were
generated.
Conclusively, this lab project allowed for increased insight into ideal and non-‐ideal systems
that were either overdamped, critically damped, or underdamped. The difference between
these types of motions may be seen below:
Brian Covello
Where the green represents underdamped motion, blue represents the overdamped and
red represents critically damped. Critically damped and overdamped very often look
similar, and three different types of plots may generally arise for these systems:
All non-‐zero solutions to non-‐ideal overdamped and critically damped spring mass systems
tend towards equilibrium as t increases. They pass through the equilibrium position at
most once. They have at most one maxima, and at most one point of inflection. These
systems tend to have solutions in the form:
Brian Covello
All non-‐zero solutions to non-‐ideal underdamped spring mass systems tend towards
equilibrium as t increases. They mas pass through the equilibrium position infinitely many
times, having infinitely many maxima. A generalized system may be seen below:
These systems are described by their imaginary roots as given in the equation below: