The document summarizes an experimental and theoretical study of a mechanical system consisting of a sliding car that bounces on an oscillating piston. The researchers were able to produce period-1, period-2, period-3 orbits, sticking solutions, and chaotic behaviors experimentally that agreed well with the theoretical model. They aim to identify parameters that produce different orbit types and find co-existing orbits by varying the piston frequency and amplitude. The experimental setup uses infrared lights and a Nintendo Wii remote to track the car's position over time.
The cubic root unscented kalman filter to estimate the position and orientat...IJECEIAES
In this paper we introduce a cubic root unscented kalman filter (CRUKF) compared to the unscented kalman filter (UKF) for calculating the covariance cubic matrix and covariance matrix within a sensor fusion algorithm to estimate the measurements of an omnidirectional mobile robot trajectory. We study the fusion of the data obtained by the position and orientation with a good precision to localize the robot in an external medium; we apply the techniques of kalman filter (KF) to the estimation of the trajectory. We suppose a movement of mobile robot on a plan in two dimensions. The sensor approach is based on the CRUKF and too on the standard UKF which are modified to handle measurements from the position and orientation. A real-time implementation is done on a three-wheeled omnidirectional mobile robot, using a dynamic model with trajectories. The algorithm is analyzed and validated with simulations.
Advances in Satellite Conjunction Analysis with OR.A.SIAntonios Arkas
As the number of the manmade objects increases in space, so does the interest and the research effort on the critical and interesting issues of collision probability assessment and decision making for cases of close approach events.
New interesting theoretical analysis has been recently published by Michael Scott Balch, Ryan Martin and Scott Ferson, on the mathematical subtleties connecting the phenomenon of probability dilution with the fundamental difference between frequentist and Bayesian approaches in statistical inference, and inspirational work has been presented from CNES by F.Laporte through his papers which describe JAC software and his approach to covariance realism.
OR.A.SI, the Flight Dynamics software for GEO and LEO that I’ve been developing for the last 17 years in C++, has been endowed since 2012 with early close approach detection based on the TLE files released from JSpOC, calculation of collision probability (S.Alfano method) based on the secondary object details found in CDM (Conjunction Data Message), Middle Man features (processing and analysis of CDM batches released for the same event) and evasive manoeuvre computation.
This new presentation exposes the latest enhancements, of the already powerful OR.A.SI routines, with all these new exiting advances. In brief the contents of the attached presentation are the following:
1. CASI (Close Approach Simulator) – Development of an analytic simulator which produces close approach events for whatever regime (LEO, MEO and GEO), and renders the probabilistic study and analysis of such events independent from the need of a CDM.
2. Computation and visualization of the probability dilution area in the two dimensional space of Kp and Ks scale factors used for the computation of the scaled probability of collision.
3. Computation of the scale factor interval in order to increase covariance realism, based on hypothesis testing with the Kolmogorov-Smirnov test (F.Laporte - CNES).
4. Computation of the effect of evasive manoeuvres, parametrized in time and velocity increment, on the scaled probability of collision.
I welcome you to the subtle but beautiful world of probabilities and inferential statistics or else how we managed to harness our ignorance to precise science!
The cubic root unscented kalman filter to estimate the position and orientat...IJECEIAES
In this paper we introduce a cubic root unscented kalman filter (CRUKF) compared to the unscented kalman filter (UKF) for calculating the covariance cubic matrix and covariance matrix within a sensor fusion algorithm to estimate the measurements of an omnidirectional mobile robot trajectory. We study the fusion of the data obtained by the position and orientation with a good precision to localize the robot in an external medium; we apply the techniques of kalman filter (KF) to the estimation of the trajectory. We suppose a movement of mobile robot on a plan in two dimensions. The sensor approach is based on the CRUKF and too on the standard UKF which are modified to handle measurements from the position and orientation. A real-time implementation is done on a three-wheeled omnidirectional mobile robot, using a dynamic model with trajectories. The algorithm is analyzed and validated with simulations.
Advances in Satellite Conjunction Analysis with OR.A.SIAntonios Arkas
As the number of the manmade objects increases in space, so does the interest and the research effort on the critical and interesting issues of collision probability assessment and decision making for cases of close approach events.
New interesting theoretical analysis has been recently published by Michael Scott Balch, Ryan Martin and Scott Ferson, on the mathematical subtleties connecting the phenomenon of probability dilution with the fundamental difference between frequentist and Bayesian approaches in statistical inference, and inspirational work has been presented from CNES by F.Laporte through his papers which describe JAC software and his approach to covariance realism.
OR.A.SI, the Flight Dynamics software for GEO and LEO that I’ve been developing for the last 17 years in C++, has been endowed since 2012 with early close approach detection based on the TLE files released from JSpOC, calculation of collision probability (S.Alfano method) based on the secondary object details found in CDM (Conjunction Data Message), Middle Man features (processing and analysis of CDM batches released for the same event) and evasive manoeuvre computation.
This new presentation exposes the latest enhancements, of the already powerful OR.A.SI routines, with all these new exiting advances. In brief the contents of the attached presentation are the following:
1. CASI (Close Approach Simulator) – Development of an analytic simulator which produces close approach events for whatever regime (LEO, MEO and GEO), and renders the probabilistic study and analysis of such events independent from the need of a CDM.
2. Computation and visualization of the probability dilution area in the two dimensional space of Kp and Ks scale factors used for the computation of the scaled probability of collision.
3. Computation of the scale factor interval in order to increase covariance realism, based on hypothesis testing with the Kolmogorov-Smirnov test (F.Laporte - CNES).
4. Computation of the effect of evasive manoeuvres, parametrized in time and velocity increment, on the scaled probability of collision.
I welcome you to the subtle but beautiful world of probabilities and inferential statistics or else how we managed to harness our ignorance to precise science!
The SpaceDrive Project - First Results on EMDrive and Mach-Effect ThrustersSérgio Sacani
Propellantless propulsion is believed to be the best option for interstellar travel. However, photon rockets or solar sails have thrusts so low that maybe only nano-scaled spacecraft may reach the next star within our lifetime using very high-power laser beams. Following into the footsteps of earlier breakthrough propulsion programs, we are investigating different concepts based on non-classical/revolutionary propulsion ideas that claim to be at least an order of magnitude more efficient in producing thrust compared to photon rockets. Our intention is to develop an excellent research infrastructure to test new ideas and measure thrusts and/or artefacts with high confidence to determine if a concept works and if it does how to scale it up. At present, we are focusing on two possible revolutionary concepts: The EMDrive and the Mach-Effect Thruster. The first concept uses microwaves in a truncated cone-shaped cavity that is claimed to produce thrust. Although it is not clear on which theoretical basis this can work, several experimental tests have been reported in the literature, which warrants a closer examination. The second concept is believed to generate mass fluctuations in a piezo-crystal stack that creates non-zero time-averaged thrusts. Here we are reporting first results of our improved thrust balance as well as EMDrive and Mach-Effect thruster models. Special attention is given to the investigation and identification of error sources that cause false thrust signals. Our results show that the magnetic interaction from not sufficiently shielded cables or thrusters are a major factor that needs to be taken into account for proper μN thrust measurements for these type of devices.
Gravitational field and potential, escape velocity, universal gravitational l...lovizabasharat
What is Escape Velocity-its derivation-examples-applications
Universal Gravitational Law-Derivation and Examples
Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples
A Numerical Integration Scheme For The Dynamic Motion Of Rigid Bodies Using T...IJRES Journal
The dynamics of rigid bodies have been studied extensively. However, a certain class of time-integration schemes were not consistent since they added vectors not belonging to the same tangent space (so3), of the Lie group (SO3) of the Special Orthogonal transformations in E3. The work of Cardona[1,2], and later Makinen[3,4], highlighted this fact using the rotation vector as the main parameter in their derivations. Some other programs in multibody dynamics, such as the work of Haug[5], rely on the Euler parameters, instead of the rotation vector, as the main variable in their formulations. For this class of programs, different time-integration schemes could be used .This paper discusses one such a scheme. As an example of application, the spinning top was used in this paper. For such a problem, the approximate change of the potential energy was found to be an upper bound to the change in the actual total energy during a time step.
The SpaceDrive Project - First Results on EMDrive and Mach-Effect ThrustersSérgio Sacani
Propellantless propulsion is believed to be the best option for interstellar travel. However, photon rockets or solar sails have thrusts so low that maybe only nano-scaled spacecraft may reach the next star within our lifetime using very high-power laser beams. Following into the footsteps of earlier breakthrough propulsion programs, we are investigating different concepts based on non-classical/revolutionary propulsion ideas that claim to be at least an order of magnitude more efficient in producing thrust compared to photon rockets. Our intention is to develop an excellent research infrastructure to test new ideas and measure thrusts and/or artefacts with high confidence to determine if a concept works and if it does how to scale it up. At present, we are focusing on two possible revolutionary concepts: The EMDrive and the Mach-Effect Thruster. The first concept uses microwaves in a truncated cone-shaped cavity that is claimed to produce thrust. Although it is not clear on which theoretical basis this can work, several experimental tests have been reported in the literature, which warrants a closer examination. The second concept is believed to generate mass fluctuations in a piezo-crystal stack that creates non-zero time-averaged thrusts. Here we are reporting first results of our improved thrust balance as well as EMDrive and Mach-Effect thruster models. Special attention is given to the investigation and identification of error sources that cause false thrust signals. Our results show that the magnetic interaction from not sufficiently shielded cables or thrusters are a major factor that needs to be taken into account for proper μN thrust measurements for these type of devices.
Gravitational field and potential, escape velocity, universal gravitational l...lovizabasharat
What is Escape Velocity-its derivation-examples-applications
Universal Gravitational Law-Derivation and Examples
Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples
A Numerical Integration Scheme For The Dynamic Motion Of Rigid Bodies Using T...IJRES Journal
The dynamics of rigid bodies have been studied extensively. However, a certain class of time-integration schemes were not consistent since they added vectors not belonging to the same tangent space (so3), of the Lie group (SO3) of the Special Orthogonal transformations in E3. The work of Cardona[1,2], and later Makinen[3,4], highlighted this fact using the rotation vector as the main parameter in their derivations. Some other programs in multibody dynamics, such as the work of Haug[5], rely on the Euler parameters, instead of the rotation vector, as the main variable in their formulations. For this class of programs, different time-integration schemes could be used .This paper discusses one such a scheme. As an example of application, the spinning top was used in this paper. For such a problem, the approximate change of the potential energy was found to be an upper bound to the change in the actual total energy during a time step.
Poster of my master\'s research presented at the Physics@FOM conference at Veldhoven on 20 januari 2010. There\'s one error in the equations, can you find it?
—This paper presents a new image based visual servoing (IBVS) control scheme for omnidirectional wheeled mobile robots with four swedish wheels. The contribution is the proposal of a scheme that consider the overall dynamic of the system; this means, we put together mechanical and electrical dynamics. The actuators are direct current (DC) motors, which imply that the system input signals are armature voltage applied to DC motors. In our control scheme the PD control law and eye-to-hand camera configuration are used to compute the armature voltages and to measure system states, respectively. Stability proof is performed via Lypunov direct method and LaSalle's invariance principle. Simulation and experimental results were performed in order to validate the theoretical proposal and to show the good performance of the posture errors. Keywords—IBVS, posture control, omnidirectional wheeled mobile robot, dynamic actuator, Lyapunov direct method.
REPORT SUMMARYVibration refers to a mechanical.docxdebishakespeare
REPORT SUMMARY
Vibration refers to a mechanical phenomenon involving oscillations about a point. These oscillations can be of any imaginable range of amplitudes and frequencies, with each combination having its own effect. These effects can be positive and purposefully induced, but they can also be unintentional and catastrophic. It's therefore imperative to understand how to classify and model vibration.
Within the classroom portion of ME 345, we discussed damped and undamped vibrations, appropriate models, and several of their properties. The purpose of Lab 3 is to give us the corresponding "hands-on" experience to cement our understanding of the theory.
As it turns out, vibration can be modeled with a simple spring-mass system (spring-mass-damper system for damped vibration). In order to create a mathematical model for our simple spring-mass system, we apply Newton's second law and sum the forces about the mass. After applying some of our knowledge of differential equations, the result is a second order linear differential equation (in vector form). This can easily be converted to the scalar version, from which it's easy to glean various properties of the vibration (i.e. natural frequency, period, etc.).
In the lab, we were provided with a PASCO motion sensor, USB link, ramp, and accompanying software. All of the aforementioned equipment was already assembled and connected. The ramp was set up at an angle with a stop on the elevated end and the motion sensor on the lower end. The sensor was connected to the USB link, which was in turn connected to the computer. We chose to use the Xplorer GLX software to interface with the sensor and record our data. After receiving our equipment, we gathered data on our spring's extension with a known load to derive a spring constant. We were provided with a small cart to which we attached weights to increase its mass. In order to model free vibration, we placed the cart on the track and attached it to the stop at the top of the ramp with a spring. After displacing the cart a certain distance from its equilibrium point, the cart was released and was allowed to oscillate on the track while we recorded its distance from the sensor. This was done with displacements of -20cm, -10cm, +10cm, and +20cm from the system's equilibrium point. After gathering this data for the "free" case, a magnet was attached to the front of the car, spaced as far from the track as possible. As the track is magnetic, this caused a slight damping effect, basically converting our spring-mass system to an underdamped spring-mass-damper system. After repeating the procedure for the "free" case, we moved the magnets as close to the track as possible (causing the system to become overdamped) and again repeated the procedure for the "free" case.
We were finally able to determine the period, phase angle, damping coefficients, and circular and cyclical frequencies for the three systems. There were similarities and differ ...
Attitude Control of Satellite Test Setup Using Reaction WheelsA. Bilal Özcan
A reaction wheel is A type of flywheel used primarily by spacecraft for attitude control without using fuel for rockets or other reaction devices.It bases on the principle of angular momentum transfer. That is Newton’s third law of action-reaction.
1st paper: https://www.researchgate.net/publication/338119144_ATTITUDE_CONTROL_OF_SATELLITE_TEST_SETUP_USING_REACTION_WHEELS
Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...IJERA Editor
Analysis of the vibrational behavior of a system is extremely important, both for the evaluation of operating conditions, as performance and safety reason. The studies on vibration concentrate their efforts on understanding the natural phenomena and the development of mathematical theories to describe the vibration of physical systems. The purpose of this study is to evaluate an undamped system with two-degrees-of-freedom and demonstrate by comparing the results obtained in the experimental, numerical and analytical modeling the characteristics that describe a structure in terms of its natural characteristics. The experiment was conducted in PUC-MG where the data were acquired to determine the natural frequency of the system. We also developed an experimental test bed for vibrations studies for graduate and undergraduate students. In analytical modeling were represented all the important aspects of the system. In order, to obtain the mathematical equations is used MATLAB to solve the equations that describe the characteristics of system behavior. For the simulation and numerical solution of the system, we use a computational tool ABAQUS. The comparison between the results obtained in the experiment and the numerical was considered satisfactory using the exact solutions. This study demonstrates that calculation of the adopted conditions on a system with two-degrees-of-freedom can be applied to complex systems with many degrees of freedom and proved to be an excellent learning tool for determining the modal analysis of a system. One of the goals is to use the developed platform to be used as a didactical experiment system for vibration and modal analysis classes at PUC Minas. The idea is to give the students an opportunity to test, play, calculate and confirm the results in vibration and modal analysis in a low-cost platform
EGME 431 Term ProjectJake Bailey, CSU FullertonSpring .docxSALU18
EGME 431 Term Project
Jake Bailey, CSU Fullerton
Spring 2016
This document serves to set forth the requirements for your term project, and the criteria
which such project submissions shall be judged. This outline should be the first point of inquiry
for any questions you may have about your project.
The project consists of a thorough investigation, analysis, and set of design improvement sug-
gestions for a simplified automobile suspension model. The dynamics of this model are rather
complex: as such, I have provided a detailed derivation of the equations of motion for this system
to you in a separate document. Your responsibility will be that of the analyst: use the provided dy-
namic models to investigate the system’s response to typical inputs, judge these responses critically,
and suggest improvements to the system.
Your project submissions shall consist of a single analysis and design report. The project
report shall be turned in no later than the final class meeting of the semester, which is May 10,
2016 at 7:00 PM. As always, late assignments will not be accepted. The report shall, at a minimum,
include:
• A description of your analysis methodology
• A summary of the important results from your analyses, including plots and data tables where
appropriate
• A thorough defense of your analysis results, including (but not limited to):
– comparison with analytical approximations
– investigation of typical results of published investigations, and
– discussion and investigation of the approximate errors accrued in your simulations
• A succinct description of the modifications you propose to improve the performance of the
system, including justification of your choices
The dynamic models which have been provided to you include both a fully coupled, non-linear
model and a simplified, linearized version. It is up to you to decide which to use for each portion
of the tasks outlined below. Note, however, that you should, at a minimum, simulate both models
under a common input. This will server as a basis for comparison.
Your specific tasks for this project are as follows:
1. Find the response of the system to a variety of inputs, including steps, impulses, and harmonic
excitation.
2. Determine the Displacement Transmissibility Ratio and Force Transmissibility Ratio of the
system over a range of input frequencies.
1
3. Using judgment, analytical techniques, and/or optimization methods, find a new set of sys-
tem parameters (stiffnesses and damping coefficients) which will improve the response of the
system to the selected inputs.
4. Finally, prepare a report which thoroughly summarizes and defends your methodology and
results.
A final word on collaboration. You are encouraged to discuss your ideas and your solution
approach with your classmates and colleagues. You are, however, expressly forbidden from sharing
simulation data, code, spreadsheets, scripts, or the like with anyone. Two students submitting
substantially similar s ...
FREQUENCY RESPONSE ANALYSIS OF 3-DOF HUMAN LOWER LIMBSIJCI JOURNAL
Frequent and prolonged expose of human body to vibrations can induce back pain and physical disorder
and degeneration of tissue. The biomechanical model of human lower limbs are modeled as a three degree
of freedom linear spring-mass-damper system to estimate forces and frequencies. Then three degree of
freedom system was analysed using state space method to find natural frequency and mode shape. A
program was develop to solve simplified equations and results were plotted and discussed in detail. The
mass, stiffness and damping coefficient of various segments are taken from references. The optimal values
of the damping ratios of the body segments are estimated, for the three degrees of freedom model. At last
resonance frequencies are found to avoid expose of lower limbs to such environment for optimum comfort.
Some experiments done on MATLAB for the course on Simulation and Modelling. Includes Model of Bouncing Ball, Model of Spring Mass System and Model of Traffic Flow.
1. Poster Design & Printing by Genigraphics® - 800.790.4001
<Andrew Miller and Ricardo Carretero>
<Nonlinear Dynamical Systems Group>
Email:andrew4@rohan.sdsu.edu
Website:
http://www.rohan.sdsu.edu/~rcarrete/
We consider, experimentally and
theoretically, a mechanical system
consisting of a sliding car on an
inclined plane that bounces on an
oscillating piston. The main aim of our
study is to reproduce the different
types of orbits displayed by nonlinear
dynamical systems. In particular, we
are looking to identify the parameters
and initial conditions for which
periodic and chaotic (irregular)
behavior are exhibited. The data in our
experimental model is collected by
using infrared lights that are monitored
by a Nintendo Wii remote which is
linked to a laptop computer via
Bluetooth. By varying the piston’s
frequency and amplitude, it is possible
to produce, for relatively small
amplitudes, periodic orbits. As the
amplitude of the piston is increased
(for a fixed piston frequency) we
observe bifurcations where the
original, stable, periodic orbit is
destabilized and replaced by a higher
order periodic orbit. For larger
amplitudes, periodic orbits are
destabilized and replaced by chaotic
trajectories. After carefully measuring
all experimental parameters, we are
able to successfully produce periodic
orbits (up to period 3), sticking
solutions (the car does not bounce, but
gets stuck to the piston), and
seemingly chaotic (irregular/
unpredictable) behavior that are in very
good agreement with the model for the
same parameter values.
Nonlinear Dynamics in the Periodically Forced Bouncing Car:
An Experimental and Theoretical Study.
Andrew Miller, R. Carretero-González, and Ricardo Nemirovsky
Nonlinear Dynamical Systems Group, Computational Sciences Research Center,
and Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, USA
• We have been able to produce experimentally
period-1, period-2, period-3, sticking solutions, and
seemingly chaotic behaviors.
• By fitting the experimental parameters we observe
good quantitative agreement between the
experimental and theoretical models as seen in
Figs.1-3.
• Some of our experimental results show
discrepancies in amplitude due to a spring attached
to the back of the sliding car. This spring adds up to
3 centimeters of error to our experimental results as
well as creates a gap between the parabolic
trajectory and oscillating sine function on the graph.
Our experimental system consists of:
- A Nintendo Wii remote
- Bluetooth reciever
- Infrared lights
- A computer
- A sliding car on an inclined plane
- An oscillating piston.
• Modeling the spring that is attached to the back of
the sliding car. This will give us a better model for
our system (it will also give us the experimental
graphs with less errors in the amplitude).
• Identifying the parameters for which there are co-
existing orbits (distinct orbits that co-exist for the
same parameter values). To accomplish this, we will
hold the amplitude constant while doing a frequency
sweep in both directions.
• Compare the basins of attraction (These are regions
of initial conditions in phase space that tend towards
specific attractors) between model and experiment.
Chaos: In mathematics, chaos is defined as seemingly
stochastic behavior occurring in deterministic systems.
This seemingly stochastic behavior is found to be
caused by sensitive dependence to initial conditions;
namely, small (infinitesimal) errors in the prescription of
the initial conditions are amplified exponentially. This
sensitive dependence to initial conditions is the major
obstacle when dealing with chaotic systems since it
precludes our ability to obtain long-term predictions.
Chaos theory, in general, tries to explain naturally
occurring phenomena such as population growth,
which displays seemingly random behavior although its
dynamics are inherently deterministic.
Our Goal: We hope to identify the parameters for
which the different types of orbits (periodic, chaotic,
etc.) in nonlinear dynamical systems are exhibited.
Furthermore, we are also working on finding,
experimentally, orbits that co-exist for the same
parameter values by conducting frequency sweeps of
our system. Understanding how and when chaos
emerges in simple systems will be very useful to
understanding other, more complex, phenomena.
INTRODUCTION
MATERIALS
Alligood, K.T., Sauer, T.D., & Yorke, J.A. (1996). Chaos: An introduction to
dynamical systems. New York: Springer-Verlag.
Kaplan, D., &Glass, L. (1995). Understanding nonlinear dynamics. New
York: Springer-Verlag.
Mello, T.M. & Tufillaro, N.B. (1986). Strange attractors of a bouncing ball.
American Journal of Physics, 55, 316-320.
Peek, B. (2009). WiimoteLib [version 15] (software). Available from
http://wiimotelib.codeplex.com/
Tufillaro, N.B., Abbot, T., & Reilly, J. (1992). An experimental approach to
nonlinear dynamics and chaos. New York: Addison Wesley.
Future Work
ResultsEquations
REFERENCES
Figure 1. Experimental results. (a) Period-1 orbit. (b) Period-2 orbit and (c) Period-3 orbit
Figure 2a.
ABSTRACT
CONTACT
Figure 2. Same as in Figure 1 for our theoretical model.
Figure 3. Impact maps corresponding to Figure 1. Red markers correspond to the initial condition, green marker/s to the
final periodic orbit, and blue markers correspond to the transient orbit. n green markers correspond to Period-n orbit.
(a) (b) (c)
θ
𝑔↓ 𝑒
𝑔
θ
(a) (b) (c)
(b)(a) (c)
The following equations were used in our theoretical model:
• The trajectory of the car, due to gravity, is given by: 𝑥(𝑡)=− 𝑔↓𝑒 ( 𝑡↑2 /2 )+ 𝑣↓0 𝑡+ 𝑥↓0 . (1)
• The piston follows the harmonic oscillation: s(𝑡)= 𝐴[sin(ω 𝑡+ θ↓0 )+1]. (2)
• The distance between the car and piston is: 𝑑(𝑡)= 𝑥(𝑡)− 𝑠(𝑡) . (3)
• Substituting 𝑥(𝑡)and 𝑠(𝑡) for an impact time ( 𝑑( 𝑡↓𝑘 )=0) yields:
• 0= 𝐴[sin(θ↓𝑘 ) +1]+ 𝑣↓𝑘 [1/ω (θ↓𝑘+1 −θ↓𝑘 )]−1/2 𝑔↓𝑒 [1/ω (θ↓𝑘+1 −θ↓𝑘 )]↑2 − 𝐴[sin(θ↓𝑘+1 ) +1].
(4)
• This equation implicitly gives θ↓𝑘+1 .
• After each collision, the velocity has to be adjusted (due to dissipation during impact) by using:
• 𝑣↓𝑘 =(1+α) 𝑢↓𝑘 −α 𝑣′↓𝑘 (where 𝑢↓𝑘 is the velocity of the piston ). (5)
• Taking the derivative of the piston’s position and the car’s position, and substituting their values into
equation (5): 𝑣↓𝑘+1 =(1+α)𝐴ωcos(θ↓𝑘+1 ) −α{ 𝑣↓𝑘 − 𝑔↓𝑒 [ 1/𝜔 (θ↓𝑘+1 −θ↓𝑘 )]} (6)
• This equation gives the next impact velocity
• From equations (4) and (6) we get ( θ↓𝑘+1 , 𝑣↓𝑘+1 ), the next impact between the car and piston.
These impacts are plotted on an impact map. A Period-n orbit will have n points on its impact map.
Amplitude=3.84
(c)
This work was supported by a STEM scholarship award
funded by the National Science Foundation grant
DUE-0850283.
Acknowledgement