1. Oscillations

2. Mass Spring

3. The period is about 6.3 seconds, the frequency is 1/period=.016 Hz, the amplitude of velocity is about 23 m/s and the amplitude of position is about 44 m. The velocity has a phase that is 90 degrees (1/4 Period) ahead of position.

4. length(t) = length(t - dt) + (Vel) * dt INIT length = 0 INFLOWS: Vel = velocity velocity(t) = velocity(t - dt) + (Acceleration) * dt INIT velocity = 10 INFLOWS: Acceleration = equilibrium_Length-length equilibrium_Length = 20

6. Assume that the units of position are mm and velocity are mm/s. For this mass spring system what is: The period of oscillation? The frequency of oscillation The amplitude of the velocity? And the phase difference between the velocity and position?

7. Assume that the units of position are mm and velocity are mm/s. For this mass spring system what is: The period of oscillation? ~3.6 second The frequency of oscillation? 1/3.6=0.28 Hz The amplitude of the velocity? About 550 mm/s And the phase difference between the velocity and position? Velocity is 90 degrees ahead of position

8. position(t) = position(t - dt) + (velocity) * dt INIT position = InitialPosition INFLOWS: velocity = VelocityStock VelocityStock(t) = VelocityStock(t - dt) + (acceleration) * dt INIT VelocityStock = InitialVelocity INFLOWS: acceleration = -gravity-position*springStiffness/mass gravity = 981 InitialPosition = 10 InitialVelocity = 0 mass = 200 springStiffness = 2000 Y_eq = -gravity*mass/springStiffness

11. Grades and Study time

12. Current_GPA(t) = Current_GPA(t - dt) + (improvement_in_grades) * dt INIT Current_GPA = 3.3 INFLOWS: improvement_in_grades = Effect_of_extra_studying_on_grades*(Study_time- Normal_Study_time) Study_time(t) = Study_time(t - dt) + (Change_in_Study_time) * dt INIT Study_time = 20 INFLOWS: Change_in_Study_time = Effect_of_grade_on_studying*(Desire_GPA-Current_GPA) Desire_GPA = 3.5 Effect_of_extra_studying_on_grades =.03 Effect_of_grade_on_studying = 4 Normal_Study_time = 20

13. Car Sales inventory

14. Resonance

15. If a system with a natural oscillation frequency (or natural frequencies) is driven by an outside influence that matches a natural frequency of the system, then the system will have a maximum amplitude oscillation at that frequency. This natural frequency is also sometimes called the resonant frequency. What frequency gives the largest output signal from a system with the smallest input signal?

16. Resonance If a system with a natural oscillation frequency (or natural frequencies) is driven by an outside influence that matches a natural frequency of the system, then the system will have a maximum amplitude oscillation at that frequency. This natural frequency is also sometimes called the resonant frequency. What frequency gives the largest output signal from a system with the smallest input signal? The resonant frequency

17. Bay of Fundy Nova Scotia

18. Resonance The system above has a natural frequency of 2.0 Hz (period 0.50 s). What is the resonant frequency of this system?

19. Resonance The system above has a natural frequency of 2.0 Hz (period 0.50 s). What is the resonant frequency of this system? 2.0 Hz, the same as the natural frequency

20. Resonance A system has four dominant natural frequencies corresponding to periods of 2.0 years, 5.0 years, 7.0 years, and 11.0 years. If an external driving influence of period 3.0 years acts on this system what sort of oscillations would one expect?

21. Resonance A system has four dominant natural frequencies corresponding to periods of 2.0 years, 5.0 years, 7.0 years, and 11.0 years. If an external driving influence of period 3.0 years acts on this system what sort of oscillations would one expect? Not much except for small amplitude random oscillations.

22. The ice age cycles have a very strong 100,000 yr component. However the solar cycles, thought to drive the ice age cycles, have a week 100,000 yr variation. This suggests that the Earth-System may be resonating at this time scale.

23. Beats (When two different oscillation combine) The beat frequency is simply the difference between the two frequencies. The time between beats (or beat period) is 1/(beat frequency). Two oscillations, one of frequency 1.01 Hz and one with frequency 1.00 Hz, combine. What is the beat frequency? And what is the time between Beats?

24. Beats (When two different oscillation combine) The beat frequency is simply the difference between the two frequencies. The time between beats (or beat period) is 1/(beat frequency). Two oscillations, one of frequency 1.01 Hz and one with frequency 1.00 Hz, combine. What is the beat frequency? 0.01 Hz And what is the time between Beats? 1/0.01 Hz=100 seconds