OSCILLATIONS - Pécsi Tudományegyetem · Harmonic oscillations: ... SUPERPOSITION All kind of periodic ... Two linear simple harmonic motions combined (same and perpendicular directions).

OSCILLATIONS

PREMED COURSE, 14/08/2015

Mechanical Optical Laser

Electrical

AC 50 Hz

Astronomical

Biological

Bunjee jumping

Metronom

ECG

Pulsar

PERIODIC MOTIONS

PERIODIC MOTIONS

Another biological exampe

PERIODIC MOTIONS

Oscillation:

A physical quantity changes periodically in space and/or time. Movement, state or change that has a periodic component.

Harmonic oscillations:

Oscillation with a single frequency. It can be described with sine (or cosine) function. Constant amplitude and period time. (Any motion that repeats itself in equal intervals of time is called periodic or

harmonic motion.)

If a particle in periodic motion moves back and forth over the same path, we call the motion oscillatory or vibratory.

SINE AND COSINE FUNCTIONS

MEASURING ANGLES IN RADIAN

𝜃 = 𝑠

𝑟

ANGULAR VELOCITY

P; Time t

X; t=0

𝜔 =𝜃

𝑡 [rad/s]

𝑣 = 𝑟𝜔

HOOKE’S LAW

When an object is bent, streched or compressed by a displacement s, the restoring force F is directly proportional to the displacement (provided the elastic limit is not exceeded). F~-s 𝐹 = −𝑘𝑠 (restoring force ~ displacement) 𝑘: elastic constant

SIMPLE HARMONIC MOTION (SHM)

A body is moving with SHM if: 1. its acceleration is directly proportional to its distance from a fixed point on its path

and 2. its acceleration is always directed towards that point.

Force + acceleration

maximum

Velocity zero

𝑎 = −𝜔2𝑠(= −𝜔2r) 𝑎: acceleration of a particle 𝑠 (𝑟): displacement of the particle from the fixed point O 𝜔2: is a constant

Any system that obeys Hooke’s law will execute SHM.

EXAMPLES OF BODIES MOVING WITH SHM

1.

2. 3.

4.

5.

KINETICS OF PERIODIC MOTIONS I. Oscillation: vertical projection of circular motion

https://www.youtube.com/watch?v=P-Umre5Np_0

https://www.youtube.com/watch?v=ipSWPnBlsD4

KINETICS OF PERIODIC MOTIONS II. Oscillation: vertical projection of circular motion

KINETICS OF PERIODIC MOTIONS III.

(hertz)

(sec)

KINETICS OF PERIODIC MOTIONS IV.

CIRCULAR MOTION OSCILLATION

𝑣𝑐 =2𝜋𝑟

𝑇

𝑣𝑐 = rω

𝑣 𝑡 = 𝑣0 cos(𝜔𝑡)

𝑣0 = 𝐴𝜔

Acceleration

𝑎𝑐 = 0 𝑎𝑐𝑝 = 𝑟𝜔

2

𝑎 𝑡 = −𝑎0sin (𝜔𝑡) 𝑎0 = −𝐴𝜔

2

𝑎 = −𝐴𝜔2 sin 𝜔𝑡 = −𝑦𝜔2

Middle-position: End-position: 𝛼 = 𝜔𝑡 = 0°, 𝑣 = 𝑣0 cos 𝜔𝑡 = 𝑣0

Middle-position: 𝛼 = 𝜔𝑡 = 0°, 𝑎 = −𝑎0 sin 𝜔𝑡 = 0

𝛼 = 𝜔𝑡 = 90°, 𝑣 = 𝑣0 cos 𝜔𝑡 = 0 End-position:

𝛼 = 𝜔𝑡 = 90°, 𝑎 = −𝑎0 sin 𝜔𝑡 = −𝑎0

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DYNAMICS OF PERIODIC MOTIONS I. Force

𝐹 = 0

𝐹𝑔 = 𝐹𝑝

𝐹 = 𝑚𝑎

𝐹 = 𝐹𝑠 − 𝐷𝑥

𝐷 =𝐹

𝑥 (𝑁

𝑚)

Spring constant (D): force necessary for spring elongation.

Harmonic oscillations: displacement is proportional to the force acting on the oscillating body but its direction is opposite.

2

D

T m

2 Dx

mr

2m

TD

Dx ma

F

ω

Oscillation frequency depends only on the mass (m) and the spring constant (D), but not from the amplitude.

THE SIMPLE PENDULUM

𝑇 = 2𝜋𝑙

𝑔

g: acceleration due to gravity; l: length

DYNAMICS OF PERIODIC MOTIONS II. Energy

Potential and kinetic energy changes as a body moves with SHM.

SUPERPOSITION

All kind of periodic and non-periodic oscillation could be described as the sum (or integral) of individual sinusoidal oscillations (with different frequence, amplitude or phase)

Combinations of Harmonic Motions-Interference: The phenomenon in which two (or more) waves superpose each other to form a resultant wave Two linear simple harmonic motions combined (same and perpendicular directions). The resulting motion is the sum of two independent oscillations.

DAMPING (DAMPED OSCILLATIONS)

In damped harmonic motion the mechanical energy approaches zero as time increases, being transformed into internal thermal energy associated with the damping mechanism.

DRIVEN OSCILLATION, RESONANCE

Free (self-) oscillation: oscillating system without external force oscillate with its self-frequency

Driven oscillation: the oscillation is driven by external, periodic force oscillation frequency = external excitation frequency

amplitude, phase could be different

f f0

A

Driving frequency

Resonance: driving frequency is in the nearby range of the self-frequency.

SUMMARY

• Oscillation is the vertical projection of a circular motion.

• The displacement, speed and acceleration changes periodically in time.

• Periodic movements can be described with sine or cosine functions.

• The self-oscillating frequency depends only from the spring constant and the mass.

• The maximal elastic energy equals the maximal kinetic energy in an oscillating system.

• Oscillations can be added: interference, superposition.

• Without external force the oscillation get damped.