Download presentation

Presentation is loading. Please wait.

1
**Lecture D31 : Linear Harmonic Oscillator**

Spring-Mass System Spring Force F = −kx, k > 0 Newton’s Second Law (Define) Natural frequency (and period) Equation of a linear harmonic oscillator

2
Solution General solution or, Initial conditions Solution, or,

3
**Graphical Representation**

Displacement, Velocity and Acceleration

4
**Energy Conservation No dissipation T + V = constant Potential Energy**

Equilibrium Position No dissipation T + V = constant Potential Energy At Equilibrium −kδst +mg = 0,

5
**Energy Conservation (cont’d)**

Kinetic Energy Conservation of energy Governing equation Above represents a very general way of de-riving equations of motion (Lagrangian Me-chanics)

6
**Energy Conservation (cont’d)**

If V = 0 at the equilibrium position,

7
**Examples Spring-mass systems Rotating machinery**

Pendulums (small amplitude) Oscillating bodies (small amplitude) Aircraft motion (Phugoid) Waves (String, Surface, Volume, etc.) Circuits . . .

8
**The Phugoid Idealized situation**

Small perturbations (h′, v′) about steady level flight (h0, v0) L = W (≡ mg) for v = v0, but L ∼ v2,

9
**The Phugoid (cont’d) Vertical momentum equation**

Energy conservation T = D (to first order) Equations of motion

10
**The Phugoid (cont’d) h′ and v′ satisfy a Harmonic Oscillator Equa-**

tion Natural frequency and Period Light aircraft v0 ∼ 150 ft/s → τ ∼ 20s Solution

11
The Phugoid (cont’d) Integrate v′ equation

Similar presentations

OK

Phy 212: General Physics II Chapter 15: Oscillations Lecture Notes.

Phy 212: General Physics II Chapter 15: Oscillations Lecture Notes.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on producers consumers and decomposers 3rd Ppt on 555 timer ic Ppt on viruses and bacteria youtube Ppt on agriculture and its types Ppt on case study of infosys Ppt on monitoring and evaluation Ppt on 98 notified sections of companies act 2013 Ppt on conservation of angular momentum Ppt on carbon and its compounds in chemistry Ppt on solid dielectrics glass