Week Eigenvalue problems for ODEs Example 1:

Slides:



Advertisements
Similar presentations
Coulomb or Dry Friction Damping.
Advertisements

MEEG 5113 Modal Analysis Set 3.
Ch 3.8: Mechanical & Electrical Vibrations
Chapter Ten Oscillatory Motion. When a block attached to a spring is set into motion, its position is a periodic function of time. When we considered.
example: four masses on springs
Chaper 15, Oscillation Simple Harmonic Motion (SHM)
Phy 212: General Physics II Chapter 15: Oscillations Lecture Notes.
The simple pendulum Energy approach q T m mg PH421:Oscillations F09
Halliday/Resnick/Walker Fundamentals of Physics 8th edition
Solving the Harmonic Oscillator
Ch 3.9: Forced Vibrations We continue the discussion of the last section, and now consider the presence of a periodic external force:
Mechanical Energy and Simple Harmonic Oscillator 8.01 Week 09D
Chapter 15 Oscillatory Motion.
13. Oscillatory Motion. Oscillatory Motion 3 If one displaces a system from a position of stable equilibrium the system will move back and forth, that.
Motion of a mass at the end of a spring Differential equation for simple harmonic oscillation Amplitude, period, frequency and angular frequency Energetics.
Harmonic Oscillation 1. If a force F acts on a spring, the length x changes. The change is proportional to the restoring force (Hooke’s Law). A spring.
Chapter 19 MECHANICAL VIBRATIONS
Describing Periodic Motion AP Physics. Hooke’s Law.
Mechanical Vibrations In many mechanical systems: The motion is an oscillation with the position of static equilibrium as the center.
Lecture 2 Differential equations
Oscillations – motions that repeat themselves Period ( T ) – the time for one complete oscillation Frequency ( f ) – the number of oscillations completed.
Physics 430: Lecture 25 Coupled Oscillations
Simple Harmonic Motion
Chapter 11 Vibrations and Waves.
Chapter 15: Oscillations
SECOND ORDER LINEAR Des WITH CONSTANT COEFFICIENTS.
SECOND-ORDER DIFFERENTIAL EQUATIONS
Vibrations & Waves. In the example of a mass on a horizontal spring, m has a value of 0.80 kg and the spring constant, k, is 180 N/m. At time t = 0 the.
Oscillatory motion (chapter twelve)
Oscillations – motions that repeat themselves Period ( T ) – the time for one complete oscillation Frequency ( f ) – the number of oscillations completed.
Vibrations and Waves Hooke’s Law Elastic Potential Energy Simple Harmonic Motion.
SIMPLE HARMONIC MOTION. STARTER MAKE A LIST OF OBJECTS THAT EXPERIENCE VIBRATIONS:
What is called vibration Analysis Design
Periodic Motions.
Ball in a Bowl: F g F N F g F N  F  F Simple Harmonic Motion (SHM) Stable Equilibrium (restoring force, not constant force)
Spring 2002 Lecture #18 Dr. Jaehoon Yu 1.Simple Harmonic Motion 2.Energy of the Simple Harmonic Oscillator 3.The Pendulum Today’s Homework Assignment.
Damped Free Oscillations
PHY 151: Lecture Motion of an Object attached to a Spring 12.2 Particle in Simple Harmonic Motion 12.3 Energy of the Simple Harmonic Oscillator.
PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2 SIMPLE HARMONIC MOTION: NEWTON’S LAW
1 Week 11 Numerical methods for ODEs 1.The basics: finite differences, meshes 2.The Euler method.
Chapter 13: Oscillatory Motion
Week 9 4. Method of variation of parameters
Vibrations in undamped linear 2-dof systems
Oscillations Simple Harmonic Motion
Dawson High School AP Physics 1
Applications of SHM and Energy
10. Harmonic oscillator Simple harmonic motion
Week 8 Second-order ODEs Second-order linear homogeneous ODEs
Differential Equations
Voronkov Vladimir Vasilyevich
Harmonic Motion (III) Physics 1D03 - Lecture 33.
PHYS 1443 – Section 003 Lecture #21
2.3 Differential Operators.
Chapter 15: Oscillations
Solving the Harmonic Oscillator
Ch 3.9: Forced Vibrations We continue the discussion of the last section, and now consider the presence of a periodic external force:
Theoretical Mechanics DYNAMICS
Oscillations Readings: Chapter 14.
Oscillatory Motion Periodic motion Spring-mass system
Signals and Systems Lecture 3
Boyce/DiPrima 10th ed, Ch 7.6: Complex Eigenvalues Elementary Differential Equations and Boundary Value Problems, 10th edition, by William E. Boyce and.
VIBRATION.
VIBRATION.
PHYS 1443 – Section 002 Lecture #25
Physics : Oscillatory Motion
Week 6 2. Solving ODEs using Fourier series (forced oscillations)
LATTICE VIBRATIONS.
Undamped Forced Oscillations
MULTI DEGREE OF FREEDOM (M-DOF)
Week 8 Second-order ODEs Second-order linear homogeneous ODEs
Presentation transcript:

Week 10 6. Eigenvalue problems for ODEs Example 1: Consider a boundary-value problem for a 2nd-order linear homogeneous ODE, (1) (2) where λ is a constant. (1)-(2) are obviously satisfied if y(x) = 0 for all x, which is called the trivial solution. It turns out that (1)-(2) may also have non-trivial solutions, but this occurs only for special values of λ.

These values and the corresponding non-trivial solutions are called the eigenvalues (EVs) and the eigenfunctions (EFs), and (1)-(2) is an eigenvalue problem (EVP). To solve (1)-(2), first solve equation (1)... (3) Substitute (3) into the boundary conditions (2): = 1 = 0 ≠ 0 Observe that B can’t be zero (why?) – hence...

hence, (4) where n = ±1, ±2, ±3... is a non-zero (why?) integer. The EFs can be found by substituting A = 0 and equality (4) into the general solution (3), (5) Comments: Typical features of eigenvalue problems Note that EVP (1)-(2) has infinitely many solutions. Note that the constant B in expression (5) for the EFs remains undetermined.

Example 2: Solve the following EVP: where λ is the eigenvalue. Example 3: Solve the following EVP:

6. Modelling: Forced oscillations Let an elastic spring be attached to the wall and a body of mass m: Let the body’s coordinate x be counted from the equilibrium position (such that the spring is neither squeezed nor stretched). The body’s motion is thus fully described by the function x(t), where t is the time (don’t get confused by the unusual notation).

Note that Newton’s Second Law states that hence, (6) where Fexternal represents human intervention, wind, earthquake, gravity from Alpha Centauri, etc. Regarding Fspring, the physicists told us that...

(Hook’s Law) where k is the spring’s modulus (to be measured by the physicists). The “–” shows that the spring force is of opposite direction to the displacement. Regarding Ffriction , the physicists told us that (7) where the “–” shows that the friction force is opposite to the velocity. (7) can be rewritten in the form

The most interesting case is that of a periodic external forcing – so let’s make it, for simplicity, sinusoidal: where F0 is the amplitude and Ω is the frequency of the forcing (if Ω is large, Fexternal changes direction very frequently). Now, equation (6) becomes (8) where ۞ ω is called the natural frequency (of an oscillating system).

First, consider the unforced undamped case (A = γ = 0), so that (8) becomes The general solution of this ODE can be easily found: and it describes oscillations of constant amplitude. If the initial displacement and velocity are both zero, i.e. (9) the solution is x = 0 for all t, i.e. the oscillator remains at rest forever.

Next, consider the forced undamped case (A ≠ 0, γ = 0), so that (8) becomes This ODE can be solved using the MoUC (the basic rule)... Given initial conditions (9) of zero displacement and velocity, one readily obtains Observe that this solution exists only if Ω ≠ ω.

If Ω = ω (i.e. if the frequency of the forcing equals the oscillating system’s natural frequency), we have to use the modification rule of the MoUC. For the initial conditions (9), we obtain This solution can be separated into the oscillatory part and the amplitude: One can see that the amplitude grows linearly with time – which physically corresponds to a phenomenon called resonance.

In manufacturing and architecture, resonances are very dangerous and should be avoided. There are two famous cases where they were not: https://www.youtube.com/watch?v=eAXVa__XWZ8 https://www.youtube.com/watch?v=XggxeuFDaDU Friction, generally, causes resonance to saturate. If, for example, we let γ ≠ 0 in equation (8), the solution will initially grow, but eventually stabilise. In terms of the MoUC, we’ll be back to the basic rule in this case. However, the amplitude at which oscillations saturate can be (and sometime is) too large. A more effective way of getting rid of a resonance is “detuning”.