Chapter 9. Vibrations (Vibeology)
Recommended Article : 【Physics】 Physics Table of Contents
1. Simple Harmonic Motion
Figure. 1. Diagram of Simple Harmonic Motion
⑴ Characteristics : Period remains constant
⑵ Solution of the Differential Equation
⑶ Frequency Domain Conversion
2. Damped Oscillation
Figure. 2. Diagram of Damped Oscillation
⑴ Governing Equation
⑵ Frequency Domain Conversion
⑶ Introduction of β and Classification of Damped Motion
① β2 - ω02 > 0 : Overdamped
② β2 - ω02 = 0 : Critically damped
③ β2 - ω02 < 0 : Underdamped
⑷ Generally, β ＜ ω0 holds true
3. Forced Oscillation
Figure. 3. Diagram of Forced Oscillation
⑴ Governing Equation
⑵ When fe is a trigonometric function
① Vibration = Natural vibration + Steady-state vibration
② Natural vibration = Homogeneous solution = Complementary solution = Transient solution
○ As time passes, converges to 0 due to e-βt
③ Steady-state vibration = Particular solution
○ Generally, only interested in particular solution
○ Assumed as complex harmony
○ (Note) ω ≪ ω0
○ (Note) ω = ω0
○ (Note) ω ≫ ω0
4. Mechanical Impedance
⑴ Definition
① Defined as Complex Driving Force ÷ Complex Velocity at Driving Point
② Concept of force magnitude for obtaining velocity
⑵ Single Degree of Freedom (SDOF) system
① Zm : Inherent properties of the material. Predicts response
② Rm : Mechanical resistance. Energy loss
③ Xm : Mechanical reactance. Energy storage
④ If impedance is known, the response can be known
⑶ ω ≪ ω0 : Zm* = -js / ω
① Spring-like (stiffness-like) impedance
② Negative imaginary
③ Inversely proportional to frequency
⑷ ω = ω0 : Zm* = Rm
⑸ ω ≫ ω0 : Zm* = jωm
① Mass-like impedance
② Positive imaginary
③ Linearly proportional to frequency
⑹ Imaginary impedance : When Rm = 0
① Force and velocity are out of phase
② No power is required to drive the system
○ Since Rm = 0, no energy is dissipated
5. Mechanical Resonance
⑴ Definition : Xm, the reactance part
⑵ Vibrational frequency when the imaginary part of impedance is 0
6. Superposition
Figure. 4. Diagram of Superposition
⑴ Assumption 1. Small amplitude response
⑵ Assumption 2. System is linear
⑶ Formulation
Input : 2019.04.09 08:59