Скачать с ютуб Damped Hamonic Motion | 2nd Order ODE, Underdamped, Overdamped, Critically Damped в хорошем качестве

Damped Hamonic Motion | 2nd Order ODE, Underdamped, Overdamped, Critically Damped

The damped harmonic oscillator 2nd Order ODE is solved and analyzed in this video. In damped harmonic motion, with a linear damping force proportional to the velocity, the oscillations exponentially decay with time. The regime of damping depends upon the difference between the natural frequency (omega 0) and the damping factor (gamma). The three regimes are underdamped (oscillating solutions), overdamped (two decaying exponential parts to the solution), critically damped (strong decay, some oscillation). The general solution procedure is shown for the 2nd Order Linear Ordinary Differential Equation using the characteristic polynomial (auxiliary equation) making use of the quadratic formula. A spring mass oscillating system is conceptualized, setting up the initial value problem (IVP) with initial position and initial velocity always given. We study and calculate aspects of the physical system and motion such as natural frequency, angular frequency, period, damping factor, maximum amplitude (Max A), We also show why changing and 2nd Order ODE to a system of two first order ODEs is so useful for numerical solving. We also show images and animations of oscillating mass and spring systems, plotting the damped spring position vs time demonstrating the amplitude decay. The phase plane of x(t) and x dot(t) trajectory for a simple harmonic oscillator is shown. Chapters / Timestamps for Video Sections: 0:00 Intro to Damped Harmonic Oscillator ODEs 1:18 Natural Frequency, Period, Damping Factor, and Damped Frequency definitions 5:33 Derivation Damped Oscillation Differential Equation 6:32 Characteristic Polynomial (Auxiliary Equation) Quadratic Formula Solution 9:28 General Solution to 2nd Order Damped ODE 11:06 Plot damped spring position vs time, amplitude decay 12:56 Underdamped, critically damped, underdamped oscillators 17:00 Maximum amplitude, underdamped example problem 23:25 Max A, critically damped worked example 29:45 Basis functions, critical damped case, omega_0 = gamma, vertical displacement 33:17 Change 2nd Order ODE to system two first order ODEs 44:43 Phase plane trajectory of simple harmonic oscillator 1:06:12 Simple Harmonic Motion, Amplitude, Phase Angle, Period 1:10:02 Summary of Damped Spring Oscilator problem 1:11:43 Take Action! --------------------- Please support us on Patreon: https://www.patreon.com/highpeakeduca... Please follow us on Facebook:   / high-peak-education-110762917338673   Please follow us on Twitter:   / highpeakeducate   Please follow us on Instagram:   / highpeakeducation   Please follow us on Reddit:   / highpeakeducation   Please follow us on TikTok:   / highpeakeducation   Interact with us on Twitch:   / highpeakeducation   Interact with us on Discord:   / discord   #HighPeakEducation #HarmonicMotion #PeriodicMotion