Using the definition of the damping ratio and the undamped...

Key Concepts

Second Order Differential Equations

Second order differential equations govern the behavior of many dynamic systems, such as mechanical or electrical oscillators. These equations typically involve parameters like mass, damping, and stiffness, and provide the framework needed to analyze and derive relationships between different system characteristics.

Damping Ratio

The damping ratio is a non-dimensional measure that indicates whether a system is underdamped, critically damped, or overdamped. It is defined as the ratio of the actual damping to the critical damping, and its value determines the speed and nature of the system’s return to equilibrium after a disturbance.

Undamped Natural Frequency

The undamped natural frequency is the inherent frequency of oscillation of a system in the absence of any damping. It is determined solely by the system’s mass and stiffness, serving as a baseline measure of the system's oscillatory behavior before any energy dissipation is considered.

Non-dimensionalization

Non-dimensionalization is the process of simplifying equations by introducing dimensionless variables and parameters. This technique reveals the fundamental behavior of a system by reducing the number of independent parameters, allowing for generalizable results like the derivation of one form of the equation from another using the damping ratio and undamped natural frequency.

Transcript

00:01 Okay, so we're going to use the definition of the damping ratio and the undamped natural frequency.

00:14 Okay.

00:18 And from those, we are going to derive equation 1 .48, so this is our goal, from 1 .47.

00:28 So start here and then using this information, get here.

00:36 Okay, dokey.

00:38 The first thing is, let's go ahead and write down the equation for the natural frequency.

00:43 Omega n.

00:46 This has two different versions.

00:49 I'm going to write on both so i can tell y 'all which one you need to use.

00:53 This can be related to your more common frequency f that we're more accustomed to.

01:02 F is measured and hurts.

01:04 But omega -n is also related to the spring constant through the square root of k over m.

01:12 And this is the relationship that we are going to want to use here.

01:20 Okay? the other thing is we will need to define a c, and this is a greek letter, and that's how you pronounce it, c.

01:32 And this equals c over 2 square root of km.

01:40 Now, because i'm looking at equation 1 .48 in the book, it's screaming that i'm going to need a, and it'll make it squared.

01:49 And so that's just k over m, because you square both sides...

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