Derive the form of λ1 and λ2 given by equation (1.31) from equation (1.28) and the definition of

Derive the form of λ1 and λ2 given by equation (1.31) from...

Key Concepts

Characteristic Equation

In the context of second?order systems, the characteristic equation is derived from the differential equation by assuming a solution of exponential form. This transformation converts the differential equation into a quadratic equation whose roots, the eigenvalues, determine the system’s transient behavior and stability.

Damping Ratio

The damping ratio is a dimensionless parameter that quantifies the amount of damping relative to the critical damping level. It is defined as the ratio between the actual damping coefficient and the critical damping coefficient, and it plays a central role in expressing the eigenvalues in relation to the natural frequency.

Natural Frequency

Natural frequency is a key parameter in second?order systems representing the oscillation frequency in the absence of damping. In the derivation of the eigenvalues, the natural frequency is combined with the damping ratio in the quadratic equation to yield a clear expression for the roots, which differentiates between underdamped, critically damped, and overdamped system responses.

Transcript

00:01 Okay, so we're going to derive the form of lambda 1 and lambda 2 given by equation 1 .31 from equation 1 .28.

00:09 Okay, so let's write down equation 1 .31 to see what we're trying to get.

00:14 Lambda 1 -2 equals negative xc, omega -n plus or minus, omega -n, square root of xc, and that's the greek letter, minus 1.

00:25 So this is what we're trying to get.

00:28 Now, what is equation 1 .28? here we go.

00:32 I can write that down right here.

00:35 It says that lambda 1 2 equals negative c over 2m plus or minus 1 over 2m square root of c squared minus 4 km.

00:49 So let's go ahead and rewrite this so that we can kind of massage it into this form.

00:56 Oh, making note that this guy right here is c over 2 square root of k.

01:04 M and omega equals the square root of k over m.

01:10 So we're going to massage this guy so that we can get factors like this and like that.

01:17 Okay, here we go.

01:21 The only way to do that is to break things apart.

01:25 So we're definitely going to have to have some square roots of k and some square roots of m.

01:29 I see m here and i know that m equals square root of m times the square root of m.

01:36 And if i don't see a k, which i don't right here, that means i'm going to have to multiply and divide by square of k over square of k because this just equals one.

01:51 So these are the tricks of the trade in this situation...

Please give Ace some feedback