1 derive the equation of motion of the spring mass pendulum...

Name: 1 derive the equation of motion of the spring mass pendulum system shown below assume small swing angles no deflection in vertical direction and negligible mass of the pendulum rod a using n 56003
Uploaded: 2025-03-15T14:49:20-08:00
Duration: 9 min 47 s
Channel: Sam Stansfield
Description: 1 derive the equation of motion of the spring mass pendulum system shown below assume small swing angles no deflection in vertical direction and negligible mass of the pendulum rod a using n 56003

1) Derive the equation of motion of the spring mass pendulum system shown below (assume small swing angles, no deflection in vertical direction and negligible mass of the pendulum rod) a. Using Newton's 2nd law (show free-body diagrams clearly) b. Using the energy method c. Using Lagrange's method

Transcript

00:01 So we have this pendulum, mass m, length l, has a spring and a damper attached to it at a distance a.

00:09 We're assuming theta is small.

00:14 So, for instance, as it oscillates, a doesn't really change.

00:23 All right, but i'm going to use a lagrangian technique.

00:35 So our kinetic energy has always got this, and then our potential energy has two parts.

00:41 The gravity part and the spring, writing it out exactly in terms of theta, but not paying attention to changes in a.

00:53 And then our gravitational part, that's sine theta.

01:18 And then we also have a dissipation function because of this.

01:27 So it's, so t or l, that's better.

02:20 All right and then our equation of motion um a squared all right and then for small angles we get this equation small angles cosine of theta is one and the sign of theta is theta so we got k a squared so there's our equation so they call this b.

04:40 There's a b that goes in here into f.

04:44 It also ends up here.

04:55 So this is a linear equation with constant coefficients.

05:27 All right.

05:28 So we know that our theta, we can write it like this as some amplitude.

05:48 I'll get to what that is in a moment.

06:02 We expect damped oscillations, but we should know whether or not they're critically damped or whatever.

06:11 So we know that what we have to do is solve the quadratic equation that we would get as a characteristic equation.

06:23 And that discriminant looks like this...