Euler’s dynamical equations for the motion of a rigid body about an axis - CSIR-NET Mathematical Sci | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

So far we have been discussing the rotation of a body fifixed at a point P . However we are often interested in the rotation of a free body suspended in space - for example, a satellite or the planets. Thankfully, this problem is identical to that of an ob ject fixed at a point. Let's show why this is the case and then go on to analyse the motion.

Figure 37:

The most general motion of a body is an overall translation superposed with a rotation. We could take this rotation to be about any point in the body (or, indeed, a point outside the body). But it is useful to consider the rotation to be about the center of mass. We can write the position of a particle in the body as

r_i(t) = R(t) + Δr_i(t)                          (11.14)

where Δr_i is the position measured from the centre of mass. Then examining the kinetic energy (which, for a free body, is all there is)

where we've used the fact that  (Omega and V are not characteristic of the ith particle so can be brought outside the middle term summed over the mass.) So we find that the dynamics separates into the motion of the centre of mass R, together with rotation about the centre of mass. (This is similar to what we derived fully in the first lecture for a collection of particles.) This is the reason that the analysis of the last section is valid for a free ob ject. Let us now consider how to transform a vector between frames.

Here we are interested in the behaviour of a vector

Figure 38:

in the body frame, for example the angular velocity vector as a function of time viewed from the space (inertial) frame. In the body frame of the ob ject the angular velocity vector is obviously zero. We must therefore be able to transfer the time derivative of a vector between frames.
Consider a displacement of a given point r in the body by rotating an innitesimal amount dφ about an axis   . From the gure, we see that
Moreover, this displacement is perpendicular to r since the distance to P is fixed by the denition of a rigid body.
So we have

dr = dφ x r                                             (11.16)

with  "Dividing" this equation by dt, we have the result

                 (11.17)

where ω = dφ/dt is the instantaneous angular velocity. In general, both the axis of rotation  and the rate of rotation dφ/dt will change over time. If we allow the point P to have a velocity with respect to the rotating system (body frame) then we must add it to the right hand side of equation 11.17:

      (11.18)

and this equation is valid for any vector, such as angular momentum for example.

11.2.1 Euler's Equations

From now on, we shall neglect the center of mass and concentrate on the rotation of the rigid body. Since the body is free, its angular momentum must be conserved.
This gives us the vector equation

         (11.19)

Let's expand this in the body frame using equation 11.18 above, working in the body frame which corrosponds to the principal axis and using , we can then write L₁ = I₁ω₁ and so on. The equations of motion are three non-linear coupled first order differential equations linking the angular velocity and principle moments of intertia (note that the principal moments of intertia do not depend on time in the body frame):

These are Euler's Equations which we derived here using conservation of angular momentum i.e. no external torques are acting. If we construct a simple coordinate system to describe rotations then we can derive these equations of motion directly from the Euler-Lagrange equations - see board notes next week.

We can extend this analysis to include a torque  . The equation of motion becomes  and we can again expand in the body frame along the principal axes to derive Euler's equations (11.20), now with the components of the torque on the RHS.

11.3 Free Tops

In this section, we'll analyse the motion of free rotating bodies (known as free tops) using Euler's equation.

We start with a trivial example: the sphere. For

this ob ject, I₁ = I₂ = I₃ which means that the angular velocity ω is parallel to the angular momentum L. Indeed, Euler's equations tell us that ω_a is a constant in this case and the sphere continues to spin around the same axis you start it on. To nd a more interesting case, we need to look at the next simplest object.

11.3.1 The Symmetric Top

The symmetric top is an ob ject with

A typical example is drawn in Figure 39. Euler's equations become

So, in this case, we see that ω₃, which is the spin about the symmetric axis, is a constant of motion. In contrast, the spins about the other two axes are time dependant and satisfy

      (11.22)

where

(11.23)

is a constant. These equations are solved by

      (11.24)

for any constant ω₀. This means that, in the body frame, the direction of the spin is not constant: it precesses about the e₃ axis with frequency Ω. The direction of the spin depends on the sign on Ω or, in other words, whether I₁ > I₃ or I₁ < I₃. This is drawn in gure 40.

Figure 40: The precession of the spin: the direction of precession depends on whether the ob ject is short and fat (I₃ > I₁ ) or tall and skinny (I₃ < I₁ )

In an inertial frame, this precession of the spin looks like a wobble. To see this, recall that L has a fixed direction. Since both ω₃ and L₃ are constant in time, the e₃ axis must stay at a fixed angle with respect to the L and ω. It rotates about the L axis as shown in gure 41. We'll examine this wobble more in the next section.

11.3.2 Example: The Earth's Wobble 

The spin of the Earth causes it to bulge at the equator so it is no longer a sphere but can be treated as a symmetric top. It is an oblate ellipsoid, with I₃ > I₁, and is spherical to roughly 1 part in 300, meaning

(11.25)

Of course, we know the magnitude of the spin ω₃: it is ω₃ = (1 day) 1. This information is enough to calculate the frequency of the Earth's wobble; from (11.23), it should be

This calculation was first performed by Euler in 1749 who predicted that the Earth completes a wobble every 300 days. Despite many searches, this effect wasn't detected until 1891 when Chandler re-analysed the data and saw a wobble with a period of 427 days. It is now known as the Chand ler wobble. It is very smallω The angular velocity ω intercepts the surface of the Earth approximately 10 metres from the North pole and precesses around it. More recent measurements place the frequency at 435 days, with the discrepancy between the predicted 300 days and observed 435 days due to the fact that the Earth is not a rigid body, but is flexible because of tidal effects. Less well undefirstood is why these same tidal effects haven't caused the wobble to dampen and disappear completely. There are various theories about what keeps the wobble alive, from Earthquakes to fluctuating pressure at the bottom of the ocean.

11.3.3 The Asymmetric Top: Stability

The most general body has no symmetries and I₁ ≠ I₂ ≠ I₃ ≠ I₁. The rotational motion is more complicated but there is a simple result that we will describe here.
Consider the case where the spin is completely about one of the principal axes, say e₁ . i.e.

     (11.27)

This solves Euler's equations (11.20). The question we want to ask is: what happens if the spin varies slightly from this direction? To answer this, consider small perturbations about the spin

 (11.28)
where η_a, a = 1, 2, 3 are all taken to be small. Substituting this into Euler's equations and ignoring terms of order η² and higher, we have

We substitute the third equation into the second to find an equation for just one of the perturbations, say η²,

 (11.31)
The fate of the small perturbation depends on the sign of the quantity A. We have two possibilities

• A < 0: In this case, the disturbance will oscillate around the constant motion.
• A > 0: In this case, the disturbance will grow exponentially.

Examining the denition of A, we nd that the motion is unstable if

I₂ < I₁ < I₃ or I₃ < I₁ < I₂                           (11.32)

with all other motions stable. In other words, a body will rotate stably about the axis with the largest or the smallest moment of inertia, but not about the intermediate axis. Pick up a tennis racket and try it for yourself !