Lectures D25-D26 :Lectures D25-D26 :3D Rigid Body Dynamics3D Rigid Body Dynamics

12 November 2004

Dynamics 16.07 Dynamics D25-D26

OutlineOutline

• Review of Equations of Motion• Rotational Motion• Equations of Motion in Rotating Coordinates• Euler Equations• Example: Stability of Torque Free Motion• Gyroscopic Motion

– Euler Angles– Steady Precession

• Steady Precession with M = 0

1

Dynamics 16.07 Dynamics D25-D26

Equations of MotionEquations of Motion

Conservation of Linear Momentum

Conservation of Angular Momentum

or

2

Dynamics 16.07 Dynamics D25-D26

Equations of Motion in Rotating CoordinatesEquations of Motion in Rotating Coordinates

Angular Momentum

Time variation- Non-rotating axes XY Z (I changes)

big problem!

- Rotating axes xyz (I constant)

3

Dynamics 16.07 Dynamics D25-D26

Equations of Motion in Rotating CoordinatesEquations of Motion in Rotating Coordinates

xyz axis can be any right-handed set of axis, but

. . . will choose xyz (Ω) to simplify analysis (e.g. I constant)

4

or,

Dynamics 16.07 Dynamics D25-D26

Example: Parallel Plane MotionExample: Parallel Plane Motion

5

Body fixed axis

Solve (3) for ωz, and then, (1) and (2) for Mx and My.

Dynamics 16.07 Dynamics D25-D26

Euler’s EquationsEuler’s Equations

If xyz are principal axes of inertia

6

Dynamics 16.07 Dynamics D25-D26

Euler’s EquationsEuler’s Equations

• Body fixed principal axes• Right-handed coordinate frame• Origin at:

– Center of mass G (possibly accelerated)– Fixed point O

• Non-linear equations . . . hard to solve• Solution gives angular velocity components . . . in

unknown directions (need to integrate ω to determine orientation).

7

Dynamics 16.07 Dynamics D25-D26

Example: Stability of Torque Free Motion Example: Stability of Torque Free Motion

Body spinning about principal axis of inertia,

Consider small perturbation

After initial perturbation M = 0

8

Small

Dynamics 16.07 Dynamics D25-D26

Example: Stability of Torque Free MotionExample: Stability of Torque Free Motion

9

From (3) constantDifferentiate (1) and substitute value from (2),

or,

Solutions,

Dynamics 16.07 Dynamics D25-D26

Example: Stability of Torque Free MotionExample: Stability of Torque Free Motion

10

Growth Unstable

Oscillatory Stable

Dynamics 16.07 Dynamics D25-D26

Gyroscopic Motion Gyroscopic Motion

• Bodies symmetric w.r.t.(spin) axis

• Origin at fixed point O (or at G)

11

Dynamics 16.07 Dynamics D25-D26

Gyroscopic MotionGyroscopic Motion

• XY Z fixed axes• x’y’z body axes — angular velocity ω• xyz “working” axes — angular velocity Ω

12

Dynamics 16.07 Dynamics D25-D26

Gyroscopic Motion Euler AnglesGyroscopic Motion Euler Angles

– position of xyz requires and

– position of x’y’z requires , θand ψ

Relation between ( ) and ω,(and Ω )

13

Precession

Nutation

Spin

Dynamics 16.07 Dynamics D25-D26

Gyroscopic Motion Euler AnglesGyroscopic Motion Euler Angles

Angular Momentum

Equation of Motion,

14

Dynamics 16.07 Dynamics D25-D26

Gyroscopic Motion Euler AnglesGyroscopic Motion Euler Angles

become

. . . not easy to solve!!

15

Dynamics 16.07 Dynamics D25-D26

Gyroscopic Motion Steady PrecessionGyroscopic Motion Steady Precession

16

Dynamics 16.07 Dynamics D25-D26

Gyroscopic Motion Steady PrecessionGyroscopic Motion Steady Precession

Also, note that

H does not change in xyz axes

External Moment

17

Dynamics 16.07 Dynamics D25-D26

Gyroscopic Motion Steady PrecessionGyroscopic Motion Steady Precession

Then,

If

18

precession velocity,spin velocity

Dynamics 16.07 Dynamics D25-D26

Steady Precession with M = 0Steady Precession with M = 0

19

constant

Dynamics 16.07 Dynamics D25-D26

Steady Precession with M = 0 DirecSteady Precession with M = 0 Direct Precessiont PrecessionFrom x-component of angular momentum equation,

20

If thensame sign as

Dynamics 16.07 Dynamics D25-D26

Steady Precession with M = 0Steady Precession with M = 0Retrograde PrecessionRetrograde Precession

21

and have opposite signs

If