Welcome to AE Resources

Converted document MOMENT EQUATION GOVERNING ROTATIONAL MOTION

MOMENT EQUATION GOVERNING ROTATIONAL MOTION

The Euler equations for a 3D body are found using the principal axes and describe the moment of a body:

(1) ⎲⎳M_cx = I^C_xxω_ẋ − (I^C_yy − I^C_zz)ω_yω_z

(2) ⎲⎳M_cy = I^C_yyω_ẏ − (I^C_zz − I^C_xx)ω_xω_z

(3) ⎲⎳M_cz = I^C_zzω_ż − (I^C_xx − I^C_yy)ω_xω_y

Notice that the equation for planar motion: ∑M_cz = I^C_zzω_ż is not extended to 3-D directly.

These equations can also be used if the body has a pivot point O. Then ‘O’ would replace ‘C’. The derivative equation can also be used to convert to a frame other than body-fixed if the moments of inertia remain fixed in the intermediate frame.

(4) ⎲⎳M_c = Ḣ^I_c = Ḣ^F_c + ω_(F)/(I) × H_c

For example, the wheel spinning and attached to a bar at C. For times when the principal axes cannot be used, the calculations with other axes can be done, though in 3D it can become complex. In planar motion (x-y), the equations become:

(5) ⎲⎳M_cx = I^C_xxω_ẋ − I^C_yzω²_z ⎲⎳M_cy = I^C_yzω_ż − I^C_xzω²_z ⎲⎳M_cz = I^C_zzω_ż