MOMENT EQUATIONS

USEFUL FORMS OF THE MOMENT EQUATIONS

The sum of moments about a point P can be written as:

(1) ⎲⎳M_p = M_p + r_pc × (ΣF)

where (ΣF) = ma_c = (d)/(dt)(mv) = (d)/(dt)(P), and ΣM_p = (dH_c)/(dt). That is, the sum of the forces equal the rate of change of linear momentum and the sum of the moments equal the rate of change of angular momentum.

(2) ΣM_p = (dH_c)/(dt) + r_pc × (dP)/(dt)

(3) ΣM_p = H_ċ + r_pc × ma_c

In general:

(4) ΣM_p = (I^c_xzα − I^c_yzω²) i + (I^c_yzα − I^c_xzω²) j + I^c_zzαk + r_pc × ma_c

If the body is symmtric so that so that the products of inertia equal to zero, or I^c_xz = I^c_yz = 0, the result is

(5) ΣM_p = I^c_zzαk + r_pc × ma_c for multiple equations

If the point of interest P lies in the plane of the mass center C:

(6) ΣM_p = I^c_zzαk + r_pc × ma_c, results in a scalar equation

MASS-CENTER FORM OF THE MOMENT EQUATIONS

Sum of the moments about an arbitrary fixed point, labeled O, is given by:

(7) ΣM_o = ΣR × ma

Also the definition of moment of the moment of momentum about a fixed point in space is:

(8) H_o = ΣR_i × mv_i

If the time derivative of angular momentum is taken, the result is:

(9) (dH_o)/(dt) = ΣR_i × ma_i(R_idoes not change with the time for a rigid body)

Combining the above expressions yields:

(10) ΣM_o = (dH_o)/(dt)

If O is the mass center, the following expression can be used:

(11) ΣM_c = (dH_c)/(dt)

For the mass center of a body (planar)

(12) H_c = r_cc × mv_c + I^c_xzωi + I^c_yzωj + I^c_zzωk

So the sum of the moments about the mass center can be written as:

(13) ΣM_c = (d)/(dt)[I^c_xzωi + I^c_yzωj + I^c_zzωk]

For a fixed set of axes, the time derivatives of these quantities are constantly computed, which is not always a good policy. A better way would be to have the frame of reference move with the body so that, I^c_xz and I^c_yz are constant.