20230212 The relationship between the moment and moment of momentum projected on the motion frame

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foreword

This paper analyzes in detail the moment M \boldsymbol{M} projected on the moving coordinate system$M$ and moment of momentumH \boldsymbol{H}The relationship between$H.$

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1. Moment $M$ and moment of momentum$What\; is\; H$ ?

Moment $M$ is the total external moment of the rigid body relative to the center of mass of the rigid body, which is related to the force and moment arm that each mass element receives without passing through the center of mass; the
moment of momentum$H$ is the total external moment of momentum of the rigid body relative to the center of mass of the rigid body, which is related to the linear velocity and moment arm of each mass element.

2. Moment $M$ and moment of momentumH \boldsymbol{H}What is the relation of$H\; ?$

1. Coordinate system

Static coordinate system $S$ : Assumed to be an inertial coordinate system;

The moving coordinate system $D$ : The origin coincides with the center of mass of the rigid body and is fixed to the rigid body.

Set the moving coordinate system $D\text{'}$ s$x$、$y$、$The\; unit\; vectors\; on\; the\; z-$ axis are$\mathbf{i}$、$\mathbf{j}$、$k$ , to describe these three vectors, it needs to be projected on a certain coordinate system. When its projection is in the static coordinate system$On\; S$ , the mathematical representation is${\mathbf{i}}^{\mathcal{S}}$、${\mathbf{j}}^{\mathcal{S}}$、${\mathbf{k}}^{S.}$ _ When its projection is in the moving coordinate system$On\; S$ , the mathematical representation is$i^{\mathcal{D}}$、$j^D$、$k^{\mathcal{D}}$。

2. Analysis

Suppose the mass element inside the rigid body is $\text{d}m$ , the radius of the mass center of the rigid body pointing to this mass element is${\mathbf{r}}_{{}_{\text{d}m}}$, then ${\mathbf{H}}^{\mathcal{S}}$ 可以表示为
$$H^{\mathcal{S}}={\int }_m{r}_{{}_{\text{d}m}}^{\mathcal{S}}\times \frac{\text{d}{\mathbf{r}}_{{}_{\text{d}m}}^S}{\text{d}t}\text{d}m$$

已知$${\mathbf{H}}^{\mathcal{S}}=h_x{\mathbf{i}}^S+h_y{\mathbf{j}}^S+h_z{\mathbf{k}}^{\mathcal{S}},H^{\mathcal{D}}=h_x{i}^D+h_y{j}^D+h_z{k}^D=[h_x,h_y,h_z{]}^T$$$${\mathbf{r}}^S=r_x{i}^S+r_y{j}^S+r_z{k}^{\mathcal{S}}$$$${\mathbf{M}}^{\mathcal{S}}=m_x{\mathbf{i}}^S+m_y{\mathbf{j}}^S+m_z{\mathbf{k}}^S$$$${oh}^S={oh}_x{i}^S+{oh}_y{j}^S+{oh}_z{k}^S$$ where,${oh}^S$ represents the action scale$D$ is relative to the static coordinate system$S$ is in the static coordinate systemS \mathcal{S}Projection in$S.$

可以得到$${\mathbf{M}}^{\mathcal{S}}=\frac{\text{d}{\mathbf{H}}^{\mathcal{S}}}{\text{d}t}=\dot{h_x}{\mathbf{i}}^S+\dot{h_y}{\mathbf{j}}^S+\dot{h_z}{\mathbf{k}}^S+h_x\frac{\text{of}{i}^S}{\text{d}t}+h_y\frac{\text{d}{\mathbf{j}}^S}{\text{d}t}+h_z\frac{\text{d}{\mathbf{k}}^S}{\text{d}t}$$

Note that the two derivation writing methods here only distinguish between vectors and scalars, since both are projected on the static coordinate system $S$ , so the two are computationally the same. If it is projected into the motion coordinate system, it cannot be calculated directly.

According to Poisson's formula ,
$$\frac{\text{of}{i}^{\mathcal{S}}}{\text{d}t}={oh}^{\mathcal{S}}\times {\mathbf{i}}^{\mathcal{S}}$$$$\frac{\text{d}{\mathbf{j}}^S}{\text{d}t}={oh}^S\times {\mathbf{j}}^{\mathcal{S}}$$$$\frac{\text{d}{\mathbf{k}}^S}{\text{d}t}={oh}^S\times {\mathbf{k}}^{\mathcal{S}}$$
则有
$${\mathbf{M}}^{\mathcal{S}}=\frac{\text{d}{\mathbf{H}}^{\mathcal{S}}}{\text{d}t}=\dot{h_x}{\mathbf{i}}^S+\dot{h_y}{\mathbf{j}}^S+\dot{h_z}{\mathbf{k}}^S+{oh}^S\times {\mathbf{H}}^S$$

Then projected to the action frame $D$，则有
$${\mathbf{M}}^{\mathcal{D}}=\dot{h_x}{\mathbf{i}}^D+\dot{h_y}{\mathbf{j}}^D+\dot{h_z}{\mathbf{k}}^D+{oh}^D\times {\mathbf{H}}^D$$即$$M^D={\dot{H}}^D+{oh}^D\times H^D$$ Note: These are all projected on the action frame.

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Summarize

The above is the moment M \boldsymbol{M} in the moving coordinate system$M$ and moment of momentum$For\; the\; detailed\; derivation\; of\; the\; relationship\; between\; H$ , the most important point is to distinguish the dynamic and static of the coordinate system.