An analysis of the putter face control mechanism in golf putting

Abstract

The B-matrix method, which systematically analyses the dynamic response of non-symmetric rigid bodies, such as golf putters, is described. The three-dimensional translational and rotational accelerations of the putter face are represented by linear equations of input forces and moments represented in matrix form. The elements of the 6 × 6 B-matrix physically represent the acceleration intensity factors. This method is applied to the analysis of the putter face control mechanism. The input forces and moments are obtained by the inverse dynamics method, whilst special attention is paid to the putter face rotation, i.e. the angular acceleration around the axis of the shaft, $\dot{w}_{ox} .$ The contribution of input forces and moments on $\dot{w}_{ox}$ is discussed quantitatively. The results demonstrate that $\dot{w}_{ox}$ is present even if the input force and moment are applied perpendicularly to the shaft. A cancelling mechanism to correct this motion is discussed based on the B-matrix method.

Appendices

Appendix 1

The time derivative of vector r ^′ _o is v ^′ _o , as shown in the left-hand diagram of Fig. 1, which leads to

$$ \varvec{v}^{'}_{\user2{o}} = \dot{\varvec{r}}^{'}_{\user2{o}} = \dot{\varvec{r}}^{'}_{\user2{g}} + \dot{\varvec{r}}^{'}_{{\user2{go}}} = \dot{\varvec{r}}^{'}_{\user2{g}} + \dot{\varvec{r}}_{{\user2{go}}}^{'*} + \user2{w}^{'}_{\user2{g}} \times \user2{r}^{'}_{{\user2{go}}} $$

(16)

where r ^′ _go is the position vector from point g to point o and $\dot{\varvec{r}}_{go}^{'*}$ in the right-hand formula is a differential relating only to the pure expansion or contraction of r ^′ _go [24]. In case of rigid bodies, $\dot{\varvec{r}}_{go}^{'*}$ is zero. Therefore, Eq. (16) becomes

$$ \user2{v}^{'}_{\user2{o}} = \user2{v}^{'}_{\user2{g}} + \user2{w}^{'}_{\user2{g}} \times \user2{r}^{'}_{{\user2{go}}} $$

(17)

and the time derivative of Eq. (17) yields the acceleration, i.e.

$$ \dot{\varvec{v}}^{'}_{o} = \dot{\varvec{v}}^{'}_{g} + \dot{\varvec{w}}^{'}_{g} \times \user2{r}^{'}_{go} + \user2{w}^{'}_{g} \times \dot{\varvec{r}}^{'}_{go} = \dot{\varvec{v}}^{'}_{g} + \dot{\varvec{w}}^{'}_{g} \times \user2{r}^{'}_{go} + \user2{w}^{'}_{g} \times \left( {\dot{\varvec{r}}_{go}^{'*} + \user2{w}^{'}_{g} \times \user2{r}^{'}_{go} } \right). $$

(18)

Consequently, the acceleration and angular acceleration $(\dot{\varvec{v}}^{'}_{o} ,\dot{\varvec{w}}^{'}_{o} )$ at point o can be expressed by $(\dot{\varvec{v}}^{'}_{g} ,\dot{\varvec{w}}^{'}_{g} )$ at point g,

$$ \dot{\varvec{v}}{}^{'}_{o} = \dot{\varvec{v}}^{'}_{g} + \dot{\varvec{w}}^{'}_{g} \times \user2{r}^{'}_{go} + \user2{w}^{'}_{g} \times (\user2{w}^{'}_{g} \times \user2{r}^{'}_{go} ), $$

(19)

$$ \dot{\varvec{w}}^{'}_{o} = \dot{\varvec{w}}^{'}_{g} . $$

(20)

Appendix 2

If the quadratic terms of w ^′ _g in Eqs. (4) and (5) are much less than the other terms, they can be neglected. As vector and tensor are frame-invariant, all of Eqs. (1)–(6) are converted to the non-dashed ones in the rigid body coordinate system. After these processings, these six equations can be described by three equations in the linear matrix form using the E, S _gi and S _go matrices:

$$ \left[ {\begin{array}{*{20}c} {\user2{F}_{g} } \\ {\user2{M}_{g} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} \user2{E} & \user2{0} \\ { - \user2{S}_{gi} } & \user2{E} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\user2{F}_{i} } \\ {\user2{M}_{i} } \\ \end{array} } \right], $$

(21)

$$ \left[ {\begin{array}{*{20}c} {\user2{F}_{g} } \\ {\user2{M}_{g} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {m\user2{E}} & \user2{0} \\ \user2{0} & {\user2{I}_{g} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\dot{\varvec{v}}_{g} } \\ {\dot{\varvec{w}}_{g} } \\ \end{array} } \right] $$

(22)

and

$$ \left[ {\begin{array}{*{20}c} {\dot{\varvec{v}}_{o} } \\ {\dot{\varvec{w}}_{o} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} \user2{E} & { + \user2{S}_{go} } \\ \user2{0} & \user2{E} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\dot{\varvec{v}}_{g} } \\ {\dot{\varvec{w}}_{g} } \\ \end{array} } \right], $$

(23)

where S _gi and S _go are skew-symmetric matrices defined using the position vectors r _gi and r _go,

$$ \user2{r}_{gi} = \left( {x_{gi} ,y_{gi} ,z_{gi} } \right)^{t} , $$

(24)

$$ \user2{S}_{gi} \equiv \left[ {\begin{array}{*{20}c} 0 & { + z_{gi} } & { - y_{gi} } \\ { - z_{gi} } & 0 & { + x_{gi} } \\ { + y_{gi} } & { - x_{gi} } & 0 \\ \end{array} } \right], $$

(25)

$$ \user2{r}_{go} = \left( {x_{go} ,y_{go} ,z_{go} } \right)^{t} , $$

(26)

$$ \user2{S}_{go} \equiv \left[ {\begin{array}{*{20}c} 0 & { + z_{go} } & { - y_{go} } \\ { - z_{go} } & 0 & { + x_{go} } \\ { + y_{go} } & { - x_{go} } & 0 \\ \end{array} } \right]. $$

(27)

An arbitrary vector V and the inertia tensor I _g in the rigid body coordinate system can be converted to these in the space coordinate system using the rotational matrix R.

$$ \user2{V}^{'} = \user2{RV,} $$

(28)

$$ \user2{I}_{g}^{'} = \user2{RI}_{g} \user2{R}^{t} . $$

(29)