Study on the wrist action in a new swing model (P1)
The dynamic model of golf swing is shown in Figure 3-1. The rotations of the arm and golf club are assumed to occur in one plane during the downswing and follow-through, and this plane is inclined with an angle θ to the ground. The assumption of the planar movement of the arm and golf club is well supported in the early work of Cochran & Stobbs and Jorgensen. Therefore, the gravity acceleration vector in the swing plane is expressed. In our model, the arm and golf grip are considered as the rigid rods, the club head as a tip mass and golf shaft is treated as an Euler-Bernoulli beam. Two co-ordinate systems in the swing plane are introduced to describe the dynamics of the golf swing: a fixed reference frame XY; and a rotational reference frame xy that attached to the end of golf grip, where its x-axis is along the undeformed configuration of golf shaft. Since the center of gravity of club head is regarded as on the central axis of golf shaft, the twisting of golf shaft is neglected and the bending flexibility of golf shaft in the swing plane is only considered. Due to the rotational motion of the system, golf shaft is stiffened by an axial force. The centrifugal stiffening of golf shaft is thus taken into account in this study.
Lagrange approach is used to derive the dynamic equations of motion for a golfer’s swing. In the co-ordinate system O−XY, r2 and rR are the position vectors of the centers of gravity of golf grip and club head, respectively; rp is the position vector of a point pon the shaft; yp is the bending displacement of a point pon the shaft in the co-ordinate system o-xy; J1 is the moment of inertia of arm about the shoulder joint O; J2 and JRare the moments of inertia of golf grip and club head, respectively; m1, m2, m3 and mR are the masses of arm, golf grip, golf shaft and club head, respectively; a1, a2 and a3 are the lengths of arm, golf grip and shaft, respectively; R is the radius of club head; ρ is the mass per unit length of golf shaft; Eis the Young’s modulus of golf shaft material; I is the area moment of inertia of golf shaft.
The following operators are denoted
The total kinetic energy of the system is given by
where T1 , T2 , T3 and T4 are the kinetic energy associated with arm, golf grip, golf shaft and club head , respectively. They are
The potential energy resulting from the gravitational forces on arm and golf grip are
The potential energy of golf shaft is written as
U31 and U32 are the potential energy due to the elastic deformation and gravitational force for golf shaft, respectively.
The axial force, resulting from the axial centripetal accelerations of golf shaft and club head, causes the potential energy U33.
fpx is the axial force for a point p on golf shaft; fpx1 fpx2, are the axial forces for the point p, resulting from the axial centripetal accelerations of golf shaft and club head, respectively. They are
where Apx1 and Apx2 are the axial accelerations for the point p and club head, respectively; both of them are directed along the x-axis.
The specific expressions of fpx1 is given by
It should be noted that the terms associated with the deformation of golf shaft yp have been ignored since the potential energy resulting from them is relatively small as compared to the other terms.
The potential energy due to the gravitational force for club head is
The total potential energy of this system can be written as
The Lagrangian L of the system can be obtained as
It is also noted that the internal structural damping in golf shaft should be considered. By using Rayleigh’s dissipation function, the dissipation energy for golf shaft is written as
where di and qi are the damping coefficient and mode amplitude associated with the ith mode of golf shaft bending vibration, respectively.
According to the assumed modes technique in Theodore & Ghosal, a finite-dimensional model of golf shaft bending displacement is written as
Where φi(x)and qi(t) are the ith assumed mode eigen function and time-varying mode amplitude, respectively. As the shaft is modeled as an Euler-Bernoulli beam with uniform density and constant flexural rigidity (EI), it satisfies with the following partial differential equation
we can obtain the general solution of Eq. 3-21:
where wi is the ith natural angular frequency.
Furthermore, φi(x) can be expressed as
The golf club has been considered as a cantilever that has a tip mass, so the following expressions associated with the boundary conditions can be obtained.
From these boundary conditions, the following results are given
The ith natural angular frequency wi can be obtained by solving the eigenvalue problem of the matrix equation (3-29), and the coefficients C1i and C2i are chosen by normalizing the mode eigen functions φi(x) such that
On the basis of the Euler-Lagrange equation
with the Lagrangian L, the dissipation energy of golf shaft ED , the generalized coordinates Qi and the corresponding generalized forces fi, the dynamic equations of motion of golf swing are achieved. Since the amplitudes of the lower modes of golf shaft bending vibration are apparently larger than those of the higher modes, m is simplified to two in this study.
The equations of motion of golf swing can be written as.
B, K and D are the inertia, stiffness and damping matrices, respectively; h is the nonlinear force vector; G is the gravity vector and τ is the input vector; q1 and q2 are the first and second time-varying mode amplitudes of the shaft bending vibration, respectively; d1 and d2 are the damping coefficients of the first and second modes of the shaft bending vibration, respectively.