On the Torque Transmission by a Cardan-Hooke Joint

In literature regarding the dynamics of the CardanHooke joint, the inertial characteristics of shafts and a cross and their influence on the joint torque transmission are often ignored (see e.g. [1]). In [2], the cross inertia and its effect on shafts bearing reactions at the constant angular velocity of the input shaft was considered. On the other hand, the problem of the torque transmission at a variable angular velocity of the input shaft and in the presence of both structural and error misalignments between the shafts is considered in [3]. Moreover, the inertial characteristic of the cross is not taken into account. In [4], only the inertia of shafts was taken into account. The aim of this paper is to form a relation between the applied and resisting torques, which will be taken into account inertial characteristics of both the cross and the shafts. In general case, the variable angular velocity of the input shaft is assumed where elasticity effects and friction are ignored.


INTRODUCTION
In literature regarding the dynamics of the Cardan-Hooke joint, the inertial characteristics of shafts and a cross and their influence on the joint torque transmission are often ignored (see e.g.[1]).In [2], the cross inertia and its effect on shafts bearing reactions at the constant angular velocity of the input shaft was considered.
On the other hand, the problem of the torque transmission at a variable angular velocity of the input shaft and in the presence of both structural and error misalignments between the shafts is considered in [3].Moreover, the inertial characteristic of the cross is not taken into account.In [4], only the inertia of shafts was taken into account.
The aim of this paper is to form a relation between the applied and resisting torques, which will be taken into account inertial characteristics of both the cross and the shafts.In general case, the variable angular velocity of the input shaft is assumed where elasticity effects and friction are ignored.

KINEMATICS OF A CARDAN-HOOKE JOINT
Various approaches have been used in kinematic analysis of the Cardan-Hooke joint, such as: the descriptive geometry approach [5,6], the transformation -matrix approach [7], the space analytic geometry approach [8], the approach based on the kinematic chain rule for angular velocity vectors [9] etc.In this paper, the kinematic analysis is based on the approach from reference [9].In authors' opinion, this approach requires minimal computation effort and is simple in application.

The case when the driving yoke plane coincides with the plane of the shafts
Figure 1 shows a Cardan-Hooke joint composed of three rigid bodies (V 1 ), (V 2 ), and (V 3 ) where (V 1 ) is the input shaft with driving yoke, (V 2 ) is the cross, and (V 3 ) is the output shaft with driven yoke.Without loss of generality it is assumed that the input shaft is placed along the y -axis of the reference frame Oxyz and that axes of the input and output shaft crossing at the point O 1 lie in the coordinate plane Oyz. e and 4 e , respectively.Let φ 1 , φ 2 , φ 3 , φ 4 and α represent, respectively, the rotating angle of the input shaft, the relative rotating angle of the cross about the axis 2 e with respect to the input shaft, the relative rotating angle of the cross about the axis 3 e with respect to the output shaft, the rotating angle of the output shaft, and the angle between axes of the shafts connected.The local coordinate frame O 1 ξηζ is fixed to the cross in the manner shown in Figure 1.In the reference configuration of the Cardan-Hooke joint shown in Figure 1 one has that φ 1 = 0, φ 2 = 0, and φ 3 = α.In this reference configuration, the driving yoke plane coincides with the plane Oyz.As in [9], applying the kinematic chain rule for angular velocity vectors [9] to the bodies (V 1 ) and (V 2 ) yields the following expression for the angular velocity of the cross, 2 ω : where 1 ϕ ɺ and 2 ϕ ɺ are, respectively, the angular speed of the input shaft and the relative angular speed of the cross with respect to the input shaft.Similarly, applying the kinematic chain rule for angular velocity vectors to the bodies (V 2 ) and (V 3 ) yields: where 3 ϕ ɺ and 4 ϕ ɺ are, respectively, the relative angular speed of the cross with respect to the output shaft and the angular speed of the output shaft.The vectors 1 e , 2 e , and 4 e may be expressed in the frame Oxyz as follows: , sin cos e j e i k where i , j and k are the unit vectors of the axes x, y, and z, respectively.By observing Figure 1, it is obvious that: Equating ( 1) with (2) yields: Premultiplying both sides of ( 6) by 4 e and 2 e one has, respectively, that: Further, solving ( 7) and ( 8) for 4 ϕ ɺ and 2 ϕ ɺ gives: sin cos cos Finally, after the integration of ( 10) it is obtained that: tan (sin tan ).

The case when the driving yoke plane is normal to the plane of the shafts
Let us now consider a Cardan-Hooke joint in which, when the joint is in the reference configuration 1 0 ϕ = , 2 ϕ α = − , and 3 0 ϕ = , the driving yoke plane is normal to the plane of the shafts as it is shown in Figure 2. Now, one has that: while the vectors 1 e and 4 e are determined by the same expressions as in ( 3) and (4).Applying the similar procedure as in previous section, the following relations are obtained: sin sin By solving (13) and ( 14) for 4 ϕ ɺ and 2 ϕ ɺ , the following expressions for angular speeds are obtained: Finally, the integration of ( 16) yields the following expression for the angle 2 ϕ : where J 1 and J 3 are the mass moments of inertia of the shafts, J 2ξ , J 2η , and J 2ζ are the mass moments of inertia about the cross principal inertia axes ξ, η and ζ, and ω 2ξ , ω 2η and ω 2ζ are the projections of the angular velocity of the cross, 2 ω , onto the axes of the local coordinate frame O 1 ξηζ.Based on (1), the projections ω 2ξ , ω 2η and ω 2ζ are determined by: Taking into account ( 9), ( 10), ( 11), (19), and (20), the kinetic energy in expended form reads: ( ) Applying Lagrange's equations of the second kind [10], the differential equation of motion of the Cardan-Hooke joint reads: where 1 Q φ is the generalized force obtained from the expression for the sum of virtual works of the applied torque T 1 acting on the input shaft and the resisting torque T 3 acting on the output shaft as follows: Now, (22) in developed form reads: In the case of a constant angular velocity of the input shaft, that is, 1 0 ϕ =

ɺɺ
, one has that: wherefrom, ignoring the inertial characteristic of cross and output shaft, the following common interdependence of the torques is obtained (see [1]):

The case when the driving yoke plane is normal to the plane of the shafts
In this case one has that: Now, the equation of motion of the Cardan-Hooke joint reads: while, for 1 0 ϕ = ɺɺ , it is obtained that: and, finally, for the inertial characteristic of cross and output shaft ignored:

NUMERICAL EXAMPLE
Let us apply everything what is considered in Section 2.1 on the Cardan-Hooke joint with constant angular velocity of the input shaft and let T II and T IV contain terms of (25) that come from the inertia of the cross and the output shaft, respectively, as follows: 1 In order to investigate the effect of the inertia of the cross and the output shaft on the torque transmission at a constant angular velocity of the input shaft, the following numerical values of the joint parameters will be used (see [11]): The plots of the quantities T II and T IV are shown in Figures 3 and 4, respectively, for various angular speed of the input shaft.Also, for the given value of resisting torque and various angular speeds of the input shaft, the plots of the applied torque are shown in Figure 5.Note that the plot of applied torque determined by means of the relation ( 26) is marked by a dash-dot line.

CONCLUSIONS
The relations obtained in this paper give an opportunity to estimate the influence of the inertia of cross and shafts on the torque transmission in Cardan-Hooke joints.By observing Figures 3 and 4 it is noted that at a constant angular speed of the input shaft, the influence of the output shaft inertia is dominant in comparison with the influence of the cross inertia.Also, from Figure 5 it can be observed that at higher values of the input shaft angular speed, the difference between results obtained by the static torque transmission relation (26) and the relation (25) is obvious.So, at higher angular speeds of the input shaft, the effect of inertia of the components of the Cardan-Hooke joint must be taken into account.The results obtained in this paper are valuable for the design as well as the stress and fatigue analysis of Cardan-Hooke joints (see e.g.[12]).

Figure 1 .
Figure 1.Cardan-Hooke joint -Variant I The cross is connected to the driving and driven yokes with the revolute joints whose axes are represented with the unit vectors2e and