A block diagram approach to characterize the kinematic and torque relationship of three-port transmission mechanism

ABSTRACT

This paper presents a kinematic and dynamic modeling of a three-port transmission mechanism using the block diagram technique. By exploiting the simplicity of the block diagram technique, unlike the more complex conventional approaches, the three-port transmission mechanism is systematically characterized to obtain the velocity kinematics and torque relationship. This is achieved by establishing a feedback connection of the three-port transmission mechanism in block diagram form, and obtain the close-loop input/output relationship using techniques known in the control community, such as Mason’s rule. Two categories of the three-port transmission mechanism were investigated including, the 1D rotation type (planetary gear) and the 2D rotation type (differential). The kinematic and dynamic expressions of both categories were shown to be easily obtained, identifying the characteristic differences. The velocity kinematics, torque relationships, and power analysis were expressed to validate the proposed modeling. Further verification using MATLAB Simulink shows a good agreement of the analytical expressions and the proposed block diagram model. To demonstrate the effective use of the proposed approach, the block diagram technique is applied in the design analysis of a south-pointing chariot device.

CO EDITOR-IN-CHIEF:

• Hsiau, Shu-San

ASSOCIATE EDITOR:

• Tsai, Jhy-Cherng

KEYWORDS:

• Kinematics
• planetary gear set
• block diagram
• differential mechanism
• three-port transmissions mechanism

Nomenclature

$B_C$	=	Viscous damping of the carrier ($Kg/s$).
$B_{Load\mathrm{\_}R}$	=	Load on the ring gear ($N.s/m$).
$B_{Load\mathrm{\_}S}$	=	Load on the sun gear ($N.s/m$).
$B_P$	=	Viscous damping of the planet gear ($Kg/s$).
$B_R$	=	Viscous damping of the ring gear ($Kg/s$).
$B_S$	=	Viscous damping of the sun gear ($Kg/s$).
$D$	=	Track width of the south pointing chariot (m).
$F_{RP}$	=	Reaction force of the planet gear meshing with the ring gear (N).
$F_{SP}$	=	Reaction force of the planet gear meshing with the sun gear (N).
$J_C$	=	Carrier moment of inertia ($kg.m^2$).
$J_P$	=	Planet gear moment of inertia ($kg.m^2$).
$J_R$	=	Ring gear moment of inertia ($kg.m^2$).
$J_S$	=	Sun gear moment of inertia ($kg.m^2$).
$K_{RP}$	=	Stiffness constant of the ring gear meshing with the planet gear ($N/m$).
$K_{SP}$	=	Stiffness constant of the sun gear meshing with the planet gear ($N/m$).
$N$	=	Gear ratio of the south pointing chariot spur gears.
$N_1$	=	Gear ratio of the first and second spur gear for the left and right sides of the south pointing chariot.
$N_2$	=	Gear ratio of the third and fourth spur gear for the left and right sides of the south pointing chariot.
$N_D$	=	Gear ration of the south pointing chariot differential.
$P_{in}$	=	Input power ($Nm/s$).
$P_{out}$	=	Output power ($Nm/s$).
$P_{out(actual)}$	=	Actual output power ($Nm/s$).
$P_{out(real)}$	=	Real output power ($Nm/s$).
$R_C$	=	Radius of the carrier ($m$).
$R_{L1\text{~}L5}$	=	Radius of the 1, 2, 3, 4, 5 left spur gears of the south pointing chariot ($m$).
$R_P$	=	Radius of the planet gear ($m$).
$R_R$	=	Radius of the ring gear ($m$).
$R_{R1\text{~}R5}$	=	Radius of the 1, 2, 3, 4, 5 right spur gears of the south pointing chariot($m$).
$R_S$	=	Radius of the sun gear ($m$).
$r_w$	=	Radius of the wheel ($m$).
${\hat{S}}_{LW}$	=	Arc length of the corresponding angular displacement for the left wheel ($m$).
${\hat{S}}_{RW}$	=	Arc length of the corresponding angular displacement for the right wheel ($m$).
$T_C$	=	External torque input to the carrier ($Nm$).
$T{{ }^{\mathrm{\prime }}}_C$	=	Reaction torque on the carrier ($Nm$).
$T{{ }^{\mathrm{\prime }}}_{C1}$	=	Reaction torque on the carrier caused by the planet gear meshing with the ring gear ($Nm$).
$T{{ }^{\mathrm{\prime }}}_{C2}$	=	Reaction torque on the carrier caused by the planet gear meshing with the sun gear ($Nm$).
$T_{Load\mathrm{\_}R}$	=	Load torque on the ring gear ($Nm$).
$T_{Load\mathrm{\_}S}$	=	Load torque on the sun gear ($Nm$).
$T{{ }^{\mathrm{\prime }}}_P$	=	Reaction torque on the planet ($Nm$).
$T{{ }^{\mathrm{\prime }}}_{P1}$	=	Reaction torque on the planet gear caused by the planet gear meshing with the ring gear ($Nm$).
$T{{ }^{\mathrm{\prime }}}_{P2}$	=	Reaction torque on the planet gear caused by the planet gear meshing with the sun gear ($Nm$).
$T_R$	=	External torque input to the ring gear ($Nm$).
$T{{ }^{\mathrm{\prime }}}_R$	=	Reaction torque on the ring gear ($Nm$).
$T_S$	=	External torque input to the sun gear ($Nm$).
$T{{ }^{\mathrm{\prime }}}_S$	=	Reaction torque on the sun gear ($Nm$).
${\alpha }_C$	=	Linear position of the carrier ($m$).
${\alpha }_P$	=	Linear position of the planet gear ($m$).
${\alpha }_R$	=	Linear position of the ring gear ($m$).
${\alpha }_{RPC}$	=	Relative linear position of the ring gear, planet gear and carrier ($m$).
${\alpha }_S$	=	Linear position of the sun gear ($m$).
${\alpha }_{SPC}$	=	Relative linear position of the sun gear, planet gear and carrier ($m$).
$\gamma$	=	South pointing chariot pointer yaw angle (rad).
${\theta }_C$	=	Angular displacement of the south pointing chariot carrier (rad).
${\theta }_{LW}$	=	Angular displacement of the left wheel (rad).
${\theta }_{RW}$	=	Angular displacement of the right wheel (rad).
${\theta }_R$	=	Angular displacement of the south pointing chariot ring gear (rad).
${\theta }_s$	=	Angular displacement of the south pointing chariot sun gear (rad).
${\omega }_C$	=	Angular velocity of the carrier (rad/s).
${\omega }_P$	=	Angular velocity of the planet gear (rad/s).
${\omega }_R$	=	Angular velocity of the ring gear (rad/s).
${\omega }_S$	=	Angular velocity of the sun gear (rad/s).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Science and Technology Council, Taiwan under [NSTC 112-2221-E-006 -116 -MY3].