Transfer function and state space model are developed for torque applied on a mass through gear train.

(input) τ1(t) = applied torque
(output) θ1(t) = top gear angle unknowns
τ2(t) = torque on mass due to τ1
θ2(t) = mass rotational position
r1 = top gear radius
r2 = bottom gear radius
J = inertia

Differential Equation

3 unknowns so 3 equations are needed. During the process of formulating these equations, a new unknown variable, f(t), is introduced; therefore, a total of 4 equations are prepared.

Transfer Function

Laplace transform (eq. 1), (eq. 2), (eq. 3), (eq. 4):

Solve for the 4 unknowns:

Solve output/input:


Block Diagram

Transfer functions for block diagram is derived using Laplace equations from previous section:

State Space Model

When a mass is present in a system, its position and velocity are commonly chosen as state variables. Similarly for the rotational case, angle and angular velocity are chosen for state variables. It can also be noted that applied torque (input), angle, and angular velocity are sufficient to determine this system's future angle (output). This confirms the selected state variables are valid for this system.

State vector:

Input vector:

Output vector:

Rewrite (eq. 1), (eq. 2), (eq. 3), (eq. 4) in these new notations:

Rearrange equations to express αΊ‹(t) and y(t) in terms of x(t) and u(t):

Rearrange in matrix format: