# Gears

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.

• Newton's 2nd law on top gear where f(t) is a new unknown variable which corresponds to the force applied to the top gear due to the bottom gear.
(eq. 1) Right hand side of the equation is zero because top gear is massless.
• Newton's 2nd law on bottom gear and mass Force on top gear is applied on bottom gear with equal but opposite direction (action-reaction). τ2(t) is not present because it is the same as the torque due to f(t). If τ2(t) were used instead, f(t) would not be present.
(eq. 2) • Definition of torque on bottom gear
(eq. 3) • Mechanism of gear (circular distance travelled is same between gears)
(eq. 4) ## Transfer Function

Laplace transform (eq. 1), (eq. 2), (eq. 3), (eq. 4): Solve for the 4 unknowns: Solve output/input:

(solution) ## 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 ẋ(t) and y(t) in terms of x(t) and u(t): Rearrange in matrix format:

(solution) 