Selecting a Rack-and-Pinion System: Selecting an L Series Product Equipped With αSTEP AZ Series

The following parameters are required when selecting rack-and-pinion systems for transporting load from A to B as shown in the figure below.

Selecting an L Series Product Equipped With αSTEP AZ Series

The required parameters are as follows:

  • Load mass (m) or thrust (F)
  • Positioning distance (L)
  • Positioning time (T)
  • Repetitive positioning accuracy
  • Maximum stroke

Among the parameters above, the thrust and positioning time can be calculated using the formulas below.

Thrust Formula

① Calculation of the Required Thrust When Accelerating a Load Attached to a Rack

\(\begin{align} F_a = m \left \{ a + g \ ( \sin \theta + \mu \cos \theta) \right \} \end{align}\)

② Calculation of Possible Thrust for Pushing and Pulling

\(\begin{align} F = F_{max} - F_a \end{align}\)

If the external force applied to the load is smaller than F, then push-pull motion is possible.

Fmax
Rack-and-pinion system maximum thrust [N]
Fa
Required thrust during acceleration/deceleration operation [N]
F
Thrust possible for pushing or pulling of external force [N]
m
Load mass to be transported on the rack [kg] + rack mass [kg]
a
Acceleration [m/s²]
g
Gravitational acceleration 9.807 [m/s²]
μ
Friction coefficient of the guide supporting the load 0.01
θ
Angle formed by the moving direction and the horizontal plane [˚]
② Calculation of Possible Thrust for Pushing and Pulling

Positioning Time Formula

Confirm that the rack-and-pinion system can be positioned in time without problems. This can be done by obtaining a rough positioning time from the graph or by obtaining a fairly accurate positioning time by calculation. Each of the confirmation procedures is explained below.
The actual operating time is subject to a small margin of error, so use the calculation only as a reference.
Example) For LM2

Positioning Time Formula

Using Formula Calculations

① Confirm the Operation Conditions

Check the following conditions:
Installation direction, load mass, positioning distance, starting speed, acceleration, operating speed

② From the Operation Conditions above, Confirm Whether the Drive Pattern is a Triangular Drive or a Trapezoidal Drive

Calculate the maximum speed of triangular drive from the positioning distance, starting speed, acceleration and operating speed. If the calculated maximum speed is equal to or below the operating speed, the operation is considered a triangular drive. If the maximum speed exceeds the operating speed, the operation is considered a trapezoidal drive.

\(\begin{align} V_{Rmax} &= \sqrt{ \frac{2 \cdot a_1 \cdot a_2 \cdot L}{a_1 + a_2} \cdot 10^3 + V_{S^2}} \\[5pt] V_{Rmax} & \leqq V_R \rightarrow \text{Triangular Drive} \\[5pt] V_{Rmax} & \gt V_R \rightarrow \text{Trapezoidal Drive}\end{align}\)

③ Calculate the Positioning Time

[In case of trapezoidal drive]

\(\begin{align}T &= T_1 + T_2 + T_3 \\[5pt] & = \frac{V_R - V_S}{a_1 \times 10^3} + \frac{V_R - V_S}{a_2 \times 10^3} + \frac{L}{V_R} - \frac{\left(a_1 + a_2 \right) \times \left(V_{R^2} - V_{S^2} \right)}{2 \times a_1 \times a_2 \times V_R \times 10^3} \end{align}\)

[In case of triangular drive]

\(\begin{align}T &= T_1 + T_2 \\[5pt] & = \frac{V_{Rmax} - V_S}{a_1 \times 10^3} + \frac{V_{Rmax} - V_S}{a_2 \times 10^3} \end{align}\)
③ Calculate the positioning time
VRmax
Maximum speed calculated for triangular drive [mm/s]
VR
Operating speed [mm/s]
Vs
Starting speed [mm/s]
L
Positioning distance [mm]
a1
Acceleration [m/s²]
a2
Deceleration [m/s²]
T
Positioning time [s]
T1
Acceleration time [s]
T2
Deceleration time [s]
T3
Constant speed time [s]

Other conversion formulas are explained below.
The pulse speed and operating speed can be converted using the formula below. Keep the operating speed below the specified maximum speed.

\(\begin{align}\text{Pulse Speed [Hz]} =\frac{\text{Operating Speed [mm/s]}}{ \text{Minimum Travel Amount [mm]}}\end{align}\)

The number of operating pulses and travel amount can be converted using the formula below.

\(\begin{align}\text{The Number of Operating Pulses [Pulse]} =\frac{\text{Travel Amount [mm]}}{ \text{Minimum Travel Amount [mm]}}\end{align}\)

The acceleration/deceleration rates and acceleration can be converted using the formula below.

\(\begin{align}\text{The Acceleration/Deceleration Rates} [\mathrm{ms/kHz}] =\frac{\text{Minimum Travel Amount} [\mathrm{mm}] \times 10^3}{ \text{Acceleration} [\mathrm{m/s^2}]}\end{align}\)

The input methods for speed, travel amount, and acceleration vary depending on the host control device. Please calculate according to your usage.

Selection Procedure and Calculation Formula