Question

In a cross lapper, if draw off speed is V$_A$ and laying speed is V$_L$, then laying angle is given by

• A. $\cot \alpha = \frac{V_A}{V_L}$
• B. $\tan \alpha = \frac{V_A}{V_L}$
• C. $\cot \alpha = \frac{V_L}{V_A}$
• D. $\tan \alpha = \frac{V_L}{V_A}$

Hint:

• Cross Lapper Lays a web (e.g., carded web) in overlapping layers onto a moving conveyor to form a cross-laid batt.
• $V_L$: Laying speed (transverse speed of the laying head).
• $V_A$: Draw-off speed (speed of the output conveyor, also called apron speed).
• The laying angle $\alpha$ (angle of the laid web strips relative to the machine direction) is given by $\tan \alpha = \frac{V_L}{V_A}$.
• This relationship comes from the vector composition of the transverse laying speed and the longitudinal conveyor speed.

Answer 1

The Correct Option is D

To solve this problem, let's carefully analyze the relationship between the draw off speed (V_A), laying speed (V_L), and the laying angle (α) in a cross lapper.

1. Understanding Cross Lapper Mechanics

A cross lapper is a textile machine that creates a layered web of fibers. The laying angle (α) is determined by the ratio between: 
- Draw off speed (V_A): The longitudinal speed of the web 
- Laying speed (V_L): The transverse speed of the carriage

2. Correct Trigonometric Relationship

The laying angle α is properly defined by: \[ \tan \alpha = \frac{V_L}{V_A} \] This is because: - V_L represents the transverse component (opposite side) - V_A represents the longitudinal component (adjacent side)

3. Why Option 4 is Correct

Option 4: tan α = V_L/V_A is correct because: 1. It properly represents the fundamental trigonometric relationship 2. In machine operation, the laying angle increases with higher transverse speed (V_L) 3. It matches the actual physical configuration where: - The carriage moves sideways at V_L - The web advances at V_A 4. This is the standard formula used in textile engineering references

4. Clarifying Common Misconceptions

While cotangent relationships (Options 1 and 3) may appear in some derivations, the fundamental definition uses tangent because: - It directly relates the observable transverse motion to longitudinal motion - The angle α is naturally defined as the angle between the fiber direction and the longitudinal axis - Practical machine adjustments are made based on this relationship

5. Practical Implications

This relationship is crucial for: - Determining web structure and uniformity - Calculating production rates - Optimizing machine settings for different materials - Ensuring proper fiber orientation in nonwoven fabrics

6. Final Answer

The correct formula for the laying angle is tan α = V_L/V_A (Option 4), as this accurately represents the relationship between the transverse and longitudinal speeds in cross lapper operation.