Lesson 1.2: Simple Stress and Strain in Actuator Systems

Learn axial stress and strain analysis through a CNC actuator shaft system, covering material selection criteria for precision mechatronic applications.

🎯 Learning Objectives

By the end of this unit, you will be able to:

1. Analyze axial stress and strain in precision actuator shafts
2. Calculate allowable loads based on yield strength criteria
3. Compare material properties (aluminum vs steel) for actuator applications
4. Solve shaft sizing problems for CNC positioning systems

🔧 Real-World System Problem: CNC Z-Axis Actuator Shaft

In a Computer Numerical Control (CNC) machine, the Z-axis actuator controls vertical tool movement. The actuator shaft must transmit precise forces while maintaining dimensional accuracy under varying loads.

System Description

CNC Z-Axis Components:

• Lead Screw (converts rotary motion to linear motion)
• Actuator Shaft (transmits axial forces to the tool)
• Linear Bearings (guide smooth motion)
• Stepper/Servo Motor (provides controlled rotation)

The Critical Challenge

During machining operations:

• The actuator shaft experiences axial tension when the tool pulls upward
• Heavy workpieces create axial compression loads
• Precision requirements demand minimal shaft deformation

Engineering Question: How do we select the right material and shaft diameter to ensure the CNC system maintains ±0.01 mm positioning accuracy under a maximum load of 5,000 N?

Why Material Selection Matters

Aluminum Shaft Benefits:

• Lightweight (reduces system inertia)
• Good machinability
• Corrosion resistance
• Lower cost

Steel Shaft Benefits:

• Higher strength
• Greater stiffness (higher E)
• Better wear resistance
• Proven reliability

The choice affects system performance, cost, and precision.

📚 Fundamental Theory: Axial Stress and Strain

Building on Lesson 1’s foundation, let’s develop the theory needed to solve our actuator shaft problem.

Axial Loading Conditions

When forces act along the centerline of a shaft:

⚡ Axial Stress Formula

Where:

• = Axial force (N)
• = Shaft diameter (m)
• = Cross-sectional area (m²)

Physical Meaning: Stress is distributed uniformly across the cross-section when forces act along the centerline.

Axial Strain and Deformation

The shaft will elongate (tension) or compress (compression) by:

📏 Axial Deformation Formula

Where:

• = Total deformation (m)
• = Shaft length (m)
• = Young’s modulus (Pa)

Physical Meaning: Deformation is proportional to force and length, but inversely proportional to cross-sectional area and material stiffness.

Design Criteria for Actuator Systems

• Strength Criterion
• Stiffness Criterion
• Material Properties

Prevent Material Failure:

🔒 Strength Design Criterion

Where:

• SF = Safety Factor (typically 2-4 for CNC systems)

Purpose: Ensures the applied stress remains well below the material’s yield strength to prevent permanent deformation.

🔧 Application: CNC Actuator Shaft Analysis

Let’s solve the actuator shaft problem step by step using our theoretical foundation.

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System Parameters:

• CNC Z-axis positioning system
• Circular actuator shaft
• Maximum axial force: 5,000 N (tension)
• Safety factor: SF = 3
• Deformation limit: δ ≤ 0.05 mm
• Shaft length: L = 400 mm

Material Options:

• Option A: Aluminum 6061-T6 (E = 70 GPa, σ_yield = 270 MPa)
• Option B: Steel 1045 (E = 200 GPa, σ_yield = 530 MPa)

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Step 1: Determine Aluminum and Steel Strengths

Click to reveal strength calculations

1. Determine Minimum Shaft Diameter (Strength) for Aluminum 6061-T6:

   Allowable stress:

   Required area:

   Minimum diameter:

2. Determine Minimum Shaft Diameter (Strength) for Steel 1045:

   Allowable stress:

   Required area:

   Minimum diameter:

Step 2: Check 10 mm diameter Shaft Stiffness Requirements

Click to reveal shaft requirements calculations

1. Aluminum shaft (d = 10 mm):

   Area:

   Deformation:

   ❌ Fails stiffness requirement (0.364 mm > 0.05 mm allowable)

2. Steel shaft (d = 10 mm):

   Deformation:

   ❌ Still fails stiffness requirement (0.127 mm > 0.05 mm allowable)

Step 3: Size for Stiffness Requirements

Click to reveal area and diameter calculations

1. Area and diameter for aluminum

   Required area for aluminum:

   Required diameter:

2. Area and diameter for steel

   Required area for steel:

   Required diameter:

Step 4: Final Design Comparison

Click to reveal design recommendations

Aluminum 6061-T6 Shaft

Required Diameter: 27 mm
Mass (L=400mm): 0.62 kg
Stress: 8.7 MPa
Deformation: 0.05 mm
Cost: Lower material cost

Steel 1045 Shaft

Required Diameter: 16 mm
Mass (L=400mm): 0.63 kg
Stress: 24.9 MPa
Deformation: 0.05 mm
Cost: Higher material cost

🎯 Material Selection Guidelines for Mechatronics

When to Choose Aluminum

When to Choose Steel

📋 Summary and Next Steps

In this unit, you learned to:

1. Analyze axial stress in actuator shafts using σ = F/A
2. Calculate deformation using δ = FL/(AE)
3. Apply both strength and stiffness criteria in design
4. Compare aluminum vs steel for precision applications

Key Engineering Insights:

• Stiffness ∝ E : Material stiffness is material-dependent
• Stiffness ∝ A : Cross-sectional area linearly affects stiffness
• Strength ∝ A : Doubling area doubles load capacity
• Deformation ∝ 1/E : Higher E dramatically reduces deflection

Coming Next: In Lesson 1.3, we’ll analyze compound bars with multiple materials, exploring how linear actuators with steel-aluminum segments share loads and deform under force.

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