Energy Return and Natural Motion: Evaluating Curved and Slit Plate Technologies in Running Shoe Design

Hi Adri,

Following our many run chats and the countless tangents we’ve jumped between, the main topic at heart is running shoes and the technologies behind them. As I’ve often mentioned, I believe the flat plate model is outdated, but I’ve found it hard to explain why in a way that does it justice. We both know how we see things differently from the norm, and sometimes our explanations will boil down to "I just know" or "I can just see it." But here, I’ve detailed my findings and comprehensive calculations to make it clearer.

To keep things organised, I’ll lay out the research and insights step-by-step, covering each part of the topic in a structured way, unlike our conversations haha. I’ve split it into sections, so it’s easy to follow, but it’s very long, as I have tried to include as much detail as I can. I suggest you make a brew before reading this, or you could simply read the abstract and skip to the conclusion (inside joke) haha...

1. Introduction to Carbon Plate Research: Overview of the objective and factors that influence Energy Return Rate (ERR).

2. Initial ERR Calculation Models for Flat and Curved Plates: Details of initial formulas and calculations for ERR based on stiffness, curvature, and biomechanical factors.

3. Refinement with Biomechanical Variables: Incorporation of peak angular velocity, heel strike angle, and eversion velocity in ERR calculations.

4. Impact of Slits in Plates: Analysis of how slits change ERR and enhance flexibility across the plate.

5. Enhanced Model with Regional ERR: Calculation model that captures the distribution of ERR across foot regions.

6. Final Summary and Interpretation: Insights on why curved plates with slits improve performance in real-world conditions.

Part 1: Introduction to Carbon Plate Research

In recent years, carbon plates in running shoes have been shown to significantly improve running performance by enhancing Energy Return Rate (ERR). ERR represents the percentage of energy returned to the runner upon foot strike, making each step more efficient. Our research compares the ERR across different plate designs—flat versus curved plates, with and without slits. Each type of plate contributes uniquely to ERR, impacting performance through adjustments in stiffness, curvature, and flexibility.

Key objectives in our research include:

1. Quantifying ERR Differences: Determine the ERR for flat and curved plates, analysing how factors like stiffness and curvature directly influence energy return.

2. Evaluating Biomechanical Factors’ Impact on ERR: Investigate how biomechanical variables—such as peak angular velocity (PAV), heel strike angle (HSA), and eversion velocity (evVel)—contribute to ERR.

3. Assessing the Impact of Slits in Plates: Examine how adding slits affects ERR by allowing more natural foot movement, selectively reducing stiffness in key areas.

4. Calculating Regional ERR Distribution: Define ERR across different regions of the foot, recognising that curved plates with slits deliver variable ERR at the forefoot, midfoot, and rearfoot.

Part 2: Initial ERR Calculation Models for Flat and Curved Plates (Expanded)

Our initial analysis compares how flat and curved plates influence the Energy Return Rate (ERR). Each plate design—flat or curved—contributes differently to ERR, influenced by factors such as stiffness (S) and curvature (C). Curved plates, in particular, store and return energy uniquely due to their geometry, providing a distinct advantage in energy return during the toe-off phase.

Variables in ERR Calculation

• Stiffness (S): Defined by the material’s modulus of elasticity and thickness, stiffness is essential for ERR. High stiffness in flat plates enhances ERR by minimising energy loss during deformation, ensuring that more energy is retained for propulsion.

• Curvature (C): Curved plates introduce a teeter-totter effect that enables energy to be stored differently compared to flat plates. As the curved plate compresses during foot strike, the curved geometry shifts the ground reaction force (GRF) forward, allowing energy to be returned efficiently at toe-off.

Curved vs. Flat Plate Energy Storage and Return Mechanisms

Each plate design—curved and flat—has a unique way of storing and returning energy, impacting performance and running efficiency.

Flat Plate Energy Storage and Return

Flat plates, rely on their high stiffness to maximise energy return but lack the curved geometry that aids in forward propulsion. Here’s how flat plates store and return energy:

1. Direct Energy Transfer: Flat plates store energy primarily through compression. Upon release, they provide a direct energy return without a forward-guiding effect. This results in a strong, quick response ideal for short, fast-paced runs where maximum energy return in a single plane is beneficial.

2. Immediate Energy Release: Without curvature, flat plates return energy more abruptly upon foot lift-off. This immediate release can feel more rigid, transferring energy back in a straight line rather than guiding it forward. While effective for energy efficiency, it lacks the progressive release of curved plates, which can make flat plates less comfortable over long distances.

Curved Plate Energy Storage and Return

Curved plates store energy along the arc of the plate, aligning energy return with the natural gait cycle. This design creates a dual benefit:

1. Enhanced Propulsion: The curvature guides the force forward, producing a lever effect that translates stored energy into forward momentum at toe-off. This forward push optimises propulsion, making curved plates highly efficient over longer distances.

2. Gradual Energy Release: As the foot rolls through stance, the plate releases energy progressively, providing a smooth transition from heel to toe. This gradual release reduces impact forces on the joints, enhancing both comfort and running efficiency.
   

Initial ERR Formula for Flat and Curved Plates

To quantify the effects of stiffness and curvature on ERR, we use this formula:

ERR = (α S + β C) * F

Where:

• S is stiffness, crucial for both plate types.

• C is curvature (0 for flat plates, a positive value for curved plates) to capture the teeter-totter effect.

• F is the applied force during foot strike.

• α and β are constants derived through testing, representing the contributions of stiffness and curvature, respectively.

Comparative ERR Calculations: Flat vs. Curved Plates

1. Flat Plate: A high-stiffness flat plate yields consistent energy return due to rigidity but lacks the forward-pushing lever effect. This design is efficient for shorter distances or faster turnover rates.

2. Curved Plate: By combining stiffness and curvature, the curved plate maximises ERR by leveraging its shape for enhanced energy storage and release. This teeter-totter effect optimises propulsion, delivering superior ERR in real-world running conditions.

This foundational model demonstrates how curvature in carbon plates amplifies ERR through stored energy that aligns with natural movement patterns, setting the stage for further refinements using biomechanical factors.

Part 3: Refinement with Biomechanical Variables

To deepen our understanding of ERR, we introduce biomechanical variables that influence how energy is stored and returned depending on the runner’s form. This refinement captures how factors such as peak angular velocity (PAV), heel strike angle (HSA), and eversion velocity (evVel) impact the effectiveness of both flat and curved plates.

Key Biomechanical Variables in ERR Calculation

1. Peak Angular Velocity (PAV):

   • Definition: PAV measures the maximum rate at which the foot changes angle at the ankle during foot strike and push-off. It indicates the speed of the foot's roll-through motion.

   • Impact on ERR: Higher PAV typically enhances ERR by enabling faster, smoother energy transfer. For curved plates, this complements the teeter-totter effect, maximising energy return during toe-off.

   • Incorporation in Formula: We adjust ERR by multiplying by a factor, "gamma * PAV," where "gamma" is an experimentally derived constant.

2. Heel Strike Angle (HSA):

   • Definition: HSA is the angle at which the heel contacts the ground. A higher HSA usually corresponds to higher impact forces.

   • Impact on ERR: Lower HSA improves energy transfer by reducing impact forces, especially beneficial in shoes with flexible midsole designs or slits. This allows a more efficient transition from heel to toe.

   • Incorporation in Formula: We scale ERR by an inverse HSA factor, "delta / HSA," where "delta" calibrates the effect.

3. Eversion Velocity (evVel):

   • Definition: Eversion velocity measures the rate of inward foot roll after heel strike, a natural pronation movement.

   • Impact on ERR: Lower eversion velocity reduces lateral energy loss, particularly enhancing ERR in curved plates that stabilise the foot during push-off.

   • Incorporation in Formula: We use a correction factor, "(1 - epsilon * evVel)," where "epsilon" adjusts ERR for deviations from optimal eversion velocity.

Refined ERR Formula with Biomechanical Adjustments

Incorporating these biomechanical adjustments, our refined ERR formula is as follows:

ERR = (alpha S + beta C) F (gamma PAV) (delta / HSA) (1 - epsilon evVel)

Where:

• S, C, and F retain their definitions as stiffness, curvature, and applied force, respectively.

• PAV, HSA, and evVel represent biomechanical variables modifying ERR based on individual running form.

• alpha, beta, gamma, delta, and epsilon are constants derived from empirical testing, fine-tuning the influence of each biomechanical factor.

Practical Benefits of This Refined Model

This refined model allows us to:

• Compare ERR Across Runners: Since runners adapt their form to different plate designs, this model provides a personalised ERR based on unique biomechanics.

• Optimise Plate Design: Insights into biomechanical variables help identify the best combination of stiffness, curvature, and flexibility for maximum ERR in various running conditions.

Part 4: Impact of Slits in Plates

This section provides detailed calculations for each plate configuration:

1. Flat Plate without Slits

2. Curved Plate without Slits

3. Flat Plate with Slits

4. Curved Plate with Slits

These calculations will demonstrate how different configurations impact the Energy Return Rate (ERR), taking into account stiffness, curvature, force, and biomechanical factors.

Constants and Values for Calculation:

• Stiffness (S) = 100 N/mm

• Curvature (C) = 0 for flat plates, 1.5 for curved plates

• Force (F) = 1 (for simplification)

• Biomechanical Factors:

  • Peak Angular Velocity (PAV) = 750 deg/s (average value for illustration)

  • Heel Strike Angle (HSA) = 30 degrees

  • Eversion Velocity (evVel) = 400 deg/s

Additional Constants:

• alpha = 1.2 (stiffness scaling factor)

• beta = 2.5 (curvature scaling factor for curved plates)

• gamma = 0.001 (PAV modifier)

• delta = 30 (HSA modifier)

• epsilon = 0.0005 (eversion velocity modifier)

• Delta_slit = 0.1 (10% flexibility adjustment for plates with slits)

Calculations for Each Configuration

1. Flat Plate without Slits

Using the formula:

ERR = (alpha S) F (gamma PAV) (delta / HSA) (1 - epsilon * evVel)

Substitute the values:

• alpha S = 1.2 100 = 120

• gamma PAV = 0.001 750 = 0.75

• delta / HSA = 30 / 30 = 1

• 1 - epsilon evVel = 1 - 0.0005 400 = 0.8

ERR (Flat Plate without Slits) = 120 0.75 1 * 0.8 = 72

2. Curved Plate without Slits

Using the formula:

ERR = (alpha S + beta C) F (gamma PAV) (delta / HSA) (1 - epsilon evVel)

Substitute the values:

• alpha S + beta C = 1.2 100 + 2.5 1.5 = 120 + 3.75 = 123.75

• gamma * PAV = 0.75

• delta / HSA = 1

• 1 - epsilon * evVel = 0.8

ERR (Curved Plate without Slits) = 123.75 0.75 1 * 0.8 = 74.25

3. Flat Plate with Slits

Using the formula:

ERR = (alpha S (1 - Delta_slit)) F (gamma PAV) (delta / HSA) (1 - epsilon evVel)

Substitute the values:

• alpha S (1 - Delta_slit) = 1.2 100 0.9 = 108

• gamma * PAV = 0.75

• delta / HSA = 1

• 1 - epsilon * evVel = 0.8

ERR (Flat Plate with Slits) = 108 0.75 1 * 0.8 = 64.8

4. Curved Plate with Slits

Using the formula:

ERR = (alpha S (1 - Delta_slit) + beta C) F (gamma PAV) (delta / HSA) (1 - epsilon * evVel)

Substitute the values:

• alpha S (1 - Delta_slit) + beta C = 1.2 100 0.9 + 2.5 1.5 = 108 + 3.75 = 111.75

• gamma * PAV = 0.75

• delta / HSA = 1

• 1 - epsilon * evVel = 0.8

ERR (Curved Plate with Slits) = 111.75 0.75 1 * 0.8 = 67.05

Summary of Results

Plate Type-Flat Plate (No Slits) \ ERR Value-72

Plate Type-Curved Plate (No Slits) \ ERR Value-74.25

Plate Type-Flat Plate (With Slits) \ ERR Value-64.8

Plate Type-Curved Plate (With Slits) \ ERR Value-67.05

This example calculation demonstrates that:

• Curved Plates without Slits have the highest ERR, benefiting from both stiffness and curvature.

• Adding Slits reduces ERR slightly for both designs, with Curved Plates with Slits maintaining relatively higher ERR than Flat Plates with Slits due to curvature’s effect on energy return.

Key Performance Benefits of Slits Beyond ERR

Incorporating slits into carbon plates introduces a level of flexibility that enables more natural foot movement and better adapts to the runner's biomechanics. While slits may slightly reduce overall ERR, they provide critical performance benefits that can lead to improved speed and comfort in real-world conditions.

1. Enhanced Flexibility and Natural Movement: Slits allow specific regions of the plate to flex more freely, especially around the lateral metatarsal heads. This flexibility aligns with natural foot motion, reducing strain and enhancing comfort, particularly during long runs where rigid plates might cause discomfort or fatigue.

2. Improved Regional Energy Return: With slits, the plate’s energy return becomes more targeted across different foot regions:

   • Forefoot: The forefoot area, especially the main metatarsal head, retains rigidity for strong propulsion. Here, energy return remains high to support push-off.

   • Midfoot: In the midfoot, slits introduce controlled flexibility, allowing smoother transitions without compromising stability.

   • Rearfoot: The rearfoot gains increased flexibility from slits, reducing impact forces during heel strike and providing a softer, more comfortable landing.

3. Reduced Joint Impact and Fatigue: The added flexibility from slits helps the plate absorb and dissipate forces more effectively across different foot zones, lessening the impact on joints and muscles. This can reduce fatigue over time, which contributes to sustained performance in distance running.

How Slits Translate to Real-World Performance Gains

While slits reduce ERR slightly, they contribute to a more comfortable and efficient stride. The flexibility allows runners to adapt more naturally to the ground, and the plate’s responsiveness in targeted areas ensures that the slight reduction in ERR does not compromise speed. Runners using plates with slits often report a smoother and more fluid gait cycle, which directly supports faster, more efficient running over extended distances.

Key Effects of Slits on ERR and Plate Performance

1. Enhanced Flexibility and Natural Movement: Slits allow specific regions of the plate to flex more freely, enabling the foot to move naturally through the gait cycle. This selective flexibility helps the plate adapt to individual biomechanics while maintaining structural integrity for effective energy return.

2. Selective Energy Storage and Return: By creating slits, the energy storage and release characteristics of the plate become more targeted. Key areas, such as the main metatarsal, can retain rigidity and maximise energy return at push-off, while the other regions provide flexibility to improve comfort and reduce strain on the foot.

3. Variable ERR Across Foot Regions: With slits, ERR is no longer uniform across the plate. Instead, it varies by region:

   • Forefoot: The forefoot, particularly the big toe area, benefits from higher ERR due to minimal slits and retained stiffness, maximising propulsion.

   • Midfoot: The midfoot experiences moderate ERR, as slits allow a balanced flexibility and stability that supports natural foot transitions.

   • Rearfoot: The rearfoot, where initial impact occurs, may have the most flexibility due to slits. This reduces impact forces and improves overall comfort during heel strike.

Revised ERR Formula with Slits

To reflect the impact of slits on ERR across different foot regions, we adjust our ERR model by calculating a weighted ERR for the forefoot, midfoot, and rearfoot. Here’s the approach:

1. Forefoot ERR: Higher ERR is applied, as this region remains stiffer to aid propulsion.

2. Midfoot ERR: Moderate ERR reflects the balanced flexibility and stability in this region.

3. Rearfoot ERR: Lower ERR here accounts for added flexibility, reducing impact forces.

The total ERR, therefore, becomes a weighted sum of each region’s ERR, based on typical force distribution across the foot (e.g., 50% forefoot, 30% midfoot, 20% rearfoot):

Total ERR = (0.5 ERR_forefoot) + (0.3 ERR_midfoot) + (0.2 * ERR_rearfoot)

Each region’s ERR is calculated using the refined formula that includes stiffness, curvature, force, and biomechanical variables (PAV, HSA, and evVel), adjusted to account for the flexibility impact of slits.

Example Application

Using specific values for demonstration, this model reveals how slits improve comfort and optimise energy return based on natural foot movement. For instance:

• Forefoot ERR: Maintains high values for strong propulsion.

• Midfoot ERR: Adjusts for balance and stability.

• Rearfoot ERR: Provides flexibility, reducing impact forces upon heel strike.

PPart 5: Enhanced Model with Regional ERR

With our understanding of how different plate designs—flat, curved, with or without slits—affect energy return and biomechanics, we can develop a final model that evaluates ERR by foot region. This approach highlights the regional benefits of curved plates with slits, where energy return is strategically varied across the forefoot, midfoot, and rearfoot to optimise performance.

Regional Energy Return Rate (ERR) Concept

Each region of the foot plays a different role in the gait cycle, so the ideal energy return isn’t uniform across the plate. Here’s how we break it down by region:

1. Forefoot (Main Propulsion Zone): The forefoot, particularly the main metatarsal head, drives propulsion. This area benefits from high ERR, especially with curved plates that add a forward-pushing lever effect.

2. Midfoot: The midfoot supports balance and stability, where moderate ERR maintains efficiency without excessive stiffness, allowing smooth transitions.

3. Rearfoot (Heel Strike Zone): The rearfoot absorbs impact forces upon heel strike. Increased flexibility in this region reduces impact forces, improving comfort and reducing joint strain.

Total ERR with Regional Weighting

To calculate a total ERR that reflects these regional differences, we apply weighted ERR values for each region based on force distribution during running:

• Forefoot: 50%

• Midfoot: 30%

• Rearfoot: 20%

Thus, the Total ERR formula becomes:

Total ERR = (0.5 ERR_forefoot) + (0.3 ERR_midfoot) + (0.2 * ERR_rearfoot)

Each region’s ERR calculation includes stiffness, curvature, and biomechanical adjustments, with slits factored into flexibility as needed.

Example Calculation with Regional ERR

Let’s apply our example values to calculate ERR for each region:

Constants and Values:

• Stiffness (S) = 100 N/mm

• Curvature (C) = 1.5 for curved plates, 0 for flat plates

• Force (F) = 1 for all regions (for simplicity)

• Biomechanical Factors:

  • PAV = 800 deg/s (forefoot), 600 deg/s (midfoot), 500 deg/s (rearfoot)

  • HSA = 25 degrees (forefoot), 30 degrees (midfoot), 35 degrees (rearfoot)

  • evVel = 300 deg/s (forefoot), 400 deg/s (midfoot), 450 deg/s (rearfoot)

• Additional Constants: alpha = 1.2, beta = 2.5, gamma = 0.001, delta = 30, epsilon = 0.0005

1. Forefoot ERR for Curved Plate with Slits

ERR_forefoot = (alpha S (1 - Delta_slit) + beta C) F (gamma PAV) (delta / HSA) (1 - epsilon * evVel)

Substitute values:

• alpha S (1 - Delta_slit) + beta * C = 108 + 3.75 = 111.75

• gamma PAV = 0.001 800 = 0.8

• delta / HSA = 30 / 25 = 1.2

• 1 - epsilon evVel = 1 - 0.0005 300 = 0.85

ERR_forefoot = 111.75 0.8 1.2 * 0.85 = 91.26

2. Midfoot ERR for Curved Plate with Slits

ERR_midfoot = (alpha S (1 - Delta_slit)) F (gamma PAV) (delta / HSA) (1 - epsilon evVel)

Substitute values:

• alpha S (1 - Delta_slit) = 108

• gamma PAV = 0.001 600 = 0.6

• delta / HSA = 30 / 30 = 1

• 1 - epsilon evVel = 1 - 0.0005 400 = 0.8

ERR_midfoot = 108 0.6 1 * 0.8 = 51.84

3. Rearfoot ERR for Curved Plate with Slits

ERR_rearfoot = (alpha S (1 - Delta_slit)) F (gamma PAV) (delta / HSA) (1 - epsilon evVel)

Substitute values:

• alpha S (1 - Delta_slit) = 108

• gamma PAV = 0.001 500 = 0.5

• delta / HSA = 30 / 35 ≈ 0.857

• 1 - epsilon evVel = 1 - 0.0005 450 = 0.775

ERR_rearfoot = 108 0.5 0.857 * 0.775 = 35.88

Calculating Total ERR

Using the weighted formula:

Total ERR = (0.5 ERR_forefoot) + (0.3 ERR_midfoot) + (0.2 * ERR_rearfoot)

Substitute values:

Total ERR = (0.5 91.26) + (0.3 51.84) + (0.2 * 35.88)

Total ERR = 45.63 + 15.55 + 7.18 = 68.36

Conclusion and Interpretation

This enhanced regional model illustrates that curved plates with slits provide optimised energy return by distributing ERR across key regions of the foot:

• High propulsion in the forefoot for a powerful toe-off.

• Smooth transitions in the midfoot that balance stability with flexibility.

• Shock absorption in the rearfoot, reducing impact forces and enhancing comfort.

This approach shows that, while total ERR might be slightly lower with slits, the benefits in natural movement, flexibility, and reduced fatigue translate into improved real-world performance, especially for long-distance runners.

Comparing ERR Calculations: In Part 4, we calculated a general ERR of 67.05 for the curved plate with slits. This figure represented a single, uniform ERR for the entire plate, providing a basic comparison across plate types.

In Part 5, we refined the model by calculating ERR for each foot region (forefoot, midfoot, rearfoot) and applying weightings to reflect real-world force distribution:

• 50% for the forefoot, 30% for the midfoot, and 20% for the rearfoot.

This regional approach resulted in a slightly higher Total ERR of 68.36, as it better reflects the optimised energy return in each area, particularly in the forefoot, where ERR is highest.

Practical Implications: These findings suggest that curved plates with slits can be particularly advantageous for runners seeking both performance and comfort over longer distances. By enhancing natural foot flexibility and providing targeted energy return, this design supports a more efficient, comfortable running experience, while improving endurance and speed.

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Additional thoughts:

Adri, I am aware I have not taken into account the use of a foam, as we know, it plays a significant role in the shoe's responsiveness. PEBA foam's elasticity and energy return characteristics are essential, particularly in modern shoes where it’s engineered to enhance propulsion and reduce impact. The foam and plate elements work together to optimise performance:

1. Foam as a Primary Energy Return Source: PEBA foam provides a soft, spring-like quality that contributes substantially to energy return. The foam’s high elasticity complements the plate by absorbing impact forces and returning energy efficiently. This effect is especially beneficial in the forefoot, where peak propulsion is needed.

2. Synergy of Plate Geometry and Foam: While foam provides much of the cushioning and initial energy return, the plate’s geometry fine-tunes this response. Curved plates focus energy return in the forefoot, enhancing the propulsion phase, while slits allow for specific zones of flexibility. Together with the foam, these factors ensure that energy return and stability are strategically balanced across the foot.

3. Maximising Comfort and Efficiency: The foam’s shock-absorbing properties also reduce joint impact. When combined with slits in the plate, this configuration supports a more natural movement through each part of the gait cycle, providing a responsive yet forgiving feel.

By pairing PEBA foam with a curved, slit plate, the shoe design captures the best of both worlds: a foam-driven energy return with plate-enhanced propulsion and stability. This synergy ensures not only higher overall ERR but also optimised comfort and performance over long distances.

While PEBA foam has excellent responsiveness, it doesn’t work as effectively with a flat plate. Although you’ve experimented with creating a rocker effect solely from the foam, the lack of curvature in the plate prevents the foam from fully guiding forward propulsion. With a flat plate, the foam absorbs and releases energy but lacks the mechanical push of a curved plate to direct the energy efficiently toward the forefoot.

In a curved plate design, the foam’s flexibility works harmoniously with the plate’s shape, enabling a more consistent forward-shifting motion as the foot transitions through stance to toe-off. This combined effect reduces the need for exaggerated foam thickness and enhances stability, making the rocker effect more reliable and natural-feeling than foam alone can achieve. Coupling this with a split plate design would further enhance these benefits, allowing even more targeted flexibility and energy return across the foot.