Technical Analysis: The Invariance Principle of Disc Spring Flatness Stiffness

A notable phenomenon in the design and application of disc springs (or slotted disc springs) is that when other geometric parameters (such as thickness, inner and outer diameters) remain constant, the flattening stiffness (the ratio of flattening load Pf to flattening displacement h₀) remains constant when only the free height (h₀) is altered. Jiangsu Sunzo Spring will explore the mechanical essence of disc spring technology to reveal the simple yet elegant mathematical relationship behind this characteristic.

1、Definition of Phenomenon: What is the Flatness Stiffness

Firstly, the concept is clarified: for disc spring, "flatness stiffness" is not the instantaneous stiffness in its working process, but a specific evaluation index, defined as:

Flatness stiffness = flat load (Pf) / flat displacement (h₀)

The flattening displacement is the free height h₀ of the disc spring, while the flattening load Pf is the force required to fully flatten the disc spring (in the theoretical state).

2、The Core Principle: Revealing the Proportional Relationship

This invariance is derived from a fundamental relationship in the disc spring theory: the flattening load Pf is proportional to the free height h₀.

i.e., Pf ∝ h₀

This proportional relationship can be derived from the classical Almanrazlo (AL) formula. In the formula, when other parameters (material E, Poisson's ratio μ, thickness t, inner/outer diameter D/d) are fixed, the expression of the flattening load Pf directly contains a term of h₀ to the first power.

3、Mathematical Derivation: The Constancy of Ratio

According to the above proportional relationship, we can express it as:

Pf = K · h₀

K is a constant that accounts for all the constant parameters, such as material properties (E, μ), thickness t, and inner/outer diameter D/d.

Thus, the flattening stiffness is defined as:

Flatness stiffness = Pf / h₀ = (K · h₀) / h₀ = K

Thus, regardless of how h₀ changes (whether increasing or decreasing), the proportional constant K remains unchanged as long as other parameters stay constant, and consequently the flatness stiffness also remains unchanged.

4、Engineering significance and enlightenment

This characteristic provides an important simplification idea and a test method for the design of disc spring.

1. Performance prediction: In the design stage, if the working stroke and load of the disc spring are adjusted by changing h₀, the "unit height flattening force" (i.e. flattening stiffness) will remain unchanged.

2. Quality inspection: In the production inspection, the flatness stiffness can be used as an indirect index to reflect the stability of the material and the core process, which is independent of the absolute size h₀.

3. The same principle is also applicable to the slotted disc spring. The slotted disc spring changes the equivalent stiffness, but the proportional relationship between the flattening load and the effective free height is still valid.

At Jiangsu Sunzo Spring, we not only master disc spring manufacturing but also deeply understand the underlying mechanical principles. This comprehensive understanding, from observable phenomena to fundamental mechanisms, is integrated into our design and process control systems. This enables us to predict product performance with greater precision, implement more effective quality control, and provide clients with professional technical consultations on parameter adjustments. We are committed to ensuring every disc spring meets specific theoretical expectations and delivers reliable engineering performance.