Symbol	Name	Unit
De	De	mm
Di	Di	mm
h0	h0	mm
material	material	—
t	t	mm
temp_C	temp_C	°C

Thermal Expansion Geometric Changes

1. Physical Mechanism

When disc springs operate at high temperatures, all geometric dimensions (outer diameter $$D_e$$ , inner diameter $$D_i$$ , thickness $$t$$ , cone height $$h_0$$ ) increase due to thermal expansion. These changes directly affect:

Installation clearance: The increase in outer diameter reduces the radial clearance with the guide sleeve, which can lead to jamming in severe cases.
Load characteristics: The increase (expansion) in cone height $$h_0$$ alters the force‑deflection curve of the disc spring, causing the force value at the same compression to deviate from the room‑temperature calibration value.
Stack stability: Dimensional changes may affect the relative positions and contact states of disc springs in series/parallel stacks.

2. Core Calculation Formulas

2.1 Basic Linear Thermal Expansion Formula

For any geometric dimension $$L$$ , the dimension $$L_T$$ after a temperature change $\Delta T$ is:

\boxed{L_T = L_0 \cdot \left[ 1 + \alpha(T) \cdot \Delta T \right]}

Where:

$$L_0$$ : Original dimension at room temperature (20 °C) (mm)
$$L_T$$ : Dimension at operating temperature $$T$$ (mm)
$\alpha(T)$ : Average coefficient of thermal expansion at operating temperature (1/K)
$\Delta T = T_{work} - 20$ : Temperature rise (K)

2.2 Thermal Expansion of Key Dimensions

Dimension	Calculation Formula	Engineering Impact
Outer diameter	$D_{e,T} = D_e \cdot (1 + \alpha \cdot \Delta T)$	Increase may reduce guide clearance, causing jamming
Inner diameter	$D_{i,T} = D_i \cdot (1 + \alpha \cdot \Delta T)$	Increase enlarges clearance with shaft, potentially causing misalignment
Thickness	$t_T = t \cdot (1 + \alpha \cdot \Delta T)$	Relatively small effect on load (thickness increase raises force)
Free cone height	$h_{0,T} \approx h_0 \cdot (1 + \alpha \cdot \Delta T)$	Directly alters the force‑deflection curve, shifting the "zero point"

3. Quantitative Impact on Disc Spring Performance

3.1 Guide Clearance Change

The room‑temperature radial single‑side clearance $$c_0$$ becomes at high temperature:

c_T = c_0 - (D_{e,T} - D_e)/2 = c_0 - \frac{D_e \cdot \alpha \cdot \Delta T}{2}

If $c_T \le 0$ , the disc spring outer diameter makes hard contact with the sleeve inner wall, leading to jamming or even preventing normal compression.

Minimum room‑temperature clearance compensation design:

c_{0,min} \ge \frac{D_e \cdot \alpha \cdot \Delta T}{2} + c_{safe}

Where $c_{safe}$ is the minimum clearance required for normal sliding (typically 0.1 ~ 0.3 mm).

3.2 Load Characteristic Shift

The change in cone height $$h_0$$ causes a slight variation in the dimensionless cone height ratio $\eta = h_0/t$ , thereby affecting the nonlinearity of the force‑deflection curve. For precision positioning or constant‑force springs, $h_{0,T}$ must be substituted into the Almen‑Laszlo formula to recalculate the working load.

Simplified estimation: Since the elastic modulus also decreases ( $E(T) < E_{20}$ ), the effects of thermal expansion geometry changes and modulus reduction on load partially offset each other, but precise calculations must still consider both simultaneously.

4. Material Thermal Expansion Data Reference

Material	$\alpha$ (20–200 °C)	$\alpha$ (20–500 °C)
51CrV4 spring steel	$11.5 \times 10^{-6} / \text{K}$	—
H13 hot work tool steel	$11.0 \times 10^{-6} / \text{K}$	$12.5 \times 10^{-6} / \text{K}$
Inconel 718	$13.0 \times 10^{-6} / \text{K}$	$14.0 \times 10^{-6} / \text{K}$

5. Calculation Example

Given: H13 disc spring, $D_e = 80\ \text{mm}$ , room temperature 20 °C, reserved guide single‑side clearance $c_0 = 0.15\ \text{mm}$ . Operating temperature 500 °C, $\alpha = 12.5 \times 10^{-6}$ .

Outer diameter expansion:

\Delta D_e = 80 \times 12.5 \times 10^{-6} \times (500 - 20) \approx 80 \times 12.5 \times 10^{-6} \times 480 \approx 0.48\ \text{mm}

i.e., single‑side radial expansion of approximately $0.24\ \text{mm}$ .

Remaining clearance at high temperature:

c_T = 0.15 - 0.24 = -0.09\ \text{mm} < 0

Conclusion: The disc spring outer diameter exceeds the sleeve inner diameter, causing severe jamming!

Corrective measure: The room‑temperature reserved clearance should be increased to at least $0.24 + 0.10 = 0.34\ \text{mm}$ , taking $0.4\ \text{mm}$ .

6. Design Guidelines

Guide design: For disc spring assemblies operating at high temperatures, the guide clearance must be checked for thermal expansion compensation based on the maximum operating temperature to ensure $$c_T > 0$$ .
Stack stability: In multi‑disc stacks, cumulative outer diameter expansion may cause overall assembly misalignment, especially with external guiding. It is recommended to maintain sufficient radial constraint even at high temperatures.
FEA verification: For precision mechanisms, thermomechanical coupled finite element analysis is recommended to automatically incorporate thermal expansion geometry changes into the model, comprehensively evaluating load and contact states.

Core conclusion: Temperature increase causes linear growth of disc spring outer diameter, cone height, and other dimensions according to $\alpha \cdot \Delta T$ . Among these, outer diameter increase is the primary cause of guide jamming. Design must reserve sufficient high‑temperature compensation clearance and recalculate the load formula using the expanded cone height.

Thermal Expansion Geometry Change

Formula Expression

Parameters

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