Design og beregninger

In nested wave springs, the total load $P_{total}$ is the sum of the loads of individual turns, but the number of waves $N$ per turn dictates the stability and the spring rate $k$. A higher $N$ increases the spring rate proportionally to $N^4$, allowing for very high loads in small axial spaces. However, a high $N$ also reduces the wave amplitude, which can lead to manufacturing tolerances having a greater percentage impact on load accuracy. For stability, $N$ must be an integer or a half-integer (e.g., 3.5, 4.5) to ensure proper crest-to-crest alignment and prevent 'shingling' where the turns slide over one another.

The maximum stress occurs at the crests of the waves and is calculated using $S = \frac{3 \pi P D_m}{4 b t^2 N^2}$. This value must be compared against the tensile strength of the material, adjusted by a safety factor for the specific application. For 17-7PH CH900, the yield strength is approximately $1,170$ MPa. If the calculated stress $S$ exceeds $80\%$ of the yield strength during operation, the spring will likely experience a permanent set. In high-cycle applications, the stress should be kept below the endurance limit defined by the Goodman relation $S_a = S_e (1 - \frac{S_m}{S_u})$ where $S_a$ is alternating stress and $S_m$ is mean stress.

As a Crest-to-Crest wave spring is compressed toward its solid height, the mean diameter $D_m$ increases due to the flattening of the waves. The standard spring rate formula $k = \frac{E b t^3 N^4}{1.23 D_m^3 Z}$ assumes a constant diameter. In reality, the increased $D_m$ leads to a non-linear softening effect initially, followed by a sharp hardening as the waves approach the solid state. Engineers must account for this by using an adjusted mean diameter $D_{adj} = D_m + (f \cdot \tan(\theta))$ where $f$ is deflection and $\theta$ is the wave angle. Failure to account for this in precision aerospace valves can lead to incorrect cracking pressures.

Spiral retaining rings are typically wound with zero gap (or a slight overlap). The radial tension $F_r$ required to expand the ring for installation is proportional to $\frac{E \cdot I}{R^2}$. A zero-gap design ensures 360-degree contact with the groove, which is essential for uniform load distribution. If a gap is introduced, the thrust capacity is reduced near the gap area. During installation, the ring is spiraled into the groove; a larger gap might make manual installation easier but reduces the 'cling' or grip the ring has on the groove bottom, which is vital for maintaining position under vibration or high-speed rotation.

Under impact loading, a spiral retaining ring may 'dish' or become conical. The dishing angle $\theta$ can be approximated by $\theta = \frac{6 \cdot P \cdot (D_o - D_i)}{E \cdot T^3 \cdot \ln(D_o/D_i)}$. Impact loads multiply the effective force $P$ by an impact factor $I_f$, which can range from 2 to 5 depending on the kinetic energy of the retained part. If $\theta$ exceeds the angle that would cause the ring to slip out of the groove (typically 10-15 degrees), the ring will fail. Multi-turn rings (2-turn or 3-turn) provide greater resistance to dishing compared to single-turn snap rings because the turns 'nest' and support each other against the bending moment.

The effective thrust capacity is highly sensitive to the groove depth $d$. A shallow groove reduces the contact area, increasing the bearing stress. Furthermore, the groove corner radius $r$ must be kept minimal (typically $< 0.1 \cdot T$) to prevent the ring from 'ramping' out of the groove. If the radius is too large, the load $P$ creates a radial component $P \cdot \tan(\phi)$ that forces the ring to expand (for internal) or contract (for external), leading to premature failure. The safety factor $S_f$ should be adjusted based on the ratio of $r/d$; as $r$ increases relative to $d$, the rated thrust capacity must be derated by as much as 50%.

For external rings, centrifugal force tends to expand the ring, potentially lifting it out of the groove. The maximum allowable speed $V$ in RPM is determined by $V = \sqrt{\frac{4 \cdot E \cdot g \cdot (d_n - d_g)}{0.0132 \cdot \rho \cdot D_m^5 \cdot (D_o - D_i)}}$, where $d_n$ is the neutral ring diameter and $d_g$ is the groove diameter. If the application speed exceeds this value, the ring must be designed with a 'self-locking' feature. This feature consists of a tab and a slot that mechanically prevents the ring from expanding beyond the groove diameter, allowing for significantly higher RPMs in high-speed rotating machinery like turbine shafts.

The thrust capacity based on ring shear $P_r$ is calculated using the formula $P_r = \frac{D \cdot T \cdot \pi \cdot S_s}{S_f}$, where $D$ is the shaft/bore diameter, $T$ is the ring thickness, $S_s$ is the shear strength of the material (approximately 0.6 times the tensile strength), and $S_f$ is a safety factor (typically 3). This calculation assumes that the groove is deep enough and the groove material is strong enough to prevent the ring from dishing. If the groove material yields before the ring shears, the capacity is limited by the groove yield formula $P_g = \frac{D \cdot d \cdot \pi \cdot S_y}{S_f}$, where $d$ is the groove depth and $S_y$ is the yield strength of the groove material.

In standard wave spring formulas, the ratio $b/t$ is assumed to be large enough for beam theory to apply. When $b/t < 8$, the spring behaves less like a simple beam and more like a curved plate. A correction factor $C_f = \frac{1}{1 - \nu^2}$, where $\nu$ is Poisson's ratio, is sometimes applied to the Modulus of Elasticity $E_{eff} = \frac{E}{1 - \nu^2}$ to account for the transverse constraint. For stainless steel ($\nu \approx 0.3$), this increases the theoretical stiffness by approximately 10%. Furthermore, for very narrow radial walls, the risk of 'twisting' or lateral-torsional buckling increases, requiring the use of a stabilization factor in the design calculations.

As a wave spring is compressed toward its solid height, the waves flatten, causing the mean diameter $D_m$ to expand. This radial expansion $\Delta D$ can be approximated by $\Delta D = 0.02 \cdot \frac{(L_0 - L_1)^2}{D_m \cdot Z}$, where $L_0$ is the free height and $L_1$ is the work height. In precision bore installations, this expansion must be accounted for to prevent binding against the housing. If the clearance is insufficient, the resulting radial friction will artificially increase the measured spring rate and may lead to premature fatigue failure due to localized stress concentrations.

The stress $S$ in a wave spring is inversely proportional to the square of the number of waves $Z$. Specifically, $S \propto \frac{1}{Z^2}$. Increasing $Z$ significantly reduces the bending stress for a given deflection, which is critical for extending fatigue life. For high-cycle fatigue (e.g., >1 million cycles), engineers must ensure the maximum stress at work height remains below the fatigue limit of the material, often referenced via a Goodman diagram. For 17-7PH CH900, the design limit is typically around 45% of the minimum tensile strength for cyclic applications.

Nested wave springs consist of multiple turns wound in parallel rather than in series. The total load capacity $P$ increases linearly with the number of nested layers $n$, following $P_{total} = n \cdot P_{single}$. This configuration allows for massive force in extremely tight radial and axial envelopes. However, the shear stress $S$ must be carefully monitored using $S = \frac{3 \cdot \pi \cdot P \cdot D_m}{4 \cdot b \cdot t^2 \cdot Z^2}$. In nested designs, friction between layers can introduce a hysteresis loop in the load-deflection curve, which is quantified by the area between the loading and unloading paths. This friction also acts as a damping mechanism in dynamic systems.

The spring rate $k$ for a multi-turn Crest-to-Crest wave spring is derived from the formula $k = \frac{E \cdot b \cdot t^3 \cdot N}{D_m^3 \cdot Z^4} \cdot \frac{4 \cdot Z}{N}$, where $E$ is the Modulus of Elasticity, $b$ is the radial wall, $t$ is the material thickness, $N$ is the number of turns, $D_m$ is the mean diameter, and $Z$ is the number of waves per turn. Linearity in the load-deflection curve is maintained between 20% and 80% of the available deflection. Beyond 80%, the 'bottoming out' effect occurs where the waves begin to touch, causes an exponential increase in $k$. Conversely, at low deflections (<20%), the rate may be lower due to the initial settling of the wave peaks against the mating surfaces.

The mathematical models for thrust capacity assume a perfectly square corner on both the groove and the retained component. In reality, manufacturing tools leave a corner radius, and components often have chamfers.

Impact and Calculation Adjustment:
1. Radius/Chamfer Effect: A radius ($r$) or chamfer ($ch$) on the retained component shifts the point of contact outward from the groove root, creating a bending moment on the retaining ring that can twist it out of the groove.
2. De-rating Factor Curve: If the chamfer ($ch$) on the mating part exceeds $0.1 \times H$ (where $H$ is the radial wall of the ring), the thrust capacity drops exponentially.
3. Thrust Load Reduction Formula:
$$P_{corrected} = P_g \times \left( 1 - \frac{r_{mating}}{d_{groove}} \right)$$
Where $r_{mating}$ is the corner radius of the retained part, and $d_{groove}$ is the groove depth. To offset this, a thicker, heavy-duty 2-turn or 3-turn spiral ring should be specified to resist twisting forces.

Spring relaxation (or taking a 'set') is the permanent reduction in free height and corresponding load loss that occurs when a wave spring is held at high stress levels for extended periods, accelerated by high temperature.

Manufacturing Mitigation (Presetting):
1. Over-coiling and Pressing: The wave spring is initially coiled to a free height ($H_0$) higher than the target design specification.
2. Presetting / Coining: The spring is compressed completely to its solid height (solid pressing) multiple times, or held at solid height for a specified duration at a elevated temperature. This intentionally induces localized plastic deformation (yielding) at the highly stressed wave crests.
3. Residual Stress Generation: This localized yielding introduces beneficial compressive residual stresses on the outer surfaces of the wave bends. When the spring is subsequently loaded in operation, these residual compressive stresses oppose the active tensile stresses, increasing fatigue life and eliminating further height loss during service.

Nested Wave Springs are wound in parallel from a single continuous flat wire ribbon, resulting in multi-layered, concentric coils where the waves are perfectly in-phase and nested together.

When to Specify Nested Wave Springs:
1. Extreme Loads with Minimal Deflection: Since the layers act as parallel springs, the combined spring rate scales proportionally with the number of turns ($N$):
$$k_{total} = N \times k_{single\_turn}$$
They produce 2x to 5x higher forces than a single-turn wave spring of identical diameter.
2. Space Constraints: Ideal for applications requiring immense forces within very tight axial envelopes (e.g., heavy-duty seals, clutches, high-pressure valves, electrical connector loading).
3. No Stacking Misalignment: Unlike stacked single-turn springs which can shift out of alignment under vibration, nested springs are physically a single component, eliminating stack-up errors and internal friction rubbing.

Wave springs in transmission systems or actuators undergo dynamic cyclic deflection between a minimum stress ($\%S_{min}$) and a maximum stress ($\%S_{max}$). The Goodman Fatigue Diagram plots Mean Stress ($S_m$) on the X-axis against Alternating Stress ($S_a$) on the Y-axis:

$$S_m = \frac{S_{max} + S_{min}}{2}$$
$$S_a = \frac{S_{max} - S_{min}}{2}$$

Fatigue Life Estimation Steps:
1. Plot the point $(S_m, S_a)$ on the modified Goodman Diagram for the specific material (e.g., carbon steel wire or 17-7PH).
2. If the operating stress point lies comfortably below the material-specific Goodman endurance limit boundary line, the spring is calculated to achieve infinite life ($> 10^6$ cycles).
3. If the point lies above the boundary line, fatigue failure is highly probable, requiring engineers to either adjust the minimum preload (reducing alternating stress $S_a$) or select a thicker, multi-turn design to distribute load and lower localized peak stresses.

The classical linear spring rate ($k$) of a multi-turn Crest-to-Crest wave spring is given by the modified Timoshenko wave spring equation:

$$k = \frac{E \cdot b \cdot t^3 \cdot N_w^4}{P_m^3 \cdot N} \times K_g$$

Where:
- $E$ is the Young's Modulus of Elasticity ($N/mm^2$)
- $b$ is the radial wall thickness ($mm$)
- $t$ is the material thickness ($mm$)
- $N_w$ is the number of active waves per turn
- $P_m$ is the mean spring diameter ($mm$), computed as $(D_{out} + D_{in}) / 2$
- $N$ is the number of active turns
- $K_g$ is a correction factor based on the expansion of diameter during deflection

Limitations of Linear Equation:
1. Friction and Hysteresis: Contact between wave crests during axial deflection generates friction, resulting in hysteresis and an increase in effective spring rate during loading vs unloading.
2. Deflection Limits: The formula is strictly linear only up to approximately 80% of its total available deflection. Beyond 80%, the waves begin to bottom out or form line-contact, exponentially increasing the spring stiffness.
3. Shim Ends: If shim ends (flat ends) are specified to distribute load evenly, their contribution must be accounted for as they increase structural rigidity and decrease total effective active turns.