Design & beräkningar

The edge margin $z$ is the distance from the groove to the end of the shaft or housing. A general engineering rule is $z \geq 3d$, where $d$ is the groove depth. If $z$ is too small, the groove wall will fail in shear before the ring reaches its rated thrust capacity. The relationship between groove depth and ring thickness is critical; as depth $d$ increases, the moment arm of the thrust load increases, potentially causing the ring to 'dish' or tilt. This 'dishing' reduces the effective contact area and can lead to premature failure. Formulaically, the groove deformation limit $P_g$ is given by $P_g = \frac{D d \pi \sigma_y K}{S_f}$, where $\sigma_y$ is the yield strength of the groove material and $S_f$ is a safety factor.

Centrifugal forces at high RPM can cause a spiral retaining ring to expand and lift out of its groove. The maximum allowable speed $N_{max}$ (in RPM) is calculated using $N_{max} = \sqrt{\frac{4.48 \times 10^{12} E I (D_g - D_i)}{\mu R_m^3 (D_g^2 - D_i^2)}}$, where $D_g$ is the groove diameter, $D_i$ is the free inside diameter, $R_m$ is the mean radius, and $\mu$ is the mass per unit length. This limit is reached when the centrifugal expansion equals the initial interference fit (pre-stress) of the ring in the groove. For high-speed gearbox applications, engineers often specify a 'heavy-duty' ring or a self-locking feature (tab and slot) that mechanically prevents the ring from expanding under these inertial loads.

The thrust load capacity based on ring shear $P_s$ is determined by the formula $P_s = \frac{D t \pi S_s}{K}$, where $D$ is the shaft or bore diameter, $t$ is the ring thickness, $S_s$ is the shear strength of the ring material, and $K$ is the safety factor (typically 3). For a material like SAE 1070 carbon steel with a shear strength of approximately $0.6 \times UTS$, the capacity is directly proportional to the cross-sectional area of the ring presented to the groove wall. However, this calculation assumes that the groove material is stronger than the ring. If the groove is made of a softer material like Aluminum 6061-T6, the groove yield becomes the limiting factor, and the calculation must shift to the bearing area of the groove wall.

The fatigue life of a wave spring is inversely related to the stress range $\Delta \sigma$ experienced during cycling. Since $\sigma \propto D_m / (b t^2 N_w^2)$, the mean diameter plays a significant role. Increasing $D_m$ for a fixed load $P$ actually increases the bending moment and the resulting stress. Using the Goodman relation, $\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = 1$ (where $\sigma_a$ is alternating stress, $\sigma_m$ is mean stress, $S_e$ is endurance limit, and $S_u$ is ultimate tensile strength), engineers must optimize the diameter to minimize the stress range. In subsea valves where long-term reliability is paramount, a larger number of waves $N_w$ is often preferred over a larger $D_m$ to reduce the stress per wave and extend the cycle life into the $10^6$ range.

Nested wave springs consist of multiple turns coiled in parallel (stacked) rather than end-to-end. The total load $P_{total}$ for a nested spring is $P_{single} \times n$, where $n$ is the number of turns, assuming a constant deflection. This configuration provides a much higher spring rate in the same radial space compared to a crest-to-crest design. Mechanically, the primary difference lies in the interaction between turns; nested springs exhibit higher internal friction and hysteresis due to the contact surfaces sliding against each other during deflection. This damping effect can be beneficial in automotive clutch assemblies to reduce vibrations, but must be factored into the load calculations using a friction coefficient $\mu$ typically ranging from 0.05 to 0.1 depending on lubrication.

As a wave spring is compressed toward its solid height, the sine-wave geometry flattens, causing a slight increase in the outer diameter ($D_{out}$). This radial expansion $\Delta D$ can be approximated by $\Delta D = \frac{0.02 f (D_{out} + D_{in})}{N_w^2}$, where $f$ is the deflection. In high-tolerance bores, failure to account for this expansion can lead to binding or interference with the housing wall, which introduces parasitic friction and alters the spring rate. Precision designs in medical devices using 316 Stainless Steel require the bore diameter to be at least $D_{out} + \Delta D + \text{clearance}$ to maintain a linear load-deflection curve and prevent wear on the housing.

The maximum fiber stress $\sigma$ at a given work height occurs at the crests of the waves and is calculated using $\sigma = \frac{3 \pi P D_m}{4 b t^2 N_w^2}$, where $P$ is the load at that height. The stress distribution is non-uniform because the wave spring functions as a series of redundant curved beams; the bending moment is highest at the contact points (crests) and decreases toward the nodes. For materials like Inconel X-750, engineers must ensure that this calculated stress does not exceed the minimum yield strength of the material at the operating temperature. If the stress exceeds approximately 80 percent of the yield strength, the spring may undergo permanent set (plastic deformation), which is often modeled using a corrected stress factor for multi-turn springs to account for the actual geometry and friction between turns.

For a crest-to-crest wave spring with shim ends, the theoretical spring rate $k$ is derived from the linear elastic deflection of a curved beam. The formula is expressed as $k = \frac{E b t^3 N_w^4}{48 I D_m^3 N}$, where $E$ is the Modulus of Elasticity, $b$ is the radial width of the material, $t$ is the material thickness, $N_w$ is the number of waves per turn, $I$ is the moment of inertia, $D_m$ is the mean diameter, and $N$ is the number of active turns. It is critical to note that the spring rate is proportional to the fourth power of the number of waves ($N_w^4$). This means that even a minor increase in the wave count significantly increases the stiffness of the spring, allowing engineers to fine-tune loads within very tight axial spaces. In high-precision aerospace applications using 17-7PH stainless steel, this relationship is used to achieve high force-to-deflection ratios where traditional coil springs would be physically too large.

During installation, the ring is expanded over a shaft or contracted into a bore. The maximum fiber stress occurs at the innermost fiber for external rings and is given by $S = \frac{E t (D_s - D_i)}{(D_s - t)(D_i + t)}$, where $D_s$ is the shaft diameter and $D_i$ is the ring ID. If $S$ exceeds the yield strength $S_y$, the ring will undergo plastic deformation (permanent set). For 302 Stainless Steel rings, the installation stress should be limited to 80 percent of $S_y$. In aerospace gearboxes, 'tapered' ring sections are sometimes used to provide more uniform stress distribution across the ring circumference.

The natural frequency $f_n$ is critical to prevent valve float. It is given by $f_n = \frac{1}{2 \pi} \sqrt{\frac{K}{m}}$, where $K$ is the spring rate and $m$ is the effective mass. For a wave spring, $m \approx \frac{1}{3} m_{spring} + m_{valve}$. If the operating frequency approaches $f_n$, the spring will undergo uncontrolled oscillations, leading to fatigue failure. By using 17-7PH CH900 material, we can increase $K$ for a given mass compared to standard carbon steels, shifting the natural frequency higher and out of the engine's primary harmonic range.

To achieve a safety factor $SF = 2.0$, the groove depth $d$ must satisfy $d \ge \frac{2 P}{\pi D S_y}$, where $P$ is the thrust load, $D$ is the shaft/bore diameter, and $S_y$ is the yield strength of the groove material. Furthermore, the edge margin (the distance from the groove to the end of the shaft) must be at least $3d$ to prevent shear-out of the material. In high-pressure hydraulic cylinders, we often specify a 'square' groove geometry with a maximum corner radius of 0.1 mm to maximize the contact area between the ring and the groove wall.

Nested wave springs are coiled in parallel from a single continuous filament. The load $P$ is proportional to the number of turns $n$, such that $P_{nested} = n \times P_{single}$. This configuration is superior to stacking individual springs because it eliminates the risk of wave misalignment (peaks overlapping peaks), which would otherwise cause a catastrophic increase in spring rate. Mathematically, nesting ensures that the moment of inertia $I = \frac{b t^3}{12}$ is effectively multiplied by $n$ without adding the friction of multiple interfaces, making it ideal for high-force, low-deflection subsea valve actuators.

A radius or chamfer on the retained part creates a moment arm that induces a 'dishing' effect on the ring. The reduced thrust capacity $P_r$ is calculated by $P_r = P \times (1 - \frac{z}{t})$, where $z$ is the chamfer size and $t$ is the ring thickness. When the chamfer exceeds 50 percent of the ring thickness, the risk of the ring being 'cammed out' of the groove increases exponentially. For heavy-duty machinery, engineers should utilize a backup washer or specify a square-edged mating surface to ensure the load is applied perpendicular to the ring face.

In a 'Gap' type spring, the ends are separated, allowing for unimpeded radial expansion during compression without the risk of binding in the bore. The gap $g$ must be larger than the circumference increase $\Delta C = \pi (D_{compressed} - D_{free})$. In an 'Overlap' type, the ends slide over each other. This design provides a more uniform 360-degree contact surface, reducing the risk of 'cocking' the load. However, the overlap increases the local thickness $2t$ at the junction, which must be accounted for in the solid height calculation $H_s = (n + 1)t$ to avoid unexpected interference.

The thrust capacity of a retaining ring assembly is the lower of the ring's shear strength and the groove's yield strength. Ring shear is $P_s = D t \pi S_s$, where $S_s$ is the shear strength of the material (approx. 0.6 times Tensile Strength). Groove yield is $P_y = D d \pi S_y$, where $d$ is the groove depth and $S_y$ is the yield strength of the housing material. If the housing is a softer material like Aluminum 6061-T6, the groove yield usually becomes the limiting factor, requiring a deeper groove or a thicker ring to distribute the load across a larger area.

Wave springs generally exhibit linear load-deflection characteristics between 20 percent and 80 percent of total available deflection. Beyond this, 'bottoming out' occurs as the waves begin to touch the mating surfaces, causing a sharp increase in the apparent spring rate $K$. This is mathematically modeled by the contact of the wave arcs, which effectively reduces the active beam length $L$ in the deflection equation $\delta \propto L^3$. For precision medical instrumentation, designers must avoid this region to prevent erratic sensor readings and excessive contact stress that could lead to galling.

Centrifugal force acts to expand an external ring, potentially leading to 'liftoff' from the groove. The limiting speed $V$ in RPM is calculated using $V = \sqrt{\frac{4 E I g (D_G - D_I)}{\gamma A R_m^3 (1 + 1/N_{turns})}}$, where $D_G$ is the groove diameter, $D_I$ is the ring inside diameter, $I$ is the moment of inertia, $\gamma$ is the material density, and $R_m$ is the mean radius. For high-speed applications like turbocharger assemblies, selecting a ring with higher wall thickness or using a 'self-locking' tab design is mandatory to increase the effective stiffness against radial expansion.

The spring rate $K$ for a Multi-Turn Crest-to-Crest wave spring is derived from the beam theory applied to curved segments. It is expressed as $K = \frac{E b t^3 N}{D_m^3 n} \times \frac{ID}{OD}$, where $E$ is the Young's Modulus, $b$ is the radial wall, $t$ is the thickness, $N$ is the number of waves per turn, $n$ is the number of turns, and $D_m$ is the mean diameter. To ensure structural integrity, the operating stress $S$ must be calculated using $S = \frac{3 \pi P D_m}{4 N b t^2}$, where $P$ is the applied load. In aerospace applications, we typically target an operating stress below 80 percent of the minimum tensile strength for 17-7PH CH900 to prevent premature fatigue failure.

The groove yield strength is the axial load at which the groove material will plastically deform. For 6061-T6 Aluminum, the yield strength $\sigma_y$ is approximately $240-270$ MPa, which is much lower than the steel of the ring. The formula is $P_g = (D \cdot d \cdot \pi \cdot \sigma_y) / S$. Because aluminum is ductile, the groove edge will tend to 'smear' or 'roll' under high loads. To maximize the capacity, the groove should be as deep as possible without compromising the shaft's structural integrity. Additionally, using a 'Multi-Turn' spiral ring helps by distributing the load over more surface area, but the fundamental limit remains the compressive yield of the aluminum. A safety factor $S$ of at least $3.0$ is recommended for soft alloys.

The bending stress $\sigma$ in a wave spring is given by $\sigma = (6 \cdot P \cdot D_m) / (n^2 \cdot b \cdot t^2)$. This formula shows that the stress is inversely proportional to the square of the number of waves $n$. By increasing the number of waves, the load $P$ is distributed across more points of contact, which significantly reduces the stress in the material for a given deflection. However, increasing $n$ also increases the spring rate $k$ by the fourth power ($k \propto n^4$), making the spring much stiffer. Designers must optimize $n$ to achieve the required force while keeping the stress level below the yield strength (for static) or the fatigue limit (for dynamic) of the material.