עיצוב וחישובים

Dish-out is the axial deflection of the ring's inner or outer edge under load, transforming the flat ring into a conical shape. As the ring dishes, the effective contact area with the groove decreases, and the load vector shifts, creating a moment that promotes further deformation. Under dynamic impact (e.g., in pneumatic hammers), dish-out can lead to the ring 'walking' out of the groove. The amount of dish is proportional to $P \cdot R^2 / (E \cdot t^3)$. To counteract this, multi-turn spiral rings (2-turn or 3-turn) are used; they provide significantly higher moment of inertia ($I$) than a single-turn stamped circlip, thereby resisting dish-out and maintaining 360-degree contact even under shock loading.

The assembly thrust capacity is the lower of the Ring Shear Strength ($P_r$) and the Groove Yield Strength ($P_g$). $P_r = \frac{D \cdot t \cdot π \cdot S_s}{K}$, where $S_s$ is the shear strength and $K$ is a safety factor (usually 3). $P_g = \frac{D \cdot d \cdot π \cdot S_y}{K}$, where $d$ is the groove depth and $S_y$ is the yield strength of the housing/shaft material. In almost all cases involving soft materials like aluminum (6061-T6), the groove is the limiting factor. The groove will 'roll' or deform plastically, causing the ring to dish out and eventually fail. Engineers must calculate the 'groove deformation' limit using the formula $P_{def} = \frac{A_s \cdot S_y}{\phi}$, where $\phi$ is a factor accounting for the angle of the load.

The centrifugal force acting on a rotating ring tends to expand it. The critical speed is reached when the centrifugal force equals the ring's radial grip on the shaft. The formula is $N_{max} = \sqrt{\frac{4.48 \cdot 10^9 \cdot E \cdot I \cdot g}{\rho \cdot A \cdot R^4 \cdot (1+n)^2}}$ where $E$ is the modulus, $I$ is the moment of inertia, $\rho$ is the density, $A$ is the cross-sectional area, and $R$ is the mean radius. For high-RPM applications like electric vehicle motors, we use 'Self-Locking' spiral rings. These rings have a small tab that locks into a notch on the adjacent turn, mechanically preventing the ring from expanding due to centrifugal forces, thus allowing speeds $3-5$ times higher than standard rings.

The number of waves $N$ is inversely proportional to the deflection and directly proportional to the cube of the load capacity. Specifically, the load $P$ is proportional to $N^4$. However, increasing $N$ reduces the arc length between peaks, which increases the stiffness but can lead to manufacturing difficulties and higher stress concentrations at the peaks. For stability, $N$ should be an integer or half-integer to ensure uniform load distribution. In applications with rotating shafts, an odd number of waves is often preferred to avoid harmonic resonance with the shaft's rotational frequency, especially when the operating speed approaches the spring's natural frequency $f_n = \frac{1}{2π}\sqrt{\frac{k}{m}}$.

Fatigue life estimation for wave springs involves calculating the alternating stress $\sigma_a$ and the mean stress $\sigma_m$. For a wave spring, the maximum stress occurs at the wave peaks and is given by $\sigma = \frac{3 \cdot π \cdot E \cdot t \cdot N^2 \cdot f}{4 \cdot D_m^2 \cdot n}$. Using the Goodman relation $\frac{\sigma_a}{\sigma_e} + \frac{\sigma_m}{\sigma_{ut}} = 1$, where $\sigma_e$ is the endurance limit and $\sigma_{ut}$ is the ultimate tensile strength (approx. 1650 MPa for 17-7PH CH900), engineers can determine if the spring will survive $10^6$ cycles. In high-cycle applications like medical pumps, we aim for a safety factor $S_f > 1.2$. Residual stresses from the coiling process must be relieved via heat treatment at $480^∘C$ to ensure the mean stress does not shift during service.

Wave springs expand radially when compressed because the arc length of the waves remains constant while the wave height decreases. The expanded outside diameter $D_{ext}$ is calculated using the formula $D_{ext} = \sqrt{D^2 + (0.6 \cdot N^2 \cdot (h^2 - h_1^2))}$ where $D$ is the mean diameter, $N$ is the number of waves, $h$ is the free height per wave, and $h_1$ is the compressed height per wave. In tight-tolerance bores, such as those found in hydraulic actuators, if the clearance between the spring OD and the bore is insufficient, the spring will bind, causing a sudden, exponential increase in the spring rate and potential galling of the housing wall.

Nested wave springs, which consist of multiple turns coiled in parallel, exhibit significant hysteresis due to inter-turn friction. When the spring is compressed, the layers slide against one another, generating frictional resistance that adds to the apparent spring load ($P_{loading} = P_{theoretical} + F_{friction}$). Upon decompression, the friction opposes the return force ($P_{unloading} = P_{theoretical} - F_{friction}$). The area within the hysteresis loop represents energy dissipation per cycle, calculated as $\oint P \, df$. This is critical in automotive damping systems. For material like 17-7PH, the coefficient of friction $\mu$ usually ranges from 0.1 to 0.2 depending on lubrication, and failure to account for this can lead to an overestimation of the return force in high-speed reciprocating valves.

The spring rate $k$ for a Crest-to-Crest wave spring is derived from the standard beam equation adapted for circular geometry. The primary formula is $k = \frac{E \cdot b \cdot t^3 \cdot N^4}{I_D^3 \cdot ID \cdot n}$ where $E$ is the Young's Modulus, $b$ is the radial wall, $t$ is the material thickness, $N$ is the number of waves per turn, and $n$ is the number of turns. However, as the spring approaches its solid height, the rate becomes non-linear due to the 'bottoming out' effect where the wave peaks begin to flatten against each other, effectively reducing the active length of the beam. To account for this, engineers apply a correction factor $K$ based on the ratio of $f/h$ (deflection to wave height). For high-precision applications in aerospace, a non-linear finite element analysis (FEA) is typically performed to map the rate change once deflection exceeds 80% of the available travel.

Impact or shock loading in drilling tools can momentarily exceed the static thrust capacity of a retaining ring by several orders of magnitude. The kinetic energy $U = \frac{1}{2} m v^2$ must be absorbed by the elastic and plastic deformation of the ring and groove. In these environments, a 'Heavy Duty' spiral ring with a larger radial wall $b$ is required. Furthermore, the groove depth $d$ should be maximized to provide more shear area. Because impact loads can cause the ring to 'hop' out of the groove due to vibration, a 3-turn ring is often preferred over a 2-turn ring as it provides more surface contact and damping. Material choice is also critical; S66286 (A286) stainless is often used for its high toughness and strength in these high-shock, corrosive environments.

Spiral retaining rings are typically 2-turn or 3-turn. The radial stress during installation is caused by the expansion (for external rings) or contraction (for internal rings) required to pass over the shaft or into the bore. The maximum stress $\sigma_{inst}$ is given by $\sigma_{inst} = \frac{E t (D_f - D_i)}{D_m^2}$, where $D_f$ is the free diameter and $D_i$ is the installation diameter. Because a 2-turn ring is thinner than a single-turn stamped ring of the same strength, it can undergo greater elastic deformation without yielding. However, if the ring is 'over-expanded' during installation, it will take a permanent set and will not seat tightly in the groove, leading to a loose fit and reduced thrust capacity. Multi-turn rings distribute the installation stress more uniformly across the coils compared to a single-turn constant section ring.

Dishing occurs when a retaining ring is subjected to thrust loads that cause the groove's edge to deform plastically. As the groove wall yields, it forms a ramp or 'radius', and the ring begins to twist or 'dish' under the moment $M = P \times (lever arm)$. The angle of dish $\phi$ is proportional to the applied load and inversely proportional to the ring's torsional stiffness. Once the dish angle exceeds a critical value (typically 10-15 degrees), the ring can no longer be contained by the groove and will fail by 'scalloping' or ejecting. To mitigate this, the corner radius at the bottom of the groove must be kept to a minimum (usually $< 0.1 \times d$), and the 'abutment' or mating component should have a sharp corner to minimize the lever arm of the applied thrust load.

External spiral retaining rings are subject to centrifugal forces that tend to expand the ring radially. At a certain rotational speed $\omega$, the ring will lose its grip on the groove bottom, known as lift-off. The critical speed $V$ (in RPM) can be approximated by $V = \sqrt{\frac{4.48 \times 10^{12} E t p^2}{D_g^3 D_r \rho}}$, where $E$ is the Modulus, $t$ is the material thickness, $p$ is the radial wall, $D_g$ is the groove diameter, $D_r$ is the ring free diameter, and $\rho$ is the density. To prevent lift-off in high-speed shafts (e.g., turbochargers), engineers can specify 'Self-Locking' rings, which feature a tab-and-slot mechanism that mechanically prevents the ring from expanding beyond a certain point, allowing the assembly to operate at significantly higher RPMs.

The thrust capacity of a spiral retaining ring is usually limited by the shear strength of the groove material rather than the ring itself. The allowable thrust load $P_g$ is calculated using the formula $P_g = \frac{D d \pi \sigma_y}{K S F}$, where $D$ is the shaft/bore diameter, $d$ is the groove depth, $\sigma_y$ is the yield strength of the groove material, and $KSF$ is a safety factor (typically 2). For the ring itself, the shear capacity is $P_r = \frac{A \tau \pi}{K S F}$, where $A$ is the shear area and $\tau$ is the shear strength of the ring material. In soft materials like aluminum, the groove will almost always fail first by 'dishing' or deforming, which allows the ring to twist and pop out. Therefore, deepening the groove or using a harder housing material is more effective than using a thicker ring.

The maximum operating stress $\sigma$ occurs at the peak of the waves and is given by $\sigma = \frac{1.5 \pi P D_m C_f}{Z^2 b t^2}$, where $C_f$ is a stress concentration factor related to the wave geometry. For 17-7PH stainless steel in the CH900 condition, the tensile strength $UTS$ is approximately $200-230$ ksi. To ensure an infinite fatigue life (over $10^6$ cycles), the operating stress at the maximum deflection must be kept below the fatigue endurance limit, typically $45-50\%$ of the $UTS$ for non-corrosive environments. If the application involves high-frequency cycling, a Goodman or Gerber criterion analysis should be performed, plotting the mean stress $\sigma_m = (\sigma_{max} + \sigma_{min})/2$ against the alternating stress $\sigma_a = (\sigma_{max} - \sigma_{min})/2$.

The solid height $H_s$ of a Crest-to-Crest wave spring is not merely the sum of material thicknesses. It is calculated as $H_s = (N \times t) + (N-1) \times t_{contact}$, where $N$ is the number of turns and $t$ is the material thickness. In practice, due to manufacturing tolerances in wave formation and the 'set' taken during the first compression, the actual solid height may be slightly higher than the theoretical sum. Engineers must ensure that the operating height $H_{op}$ is always greater than $H_s$ by a safety margin of at least $10\%$ of the wave height to prevent the spring from acting as a solid shim, which would transfer the full load through the material in compression rather than bending, potentially exceeding the material's compressive yield strength.

Nested wave springs consist of multiple turns coiled in parallel (stacked) rather than in series. The primary benefit is that the load capacity increases proportionally with the number of turns $n$, such that $P_{total} = n \times P_{single}$. This is particularly useful in subsea valve actuators where high force is required in a small radial envelope. In contrast, Crest-to-Crest springs are used to increase the total deflection while maintaining a lower spring rate, where the total rate $k_{total} = \frac{k_{turn}}{N}$. The nested design maintains a constant spring rate throughout its stroke, but requires careful lubrication between layers to prevent frictional hysteresis and fretting wear under high-frequency oscillation.

The number of waves $Z$ significantly dictates the stiffness and the stress distribution. As $Z$ increases, the spring rate increases by a factor of $Z^4$. However, in aerospace applications where space is constrained, increasing $Z$ reduces the arc length of each wave, leading to higher localized bending stresses $\sigma = \frac{3 \pi P D_m}{2 Z^2 b t^2}$. If $Z$ is too low (e.g., $Z < 3$), the spring may become unstable and tip within the bore. To maintain linearity, the deflection $f$ should not exceed $80\%$ of the available travel to avoid the 'bottoming out' effect where the wave peaks flatten against the contact surfaces, causing an exponential increase in the apparent spring rate.

The spring rate $k$ for a multi-turn Crest-to-Crest wave spring is derived from the deformation of curved beams. For a spring with $N$ active turns and $Z$ waves per turn, the rate is defined as $k = \frac{E b t^3 Z^4}{1.5 \pi D_m^3 N} \frac{I_D}{O_D}$, where $E$ is the Modulus of Elasticity, $b$ is the radial wall, $t$ is the material thickness, and $D_m$ is the mean diameter. When shim ends are added, the effective number of turns increases slightly because the shim acts as a rigid boundary, reducing the active deflection length. For high-precision applications, the correction factor for the mean diameter $D_m = (D_{outer} + D_{inner})/2$ must account for the radial expansion during compression, as $D_m$ increases slightly, potentially leading to a non-linear stiffening effect as the spring approaches its solid height $H_s$.

When the housing material (e.g., Magnesium or Plastic) has a lower yield strength than the ring, the assembly's capacity is governed by the 'Groove Yield'. The allowable thrust load $P_a = \frac{D d π σ_y}{S}$, where $\sigma_y$ is the yield strength of the housing material. If the calculated $P_a$ is less than the required design load, simply increasing the ring's strength will not help. Instead, the designer must increase the 'Groove Depth' $d$ or the 'Groove Diameter' $D$. Another advanced technique is to use a 'Load-Spreading Washer' between the ring and the housing to distribute the axial force over a larger area, effectively reducing the localized stress on the groove wall and preventing the 'dishing' failure mode.

The radial wall $b$ (the width of the material cross-section) has a linear relationship with the spring rate $k$ and an inverse relationship with the stress $\sigma$. According to $k ∝ b$ and $\sigma ∝ 1/b$, doubling the radial wall will double the spring's stiffness but halve the stress for the same deflection. In space-constrained designs, increasing $b$ is often the only way to increase the load capacity without increasing the axial height (thickness $t$). However, a wider radial wall increases the risk of 'clashing' or interference with the bore or shaft during expansion. The designer must ensure that $O.D._{max} = O.D._{free} + \text{expansion}$ is always less than the Bore Diameter. If the wall is too wide, the spring may also become too stiff to install without permanent set.