Tervezés és számítások

Installation stress occurs when the ring is expanded over a shaft or contracted into a bore. The stress is calculated as $\sigma_{inst} = _x000c_rac{4 imes E imes t imes ext{Expansion}}{D_m^2}$. This stress must not exceed the yield strength of the material to avoid permanent deformation. The 'Modulus of Resilience' ($U_r = _x000c_rac{\sigma_y^2}{2E}$) represents the material's ability to absorb energy elastically. A material with a high $U_r$, like 17-7PH, can be expanded significantly more than a material with a lower $U_r$, like 316 Stainless, before taking a set. If the required expansion for installation exceeds the material's elastic limit, the ring must be designed with a larger free diameter or a multi-turn configuration to distribute the strain.

The groove depth $d$ must be sufficient to seat the ring securely while accounting for the corner radius ($r$) of the retained part. If the retained part has a large radius, it will contact the ring further out, creating a 'lever arm' that can twist the ring out of the groove (the 'dishing' effect). The effective groove depth $d_{eff} = d - (r imes 0.707)$ is often used as a conservative estimate. The minimum groove depth is typically 1/3 of the ring's radial wall width. For high-thrust applications, the groove depth is increased, but this must be balanced against the stress concentration factor $K_t$ introduced to the shaft, which is calculated as $K_t = f(d, r_{groove})$.

The thrust capacity is the lower of two values: Ring Shear and Groove Deformation. Ring shear strength is calculated as $P_r = _x000c_rac{D imes t imes au imes _x0008_eta}{S}$, where $D$ is the shaft/bore diameter, $t$ is the ring thickness, $\tau$ is the shear strength of the ring material, and $\beta$ is a safety factor. Groove deformation (yielding) is usually the limiting factor when the housing is made of softer materials like aluminum. It is calculated based on the groove depth ($d$) and the yield strength of the housing material ($\sigma_y$): $P_g = _x000c_rac{D imes d imes \sigma_y imes an(\alpha)}{K}$, where $\alpha$ is the contact angle. In most engineering designs, a safety factor of 2 or 3 is applied to ensure the groove does not 'roll over' and eject the ring under peak loads.

During the cold-coiling of 302 or 316 stainless steel, the material undergoes work hardening, which increases its yield strength but can also slightly increase the effective Young's Modulus ($E$) due to the formation of strain-induced martensite. This can result in a spring rate that is 2-5% higher than calculated using the nominal $E$ values of annealed material. Engineers must use the 'as-coiled' or 'work-hardened' property values in their FEA models. Additionally, a low-temperature 'stress-relief' heat treatment (approx. $600-750^{\circ}F$) is usually performed after coiling to stabilize the dimensions and ensure the calculated rate is maintained throughout the spring's service life.

The ratio $b/t$ is a primary factor in preventing 'twisting' or 'buckling' of the wave cross-section. A high $b/t$ ratio (wide radial wall, thin material) provides excellent load capacity but increases the risk of the spring tilting within the housing if the waves are not perfectly formed. Conversely, a low $b/t$ ratio (narrow wall, thick material) mimics a wire spring and is more stable against lateral forces but provides less surface contact area. For optimal stability, Smalley and other standards typically suggest a $b/t$ ratio between 8 and 15. If the ratio exceeds 20, the spring becomes susceptible to 'dish' distortion, where the inner and outer diameters deflect at different rates under load.

A 'gap' type spring has ends that do not touch, allowing the spring to expand radially without interference. Its load capacity is purely a function of the wave geometry. An 'overlap' type has ends that slide over each other. During compression, the overlapping ends provide a more continuous 360-degree contact surface, which can lead to a slight increase in 'apparent' stiffness due to friction between the ends. However, the overlap design is primarily used to prevent tangential interference (clashing) in tight radial spaces. Theoretically, the load $P = _x000c_rac{k imes \delta}{Z}$ remains the same, but the overlap version is more robust against radial misalignment in high-vibration environments where a gap spring might shift and snag.

The maximum stress in a single-turn wave spring is located at the inner diameter ($D_i$) at the peak of each wave. The stress $\sigma$ is calculated as $\sigma = _x000c_rac{6 imes P imes D_m}{b imes t^2 imes N^2} imes K$, where $P$ is the load and $K$ is a stress concentration factor. This is a bending stress calculation based on the beam theory where the wave is treated as a curved segment. For carbon steel, the calculated stress should not exceed 80% of the minimum tensile strength for static applications, or 50% for cyclic applications. Exceeding these limits leads to permanent set, where the material transitions from elastic to plastic deformation zones, significantly altering the spring's load-deflection curve.

The spring rate for a multi-turn wave spring is inversely proportional to the number of turns ($Z$) and sensitive to the number of waves per turn ($N$). The fundamental formula is $k = _x000c_rac{E imes b imes t^3 imes N^4}{ID_m^3 imes Z} imes f$, where $E$ is the Young's Modulus, $b$ is the radial wall, $t$ is the material thickness, $D_m$ is the mean diameter, and $f$ is a correction factor for nonlinearity. As $N$ increases, the spring rate increases by the fourth power, making it a critical design parameter. Linearity is typically maintained between 20% and 80% of the total available deflection. Beyond this, 'bottoming out' occurs as the waves flatten, leading to an exponential increase in rate due to the reduction of effective moment arms.

Groove depth $d$ is a critical parameter. It is defined as $d = \frac{D_{shaft} - D_{groove}}{2}$ (for shafts) or $d = \frac{D_{groove} - D_{bore}}{2}$ (for bores). The stability of the ring is dependent on the 'percent engagement', which is the ratio of the ring's radial wall $b$ that sits inside the groove. A deeper groove increases thrust capacity but makes installation harder and increases the risk of 'over-stressing' the ring during assembly. The optimal depth $d$ ensures that even under the maximum tolerances (worst-case scenario), the ring maintains at least $50\%$ engagement. For precision medical devices, $d$ is often kept small to minimize the footprint, requiring the use of Beryllium Copper rings to maintain high spring force at low deflections.

The edge margin is the distance from the edge of the groove to the end of the shaft or bore. If this margin is too small, the material will fail via 'shear-out' or 'blow-out'. The rule of thumb is that the edge margin $z$ should be at least $3 \cdot d$ (three times the groove depth). The calculation for the maximum load $P_z$ before edge failure is $P_z = \frac{2 π D z τ_y}{S}$. In subsea connectors where space is at a premium, engineers often use high-strength alloys like Inconel 718 for the shaft to reduce the required edge margin, allowing for a more compact design without sacrificing the $P_z$ rating.

When the groove material is sufficiently hard, the failure mode shifts to the shearing of the ring itself. The thrust load capacity based on ring shear $P_r$ is calculated as $P_r = \frac{D \cdot t \cdot π τ_{ult}}{S}$, where $t$ is the ring thickness and $τ_{ult}$ is the ultimate shear strength of the material (typically $τ_{ult} \approx 0.6 \cdot σ_{uts}$). For a spiral ring made of SAE 1070 carbon steel with a tensile strength of $200$ ksi, the shear strength would be approximately $120$ ksi. It is vital to note that spiral rings are often multi-turn; however, the shear calculation usually focuses on the thickness of a single turn unless the turns are perfectly synchronized in load sharing. For Smalley-style 2-turn rings, the effective $t$ is the sum of the turn thicknesses.

In many cases, the housing or shaft material is softer than the retaining ring, meaning the groove will fail before the ring shears. The allowable thrust load $P_g$ based on groove deformation is $P_g = \frac{D \cdot d \cdot \pi \cdot σ_y}{S}$, where $D$ is the shaft/bore diameter, $d$ is the groove depth, $σ_y$ is the yield strength of the groove material, and $S$ is a safety factor (typically $2$). If the load exceeds $P_g$, the groove wall will 'dish', causing the ring to tilt and eventually pop out. For high-load aerospace gearboxes, the groove is often hardened or the depth $d$ is increased. However, increasing $d$ also increases the stress concentration factor $K_t$ for the shaft, which must be balanced in the overall fatigue analysis.

For a spiral retaining ring installed on a shaft, centrifugal force tends to expand the ring, potentially lifting it out of the groove. The critical speed $V$ at which the ring begins to lose contact is calculated using: $V = \sqrt{\frac{4 E g (D_g - D_i)}{\rho D_m^3}}$, where $E$ is the modulus, $g$ is gravity, $D_g$ is the groove diameter, $D_i$ is the free inside diameter, $\rho$ is the material density, and $D_m$ is the mean diameter. For high-speed applications like turbochargers, designers must use a 'Self-Locking' feature. This involves a tab-and-slot mechanism that mechanically prevents the ring from expanding. Without this, the ring could fail at speeds exceeding $30,000$ RPM, even if the material is $17-7PH$ CH900.

Bearing preloading often utilizes single-turn wave springs to take up axial play. The load $P$ at a given deflection $f$ is $P = _x000c_rac{f E b t^3 N^4}{0.583 D_m^3}$. For high-speed applications, the preload must be sufficient to prevent ball skidding, which occurs if $P < P_{min}$. $P_{min}$ is typically defined by the bearing manufacturer based on the kinematic friction of the lubricant. If the spring rate is too high, it leads to excessive heat generation and reduced bearing life. Using materials like Beryllium Copper (CuBe) provides high conductivity and corrosion resistance while maintaining a stable spring rate across varying temperatures in medical imaging equipment.

The solid height $H_s$ of a Crest-to-Crest wave spring is defined as $H_s = n \cdot t$, where $n$ is the total number of turns and $t$ is the material thickness. The maximum allowable deflection $f_{max}$ is $f_{max} = H_f - H_s$. However, operating a spring near its solid height is discouraged due to the 'Bottoming Out' effect, where the rate becomes infinite as the waves make full contact. For linear performance, the working height $H_w$ should generally be no less than $20\%$ above $H_s$. The stress at solid height $\sigma_s$ must be checked against the material's elastic limit to prevent permanent set. For applications using A286 superalloy, the allowable stress is higher, but the $E$ modulus remains relatively stable at $200$ GPa.

As a wave spring is compressed from its free height to a working height, the waves flatten, causing an increase in the outer diameter ($O.D.$). This radial expansion $\Delta D$ can be approximated by the formula $\Delta D = 0.02 \cdot \frac{(W_f - W_h) \cdot N^2}{D_m}$, where $W_f$ is free height and $W_h$ is work height. If the spring is housed in a bore with insufficient clearance, it will bind against the wall, causing a non-linear spike in the spring rate and localized wear. Designers must ensure that $Bore_{min} > O.D._{max} + \Delta D$. In high-temperature environments using $17-7PH$ stainless steel, the thermal expansion coefficient $\alpha$ must also be added to the radial expansion calculation to prevent catastrophic interference at the operating temperature $T_{op}$.

Nested wave springs, which consist of multiple turns wound in parallel, exhibit a distinct hysteresis loop during loading and unloading cycles. This is primarily caused by inter-turn friction. The total load $P_{total} = P_{theoretical} \pm P_{friction}$. During the compression stroke, the friction between coils increases the apparent spring rate, while during the return stroke, friction opposes the spring's restorative force. For a nested spring with $n$ turns, the theoretical rate is $n$ times that of a single turn: $K_{nested} = n \cdot \frac{E b t^3 N^4}{I D_m^3}$. However, engineers must account for a $3\%$ to $5\%$ variation in load due to surface finish and lubrication. In subsea valves, where SAE 1070 carbon steel or Inconel X-750 is used, the coefficient of friction $\mu$ significantly shifts the $P-f$ curve, requiring precise characterization to avoid actuator lag.

For a Crest-to-Crest wave spring, the spring rate $K$ is determined by the material properties and geometric configuration. The formula is $K = \frac{E b t^3 N^4}{K_w D_m^3 n}$, 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, $n$ is the number of active turns, and $D_m$ is the mean diameter. The factor $K_w$ accounts for the curvature effect. To calculate the operating stress $\sigma$, we use $\sigma = \frac{3 π P D_m}{4 b t^2 N^2}$, where $P$ is the applied load. It is critical to ensure that $\sigma$ does not exceed the minimum yield strength of the material, typically $17-7PH$ CH900, after accounting for the safety factor. In high-precision aerospace applications, $D_m$ must be calculated at the work height due to radial expansion.

Dishing occurs when an axial load $P$ causes the ring to deflect into a conical shape. This happens because the load is typically applied at a point away from the groove support, creating a bending moment $M = P · e$, where $e$ is the eccentricity. The ring's resistance to dishing is a function of its radial wall $b$ and thickness $t$. The maximum deflection before the ring 'walks' out of the groove is critical. Dishing is aggravated by large radii on the retained part. To minimize dishing, designers should ensure the retained part has a square corner or use a backing washer to distribute the load evenly across the ring surface.

Edge margin is the axial distance from the groove to the end of the shaft or housing. If this margin is too small, the material between the groove and the end of the part will fail in shear (blowout). The required edge margin $Y$ can be estimated as $Y = \frac{3 · P · S_f}{π · D · σ_y}$, where $P$ is the thrust load. A general rule of thumb is $Y ≥ 3d$, where $d$ is the groove depth. For brittle materials like cast iron, this should be increased to $5d$. If space is limited, the edge margin can be reinforced with a hardened collar, but this adds complexity and cost to the assembly.