Garter Spring

garter spring

Revision for “Garter Spring” created on May 26, 2022 @ 09:08:42

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Garter Spring
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<p style="text-align: justify;">A Garter spring is a coiled steel spring that is connected at each end to create a circular shape, and is used in oil seals, shaft seals, belt-driven motors, and electrical connectors. There are several ways to connect the ends depending on the application: connector ends, loop connection and tapered end connection. Garter springs are generally divided in compression and extensions springs according to the direction of the radial load exerted, i.e., inwards, or outwards.</p> <p style="text-align: center;"><img class="alignnone wp-image-58348" src="https://www.tribonet.org/wp-content/uploads/2022/03/a-close-up-of-a-ring-description-automatically-ge.jpeg" alt="A close-up of a ring Description automatically generated with low confidence" width="255" height="167" /></p> <p style="text-align: justify;"><img class="wp-image-58349 aligncenter" src="https://www.tribonet.org/wp-content/uploads/2022/03/word-image.png" width="684" height="257" /></p> <p style="text-align: center;"><em>Figure 1. Garter spring sketch.</em></p> <p style="text-align: justify;">Many lip seals count with a Garter spring for loading the seal tip against the shaft. The extra load provided by the Garter spring ensures that the stress relaxation of the seal material does not decay beyond the percolation limit of the contact. Because the load exerted by Garter spring is generally constant, i.e., the spring does not generally change its elongation while operating, Garter springs for shaft seals are often referred to as “constant load springs”.</p> <p style="text-align: justify;"><strong>Theoretical background</strong></p> <p style="text-align: justify;">There is little written on garter springs. The following procedure estimates the radial load exerted by an extension garter spring when working under linear deflection:</p> <table style="height: 238px;" width="1005"> <tbody> <tr> <td> <p style="text-align: center;">1</p> </td> <td style="text-align: center;">Wire diameter</td> <td style="text-align: center;">[math]d[/math]</td> </tr> <tr> <td> <p style="text-align: center;">2</p> </td> <td style="text-align: center;">Mean diameter of the helix</td> <td style="text-align: center;">[math]D[/math]</td> </tr> <tr> <td> <p style="text-align: center;">3</p> </td> <td style="text-align: center;"> <p style="text-align: center;">Shaft diameter</p> </td> <td style="text-align: center;">[math]D_s[/math]</td> </tr> <tr> <td style="text-align: center;">4</td> <td> <p style="text-align: center;">Initial tension</p> </td> <td style="text-align: center;">[math]P_I[/math]</td> </tr> <tr> <td style="text-align: center;">5</td> <td> <p style="text-align: center;">Spring Working Length</p> </td> <td style="text-align: center;">[math]L_0[/math]</td> </tr> <tr> <td style="text-align: center;">6</td> <td> <p style="text-align: center;">Young's modulus</p> </td> <td style="text-align: center;">[math]E[/math]</td> </tr> <tr> <td style="text-align: center;">7</td> <td style="text-align: center;"> <p style="text-align: center;">Rigidity modulus</p> </td> <td style="text-align: center;">[math]G[/math]</td> </tr> </tbody> </table> <p style="text-align: justify;">The following parameters can be deduced from the ones above:</p> <table style="height: 698px;" width="1001"> <tbody> <tr> <td style="text-align: center;">8</td> <td style="text-align: center;">Number of working coils</td> <td style="text-align: center;">[math]n[/math]</td> </tr> <tr> <td style="text-align: center;">9</td> <td style="text-align: center;">Inside diameter of the Garter spring</td> <td style="text-align: center;">[math]D_r_i[/math]</td> </tr> <tr> <td style="text-align: center;">10</td> <td style="text-align: center;">Increase in diameter when fitted [math] \Delta D = D_s - D_r_i [/math]</td> <td style="text-align: center;">[math]\Delta D[/math]</td> </tr> <tr> <td style="text-align: center;">11</td> <td style="text-align: center;">Spring rate</td> <td style="text-align: center;">[math]S[/math]</td> </tr> <tr> <td style="text-align: center;">12</td> <td style="text-align: center;">Spring Index [math]c = D/d[/math]</td> <td style="text-align: center;">[math]c[/math]</td> </tr> <tr> <td style="text-align: center;">13</td> <td style="text-align: center;">Correction Factor [math] k =(c + 0.2)/(c-1) [/math]</td> <td style="text-align: center;">[math]k[/math]</td> </tr> <tr> <td style="text-align: center;">14</td> <td style="text-align: center;">Combined stress due to bending and torsion</td> <td style="text-align: center;">[math]q[/math]</td> </tr> <tr> <td style="text-align: center;">15</td> <td style="text-align: center;">Additional stress due to initial tension</td> <td style="text-align: center;">[math]q_I[/math]</td> </tr> <tr> <td style="text-align: center;">16</td> <td style="text-align: center;">Total stress  [math] q_t = q_I + q [/math]</td> <td style="text-align: center;">[math]q_t[/math]</td> </tr> <tr> <td style="text-align: center;">17</td> <td style="text-align: center;">Circumferential force</td> <td style="text-align: center;">[math]P_c[/math]</td> </tr> <tr> <td style="text-align: center;">18</td> <td style="text-align: center;">Radial force unit per length</td> <td style="text-align: center;">[math]P_r[/math]</td> </tr> </tbody> </table> <p style="text-align: justify;">The number of working coils can be deduced from the working spring length [math]L[/math]:</p> <p style="text-align: center;">[math]n = \frac{L_0}{d}[/math]</p> <p style="text-align: justify;">Therefore, the inside diameter of the garter spring [math]D_r_i[/math] (when both edges are attached together):</p> <p style="text-align: center;">[math]D_r_i =\frac{L_0}{\pi } = \frac{nd}{\pi } - d^*[/math]</p> <p style="text-align: center;"><em>(*sometimes [math]d/2 [/math] is used instead. The difference between both option is usually not significant.)</em></p> <p style="text-align: justify;">The spring rate [math]S[/math] is obtained from the following formula:</p> <p style="text-align: center;">[math] S =\frac{d^4G}{8nD^3} = \frac{dG}{8nc^3}[/math]</p> <p style="text-align: justify;">The circumferential and radial loads are related by the following expression (see section below):</p> <p style="text-align: center;">[math] P_r =\frac{2P_c}{D_s}[/math]</p> <p style="text-align: justify;">The force exerted radially by the garter spring when mounted is defined as follows:</p> <p style="text-align: center;">[math] P{_c__{extension}} = P_I + S(\pi D_s-\pi D_r_i) [/math]</p> <p style="text-align: center;">[math] P{_r__{extension}} = 2(\frac{P_I}{D_s}+\pi S(1-\frac{D_r_i}{D_s})) [/math]</p> <p style="text-align: center;">[math] P{_c__{compression}} = S(\pi D_s-\pi D_r_i) [/math]</p> <p style="text-align: center;">[math] P{_r__{compression}} = 2(\pi S(\frac{D_r_i}{D_s}-1)) [/math]</p> <p style="text-align: justify;">Most extension springs are wound with initial tension [math] P_I [/math].  This is an internal force that holds the coils tightly together.  Unlike a compression springs, which has zero load at zero deflection, an extension spring have an initial tension [math] P_I [/math].</p> <img class="wp-image-58351 aligncenter" src="https://www.tribonet.org/wp-content/uploads/2022/03/word-image-1.png" width="442" height="361" /> <p style="text-align: center;"><em>Figure 2. Load-deflection curve for helical spring with initial tension [math]P_I[/math].</em></p> <p style="text-align: justify;">It is important to estimate the total stress imposed on the spring due to torsion and bending [math] q_t [/math]. By comparing it with the yield strength of the spring material [math] \sigma_y [/math] it is possible to check if the spring works in the elastic deformation range.</p> <p style="text-align: justify;">The stress resultant from the spring elongation [math] q [/math] is estimated as follows:</p> <p style="text-align: center;">[math]q=(\frac{D_s-D_r_i}{D}+\frac{2}{1+\frac{2G}{E}}) \frac{Gk}{nc}[/math]</p> <p style="text-align: justify;">It is necessary to include the stress induced by the initial tension [math] q_I [/math]:</p> <p style="text-align: center;">[math] q_I = \frac{8cP_Ik}{\pi d^2}[/math]:</p> <p style="text-align: justify;">The relaxation of the spring material is usually neglected but it sould be considered when operating at elevated temperatures.</p> <p style="text-align: justify;"><strong>Relationship between the circumferential and radial loads</strong></p> <strong><img class="wp-image-58352 aligncenter" src="https://www.tribonet.org/wp-content/uploads/2022/03/word-image-2.png" width="509" height="288" /></strong> <p style="text-align: center;"><em>Figure 3. Radial and circumferential load decomposition.</em></p> <p style="text-align: justify;">To calculate the radial load exerted by a Garter spring, the Hooke’s law is applied in the circumferential direction and the radial load is obtained from forces decomposition:</p> <p style="text-align: center;">[math] P_r [\frac{N}{m}] = \frac{2P_c}{D_2} = \frac{2}{D_2}(P_I+k\pi (D_2-D_1)) [/math]</p> <p style="text-align: justify;"><strong>Realtionship between radial [math] P_r [\frac{N}{m}] [/math] and circumferential [math] P_c [N] [/math] forces</strong></p> <p style="text-align: center;">[math] P_c [N] = \(\int_{s_1}^{s_2} P_r sin(\theta) \, ds\) [/math]</p> <p style="text-align: center;">[math] P_c [N] = \(\int_{0}^{\pi/2} P_r sin(\theta)  \frac{D}{2}\, d\theta\) [/math]</p> <p style="text-align: center;">[math] P_c [N] = P_r \frac{D}{2} [-cos (\theta)]_0^{\pi/2} = P_r \frac{D}{2}[/math]</p> <p style="text-align: center;">[math] P{_{r__{tot}}} [N]= P_r [\frac{N}{m}] \pi D [m] [/math]</p>
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Industrial Engineer with focus on Tribology and Sealing Technology. Team player with an open-minded mentality author of several scientific publications and an industrial patent. Interested in Lean Management, Innovation, Circular Economy, Additive Manufacturing and Connected Objects Technology.