Force level

Burstone26 advised that “T-loops” (0.017 × 0.025 inch, TMA) attraction springs must be kept below 300 g to minimize anterior retraction and produce posterior protraction. Braun et al6 studied forces developed by centered T-loops (0.017 × 0.025 inch, TMA) and reported forces that varied from 50 to 300 g. Braun and Marcotte16 also studied centered T-loop (0.017 × 0.025 inch, TMA) and reported forces that varied from 50 to 350 g.

Moment-to-force ratios (M:F)

According to Braun et al,6 an M:F ratio of approximately 10 mm must be applied to the anterior and posterior teeth to achieve a translatory force system. To achieve posterior protraction, a larger moment should be applied to the anterior teeth (M:F approximating 12–13 mm), and reversing the force system would produce anterior retraction. Siatkowski11 reported an M:F ratio of 8.4 mm, capable of producing translation of the maxillary anterior teeth and an M: F varying from 8.2 mm to 8.5 mm for the maxillary posterior teeth.

Spring rate

Fryar8 studied a single “vertical helical torsion” spring (0.008 × 0.020 inch, SS) and reported that the load-deflection rate averaged 12 g/mm (10 mm height). Burstone et al17 investigated centered T-loops (0.017 × 0.025 inch, TMA) and found a mean spring rate of 55 g/mm. Yang and Baldwin19 analyzed the behavior of “Bull” and “Vertical” loops (0.017 × 0.022 inch, SS) and found a mean spring rate of 57 and 114 g/mm, respectively. Gjessing27 developed a spring (0.016 × 0.022 inch, SS) with a spring rate of 45 g/mm. Bench et al28 verifying “Double Vertical Helical” closing loops (0.016 × 0.016 inch, Co-Cr) found a spring rate of 75 g/mm.

Sample

The sample consisted of 36 springs made in titanium-molybdenum wire (TMA, Ormco Corp, Glendora, Calif) divided into three spring groups (Alpha Spring Group— ASG, Beta Spring Group—BSG, and Centered Spring Group—CSG). Each group presented 12 springs, with different end angles (120° and 130°) and different CSs, 0.016 × 0.022 inch and 0.017 × 0.025 inch. Each spring was submitted to 10 measurements, involving 30 repetitions (Table 1). The spring geometry consists of a delta form with one-and a half-turn helix in its superior portion and gable bends (120° or 130°). Figure 1 illustrates schematically the described geometry.

TABLE 1.  Spring Groups, Theta Angles, Cross Sections, and Samples

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Table 2 shows wire material properties. The spring features were inspected in relation to their reproducibility in a profile projection device (Projection Screen 560 × 460 mm, accuracy of magnification 0.5%; working scale ±0.1 mm, Henri Hauser SA, Bienne, Suisse). Only the springs whose angular and linear dimensions were within the established tolerance limit were tested (Table 3). The following geometrical parameters were defined.

• CS.

• Theta Angle (θA) formed between the alpha and beta extremities.

• Delta position (ΔP).

TABLE 2.  Wire Material, Cross Section, Yield Strength (σθ), and Elastic Modulus (E)^a

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TABLE 3.  Accepted Deviations and Spring Dimensions

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Apparatus

A platform for measuring forces and moments has been developed jointly by the Integrated Laboratory of Materials, Federal Center of Technological Education of Paraná (CEFET-PR) and the Laboratory of Applied Mechanics, Federal University of Rio Grande do Sul (UFRGS). The platform is capable of measuring forces and moments produced by a closing loop. In this study, it was established that Fy forces are caused by the activation of the spring in the horizontal direction, whereas Fx forces indicate the vertical forces, which are perpendicular to Fy. Finally, the Mz moment represents a rotational tendency due to the inclinations that occur in the extremities of the spring. The platform (Figure 2 shows the platform and data acquisition set), a home-made aluminum structure (AI2024T3), is made up of a cross-shaped beam tied to its internal part, with 12 strain gauges (SG). Figure 3 shows the SG, in the cross-shaped beam, for the measurement of Fx, Fy, and Mz. The platform provides a scale rate of 1 kgf, a supply source of 5 V, and data acquisition system (A/D) of ±5 V. SG of Mz and Fx are double (model EA-13-060PB-120, Texas Measurements Inc, Dallas, TX), and the SG of Fy are simple uniaxial (model KFG-2N-120-CI-23, Soltec Corporation, San Fernando, CA).The system was calibrated with dead weights up to 500 g.

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Procedure

The springs were adapted for testing (the spring being mounted in one extremity and freely dislocated in the other) in the platform in the following way.

The delta was dislocated three mm toward the responsive fitting of the device to study the force system acting on the alpha extremity in an asymmetrical approach (ASG) (Figure 4).

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The spring was reversed and the delta was dislocated three mm toward the nonresponsive fitting of the device to study the force system acting on the beta extremity in an asymmetrical approach (BSG) (Figure 5).

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The delta was reset equidistant from the fixing points (CSG) (Figure 6).

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Statistical analysis

To identify differences between the averages produced by three possible variation sources (independent variables) and the interactions between them (CS, ΔP, and θA), a three-factor analysis of variance (ANOVA) was performed (complete factorial) (Table 4). Tables 5–7 show the results of ANOVA. The Tukey honestly significant difference test was also used to determine multiple comparisons between the main effects and their interactions.

TABLE 4.  Descriptive Statistics, Means of Fy, Fx, and Mz by CS, θA, and ΔP

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TABLE 5.  Three-Way ANOVA Summary Table for the Data Variables Fy^a

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General comparison of the samples

For Fy at rest and Fy at six mm activation, only the θA (120° or 130°) variable has not shown any influence on the results obtained. For Fy at three mm activation, it was observed that there was no statistically significant difference for the θA variable, and there was no interaction between the variables ΔP − θA. For Fx at rest, there was no interaction between CS and θA. For Fx at three mm, there was no interaction between CS − ΔP, CS − θA, and between CS − ΔP − θA. For Fx six mm, there was no interaction between CS − θA and between CS − ΔP − θA. Toward moment Mz, all variables proved to be significant to explain the variations at Mz. (P ≤ .05) (Tables 5–7).

Comparison of the samples through Tukey test

There was a statistically significant difference between the variables studied for Fy, Fx, and Mz at rest and at three and six mm activation through Tukey analysis (Table 4).

It has been noticed that

• the CS variable was relevant for Fx, Fy, and Mz forces for the activation values studied;

• the angle variable (θA) was relevant for Fx and Mz for the activation values studied;

• the delta position (ΔP) variable was relevant for Fx (rest) forces between ASG and BSG and between BSG and CSG; for Fy rest and Fx (three and six mm), it was relevant among the spring groups; for Fy (three and six mm) was relevant for ASG and CSG; and for Mz (three and six mm), it has been relevant for all the spring groups. The value of Mz (rest) was relevant between ASG and CSG and between BSG and CSG. The behavior of geometrical variables interaction of the spring groups studied is shown graphically in Figures 7–13.

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Force level

The force levels found in all the spring groups studied have varied from an average of 34 gf to 230 gf (one to six mm activation, respectively). The negative values found for Fy (rest) reveal compressive forces over the springs,1 due to friction, because they are adapted into the transducer. However, they are of low magnitude and can be disregarded (Table 8).

TABLE 8.  Average Force Level

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The M:F ratio

Concerning ASG (0.016 × 0.022 inch, 130°), the Mα: Fy at four mm activation is equal to 7.7 mm, and Mα:Fy is 8.2 mm at three mm activation. Finally, Mα:Fy reaches 14.4 mm at two mm activation. At five mm activation, the Mα:Fy ratio is 5.2 mm. For ASG (0.017 × 0.025 inch, 120°), the Mα:Fy has varied from 6.1 to 6.7 mm at six and five mm activation, respectively. The Mα:Fy increases from 8.4 to 15.0 mm when the spring is deactivated at three mm and becomes stable at two mm activation (Mα:Fy is 15.4 mm). For BSG (0.017 × 0.025 inch, 120° and 0.017 × 0.025 inch, 130°), the Mβ:Fy ratios have been shown to be low, from three mm activation up to six mm. The CSG (0.017 × 0.025 inch) with 120° end angles has shown that the Mc:Fy ratio is altered from 6.9 mm at six mm activation to 9.6 mm at five mm activation and to 11.5 mm at four mm activation. Springs with 130° end angles have shown less sudden alterations during deactivation (Table 9).

TABLE 9.  Spring Groups M:F Ratios Vs Activation

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Spring rate

This study has shown a spring rate variation from 33.1 to 43.9 gf/mm depending on the spring group studied (Table 10).

TABLE 10.  Estimated Coefficient of Spring Rate and Standard Error

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Geometry

An apical helix has been included inside the delta to reduce a concentration of efforts and to produce a greater spring effect. When the spring is activated, the delta sides come closer, becoming almost parallel at the maximum activation studied (six mm). In this way, a greater range of activation can be found because the spring is activated in the same direction as the original bending.18

Force level

The average values found have shown a force magnitude varying from 34 gf to 230 gf, which is slightly lower than the forces described by Braun et al.6 The ASG 0.016 × 0.022 inch, 120°, has stored 119 gf at three mm activation, whereas ASG 0.017 × 0.025 inch, 130°, has stored 118 gf. These forces agree with Burstone et al17 (composite spring 0.017 × 0.025 inch, TMA, long). For T-loops with asymmetrical angulations to produce posterior protraction, they found 230 g (three mm activation) and 466 g (six mm activation). The BSG 0.016 × 0.022 inch, 120°, has stored 110.5 gf (three mm activation) and 197 gf (six mm activation). Braun and Marcotte17 (T-loops, 0.017 × 0.025 inch, TMA) found forces that varied from 50 g (one mm activation) to 350 g (eight mm activation), whereas this study shows average force levels that are slightly lower (for CSG 0.017 × 0.025 inch, 130°). Similar values were found for CSG 0.017 × 0.025 inch, 120°.

The M:F ratio

The data obtained were the result of an experiment, so they may not reflect clinical conditions exactly. However, the spring groups studied might be analyzed in relation to the M:F developed by them. The ASG (0.017 × 0.025 inch, 130°) produced an Mα:F compatible with translation at three mm activation and an Mα:F compatible with uncontrolled tipping at six mm activation. At two and four mm activation, these springs showed an Mα:F equal to 16.0 and 8.8 mm, respectively. The values of ASG (0.017 × 0.025 inch, 120° and 130°) are slightly above the ones found by Burstone26 (Table 11). The BSG (0.016 × 0.022 inch, 130°; 0.017 × 0.025 inch, 120°; and 0.017 × 0.025 inch, 130°) has shown very low Mβ:Fy ratios, which makes these groups undesirable for protraction movements.17 This is something that has not taken place in relation to 0.016 × 0.022 inch, 120° springs, which have proven to be appropriate at five mm activation and reactivated before two mm (Table 12). In this study, the Mc:F ratio is compatible with translation (from five to four mm activation) and with root movement (at three mm activation) for 0.017 × 0.025 inch, 120° and 130° springs (Table 13).

TABLE 11.  Mα:F Ratio Results Vs Literature Review

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TABLE 12.  Mβ:F Ratio Results Vs Literature Review

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TABLE 13.  Mc:F Ratio Results Vs Literature Review

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Spring rate

The ASG (0.017 × 0.025 inch) has shown the highest spring rates (42.8 gf/m, average), which is slightly above the value found by Burstone26 (33 g/mm), for the composite T-loop and which is slightly lower in relation to the centered T-loops (0.017 × 0.025 inch, TMA) studied by Burstone et al17 (approximately 55 g/mm). The canine-retraction developed by Gjessing27 stored 45 g/mm. Yang and Baldwin19 found 57 g/mm (“Bull loop”), whereas BSG (0.017 × 0.025 inch) has shown 38 gf/mm. The CSG (0.017 × 0.025 inch) has shown a mean 37 gf/mm. The values found in the spring groups studied are within compatible levels. While comparing such outcomes, it is important to emphasize that the cross section, wire material, and loop geometry have strong influence on the spring rate definition.