Effect of horizontally shifting the center of magnetic assembly and that ofkeeper on the retentive force of cup-yoke type of dental magneticattachments
M. Takahashi, H. Sakatsume, M. Kanyi and Y. Takada
Division of Dental Biomaterials, Tohoku University Graduate School of Dentistry

Introduction
It is well known that the retentive force of a dental magnetic attachment decreases with the presence of avertical gap between mating surfaces, which may occur due to an air gap erroneously introduced in themagnetic assembly during the fixation of a magnetic attachment into a removable denture1). Threedimensional finite element method of analysis and some rough measurement show that even in the absenceof a gap between the mating surfaces, any horizontal displacement between magnetic assembly and keeper
reduces the retentive force2,3). However, there is no available research showing the detailed relationshipbetween the horizontal displacement and retentive force.
There are two types of magnetic attachments based on magnetic circuits: open and closed. The open
magnetic circuit type has a retentive force that is directly proportional to the area of mating faces in contact;
however, this type of attachment is hardly used. In contrast, the closed magnetic circuit type does not have aretentive force that is directly proportional to the contact surface area. This is due to the influence of themagnetic flux direction on the retentive force; this type of attachment is mainly used in Japanese products. A
proper understanding of the retentive force change mechanism is useful for developing a magnetic attachment
and the appropriate technique for clinical purposes.

ObjectiveIn this study, the retentive forces of the cup-yoke type dental magnetic attachments were measured indifferent horizontal positions and the relationship between the retentive force and horizontal displacement
was investigated. The mechanism leading to the changes in retentive force was studied by means of modeling.

Materials and Methods
Three dental magnetic attachments were used: GIGAUSS D600 (GC), HYPER SLIM 3513 (Morita), andan experimental set (hereafter named Hollow Keeper) based on HYPER SLIM magnetic assembly pairedwith a keeper ( 3.5 mm) made of SUS XM27 bearing a  1.4 mm hole at the center. A measuring device
that matched standard ISO 13017:2012/Amd.1:20154,5) was connected to a digital force gauge (ZPS, Imada).
Retentive forces were measured at a crosshead speed of 5 mm/min according to standard ISO 13017 invarious positions wherein the center of magnetic assembly and that of the keeper was shifted horizontally(horizontal displacement) at intervals of 100 µm.
Simplified models of the magnetic attachments were used to assist in the calculation of the contact surface
area between the cup-yoke and keeper, and between the disk-yoke and keeper in different positions. Thevalues derived from mathematical projections and calculated by formulas were compared with the actualones.

Results
Dependencies of the retentive force on the horizontal displacement are shown in Fig. 1 for GIGAUSS,HYPER SLIM and Hollow Keeper. Retentive forces for all the three types decreases with horizontal
displacement increasing. The dependency was not linear and had several inflection points. The curve lines
for retentive force versus displacement for GIGAUSS and HYPER SLIM were smooth in contrast to that for
Hollow Keeper, which had a single odd point slightly above 2 mm of displacement.

6.0
Retentive force (N)Retentive force (N)

6.0
4.0
2.0
0.0

4.0
2.0
0.0

0.0

1.0

2.0

3.0

4.0

0.0

Horizontal displacement (mm)a: GIGAUSS

1.0
2.0
3.0
4.0
Horizontal displacement (mm)b: HYPER SLIM

Retentive force (N)

6.0
4.0

2.0
0.0
0.0

1.0
2.0
3.0
4.0
Horizontal displacement (mm)c: Hollow Keeper
Fig. 1 Retentive force against the horizontal displacement

Discussion
1. Internal structure of dental magnetic attachments
In order to analyze the retentive force of a magnetic attachment, it is important to know the internal
structure of magnetic attachments beyond the information disclosed by the manufacturer. A simple model ofmagnetic attachments for illustration of internal structure is shown in Fig. 2, where r is the radius of themagnetic assembly and keeper, a is the radius of the disk-yoke, b is the thickness of the shield ring (spacer),c = a + b is the radius of the disk-yoke and thickness of shield ring, and d is the radius of the hole in theHollow Keeper.

ardbrrc

a: magnetic attachment
Transverse view

b: magnetic assembly

c: keeper
surface viewFig. 2 simple model of magnetic attachments

d: hollow keeper

The magnetic attractive force (F) is calculated using equation (1), where C is constant, B is the magnetic flux
density, and S is the surface area. Increasing of magnetic flux density enhances the magnetic attractive force
compared to a similar increasing of surface area of a mating surface.
𝐹 = 𝐶 × 𝐵2 × 𝑆(1)

Since in all the sets in this study a strong neodymium magnet was used, the magnetic flux density at a cupyoke surface was predominantly saturated. Therefore, the cup-yoke surface area was the main determinantof the retentive force. Magnetic flux begins on the cup-yoke and returns to the magnet through the disk-yoke.
To optimize this effect, the design should be geared toward a cup-yoke, which has a surface area (Scup) equal
or slightly smaller than that of a disk-yoke (Sdisk).
𝑆𝑐𝑢𝑝 = 𝜋𝑟 2 − 𝜋(𝑎 + 𝑏)2𝑆𝑑𝑖𝑠𝑘 = 𝜋𝑎2
𝑆𝑐𝑢𝑝 ≤ 𝑆𝑑𝑖𝑠𝑘(2)(3)(4)

As an illustration GIGAUSS has an outer diameter of 3.6 mm (radius is1.8 mm), and a shield ring generally
has thickness of 0.15~0.2 mm2, 3). Thus r = 1.8 and b = 0.2. After substitution of the above mentioned values
into equations (2-4), one obtains a ≥ 1.169. The thickness of the cup-yoke {r - (a + b)} is 0.4 mm whereas
the radius of the disk-yoke is 1.2 mm. In a similar way for HYPER SLIM (r = 1.75 and b = 0.2) one cancalculate thickness of the cup-yoke as 0.4 mm and the radius of the disk-yoke as 1.15 mm.
2. Calculation of the contact surface area between magnetic assembly and keeper
Since the retentive force of magnetic attachment was dependent on the contact surface area between
magnetic assembly and keeper, we calculated the contact surface area for the horizontal displacement.
Horizontal displacement (t) represents the distance between the center of magnetic assembly and that of thekeeper which was calculated using equation (5). The central angle θ is formed by the intersection of two lines
drawn from the center of either keeper or magnetic assembly to each of the two points where the outermargins of keeper and assembly intersects (Fig. 3a).
𝜃𝑡 = 2𝑟 cos 2(5)t and t’ in Fig 3a correspond to the shifting movement of the assembly or keeper. Wherein 𝜃: π → 0 in
equation (5), 𝑡: 0 → 2𝑟. With overlap decreasing the assembly and keeper eventually become completely
separated (Fig. 3a).
Two additional equations were derived in order to calculate the contact surface area of the magnetic
assembly in contact with the keeper at the different positions based on displacement. Equation (6) is basedon Fig. 3b, which illustrates a certain position when the area Srr is formed by two components with sameradii (r) at distance t.
𝜃𝑆𝑟𝑟 = 𝜋𝑟 2 2𝜋 × 2 − 𝑟 ∙ 𝑟 sin 𝜃 = 𝑟 2 (𝜃 − sin 𝜃)(6)Equation (7) is based on Fig. 3c, which illustrates a certain position in which the area Scr was formed by twocircles with different radii (r, c) at distance t.
𝑡 2 +𝑐 2 −𝑟 2𝑡 2 −𝑐 2 +𝑟 2

𝑆𝑐𝑟 = 𝑐 2 𝜃 ′ + 𝑟 2 𝜃 ′′ − 𝑡 × 𝑐 sin 𝜃′ = 𝑐 2 cos−1 ( 2𝑡𝑐 ) + 𝑟 2 cos−1 ( 2𝑡𝑟 ) −𝑡2 + 𝑐2 − 𝑟2𝑡2 − 𝑐2 + 𝑟2
∵ cos 𝜃′ =,cos 𝜃′′ =,sin2 𝜃′ + cos 2 𝜃′ = 12𝑡𝑐2𝑡𝑟r

r
θt't

Srrtr

cθ'

√4𝑡 2 𝑐 2 −(𝑡 2 +𝑐 2 −𝑟 2 )22Scrrθ''t

a: horizontal displacement t
b: same radius size
c: different radius sizeand central angle θFig. 3 Calculation of the contact surface area in various positions of displacement(7)3. Contact surface area of GIGAUSS and HYPER SLIM
Since the retentive force of the cup-yoke type magnetic attachment was relatively dependent on the areaof the cup-yoke surface in contact with the keeper, the area values were calculated at three different positionsof horizontal displacement as a result of shifting either the keeper or assembly laterally. Fig. 4 shows
equations for calculating surface area Scup1-3 corresponding to the horizontal displacement t1-3.
𝑆𝑐𝑢𝑝 1 = 𝑟 2 (𝜃 − sin 𝜃) − 𝜋𝑐 2(8)√4𝑡 2 𝑐 2 −(𝑡 2 +𝑐 2 −𝑟 2 )2𝑡 2 +𝑐 2 −𝑟 2𝑡 2 −𝑐 2 +𝑟 2
𝑆𝑐𝑢𝑝 2 = 𝑟 2 (𝜃 − sin 𝜃) − {𝑐 2 cos −1 ( 2𝑡𝑐 ) + 𝑟 2 cos−1 ( 2𝑡𝑟 ) −}22 (𝜃𝑆𝑐𝑢𝑝 3 = 𝑟− sin 𝜃)

t1

t2t3

(9)
(10)

Scup1 =

Srr

−Sc

Scup2 =

Srr

−

Scr

Scup3 =

Srr

Fig. 4 Schematic diagram for calculation of surface area of cup-yoke in contact with a keeper in different
positions.
Values obtained from equations (2-4) for both GIGAUS and HYPER SLIM were substituted into equations(8-10), and the obtained values were plotted on graph as shown in Fig. 5a or 5b (blue lines).
5.0

Contact surface area
against Keeper (mm2)Contact surface area
against Keeper (mm2)5.0
Cup-yoke

4.0

Disk-yoke

3.0
2.0
1.0

0.0
0.0

1.0

2.0

3.0

4.0

Horizontal displacement, t (mm)Cup-yoke

4.0

Disk-yoke

3.0
2.0
1.0

0.0
0.0

1.0

2.0

3.0

4.0

Horizontal displacement, t (mm)a: GIGAUSS
b: HYPER SLIM
Fig. 5 Contact surface areas against the horizontal displacement
Although the general view of the cup-yoke curve lines is similar to the retentive force in Fig. 1 for values
less than 2 mm, the difference becomes apparent for values above 2 mm. This is due to the fact that thesurface area between the cup-yoke and keeper is not the only factor that determines the functionality of aclosed magnetic circuit. Therefore, dependence of the area of the disk-yoke surface in contact with keeper(Sdisk1-3) on the horizontal displacement (t1-3) was determined as shown in Fig. 6, using the following
equations.
𝑆𝑑𝑖𝑠𝑘 1 = 𝜋𝑎2√4𝑡 2 𝑎2 −(𝑡 2 +𝑎2 −𝑟 2 )2𝑡 2 +𝑎2 −𝑟 2𝑡 2 −𝑎2 +𝑟 2
𝑆𝑑𝑖𝑠𝑘 2 = 𝑎2 cos−1 ( 2𝑡𝑎 ) + 𝑟 2 cos−1 ( 2𝑡𝑟 ) −2𝑆𝑑𝑖𝑠𝑘 3 = 0

(11)
(12)
(13)

The values obtained for GIGAUS and HYPER SLIM were substituted into equations (11-13), and obtained
values were plotted on a graph as shown in Fig. 5a or 5b (magenta lines).

Sdisk1 =Sa

Scup2 =Sar

Scup3 = 0
t1

t2t3

Fig. 6 Schematic diagram for calculation of surface areas of disk-yoke in contact with the keeper
Magnetic flux from the cup-yoke passes through the keeper and returns to the magnet via the disk-yoke ina closed magnetic circuit system. When the surface area of the contact between the disk-yoke and keeper issmaller than that between the cup-yoke and keeper, the magnetic flux that exits through the cup-yoke does
not return back as in case of closed magnetic circuit. In such case, the total retentive force is a function ofboth closed and open magnetic circuit systems. Therefore, the effect of surface area on retentive force should
be considered in a relative manner. The resulting surface area related to the total magnetic circuit effect isdesignated as “Smag”. When the contact surface area between the cup-yoke and keeper is smaller than thatbetween the disk-yoke and keeper, the magnetic circuit works absolutely as a closed magnetic circuit, andSmag is equal to Scup. The value of retentive force in closed magnetic circuit is 2-4 times larger than that of anopen magnetic circuit. The influence area has on the retentive force in an open magnetic circuit is 1/3 of aclosed magnetic circuit. Therefore Smag can be calculated using equation (14) in a case when the surface areaof contact between the cup-yoke and keeper is larger than that between the disk-yoke and keeper.
𝑆𝑚𝑎𝑔 = 𝑆𝑑𝑖𝑠𝑘 +𝑆𝑐𝑢𝑝 −𝑆𝑑𝑖𝑠𝑘

(14)

3

The resulting surface area (Smag) values were converted to retentive force. The conversion was based on
Smag value and actual measured force value at position where t = 0 (without displacement), then plotted asshown in Fig. 7. The similarity in shape of curves indicates that the retentive force is dependent on the contact
surface area, which influences the magnetic circuit. In general, the actual measured values were higher thanthose that were generated using values from projected calculations. This is probably due to leakage ofmagnetic flux from the cup-yoke through a portion of an area that does not contact with the keeper.
6.0

Actual measured value
Values derived frommathematical formulas

4.0

2.0
0.0

Retentive force (N)Retentive force (N)6.0

Actual measured value

Values derived frommathematical formulas

4.0
2.0
0.0

0.0

1.0
2.0
3.0
4.0
Horizontal displacement (mm)

0.0

1.0
2.0
3.0
4.0
Horizontal displacement (mm)a: GIGAUSS
b: HYPER SLIM
Fig. 7 Retentive force against the horizontal displacement

4. Contact surface area of Hollow Keeper
Calculation of contact surface area in case of Hollow Keeper is quite complex as illustrated in Fig. 8. The
contact surface areas between magnetic assembly and keeper in various positions of displacement wereplotted on a graph and shown in Fig. 9a (blue and magenta) lines for the cup-yoke and disk-yoke, respectively.
It is remarkable that at certain positions of displacement, the contact surface area of either the cup-yoke or
disk-yoke against keeper increased with an increase in the horizontal gap. Similarly, when the area of diskyoke in contact with the keeper was smaller than that of the cup-yoke in contact with the keeper, the resulting
surface area related to the total magnetic circuit (Smag) was calculated using equation (14). Both derived andactual values were represented on a graph and shown in Fig. 9b. In general, the shapes of the two curves aresimilar. There was a peculiar point (≈ 2.1 mm) on the graph when the curve based on derived values showeda spike. This represents the position where the retentive force based on Smag increased with an increase in thehorizontal gap. The spike corresponded with a previously noted odd point on the curve of actual measured
values. This proves that retentive force is influenced by the resulting contact surface area related to a magnetic
circuit. The reason for the large difference between actual and calculated values at t = 1.0~2.0 probably liesin the effect of the magnetic flux leakage through the hole that exists in the keeper in that displacement range.

Scup =−

Srr

Scr−

Sd−－S

cdFig. 8 Schematic diagram for calculation of surface areas of cup-yoke in contact with the hollow keeper
6.0
Cup-yoke

4.0

Retentive force (N)Contact surface area
against Keeper (mm2)5.0

Disk-yoke

3.0
2.0

1.0
0.0

Actual measured value
Values derived frommathematical formulas

4.0
2.0
0.0

0.0

1.0

2.0

3.0

4.0

0.0

Horizontal displacement, t (mm)

1.0
2.0
3.0
4.0Horizontal displacement (mm)a: contact surface areab: retentive force
Fig. 9 Contact surface areas and retentive force of Hollow Keeper against horizontal displacement

Conclusion
Retentive forces of all the sets decreases with increasing of horizontal displacement. The decrease was not
uniform and the curve had several inflection points. Analysis carried out by means of mathematical formulas
demonstrates that retentive force is influenced by the contact surface area between the cup-yoke and keeper,
as well as disk-yoke and keeper, which determine the resulting contact surface area in the magnetic circuit.

References
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4.
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Y. Tanaka, New dental magnetic attachment, pp 2-90, Ishiyaku Publishers, Inc, Tokyo, 2016.
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ISO 13017:2012/Amd.1:2015 Dentistry — Magnetic attachments.