Linear springs constructed with permanent magnets can behave in interesting ways. In levitation contexts, the natural gravity-opposing characteristic with zero power input makes permanent magnets an appealing choice for force generation (e.g., see [Íñiguez 2010]). In the context of supporting a variable-mass load for vibration isolation, the nonlinear force versus displacement characteristic decreases the amount of variability in the resonance frequency of the structure, since as the mass increases and closes the gap between the magnets, the stiffness also increases [Bonisoli 2007]. When used in attraction, the negative stiffness can be used to decrease the resonance frequencies of a supported mass, applicable for ‘‘high-static-low-dynamic’’ or ‘‘quasi-zero stiffness’’ springs [Carrella 2008, Robertson 2009].

With well-known closed form solutions for calculating the forces between cuboid permanent magnets of parallel magnetization [Akoun 1984] and solutions for the forces with orthogonal magnetizations recently published [Janssen 2009, Allag 2009b], it is now possible to analyze a wide variety of magnet configurations that previously required semianalytical or finite-element analysis techniques. This letter investigates the force characteristics between linear multipole magnet arrays as a function of array size and magnet arrangement using cuboid-shaped magnets. The forces are calculated using the force equations derived by the aforementioned researchers.

The results presented in this letter are reproducible [Buckheit 1995] with code located at http://www.github.com/wspr/magcode. This is a MATLAB software package written by the authors for calculating the forces between magnets and multipole arrays of magnets, and is freely available to be used by the public. The directory examples/magspring/ contains the code that has been used to directly generate the figures in this letter. Each figure caption refers to the file from which it is generated.

SECTION II

## FORCES BETWEEN SINGLE MAGNETS

Consider two cuboid-shaped permanent magnets of equal size, facing in repulsion, with vertical magnetization and with displacement between them only in the vertical direction. The shape of the cuboid has a dramatic effect on the force versus displacement characteristic of the system. Assume square facing sides and a height-to-length ratio of *α*.For fixed magnet volume *V*, the height of the cuboid is , and the face size length is . The force versus displacement characteristic between two such magnets each with constant volume *V* = 1000 mm^{3} over a range of aspect ratios αis shown in Fig. 1. Displacement is measured between the near faces of the magnets. (Note that all forces in this letter are calculated assuming a magnetization of 1 T for all magnets.This value has been chosen to in-effect normalize the output forces by the magnetization strength.) These results show that for a constant volume of magnetic material, a greater facing area yields greater forces.

SECTION IV

## GEOMETRY AND MAGNETS ARRANGEMENT

In this letter, we will consider a linear stack of magnets with *N* magnets aligned along a horizontal axis (often referred to as a linear Halbach array); planar and volumetric stacks that have multiple magnets in the other directions will not be considered here. As the number of magnets per wavelength of magnetization *R* increases, the magnetization pattern of the array more closely approximates true sinusoidal magnetization, as shown in Fig. 2.

The magnetic flux pattern of the array is also dependent on the wavelength of magnetization *λ*. As the wavelength of magnetization decreases, the total number of magnets used in the array increases, for a fixed array length *I*. The relationship between wavelength, array length, and number of magnets is shown in Fig. 3.Note the one extra magnet included in Figs. 2 and 3 such that the total number of wavelengths *W* = (*l* − *m*)/*λ* and the total number of magnets *N* = *WR* + 1. This magnet is necessary to balance the forces in the horizontal direction such that only vertical forces are generated between the arrays.

There are two independent variables to consider when choosing the parameters for a linear Halbach array of a certain size: number of magnets per wavelength *R*, and total number of wavelengths *W* in the array.

### A. Varying Magnetization Discretization and Wavelength

Consider two linear Halbach arrays of equal size with height *h* = 10 mm, square cross section, and length *l* = 100 mm. Their strong sides are aligned towards each other and their magnetization pattern is such that there is a repulsive force between them. The vertical displacement between their centers is * δ*, which can be normalized by the height of the arrays; *δ */*h* = 1 corresponds to the position at which the faces of the two arrays are touching. By calculating the forces between the arrays using superposition of the forces between each permutation of magnet pairs in the two arrays [Allag 2009a], the force versus normalized vertical displacement was calculated for the number of magnets per wavelength *R* ∊ {2, 4, 8} and the number of wavelengths *W* ∊ {1, 2, 4} and compared with the forces generated between a pair of equivalently sized magnets of homogeneous magnetization. These results are shown in Fig. 4.

It can be seen in Fig. 4(a) that for a small number of wavelengths, the discretization of the magnetization makes little difference to the force characteristic. But as the number of wavelengths increases the number of magnets per wavelength has an increasing effect. As seen most prominently in Fig. 4(c), increasing the number of magnets per wavelength *R* increases the forces over all values of displacement considered.

Therefore, as a general design guideline, it is only necessary to use a large number of magnets per wavelength if there are at least several wavelengths of magnetization in total in the array. In the results shown in Fig. 4, the ratio in forces between *R* = 2 and *R* = 4 is greater than the ratio in forces between *R* = 4 and *R* = 8; most of the benefit of increasing the number of magnets is realized using four magnets per wavelength of magnetization (i.e., 90°rotations between successive magnets such as shown in Fig. 3).In the cases where there are many more wavelengths of magnetization again(as shown in Fig. 5 later), there is a greater advantage to using *R* = 8 over *R* = 4. Therefore, the greater the number of wavelengths of magnetization, the greater the force improvement in increasing the number of magnets per wavelength.

The use of multipole arrays can have a significant effect on the useful range of the force/displacement characteristic. As the number of wavelengths increases, the magnetic field of each array becomes stronger but the magnetic field lines exhibit smaller excursions outside the magnet array before returning.Thus, the forces become stronger but over a smaller displacement, and therefore the stiffness of the magnetic spring is increased as well. For some purposes and in some cases, this can be detrimental in that it can increase the resonance frequency of the system, resulting in poorer vibration isolation properties.

### B. Constant Number of Magnets

The results shown previously have in general indicated that improvements to the force characteristic are seen with a greater number of magnets. However, given a minimum magnet thickness that can be fabricated, and hence, for a given array length a maximum number of magnets in total, the question arises:is it better to maximize the number of wavelengths *W* or the number of magnets per wavelength *R*? Consider an array of the same outer dimensions as the previous example composed of magnets each of length *m* = 2 mm and of cross-sectional area 10 mm × 10 mm, such that there are 50 magnets in the array. The force characteristic for this system, again with *R* ∊{2, 4, 8}, is shown in Fig. 5. In this extreme example with large *W*, the strong region of the field is close to the surfaces of the arrays and there is a considerable difference in the curves for each value of *R*; maximizing *R* produces stronger results providing there are sufficiently many wavelengths of magnetization along the length of the array. When *W* is small (e.g., *W* < 5) for a fixed magnet size, these general results do not hold and the design possibilities must be evaluated on a case-by-case basis.

### C. Unequal Magnet Sizes

While the force characteristic of an eight-magnet wavelength (*R* = 8) array can outperform the four-magnet (*R* = 4) array, the latter can be improved in some cases by adjusting the relative sizes of the magnets in the array. Consider the four-magnet array shown in Fig. 6 in which the horizontally polarized magnets of length *b* are smaller than the vertically polarized magnets of length *a*. Magnet size ratio γ = *b*/*a* is the measure used here to compare different array configurations, for which γ = 0 corresponds to an array composed only of vertically oriented magnets, and *γ* = 1 corresponds to equally sized magnets of both horizontal and vertical magnetizations(as considered previously in this letter).

Fig. 7 compares the force characteristic with a variety of magnet size ratios for arrays composed of nine magnets (i.e., two wavelengths of magnetization with a symmetry magnet), of length *l* =100 mm, and of cross-sectional area 10 mm × 10 mm. As expected from the previous results, *γ* =0 results in smaller forces than for *γ* =1; however, γ = 0.5 results in slightly greater forces again: an increase of *5* % at a displacement of approximately *δ* = 1.3 *h*, tapering off as the displacement increases. (This value of *γ* is close to optimum for this system; see Fig. 8.)This result can be justified intuitively with the recognition that there is a stronger vertical force between opposing vertically polarized magnets than between horizontally polarized magnets; dedicating a greater proportion of the magnet volume to the vertical magnets yields an increase in the total force.

However, as the number of wavelengths of magnetization *W* increases, there is a decrease in the improvement offered by reducing the size of the horizontal magnets. This can be quantified by comparing the integral of force over the displacement range of interest for a variety of magnet length ratios *γ*.In Fig. 8, such results are shown(for the same arrays discussed previously) comparing the relative difference of the force-displacement integral as a function of the magnet length ratio, normalized by the integral results for *γ* =1. Since the force improvement with adjusting *γ* is only significant for low numbers of wavelength of magnetization, this technique is only suitable for increasing the forces when a small total number of magnets are to be used, perhaps for ease of construction of the magnet array. Otherwise, it is more efficient simply to increase the number of magnets than to change the magnet size ratio.