Rolling element bearings are crucial supporting parts in a variety of mechanical systems, such as precision instruments, transportation, and aerospace, for transferring motions and bearing loads. Rolling bearings are usually composed of four parts: an outer ring, an inner ring, rolling elements and a cage. Among them, the bearing cage plays an important role in separating and guiding the rolling elements, reducing friction and heating, making the load evenly distributed and storing grease [1]. During operation of bearings, especially when they rotate under complex loads and high speeds, the cage will suffer huge centrifugal force, impact and vibration [2]. Moreover, the sliding friction between the cage and rolling elements is not negligible and generates a large amount of heat [3]. Excellent performance is necessary for the cage under the complex interaction of force and heat.
For rolling bearings working under high speeds and light loads, wear is the main cause of bearing failures. Therefore, in addition to having enough strength, bearing cages need to reduce friction as much as possible. In industry, there are two kinds of bearing cage materials: metal (steel, brass, aluminum alloy, etc.) and composite materials (nylon, phenolic resin, nylon reinforced by glass fiber, etc.) [4]. For bearing cages made of metal materials, grease is easy to lose, while porous polymer materials can restore more grease amount and lead to lower inertial and centrifugal forces than metal cages, but their mechanical properties are relatively poor [5]. Therefore, how to manufacture bearing cages with excellent dynamic properties, less friction and light weight is still a great challenge.
Metamaterials are a new type of functional materials with unique macro properties due to the periodic arrangement of artificially designed microstructure units [6]. Mechanical metamaterials are structures that contain unnatural mechanical constants. For example, Poisson's ratios for general materials are usually positive, but with proper structural designs, metamaterials can have negative Poisson's ratios. In general, the properties of metamaterials are evaluated by the relationship between volume modulus, shear modulus and Poisson's ratio [7]:
$$\frac{B}{G}=\frac{v+1}{3\left(\frac{1}{2}-v\right)}$$
1
where, the shear modulus G is a material constant that describes the degree of deformation of an object subjected to shear stress. The bulk modulus B describes the compressibility of the object. Poisson's ratio (ν) describes the transverse contraction of a body subjected to normal stress. For conventional isotropic materials, the B⁄G ratio should be greater than zero, and, therefore, the Poisson's ratio should be in the range from − 1 to 0.5. In fact, for natural isotropic materials it is generally valued at [0, 0.5]. For example, Poisson's ratio of steel is 0.25 ~ 0.3.
The B/G ratio mentioned above is usually called metamaterial’s quality factor (FOM = B/G). The larger the quality factor is, the greater the bulk modulus is than the shear modulus, indicating that it is more difficult to change the volume than shape, similar to the behavior of fluids.
In 1995, Milton and Cherkaev proposed a pentamode metamaterial structure [8] as shown in Fig. 1(a). It was composed of double-cone elements with total length h, and has an artificial diamond crystal with lattice constant a. The shear modulus of this metamaterial was almost zero, which meant that the B⁄G ratio approached infinity, and hence the Poisson's ratio approached 0.5. This metamaterial behaved like a liquid, hence was also termed as superfluid. However, the pentamode metamaterial proposed by Milton could not be used in practice due to the instability caused by an infinitely small connection point.
Figure 1(b) exhibits a simplified pentamode metamaterial structure suggested by Kadic et al. in 2012 [9]. In contrast to (a), the connection regions of touching cones have a finite diameter d. The diameter of the thick end of the cones, D, and the total double-cone length, h, are also indicated. Therefore, the pentamode metamaterial was manufactured for the first time by 3D printing technology. This simplification changed the shear modulus of the model from nearly infinite to finite. However, simulations and experiments have shown that the B/G ratio of the new metamaterial is much higher than that of conventional materials.
Based on static continuum mechanics, Kadic et al. made extensive numerical calculations for the pentamode structure [10]. The results showed that its equivalent shear modulus was not completely isotropic, and an empirical conversion formula between dimension of micro units and quality factor was obtained. In the case of d < D < < a, the formula is:
$$FOM=\frac{B}{C}\approx 0.63{\left(\frac{ℎ}{a}\right)}^{2}\sqrt{\frac{ℎ}{D}}$$
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The pentamode structure proposed by Milton was sensitive to scale due to its lack of central symmetry. Huang et al. proposed a pentamode structure of body-centered cubic micro units to remedy this shortcoming [11], as shown in Fig. 2.
Recently, 3D printing technology has developed vigorously and become more and more mature. Through the principle of discrete stacking, structures that are difficult to be manufactured by traditional processing methods can be produced, which provides us a new idea for the manufacturing of bearing cage, as shown in Fig. 3. In this paper, SLS technology and nylon 66 were selected to print a metamaterial bearing cage. The pressure tests and triaxial torsion tests were carried out to compare the quality factor of the original cage and the 3D printed metamaterial cage. Subsequently, digital image correlation (DIC) and finite element analysis were combined to further study the mechanical properties of the metamaterial cage. Finally, the bearing test platform wa built and the rotation tests were carried out to verify the feasibility of the 3D printed metamaterial cage.