Nonlinear Elements with Hardening Characteristics

Deformation pattern of a variable coil diameter spring is shown inFig. 3.4.Since axial compression of a coil is proportional to the third power of its diameter, the larger coils would deform more than the smaller coils and thus would eventually flatten and reduce the effective number of coils. As a result, stiffness of the spring increases with load (hardening characteristic). In some designs, each coil can fit inside the preceding larger coil, so that the spring would become totally flat at its ultimate load.

In a variable pitch spring in Fig. 3.3e, coils with the smaller pitch gradually touch each other and thus reduce the effective number of coils, while the coils with the larger pitch are still operational. The result is also a hardening load-deflection characteristic.

Figure 3.3f shows a nonlinear coil spring 1 loaded in torsion. The applied torque causes reduction of its diameter, and variable diameter core 2 allows one

Figure 3.3 Coil springs with (a) linear and (b–g) nonlinear characteristics: a⫽ cylindri-cal spring with constant pitch and constant diameter; b, c, d⫽ variable diameter springs (b⫽ conical; c ⫽ barrel; d ⫽ hourglass); e ⫽ constant diameter/variable pitch spring;

f⫽ nonlinear torsion spring; g ⫽ nonlinear flat spring having a shaped support surface.

Figure 3.4 Deformation of conical spring.

to change the number of active coils while the torque is increasing, thus creating a hardening characteristic ‘‘torque-twist angle.’’ This is a typical example of an elastic element whose contact surfaces with other structural components are changing with increasing load. A similar concept is used for a nonlinear flat spring shown inFig. 3.3g,whose effective length is decreasing (and thus stiffness is increasing) with increasing deformation. Another example of achieving nonlin-ear load-deflection characteristic by changing contact surfaces is presented by deformable bodies with curvilinear external surfaces. Frequently, rubber elastic elements are designed in such shapes that result in changing their ‘‘footprint’’

with changing load (e.g., spherical or cylindrical elements [3–5]).

Rubber-like (or elastomeric) materials have unique deformation characteris-tics because their Poisson’s ratioν⫽ 0.49 ⫺ 0.4995 ⬇ 0.5. Since the modulus of volumetric compressibility is

K⫽ G/(1 ⫺ 2ν) (3.4)

then it is approaching infinity when ν is approaching 0.5. Materials withν⫽ 0.5 are not changing their volume under compression, thus rubber is practically, a ‘‘volumetric incompressible material.’’ Change of volume of a compressed rubber component can occur only due to minor deviations of the Poisson’s ratio fromν⫽ 0.5. Accordingly, compressive deformation of a rubber component can occur only if it has free surfaces so that bulging on these free surfaces would compensate the deformation in compression. Thus, compression deformation un-der a compression force Pz of cylindrical rubber element 1 inFig. 3.5a,which is bonded to metal end plates 2 and 3, can develop only at the expense of the bulging of element 1 on its free surfaces. If an intermediate metal layer 4 is placed in the middle of and bonded to rubber element 1 as in Fig. 3.5b, thus dividing it into two layers 1′and 1″, the bulging becomes restricted and compres-sion deformation under the same force Pzis significantly reduced. For a not very thin layer, dⱕ 5–10h, the apparent compression (Young’s) modulus of a cylin-drical rubber element bonded between the parallel metal plates can be calculated as [3]

E⫽ 3mG(1 ⫹ kS²) (3.5)

where S⫽ so-called shape factor defined as the ratio of loaded surface area Al

to free surface areas Afof the element; G⫽ shear modulus; k and m ⫽ coefficients depending on the hardness (durometer) H of rubber, with k⫽ 0.93 and m ⫽ 1 at H⫽ 30, k ⫽ 0.73 and m ⫽ 1.15 at H ⫽ 50, and k ⫽ 0.53 and m ⫽ 1.42 at H⫽ 70. For an axially loaded rubber cylinder having diameter d and height h S⫽ Al/Af⫽ (πd²/4)/πdh⫽ d/4h (3.6)

Figure 3.5 Shape factor influence on compression deformation of bonded rubber ele-ment.

The stiffening nonlinearity of the compressed rubber elements can be en-hanced if elements with cross sections varying along the line of force application are used. The most useful elements are cylindrical elements loaded in the radial direction as well as spherical and ellipsoidal elements. Such streamlined shapes result in the lowest stresses for given loads/deformations, and in reduced creep, as it was shown in [4], [5]. These features also contribute to enhancement of fatigue life of the rubber components. Stiffness of a cylindrical or spherical rubber element under radial compression (Fig. 3.6) increases with increasing load due to three effects contributing to a gradual increase of the shape factor S: increasing

‘‘footprint’’ or the loaded surface area; increasing cross-sectional area; and de-creasing free surface on the sides due to reduced height.Figure 3.7a[4] shows the load-deflection characteristic of a rubber cylinder L⫽ D ⫽ 1.25 in. (38 mm) under radial compression (line 2). This load-deflection characteristic can be com-pared with line 1, which is the load-deflection characteristic of the same rubber cylinder (whose faces are bonded to the loading surfaces) under axial compres-sion, in which case the contact areas are not changing.

It can be shown (see Article 1) that performance of vibration isolators significantly improves if they have the so-called constant natural frequency (CNF) characteristic when the natural frequency of an object mounted on the isolators does not depend on its weight W. The natural frequency fn of a mass (m)⫺ spring (k) system is

Figure 3.6 Compression of spherical rubber element.

fn⫽ 1

2π

√

√

where g ⫽ acceleration of gravity. Accordingly, in order to assure that fn ⫽ a constant for any weight, stiffness k must increase proportionally to the weight load W on the isolator. It represents a nonlinear elastic element with a special hardening characteristic for which

∆P/∆x⫽ k ⫽ AW (3.8)

where A⫽ a constant. Such a characteristic can be achieved by radial compres-sion of cylindrical or spherical rubber elements as illustrated inFig. 3.7b.The load range within which this characteristic occurs is described in ratio of the maximum Pmaxand minimum Pminloads of this range. For cylindrical/spherical rubber elements Pmax/Pmin⫽ 3 to 5. Although this is adequate in many applica-tions, in some cases (such as for isolating mounts for industrial machinery), a broader range is desirable.

A much broader range Pmax/Pminin which the CNF characteristic is realized can be achieved by designing a system in which bulging of the rubber element during compression is judiciously restrained. Bulging of the rubber specimen on the side surfaces can be restrained by designing interference of two bulging sur-faces within the rubber element and/or by providing rigid walls. Both approaches were used in the design of a popular nonlinear vibration isolator for industrial machinery shown in Fig. 3.8 [6] (see also Article 1). Its elastic element 3 is comprised of two rubber rings 3′(external) and 3″ (internal), separated by an

Figure 3.7 (a) Deformation and (b) load-natural frequency characteristics of rubber cylinder (D⫽ L ⫽ 1.25 in.) when loaded in (1) axial and (2) radial directions.

Figure 3.8 Constant natural frequency vibration isolator: 1⫽ bottom cover; 2 ⫽ top cover; 2′⫽ lid; 3 ⫽ rubber elastic element; 4 ⫽ transversal reinforcing ring (rib); 5 ⫽ viscous damper; 6 ⫽ rubber friction rings; 7 ⫽ level adjustment unit; 8 ⫽ foot of the installed machine. (All dimensions in millimeters.)

annular clearance∆2. Rings 3′and 3″are bonded to lower 1 and upper 2 metal covers. When the axial load Pz(weight of the installed machine 8) is small, each ring is compressing independently and rubber can freely bulge on the inner and outer side surfaces of both rings. At a certain magnitude of Pz, the bulges on the inner surface of ring 3′and on the outer surface of ring 3″ are touching each other, and further bulging on these surfaces is restrained. At another magnitude of Pz, the bulge on the outer surface of ring 3′touches lid 2′of top cover 2, thus also restraining bulging. Initially, there is an annular clearance∆1 between lid

2′and the external surface of ring 3′. Both restraints result in a hardening nonlin-ear load-deflection characteristic whose behavior can be tailored by designing the clearances∆2between rings 3′and 3″, and∆1between the outer surface of ring 3′and lid 2′. Plots inFig. 3.8billustrate load-natural frequency characteris-tics of several commercially realized CNF isolators of such design. It can be seen that the ratios Pmax/Pminas great as 100 : 1 have been realized.

While the load range within which the CNF characteristic occurs for the streamlined rubber elements, as inFig. 3.7b,is not as wide as in Fig. 3.8b, the design is much simpler and easier to realize than for a mount in Fig. 3.8a.

Thin-layered rubber-metal laminates have nonlinear properties which are very interesting and important for practical applications [7] (see Article 3). Fur-ther splitting and laminating of the block inFig. 3.5leads to even higher stiffness.

When the layers become very thin, on the order of 0.05–2 mm (0.002–0.08 in.), compression stiffness becomes extremely high and highly nonlinear, as shown in Fig. 3.9. Both stiffness and strength (the ultimate compressive load) are greatly

Figure 3.9 Compression modulus of ultrathin-layered rubber-metal laminates. Rubber layer thickness: (1) 0.16 mm; (2) 0.33 mm; (3) 0.39 mm; (4) 0.53 mm; (5) 0.58 mm; (6) 0.106 mm; (7) 0.28 mm; (8) 0.44 mm. Test samples (1)–(5) have brass intermediate layers;

(6)–(8) have steel interlayers.

influenced by deformations and strength of the rigid laminating layers (usually metal). The thin-layered rubber-metal laminates fail under high compression forces when the yield strength of the metal layers is exceeded and the metal disintegrates. When the rigid laminating layers made of a high strength steel were used, static strength values as high as 500–600 MPa (75,000–90,000 psi) have been realized for rubber layer thickness of 0.5–1.0 mm (0.006–0.04 in.). This unique material changes its stiffness by a factor of 10–50 during compressive deformation of only 10–20 µm.

Another special feature of the rubber-metal laminates is anisotropy of their stiffness characteristics. Since shear deformation (under force PxinFig. 3.5) is not associated with a volume change, shear stiffness does not depend on the design of the rubber block, only on its height and cross-sectional area. As a result of this fact, shear stiffness of the laminates stays very low while the compression stiffness increases with the thinning of the rubber layers. Ratios of stiffness in compression and shear exceeding 3000–5000 are not difficult to achieve.