S.D. Campbell/Kinetics Noise Control
ANALYTICAL SOLUTION OF BEARING BEHAVIOR
The design of laminated bearing pads is a quite complex undertaking. The size of the pad, number and thickness of the elastomer layers, durometer of the elastomer, and reinforcing thickness all need to be determined. Several, often conflicting, criteria must be satisfied including limitations on the pad size and total thickness, natural frequency at the design load, appropriate safety factors when overloaded, and the changes in properties under long-term or cyclic loads. It is therefore essential that an analytical model of the bearings be developed in order to understand the behavior and correctly design the pads.
Solutions are readily available for elastomeric pads restrained top and bottom by rigid supports [2] and are used for analysis of steel plate-reinforced bearings. The carbon fiber mesh used as reinforcement in the new pads is not rigid but can deform slightly along with the elastomer, requiring the reinforcement flexibility to be considered in the design. A brief overview of the analytical model derivation and the resulting solution is presented below. In common with the rigid support solutions, it is assumed that the load is applied vertically, that horizontal planes remain horizontal, and that points lying on a vertical line before loading lie along a parabola after loading.
The total horizontal displacement in the x-direction at any point, u(x,y), is the sum of the elastomer displacement assuming rigid boundaries on both horizontal surfaces, u0(x,y), and the displacement of the reinforcement on the boundaries, u1(x,y). The solution begins with a compatibility equation, including the compressibility of the elastomer, relating the strains in the bearing to the applied pressure and bulk modulus of the elastomer as
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where Exx, Eyy, and Ezz are the strains along the principle axes, p is the vertical pressure in the pad, and K is the bulk modulus of the elastomer. Combining the compatibility equation with equilibrium equations for the shear stress, boundary conditions, and symmetry requirements, the governing differential equation can be derived as
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where G is the elastomer shear modulus, t is the thickness of the elastomer layer, Ef and tf are the modulus and effective thickness of the reinforcing layer, and Ec is the average compressive strain in the elastomer.
Given the value of average compressive strain the equation can be solved for the pressure. For a pad with dimension a in the x-direction and b in the y-direction, the result is
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where
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The x-direction displacement of the rubber and the reinforcement can then be derived as
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with similar expressions for the y-direction displacements. The normalized predicted pressure distribution for a typical pad is shown in Figure 1.
Determination of the stress in the reinforcing is critical for selecting the required thickness of the mesh and for provided adequate safety against debonding of the reinforcement and elastomer. The reinforcing force per unit length is given by
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In this example, the maximum force in the reinforcement (and pressure) is almost twice the average value. This is the case for most bearings in typical sizes.
Derivation of the solution for an actual problem is an iterative process since the total applied load (integral of the pressure over the pad area) is typically the known value and the material properties for the elastomer depend upon the strain. Thus, a value of compressive strain is assumed, material properties are determined, and the total applied load is calculated from the pressure prediction as shown above. Based on the calculated load and the actual load, a new value of compressive strain is estimated and the solution proceeds until it has converged. For a laminated section this process is repeated for each layer thickness, and the results are combined to determine the overall bearing displacement and hence stiffness and natural frequency.