Design sensitivity analysis of a rubber bushing

This example demonstrates how design sensitivity analysis can be used to improve the design of a rubber bushing. The objective is to alter the bushing geometry to lower the maximum axial stress, thus increasing the service life. Design sensitivity analysis provides a means of predicting the effect of a change in the geometry on the stress concentration, thereby aiding in identifying the important design parameters and determining an appropriate design change.

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Geometry, model properties, and design parameters

The bushing consists of inner and outer steel tubes that are bonded to a central rubber cylinder (Figure 1). It is assumed that the outer perimeter of the bushing is fully fixed. The bushing is 457.2 mm (18.0 in) long, with an outside diameter of 508.0 mm (20.0 in) and an inside diameter of 228.6 mm (9.0 in). The steel is elastic with Young's modulus = 206.0 GPa (3.0 × 107 psi) and Poisson's ratio = 0.3. The rubber is modeled as a fully incompressible hyperelastic material that at all strain levels is relatively soft compared to the steel. The nonlinear elastic behavior of the rubber is described by a strain energy function that is a second-order polynomial in the strain invariants.

The model is discretized with standard axisymmetric elements since the axial loading results in pure axisymmetric deformation. CAX4 elements are used for the steel components, and CAX4H elements are used for the rubber component. Rigid elements (element type RAX2) are attached to the inside of the bushing in both models to represent the relatively stiff shaft. The use of these elements also simplifies the application of the loading conditions. The axisymmetric finite element mesh is shown in Figure 1. An axial force of magnitude 10675.0 N (2400.0 lbs) is applied to the rigid body reference node, while the outer steel tube is fully fixed.

Two design parameters are considered for the design sensitivity analysis: the thickness, t, and the fillet radius, r, of the rubber bushing at the top and bottom ends where it is bonded to the inner steel tube, which is shown in Figure 1. These parameters represent typical geometry properties that may be considered during design evaluation.

Design sensitivity analysis and design parameters

To carry out a design sensitivity analysis with respect to a shape design parameter, the gradients of the nodal coordinates with respect to the design parameter must be specified with parameter shape variation. One simple approach to obtaining these gradients is to perturb the shape design parameters r and t one at a time and to record the perturbed coordinates. The gradients are then found by numerically differencing the initial and perturbed nodal coordinates. In the current study the constraint is imposed that a change in the thickness causes the line of nodes connecting the thickness dimension to the fillet radius to rotate about the point of tangency of this line to the fillet radius.

Results and discussion

The deformed mesh is shown in Figure 2. Figure 3 shows the contours of axial stress in the rubber part of the bushing at the end of the axisymmetric analysis. The maximum stress occurs near the top fillet close to the axis. Figure 4 and Figure 5 show the contours of the sensitivities of the axial stress for the shape design variables r and t, respectively. Table 1 shows the normalized sensitivities of the maximum axial stress, σmax=​ 0.17 MPa (24.55 psi), with respect to the shape design variables. The normalization has been carried out by multiplying the sensitivities by a characteristic dimension (initial fillet radius r0=​ 12.7 mm (0.5 in) and initial thickness t0=​ 15.24 mm (0.6 in)) and dividing by the maximum stress. As can be inferred from this table, a change in the fillet radius influences the maximum stress to a larger extent than a change in the thickness of the rubber. Hence, it is desirable to change r to modify the stresses. To obtain approximately a 10% reduction in the maximum stress in the axial direction, the fillet radius is increased by

Δ⁢r=Δ⁢σmaxd⁢σmaxd⁢r.

Substituting Δ⁢σmax=0.1⁢σmax and d⁢σmax/d⁢r=​ 0.008 MPa/mm (28.75 psi/in) (see Figure 4) gives Δ⁢r=​ 2.25 mm (0.09049 in). A reanalysis of the problem with the radius changed to r0+Δ⁢r=​ 14.99 mm (0.59049 in) yields a reduction of 8.8% in the maximum axial stress, which is slightly less than the goal of 10%. This is expected because of the nonlinearity of the problem; to achieve the 10% reduction, this process would have to be repeated, which is essentially an optimization problem.

Tables

Table 1. Normalized sensitivities of the maximum stress.
Parameterr0σmax⁢d⁢σmaxd⁢rt0σmax⁢d⁢σmaxd⁢t
r −0.58 —
t — −0.11

Figures

Figure 1. Axisymmetric cross-section.

Figure 2. Deformed mesh after axial loading.

Figure 3. Variation of axial stress in the rubber after axial loading.

Figure 4. Variation of the sensitivity of the axial stress with respect to an increase in the radius of the fillet, r.

Figure 5. Variation of the sensitivity of the axial stress with respect to a decrease in the thickness of the rubber, t.