The design variables are chosen to define the geometry of the belt drive. This includes the radii of the driving and the driven pulley, and the pitch length of the V-belt. Figure 1 provides a schematic drawing of the belt drive showing the variables employed.

The basic characteristics essential for the design of the belt drive transmission are the input and output power, , and the rotational speed,. The objective of this study is to maximize the power transmission of the drive. The input power to the belt from the driving pulley could be written as;

And the output power received from the belt by the driven pulley could be written as;

where is the torque that pulley-A delivers on the belt and is the torque that pulley-B received from the belt.

Belt drives depend for their operation on friction between the belt and the pulleys. The relation between the belt friction coefficient m and the transmission torque t could be obtained for the driving and driven pulleys respectively as follows;

and (3)

To develop adequate traction, the belt must be given a certain amount of pretension,. As the torques are transmitted by the system a tension difference between the tight and slack sides will occur and induce friction on the contact surfaces between the V-belt and the pulleys. The frictional contact behavior on the frictional surfaces and the tension existing in the V-belt is well discussed in [6] . The changing in the belt tension causes changes in strain in the elastic belt which make the belt move relative to the rotating pulleys [7] . A speed differential known as creep is then developed between the belt and the pulley causing the belt span at the tight side to travel slightly faster than the belt span at the slack side. This creep phenomenon may be expressed as a ratio of the speeds [30] ,

where the slippage factor ε lies between the values 0.005 to 0.03. Denoting an assumed constant belt speed by v_b, the output power by the belt can be obtained as expressed;

The tensions and transmitting the power is defined by considering the influence of the centrifugal force on the belt system in operation as follows [31] ;

and (6)

where and are the tensions on the tight side and slack side of the belt respectively. is the centrifugal force. The maximum torque that can be transmitted between the belt and the pulley occurs when limiting friction is developed around the arc of contact. The forces acting on a short length of the belt which subtends an angle ofdqat the centre of the pulley are shown in Figure 2. Consequently the maximum possible ratio of tensions between the “tight” and “slack” sides of the belt for a drive between pulleys of unequal radius was determined as [32] ;

b is the V-half angle of groove in degree. The power output for the drive is obtained as expressed in Equation (8).

An accurate running tension can be achieve for the drive by setting an initial installation tension to which

usually depends upon the belt length, belt elasticity, pulley geometry, and the center to center distance of the pulleys. Assuming that the length of the belt is constant and the belt elasticity is linear, the following equation applies [10] ;

The effect of the initial tension as determined in Equation (9) could be obtain with the re-expression of the power output Equation (8) as obtained in Equation (10).

where R is the design vector of the system taken as follows:

Assuming that m, b, and are constants, then the parameters T₀ and provides the measure of the power transmitted and are used as the basis for the objective function in the problem formulation.

In the constraint set, a condition for maximum power transmission could be obtained by the limitation on the initial tension in the belt expressed as [33] ;

A geometrical relation could be captured with coordinate system as defined in Figure 2 and the following geometrical relations are obtained for the angle of lap on the driven pulley;

where and are the radii measured from the pulley center to the centroidal axis of the V-belt. C is the center-to-center distance of the drive system.

The calculations involving the belt length are usually based on the pitch length, of the drive. For any given belt section, the pitch length and center-to-center distance are obtained as [8] ;

The center-to-center distance of the V-belt drive is limited to the following inequalities to prevent excessive vibration of the slack side [9] ;

and (17)

The constraint on the ratio of tension could be obtained from Equations (7), (13) and (14) as follows:

Combining Equations (14) and (17) the constraint on the center-to-center distance can be obtained as:

and

The optimization problem could therefore be stated as follows:

Maximize

Subject to

The above system of the objective function equation and the constraints are nonlinear functions of the design variables. Response surface methodology was used for the optimization of the power output. FORTRAN 90 code was developed to simulate the objective function subject to the constraints as shown in flowchart Figure 3. The actual values from the FORTRAN 90 code simulation for the model Equations (10)-(20) were used directly for the RSM experiment design. The behavior of the power output, P, obtained in the simulation data were modeled as functions of the driving pulley radius, R_A, and driven pulley radius, R_B using the Response Surface Methodology (RSM). The response surface methodology was obtained from the design expert software version 7.0.0.

Response surface methodology usually aim at determining the optimum settings for the variables and to see how the variables perform over the whole experimental domain, including any interactions such as the simultaneous influence of the deriving and driven pulley parameters on the power output of the V-belt drive system. The driving pulley and driven pulley radius were taking as two independent variables which determine the response of the power output P of the belt drive. The experimental design and statistical analysis were performed according to the response surface analysis method using Design Expert 7.0.0 software. Historical data obtained from the FORTRAN simulation was employed to study the combined effect of the pulley radii R_A and R_B. The dependent variable (P) measured is the power output of the belt drive system.

A cubic order three dimensional surface model was obtained to describe the relationship between the power output and the pulley radii. The model was able to account for the curvature of the response and the interaction of the independent variables in the response surface. The data point (P, R_A, R_B) defines a curved surface in 3D space represented by the following polynomial [33] .

The parameters are constant coefficients known as the regression coefficients. These coefficients measure the expected change in the response P per unit increase in Ra when the Rb is held constant and vice versa and is established by regression analysis in the RSM programme. is the main effect, and are the curvature, is the interaction and e is the error. All the coefficients were obtained by the use of the Design Expert software package. The goodness of fit for each property model was confirmed by the R2 values and the probability obtained from the analysis of variance (ANOVA). The optimum values of the power output was obtained from the numerical analysis of the RSM package.

The simulation results as obtained from the FORTRAN 90 code is shown in Table 1.

Table 1 shows the dependency of the power output on the Pulley radii for the combined influence of the pulley radii the power output of the V-belt drive. The data in the table was populated in the RSM actual-design value frame for the 100 observations obtained. The RSM capable of developing model fits for the data was used to develop the model describing the relationship of the power output on the combined influence of the driving and driven pulley radii. Table 2 shows the result of the model fit for the data as analyzed using the RSM.

The results suggest cubic order polynomials for the description of the power output relationship with the pulley radii. This was obtained by focusing on the model that maximizes the adjusted and predicted R-square values for the system. The cubic order model compared to the other models has the lowest standard deviation, higher R² values and low predicted residual sum (PRESS) of squares for the drive indicating that the cubic model is the most suitable for describing the power output of the system.

The Analysis of Variance (ANOVA) for the response surface cubic model of the power output is as shown in Table 3 with estimated values of the regression coefficients. The ANOVA is employed in order to determine which of the pulley parameter is significant in describing the behavior of the power output. The R2 values were obtained from the F-test. The significant parameters are as shown in Table 3. Parameters with P-values less than 0.05 are significant to the description of the power output.

The F-values for the model with p-values <0.0001 implied that such model is significant. This means that there is only 0.01% chance that the model F-values as large as obtained could occur due to noise.

The model terms with p-values less than 0.05 are considered to be significant and influence the power output considerably. The smaller values of the p-values suggest that there is curvature in the response surfaces. This is in line with rejecting the hypothesis that a particular regression coefficient does not influence the power output. The driven pulley main effect B, the curvature term B2 and the driving pulley main effect term A has the most significant influence on the power output of the drive with F-value of 31.46, 25.70 and 23.96 respectively. This is followed by the curvature term B3 of the driven pulley and interaction term AB with F-value of 19.08 and 8.46.

The normal probability of the power output is shown in Figure 4.

It was observed that the residuals tend to aligned with the normal distribution assumptions as defined by the straight lines. This implies that the errors are normally distributed. Predicted values from the model could therefore be considered useful for information extraction for the belt drive.

The surface response and the contour plots of the power output are as shown in Figure 5 and Figure 6.

The plots show the combined influence of the pulley radii on the power output of the V-belt drive. The contour plots of the power output of the drive shows increasing power output with increasing driving pulley radius and reducing driven pulley radius. The power output however reduces as the driving pulley radius increases above 550 mm for small driven pulley radius. This phenomenon places limit on the pulleys parameter. Improved power output could therefore be obtained from driven pulley radius of 225 mm. Also the power output is limited to a maximum driving pulley radius of about 900 mm beyond which the power output of the drive reduces. The contour range of optimized power output could therefore be obtained at driven pulley radius range of 250 - 500 mm and driving pulley radius range of 550 - 900 mm. the limitation could be explained as due to the design constraint of the minimum range pulley belt-pulley lap angle which should be less than 180 degrees as obtained in theoretical models.

The power output region for optimized value is exposed on the contour plot of Figure 6. It is cleared from the plot that the optimum power output could be obtained at 550 < Ra < 900 mm and 250 < Rb < 500. The predicted optimum power output is 1418.76 kW at driving and driven pulley radius of 846 and 486 mm respectively. The standard pulley radius for the belt drive operation could be selected within these limits for optimum power output delivery for such drive.