Chapter 10, Problem 1P

Within the range of recommended values of the spring index, C, determine the maximum and minimum percentage difference between the Bergsträsser factor, K_B, and the Wahl factor, K_W.

To determine

The maximum percentage difference between Bergstrasser factor and Wahl factor.

The minimum percentage difference between Bergstrasser factor, $K_B$ and Wahl factor, $K_W$.

Answer to Problem 1P

The maximum percentage difference between $K_B$ and $K_W$ is $1.36$.

The minimum percentage difference between $K_B$ and $K_W$ is $0.741$.

Explanation of Solution

Write the expression for Bergstrasser factor.

$K_B=\frac{4C+2}{4C-3}$ (I)

Here, Bergstrasser factor is $K_B$ and the spring index is $C$.

Write the expression for Wahl factor $K_W$.

$K_W=\frac{4C-1}{4C-4}+\frac{0.615}{C}$ (II)

Here, the Wahl factor is $K_W$.

Write the expression for the minimum percentage difference between $K_B$ and $K_W$.

$\text{Min}=\left(\frac{K_W-K_B}{K_W}\right)\times 100$ (III)

Write the expression for Maximum percentage difference between $K_B$ and $K_W$.

$\text{Max}=\left(\frac{K_W-K_B}{K_W}\right)\times 100$ (IV)

Conclusion:

Substitute $12$ for $C$ in Equation (I).

$\begin{array}{c}{K}_B=\frac{4\times 12+2}{4\times 12-3}\\ =\frac{50}{45}\\ =1.1111\end{array}$

Substitute $12$ for $C$ in Equation (II).

$\begin{array}{c}{K}_W=\frac{4\times 12-1}{4\times 12-4}+\frac{0.615}{12}\\ =\frac{47}{44}+0.05125\\ =1.068+0.05125\\ =1.1194\end{array}$

Substitute $1.1111$ for $K_B$ and $1.1194$ for $K_W$ in Equation (III).

$\begin{array}{c}\text{Min}=\left(\frac{1.1194-1.1111}{1.1194}\right)\times 100\\ =\frac{0.0083\times 100}{1.1194}\\ =0.741\%\end{array}$

Thus, the minimum percentage difference between $K_B$ and $K_W$ is $0.741$.

Substitute $4$ for $C$ in the Equation (I)

$\begin{array}{c}{K}_B=\frac{4\times 4+2}{4\times 4-3}\\ =\frac{18}{13}\\ =1.3846\end{array}$

Substitute $4$ for $C$ in the Equation (II).

$\begin{array}{c}{K}_W=\frac{4\times 4-1}{4\times 4-4}+\frac{0.615}{4}\\ =\frac{15}{12}+0.1537\\ =1.25+0.1537\\ =1.4037\end{array}$

Substitute $1.3846$ for $K_B$ and $1.4037$ for $K_C$ in the Equation (IV).

$\begin{array}{c}\text{Max}=\left(\frac{1.4037-1.3846}{1.4037}\right)\times 100\\ =\frac{0.0191\times 100}{1.4037}\\ =1.36\end{array}$

Thus, the maximum percentage difference between $K_B$ and $K_W$ is $1.36$