Series Spring Stiffness Explained

This section explains how to determine the equivalent stiffness when two springs, each possessing a stiffness constant denoted by k, are connected in series.

Understanding Series Spring Connections

When springs are arranged in series, they are linked one after another, forming a chain. In this configuration, the total force applied to the spring system is distributed equally across each individual spring. Conversely, the total elongation (or compression) of the system is the sum of the elongations experienced by each spring separately.

Series Spring Formula

The general formula used to calculate the equivalent stiffness ($k_{eq}$) of multiple springs connected in series is based on the reciprocals of their individual stiffness constants ($k_1, k_2, \dots$):

$$ \frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \dots $$

Calculating Series Spring Stiffness

For this problem, we are given two springs, and both have the same stiffness, k. Therefore:

• Stiffness of the first spring: $k_1 = k$
• Stiffness of the second spring: $k_2 = k$

Substitute these values into the series formula:

$$ \frac{1}{k_{eq}} = \frac{1}{k} + \frac{1}{k} $$

Combine the fractions on the right side by adding them:

$$ \frac{1}{k_{eq}} = \frac{1 + 1}{k} $$

$$ \frac{1}{k_{eq}} = \frac{2}{k} $$

To find the equivalent stiffness ($k_{eq}$), we invert both sides of the equation:

$$ k_{eq} = \frac{k}{2} $$

Final Equivalent Stiffness Value

The calculation shows that the equivalent stiffness of two springs, each with stiffness k, connected in series is k/2.