A linear spring stores elastic energy when displaced by from its relaxed position. The external force maintaining that displacement is ; its restoring force on the moving attachment has the opposite sign. Its spring constant has units force per length.
A zero-rest-length spring has quadratic potential energy in its actual length , with no subtracted natural length. It is a useful idealization, not an ordinary unstretched spring of arbitrary length. For an end-to-end spring on a two-link pendulum, changing the natural length changes the quadratic bending stiffness about the straight configuration.
Identical linear springs of stiffness have equivalent stiffness in parallel and in series. Parallel springs have equal displacement and add forces; series springs carry equal force and add extensions. A mass attached to an equivalent massless spring has angular frequency .
For point masses joined by unstretched central springs obeying Hooke's law, let be the equilibrium unit vector along an edge and the small displacement of its endpoint. The leading extension is , so its quadratic elastic potential energy is . Consequently the block stiffness matrix has off-diagonal blocks and diagonal blocks , with absent edges contributing zero. It is symmetric and positive semidefinite. The normal modes solve , where is the mass matrix. Rigid translations and infinitesimal rotations are rigid-motion zero modes of a spring framework. A prestressed spring can add transverse quadratic terms, so the unstretched assumption matters.
An ideal linear torsion spring exerts a restoring torque proportional to the relative angle between its two ends. Integrating the torque gives its elastic potential energy. A spring connecting two rotating bodies contributes to the potential energy; a spring connected to a fixed frame is obtained by setting one angle to zero. The coefficient is the torsional spring constant.
The derivative of an axial torque with respect to relative end-rotation angle, evaluated at the operating configuration. In a linear torsion spring, .
For a uniform rod of length and twist per unit length , the end-rotation angle is . The chain rule gives , where is the torsional stiffness. If a source defines a normalized torque , its derivative is .
A circular shaft twisted uniformly through small end angle over length has local engineering shear strain at radius . Its shear stress is , so the integrated torque is , using the polar second moment of area. The restoring torque has the opposite sign to the imposed twist. The stored elastic energy is . The circular cross-section is important: general noncircular sections can warp and have a different torsion constant.
The spring constant is the positive proportionality coefficient in the ideal spring restoring force . Its SI unit is newtons per metre. With a constant attached mass , it sets the angular frequency of simple harmonic motion. Doubling doubles the restoring force at a fixed displacement, but increases that angular frequency only by a factor .
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