Design and optimization of a general planar zero free length spring

Arnoud A.T.M. Delissen*, Giuseppe Radaelli, Just L. Herder

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

13 Citations (Scopus)
48 Downloads (Pure)


A zero free length (ZFL) spring is a spring with special properties, which is commonly used in static balancing. Existing methods to create ZFL springs all have their specific drawbacks, which rises to the need of a new method to create such a spring. A method is proposed to design planar ZFL springs with specified stiffness (250–750 N/m) within a certain range (up to 20 mm of displacement). Geometric non-linearities of a curved leaf spring are exploited by changing its shape. The shape is determined by a non-linear least squares algorithm, minimizing the force residuals from a non-linear numerical analysis. Constraints are introduced to help in preventing the spring from intersecting itself during deformation. For three types of springs with different boundary conditions, designs are found with characteristic shapes and maximum force errors less than 1%. A trend is observed between spring size, maximum stress and desired stiffness. New type of ZFL springs can now be designed, which can not only be used in existing applications, but also enables the use of ZFL springs in micro mechanisms.

Original languageEnglish
Pages (from-to)56-77
JournalMechanism and Machine Theory
Publication statusPublished - 2017

Bibliographical note

Accepted Author Manuscript


  • Curved flexure
  • Desired force-displacement
  • Ideal spring
  • Null-length spring
  • Shape optimization
  • Zero free length spring


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