Leaf spring/flexure: Reinforced

Construction Design & Examples


Flexures or leaf springs can be used for play and friction free motion. A downside is stiffness and to minimize the needed force, the flexures are made slender and thin. Smart reinforcement of flexures and leaf springs can help to keep the needed motion-force minimal while the flexure or leaf spring is made thicker, which is beneficial for its carrying stiffness and easier to manufacture which will decrease the manufacturing costs.

Design parameters







Deformation characteristics

$u_z=\frac{F_z}{C_z} $
$u_x=\frac{u_z^2}{2L\left(1-\lambda\right)}$ (s-shape deformation)

Stiffness for s- and c-shape deformation

$C_y=\frac{1}{2\lambda\left(4\lambda^2-6\lambda+3\right)\left(1-\gamma\right)+\gamma}\cdot\frac{Etb^3}{L^3}$ (s-shape deformation)
$C_y=\frac{1}{2\lambda\left(4\lambda^2-6\lambda+3\right)\left(1-\gamma\right)+\gamma}\cdot\frac{Etb^3}{4L^3}$ (c-shape deformation)
$C_z=\frac{1}{2\lambda\left(4\lambda^2-6\lambda+3\right)\left(1-\gamma^3\right)+\gamma^3}\cdot\frac{Ebt^3}{{L}^3}$ (s-shape deformation)
$C_z=\frac{1}{2\lambda\left(4\lambda^2-6\lambda+3\right)\left(1-\gamma^3\right)+\gamma^3}\cdot\frac{Ebt^3}{4{L}^3}$ (c-shape deformation)

$K_y=\frac{1}{2\lambda\left(1-\gamma^3\right)+\gamma^3}\cdot\frac{Ebt^3}{12L}$ (c-shape deformation)
$K_z=\frac{1}{2\lambda\left(1-\gamma\right)+\gamma}\cdot\frac{Etb^3}{12L}$ (c-shape deformation)

Leaf spring - Reinforced
S-shape deformation (top) and c-shape deformation (bottom) of a reinforced leaf spring with y-dimension b.

Force limits (buckling)

When a force in x-direction is applied buckling can occur, for equations to calculate the buckling load see Beam Theory: Buckling.

Design guidelines

Keep $\frac{1}{10}<\lambda<\frac{1}{3}$ and $\frac{1}{10}<\gamma<\frac{1}{2}$

Typical $\lambda=\frac{1}{6}$ and $\gamma=\frac{1}{5}$

$C_y=1.3\cdot\frac{Etb^3}{L^3}$ (s-shape / c-shape)
$C_z=1.4\cdot\frac{Ebt^3}{L^3}$ (s-shape / c-shape)

$K_y=3.0\cdot\frac{Ebt^3}{12L}$ (c-shape)
$K_z=2.1\cdot\frac{Etb^3}{12L}$ (c-shape)

Normalized stiffness increase due to reinforcement

The graphs shown below indicate the normalized stiffness increase with respect to the non-reinforced case ($\lambda=0.5$).

Normalized stiffness of leaf spring/flexure reinforced

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