FLEXURE ENGINEERING FUNDAMENTAL – ROD SPRING

Construction fundamentals

INTRODUCTION

A rod flexure of ‘rod spring’ is one of the most fundamental features in flexure engineering and can be used in monolithic structures to fix 1 DOF of an element. The plate flexure can be set in S-mode or in C-mode, dependent on other kinematic constraints. C-mode comprise different flexure stiffness and flexure motion than the S-mode. This Precision Point sheet allows you to compute these to benefit your design.

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S-shape stiffness

C_x=C_y=\frac{12EI}{L^3}=\frac{3\pi Ed^4}{16L^3}
C_z=\frac{EA}{L} only if u_x=0
C_z=\frac {1}{\frac {L}{EA}+\frac{u_x^2L}{700EI}} for u_x{\neq}0

K_x=K_y=\frac{EI}{L}=\frac{\pi Ed^4}{64L}
K_z=\frac{G\pi d^4}{32L}=\frac{E\pi d^4}{64(1+\nu )L}

S-shape motion characteristics

u_x=\frac{F_x}{C_x}
u_{xmax}=\frac {1}{3}\frac{L^2}{Ed}\sigma_{max}
u_z=\frac{3}{5}\ast \frac {u_x^2}{L}

S-shape force limits

F_{z buckling}=4\frac{\pi ^2EI}{L^2} only if u_x=0
\sigma _{max}=\frac {M_{max}}{I}\ast \frac {1}{2}d=\frac{\frac{F_xL}{2}}{ I}}\ast \frac {1}{2}d=\frac{F_xLd}{4I}

dynamic movements:\sigma _{max}< fatigue stress limit
static deformation:\sigma _{max}< yield stress limit (\sigma _{0.2})

C-shape stiffness

C_x=C_y=\frac {3EI}{L^3}
C_z=\frac {L}{EA} only if u_x=0

K_x=K_y=\frac {EI}{L}=\frac {\pi Ed^4}{64L}
K_z=\frac {G\pi d^4}{32L}=\frac {E\pi d^4}{64(1+\nu )L}

C-shape motion characteristics

u_x=\frac{F_x}{C_x}
u_{xmax}=\frac {2}{3}\frac{L^2}{Ed}\sigma_{max}
u_z=\frac{3}{5}\ast \frac {u_x^2}{L}

C-shape force limits

F_{z buckling}=2.045\frac{\pi ^2EI}{L^2} only if u_x=0
\sigma _{max}=\frac {M_{max}}{I}\ast \frac {1}{2}d=\frac {F_xL}{I}\ast \frac {1}{2}d=\frac{F_xLd}{2I}

dynamic movements:\sigma _{max}< fatigue stress limit
static deformation:\sigma _{max}< yield stress limit (\sigma _{0.2})

Rod Spring S-Shape Deformation

Rod spring in s-shape deformation:I=\frac {\pi d^4}{64}

Rod Spring C-Shape Deformation

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