# Spring Constant

Written by Jerry Ratzlaff on . Posted in Constants

Spring force constant, abbreviated as $$k_s$$, also called spring constant, is the ratio of opposing force to the displacement from the origional position or how much force is needed to change a springs distance.

## Spring Constant formulas

 $$\large{ k_s = - F \; d_s }$$ $$\large{ k_s = \frac {F} {d_s} }$$ $$\large{ k_s = \frac {2 \; E}{ d^2 } }$$ $$\large{ k_s = \frac { F }{ x \;-\; x_0 } }$$ $$\large{ k_s = \frac { 2 \; PE_s }{ x^2 } }$$ $$\large{ k_s = \frac {G \; d^4} {8 \; n_a \; D^3} }$$ $$\large{ k_s = \frac { 2 \; C \;+\; 1 }{ 2 \; C } }$$     $$\large{ C = \frac{ D }{ d_w } }$$

### Where:

 Units English Metric $$\large{ k_s }$$ = spring force constant $$\large{lbf}$$ $$\large{N}$$ $$\large{ x }$$ = distance from equilibrium $$\large{in}$$ $$\large{mm}$$ $$\large{ D }$$ = mean coil diameter $$\large{in}$$ $$\large{mm}$$ $$\large{ n_a }$$ = number of active coils $$\large{dimensionless}$$ $$\large{ G }$$ = shear modulus of material $$\large{\frac{lbf}{in^2}}$$ $$\large{MPa}$$ $$\large{ d_s }$$ = spring displacement $$\large{in}$$ $$\large{mm}$$ $$\large{ E }$$ = spring energy $$\large{lbf-ft}$$ $$\large{J}$$ $$\large{ F }$$ = spring force $$\large{lbf}$$ $$\large{N}$$ $$\large{ x_0 }$$ = spring equilibrium position $$\large{in}$$ $$\large{mm}$$ $$\large{ PE_s }$$ = spring potential energy $$\large{lbf-ft}$$ $$\large{J}$$ $$\large{ d_w }$$ = wire diameter $$\large{in}$$ $$\large{mm}$$ $$\large{ d }$$ = wire size $$\large{in}$$ $$\large{mm}$$ $$\large{ D/N }$$ = index correction  $$\large{ G }$$ value for common spring materials Copper = 6.5 x 10^6 Beryllium Copper = 6.9 x 10^6 Inconel = 11.5 x 10^6 Monel = 9.6 x 10^6 Music Wire = 11.5 x 10^6 Phospher Bronze = 5.9 x 10^6 Stainless Steel = 11.2 x 10^6 $$\large{dimensionless}$$