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Math for Mass Spring System

Problem Setup

Two nodes in 3D space

vector p1 = [x1, y1, z1]

vector p2 = [x2, y2, z2]

One spring

Stiffness $k_s$

Rest length L

Notation

$||p1 - p2||$ is the Euclidean norm

$V$ is potential energy scalar

$F$ is the force vector

$K$ is the stiffness matrix

Spring Potential Energy

$V = \frac{1}{2}k_s (\Delta x)^2$

where $\Delta x = ||p1 - p2|| - L$

V = \frac{1}{2}k_s(||p1 - p2|| - L)^2

Differentiate for forces:

$\nabla V = 2\frac{1}{2}k_s(||p1 - p2|| - L)*\nabla (||p1-p2|| -L)$ by the chain rule

$= \nabla V = 2\frac{1}{2}k_s(||p1 - p2|| - L)*(\nabla||p1-p2|| - 0)$

$= k_s (||p1 - p2|| - L)\frac{(p1 - p2)}{||p1 - p2||}$ since $\nabla ||v|| = \frac{v}{||v||}$

Spring Force

F = -\nabla V = k_s(||p1 - p2|| - L)\frac{(p1 - p2)}{||p1 - p2||}

where

$k_s(||p1 - p2|| - L)$ is the magnitude of F

$\frac{(p1 - p2)}{||p1 - p2||}$ is the direction of F

so

$F_{12} = -F$ force on $p1$ by $p2$

$F_{21} = F$ force on $p2$ by $p1$

Differentiate for Stiffness

$\nabla F = \nabla k_s(||p1 - p2|| - L)\frac{(p1 - p2)}{||p1 - p2||}$

$= k_s(\nabla (p1 - p2) - L\nabla \frac{(p1 - p2)}{||p1 - p2||})$ multiply and simplify

$= k_s(I - L*(\frac{||p1 - p2||\nabla (p1 - p2) - (p1 - p2)\nabla||p1 - p2||}{||p1 - p2||^2}))$ division rule

$= k_s (I - L*(\frac{I}{||p1 - p2||} - \frac{(p1 - p2)(p1 - p2)^T}{||p1 - p2||^3}))$ simplify

Stiffness: Jacobian of Forces

K = \nabla F = k_s (I - L*(\frac{I}{||p1 - p2||} - \frac{(p1 - p2)(p1 - p2)^T}{||p1 - p2||^3}))

so

$K_{11} = -K$

$K_{12} = K$

$K_{21} = K$