This simulation shows two objects connected by springs and suspended from an anchor
point. The objects are able to move in 2 dimensions and gravity operates. The anchor
point is moveable.
You can drag either mass with your mouse. You can also drag the top anchor point. Click
the "reset" button to put the masses in a resting equilibrium. Try changing parameters
such as gravity, mass, spring stiffness, and friction (damping).
The math behind the simulation is shown below. Also available are:
open source code,
documentation and a
simple-compiled version
which is more customizable.
Physics
An immoveable (but draggable) anchor point has two spring2 and bobs hanging below
and swinging in two dimensions. We regard the bobs as point masses. We label the upper
spring and bob as number 1, the lower spring and bob as number 2.

Double 2D Spring
Variables
Define the following variables:
-
θ =
angle (
0=
vertical)
-
S =
spring stretch (displacement from rest length)
-
L =
length of spring
-
u =
position of bob
-
v = u'=
velocity of bob
-
a = u''=
acceleration of bob
-
F =
net force on a bob
Define some constants:
-
R =
rest length of spring
-
T =
position of anchor point
-
m =
mass of bob
-
k =
spring constant
-
b =
damping constant
-
g =
gravitational constant
Note that for this simulation the vertical dimension increases downwards.
Here are the equations of motion. The derivation is similar to that given for the
Single 2D Spring.
F1x = m1 a1x = −k1 S1 sin θ1 − b1 v1x + k2 S2 sin θ2
F1y = m1 a1y = −k1 S1 cos θ1 − b1 v1y + k2 S2 cos θ2 + m1 g
F2x = m2 a2x = −k2 S2 sin θ2 − b2 v2x
F2y = m2 a2y = −k2 S2 cos θ2 − b2 v2y + m2 g
The spring stretch
Sn
and angles
θn
are
functions of the positions
un
of the bobs as follows:
L1 = √((u1x − Tx)2 + (u1y − Ty)2)
L2 = √((u2x − u1x)2 + (u2y − u1x)2)
S1 = L1 − R1
S2 = L2 − R2
cos θ1 = (u1y − Ty)/L1
sin θ1 = (u1x − Tx)/L1
cos θ2 = (u2y − u1y)/L2
sin θ2 = (u2x − u1x)/L2
Numerical Solution
To solve the equations of motion numerically, so that we can drive the simulation,
we use the Runge Kutta method
for solving sets of ordinary differential equations. We need to convert the four second
order equations of motion to eight first order equations.
u1x' = v1x
u1y' = v1y
u2x' = v2x
u2y' = v2y
v1x' = −(k1/m1) S1 sin θ1 − (b1/m1) v1x + (k2/m1) S2 sin θ2
v1y' = −(k1/m1) S1 cos θ1 − (b1/m1) v1y + (k2/m1) S2 cos θ2 + g
v2x' = −(k2/m2) S2 sin θ2 − (b2/m2) v2x
v2y' = −(k2/m2) S2 cos θ2 − (b2/m2) v2y + g
We keep in mind that the spring stretch
Sn
and angles
θn
are functions of the position of the bob
un
as given previously.
This web page was first published April 2001.