I'm not going to muck about with whatever problems you're having with your pendulums.
I'm only going to address your "How can I refresh the background without also blowing away everything?" issue.

It is possible, but it's a real trick. You have to draw the lines caused by the movement to an off-screen image, and then draw that image over your main window. Here's a demo sketch:

1. PGraphics h;
2. void setup(){
3.   size(400,400);
4.   h = createGraphics( width, height, P2D);
5. }
6. void draw(){
7.   background(0);
8.   h.beginDraw();
9.   h.stroke(0,255,255);
10.   h.line(width-pmouseX, height-pmouseY, mouseX, mouseY);
11.   h.endDraw();
12.   fill(255,0,0);
13.   rect(mouseX - 50, mouseY - 50, 100,100);
14.   image( h, 0, 0 );
15. }

Notice that the red square doesn't leave a trail as it moves (because of the call to background()), but the blue lines aren't removed by that call to background().
That's the effect you want, right?

I think i get you, i'll try it this afternoon. Also considered just drawing a background between drawing pendulum frames, might try that as well.

did you read
http://www.jonathan.lansey.net/pastimes/pendulum/index.html
and
(I can't get the packed zip files to open though)

http://www.generative-gestaltung.de/M_2_4_01

here is my Code.

Greetings, Chrisir  

1. // from http://www.jonathan.lansey.net/pastimes/pendulum/index.html
2. // and http://forum.processing.org/topic/pendulum-class
3. float x;
4. float y;
5. // amplitude
6. float a= .3;
7. float r= 2.0;
8. // time
9. float t=300; // time
10. float b=.2;
11. float k=.9; // damping I
12. float j=.53; // damping II
13. float f=1.86;
14. float m=1;
15. PVector previous;
16. void setup() {
17.   size (900, 900);
18.   previous=new PVector(-1, -1);
19.   stroke(0);
20. }
21. void draw () {
22.   if (t>=0) {
23.     //
24.     // 3 different forms:
25.     // x=a*cos(t*b)*(k*t);
26.     // y=a*cos(t)*(k*t);
27.     /*
28.      x=a*cos(t*b)*(k*t)+r*cos(cos(t*f)*(j*t))*m;
29.      y=a*cos(t)*(k*t)+r*sin(cos(t*f)*(j*t))*m;
30.      */
31.     //
32.     x=cos(t*1.01)*2*t-25*sin(cos(t*2.06)*0.001*t)*5;
33.     y=cos(t)*2*t+25*cos(cos(t*2.06)*0.001*t)*5;
34.     x+=400;
35.     y+=400; 
36.     if (previous.x!=-1)
37.       line (x, y, previous.x, previous.y);
38.     previous.set(x, y, 0);
39.     point (x, y);
40.     // ellipse (x, y, 3, 3);
41.     t-=.1;
42.   }
43.   else {
44.     println ("Done");
45.   }
46. }