Brian  Covello  

Project  2:  Mass-­‐Spring  System  With  Rubber  Band      

Background:         Second  order  differential  equations  are  frequently  used  to  model  the  physical  world.  

This  project  aims  to  model  the  behavior  of  the  model  below:    

 

 

   

For  ease  of  calculations  and  conversion  to  maple  format,  the  second-­‐order  differential  

equations  were  reduced  to  a  series  of  first  order  differential  equations.    

1) !"!"= 𝑥, !"

!"= 10 − 𝑘!𝑦  

2) !"!"= 𝑥, !"

!"= 10 − 𝑘!𝑦 − 𝑏𝑥  

3) !"!"= 𝑥, !"

!"= 10 − 𝑘!𝑦 − 𝑘!ℎ 𝑦  𝑓𝑜𝑟  𝑦 > 0  ;  !"

!"= 𝑥, !"

!"= 10 − 𝑘!𝑦 − 0  𝑓𝑜𝑟  , 𝑦 < 0  

4) !"!"= 𝑥, !"

!"= 10 − 𝑘!𝑦 − 𝑘!ℎ 𝑦 − 𝑏𝑥  

For  representation  of  the  piecewise  function  in  (3),  the  heaviside  function  was  utilized.  

This  function  can  accurately  describe  model  the  restoring  force  of  a  rubber  band  when  it  is  

stretched  and  the  lack  of  force  by  the  rubber  band  when  it  is  compressed.    

To  this  end,  this  project  begins  with  an  elementary  

analysis  of  simple  harmonic  oscillator  motion  as  defined  

by  gravity.  From  there,  the  effect  of  a  dampener  will  be  

examined  with  respect  to  the  oscillatory  motion.  The  

dampener  will  then  be  taken  away,  and  a  rubber  band  

will  be  added  to  the  system.  Finally,  the  combined  effects  

of  a  rubber  band  and  a  dampened  system  are  analyzed.  

We  assume  m=1  and  gravitational  force  =10.    

1) Simple  harmonic  oscillator    2) Harmonic  oscillator  with  damping  

 3) Oscillator  and  rubber  band,  no  

damping    

4) Oscillator  and  rubber  band  with  damping  

Brian  Covello  

Calculations  and  Data:    

Ideal  Mass  Spring  System:  

  We  begin  by  modeling  an  ideal  mass-­‐spring  system  with  no  rubber  band  given  by  the  

equation:  !"!"= 𝑥, !"

!"= 10 − 𝑘!𝑦  ,  where  k1=12.5.  In  this  model,  the  restoring  force  is  supplied  by  

the  spring.    The  phase  portrait  is  represented  on  the  left,  and  the  time  plots  are  represented  on  the  

right  with  x  as  red  and  y  as  blue.    

    In  the  above  graphs,  y(0)=2  and  x(0)=0.  Thus,  initial  velocity  is  zero,  and  the  

displacement  from  equilibrium  is  +2.  Notice  that  when  the  system  is  displaced  from  

equilibrium  2  units  in  the  downward  direction,  the  velocity  oscillates  between  -­‐4  and  4.  

The  oscillations  remain  constant  towards  infinity  in  time.  Below  is  a  representation  of  the  

system  with  no  initial  displacement  from  equilibrium  and  no  initial  velocity.  This  system  

represents  the  effects  of  gravity  upon  the  system  starting  at  the  equilibrium  position.    

Notice  that  gravity  alone  is  insufficient  to  compress  the  spring.    

 

Brian  Covello  

  These  simple  harmonic  oscillators  have  sinusoidal  solutions  as  expected.  The  lack  of  

dampening  is  manifest  in  the  constant  amplitude  of  the  time  plot  solutions.  Lastly,  

frequency  of  simple  harmonic  oscillators  is  represented  by  𝜔 = 𝑘!  and  𝑓 =!!!   ;𝑇 = !

!.  

Thus  period  is  a  constant  1.77  any  initial  values.    This  system  provides  a  good  basis  for  

continuing  our  exploration,  but  it  fails  to  provide  a  realistic  representation  of  the  system  

under  realistic  conditions.  Real  systems  are  dampened,  thus  we  introduce  the  damping  

coefficient  b  and  a  new  model.    

 

Damped  Mass  Spring  System:  

  This  system  can  be  modeled  by  the  equation  !"!"= 𝑥, !"

!"= 10− 𝑘!𝑦 − 𝑏𝑥.  Where  k1  

remains  the  12.5  value  from  the  previous  model  and  b  is  the  damping  coefficient.  The  

spring  mass  system  given  by:    𝑚 !!!!!!

+ 𝑏 !"!"+ 𝑘!𝑦 = 0  leads  to  three  different  cases  of  

solutions.  This  reformation  is  possible  due  to  static  equilibrium  conditions.    

Case  1  (Over-­‐damped):  b2-­‐4mk>0  

Case  2  (Critically  Damped):  b2-­‐4mk=0  

Case  3  (Under  damped):  b2-­‐4mk<0  

It  is  highly  unlikely  that  a  natural  system  will  be  critically  damped,  however,  case  2  

provides  a  basis  for  understanding  the  differences  in  the  following  behaviors  when  b  is  

small  compared  to  when  b  is  large.  Specifically,  one  expects  a  bifurcation  value  to  occur  at  

𝑏 = 4𝑚𝑘 = 4 ∗ 12.5.  We  begin  by  analyzing  the  under  damped  case  where  b=1  at  an  

initial  velocity  of  zero,  and  a  displacement  of  4.  

 

Brian  Covello  

Here  the  phase  plot  spirals  in  to  zero  and  the  frequency  is  lower.  Specifically,  the  frequency  

is  now  reduced,  having  the  value  𝜔! =|!!!!!"|!!

=  3.5,  and  T=1.795s.  There  is  an  

exponentially  decaying  amplitude  and  solutions  with  sine  and  cosine  characteristics.  Even  

with  different  initial  conditions  these  characteristics  remain.    

 We  now  turn  our  attention  to  the  over-­‐damped  case  where  b=10.  Notice  the  time  it  takes  

for  the  system  to  decay  is  substantially  lower.    

 Here  we  notice  no  oscillatory  of  the  system.  At  different  initial  conditions  some  initial  

increase  in  displacement  followed  by  continuous  decaying  amplitude.      

Brian  Covello  

 Mass-­‐Spring  and  Rubber  Band  without  Damping:  

  The  third  situation  may  be  modeled  by  the  piece  wise  equations  !"!"= 𝑥, !"

!"= 10 −

𝑘!𝑦 − 𝑘!ℎ 𝑦  𝑓𝑜𝑟  𝑦 > 0  ; 𝑎𝑛𝑑      !"!"= 𝑥, !"

!"= 10 − 𝑘!𝑦 − 0  𝑓𝑜𝑟  , 𝑦 < 0.  This  was  modeled  using  the  

Heaviside  function  in  maple.  This  function  allows  for  modeling  of  the  force  exerted  by  the  rubber  

band  as  it  is  stretched  AND  the  lack  of  force  by  the  rubber  band  as  it  is  compressed.  We  chose  

k2=4.7,  y(0)=4  and  x(0)=0.    

 These  graphs  indicate  a  return  to  the  ordinary  oscillatory  behavior  of  the  ideal  system.  We  

notice  the  period  is  once  again  1.77.Below  is  initial  conditions  x(0)=0,  y(0)=0.  Notice  there  

is  consistent  behavior  even  with  different  initial  conditions.  With  a  lack  of  conservative  

forces,  the  system  will  continue  to  oscillate  as  t  approaches  infinity.    

 

Brian  Covello  

 In  these  models,  K2  represents  the  elasticity  constant  of  the  rubber  band.    

Mass-­‐Spring  with  Rubber  Band  and  Damping  

  This  final  model  brings  all  additional  models  to  focus.  In  this  case  the  Heaviside  

function  remains,  and  the  system  is  damped.  This  model  is  represented  by  the  equation  !"!"= 𝑥, !"

!"= 10 − 𝑘!𝑦 − 𝑘!ℎ 𝑦 − 𝑏𝑥  ,  where  b  is  the  damping  coefficient,  K2  is  the  elasticity  

constant  of  the  rubber  band  and  k1  is  the  spring  constant.  We  will  let  K1=12.5  and  K2=4.7,  varying  

the  damping  coefficient.  In  the  graphs  below,  y(0)=0,  x(0)=0,  and  b=1.    

 Below  are  the  graphs  for  initial  conditions  x(0),  y(0)=4,  and  b=1.  Here  the  conservative  

forces  act  to  decay  exponentially  the  amplitude  for  larger  values  of  t.  The  system  is  tending  

towards  its  equilibrium  position.    

Brian  Covello  

    Note  the  similarities  between  these  graphs  and  the  under  damped  case  as  portrayed  

in  the  damped  mass  spring  system  above.  Likewise,  one  may  also  notice  the  over  damped  

behavior  when  b=10  as  depicted  below:    

 

 

 One  interesting  case  occurs  as  the  damping  coefficient  becomes  small.  When  b  becomes  

smaller,  the  system  approaches  resonance  frequency.  Below  y(0)=10,  x(0)=0,  and  

b=0.0001.    

Brian  Covello  

    Notice  that  near  resonance  frequencies,  the  system  is  continuously  in  motion  as  time  

approaches  infinity  with  varying  amplitudes  giving  rise  to  their  own  frequency  ranges.  

Resonance  frequencies  have  long  been  associated  with  collapse  of  bridges  and  occur  quite  

often  in  nature.    

  One  expects  the  same  type  of  consistent  behavior  to  have  a  bifurcation  value  near  that  

of  the  damped  mass  spring  system,  portraying  critical  dampness  around  7.07.  At  starting  

conditions  y(0)=4,  x(0)=0,  with  b= 50,  k1=12.5,  k2=4.7  the  following  graphs  were  

generated.    

 Conclusively,  this  lab  project  allowed  for  increased  insight  into  ideal  and  non-­‐ideal  systems  

that  were  either  overdamped,  critically  damped,  or  underdamped.  The  difference  between  

these  types  of  motions  may  be  seen  below:    

 

Brian  Covello  

 Where  the  green  represents  underdamped  motion,  blue  represents  the  overdamped  and  

red  represents  critically  damped.  Critically  damped  and  overdamped  very  often  look  

similar,  and  three  different  types  of  plots  may  generally  arise  for  these  systems:    

 All  non-­‐zero  solutions  to  non-­‐ideal  overdamped  and  critically  damped  spring  mass  systems  

tend  towards  equilibrium  as  t  increases.  They  pass  through  the  equilibrium  position  at  

most  once.  They  have  at  most  one  maxima,  and  at  most  one  point  of  inflection.  These  

systems  tend  to  have  solutions  in  the  form:    

 

Brian  Covello  

  All  non-­‐zero  solutions  to  non-­‐ideal  underdamped  spring  mass  systems  tend  towards  

equilibrium  as  t  increases.  They  mas  pass  through  the  equilibrium  position  infinitely  many  

times,  having  infinitely  many  maxima.  A  generalized  system  may  be  seen  below:    

 These  systems  are  described  by  their  imaginary  roots  as  given  in  the  equation  below: