Module

for

Spring-Mass Systems

 

Background

    Consider the system of two masses  [Graphics:Images/SpringMassMod_gr_1.gif]  and two springs with no external force.  Visualize a wall on the left and to the right a spring , a mass, a spring and another mass.  Assume that the spring constants are  [Graphics:Images/SpringMassMod_gr_2.gif].   See Figure 1 below.  

      

[Graphics:Images/SpringMassMod_gr_3.gif]

  
      Figure 1. Coupled masses with spring attached to the wall at the left.  

    Assume that the masses slide on a frictionless surface and that the functions  [Graphics:Images/SpringMassMod_gr_4.gif]  denote the displacement from static equilibrium of the masses  [Graphics:Images/SpringMassMod_gr_5.gif], respectively.  It can be shown by using Newton's second law and Hooke's law that the system of  D. E.'s for  [Graphics:Images/SpringMassMod_gr_6.gif]  is  

        [Graphics:Images/SpringMassMod_gr_7.gif]   

Remark.  The eigenfrequencies can be obtained by taking the square root of the eigenvalues of the matrix [Graphics:Images/SpringMassMod_gr_8.gif].

 

Example 1.  Equal Masses, Unequal spring constants.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_9.gif]   

Solution 1.

 

Example 2.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_53.gif]   

Solution 2.

 

Example 3.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_90.gif]   

Solution 3.

 

Example 4.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_127.gif]   

Solution 4.

 

Example 5.  Unequal Masses, Equal spring constants.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_160.gif]   

Solution 5.

 

 

More Background

    Consider the system of two masses  [Graphics:Images/SpringMassMod_gr_193.gif] and three springs with no external force.  Visualize a wall on the left and to the right a spring , a mass, a spring, a mass, a spring and another wall.  Assume that the spring constants are  [Graphics:Images/SpringMassMod_gr_194.gif].   See Figure 2 below.  

    

[Graphics:Images/SpringMassMod_gr_195.gif]

  
    Figure 2. Coupled masses with springs attached to walls at the left and right.  

    Assume that the masses slide on a frictionless surface and that the functions  [Graphics:Images/SpringMassMod_gr_196.gif]  denote the displacement from static equilibrium of the masses  [Graphics:Images/SpringMassMod_gr_197.gif], respectively.  It can be shown by using Newton's second law and Hooke's law that the system of  D. E.'s for  [Graphics:Images/SpringMassMod_gr_198.gif]  is  

        [Graphics:Images/SpringMassMod_gr_199.gif]  

Remark.  The eigenfrequencies can be obtained by taking the square root of the eigenvalues of the matrix [Graphics:Images/SpringMassMod_gr_200.gif].

 

Example 6.  Equal Masses, Unequal spring constants.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_201.gif]   

Solution 6.

 

Example 7.  Equal Masses, Unequal spring constants.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_245.gif]   

Solution 7.

 

Example 8.  Equal Masses, Equal spring constants.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_282.gif]   

Solution 8.

 

Example 9.  Equal Masses, Unequal spring constants.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_315.gif]   

Solution 9.

 

Example 10.  Unequal Masses, Equal spring constants.  Find the general solution to the system of D. E.'s  and plot the solution curves.

    [Graphics:Images/SpringMassMod_gr_348.gif]   

Solution 10.

 

Old Lab Project (Spring Mass Systems  Spring Mass Oscillations).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

 

Spring Mass Oscillations  Spring Mass Oscillations  Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook Spring-Mass Systems

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2005