Spring-Mass Systems

Background

    Consider the system of two masses  
  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  
.   See Figure 1 below.  

      

  
      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  
  denote the displacement from static equilibrium of the masses  
, respectively.  It can be shown by using Newton's second law and Hooke's law that the system of  D. E.'s for  
  is  

        
   

Remark.  The eigenfrequencies can be obtained by taking the square root of the eigenvalues of the matrix
.

More Background

    Consider the system of two masses  
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  
.   See Figure 2 below.  

    

  
    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  
  denote the displacement from static equilibrium of the masses  
, respectively.  It can be shown by using Newton's second law and Hooke's law that the system of  D. E.'s for  
  is  

        
  

Remark.  The eigenfrequencies can be obtained by taking the square root of the eigenvalues of the matrix
.