**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 .

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 .

Eigen Frequencies:

Consider three springs in parallel, with two of the springs having spring constant *k* and attached to two walls on either end, and the third spring of spring constant *k* placed between two equal masses *m*. To solve for the motion of the masses using the normal formalism, equate forces

(1) | |||

(2) |

(3) |

To find the equations of motion from energy considerations, note that the kinetic energy is defined by

(4) |

so the rate of change of *K* is

(5) |

Similarly, the potential energy is defined by

(6) |

so the rate of change of

*U*is

(7) |

But energy is conserved, so

(8) |

and

(9) |

Now, this equation must hold for arbitrary and

(10) |

(11) |

This has solutions when the determinant is 0, so

(12) |

(13) |

(14) |

To find the eigenvectors, plug back in. For

(15) |

so and the first eigenvalue and its associated eigenvector are

(16) |

This corresponds to masses moving in opposite directions. For

(17) |

(18) |

This corresponds to masses moving in the same direction.

If the middle spring has the same spring constant as those on either side, then

(19) | |||

(20) |

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