Monday, February 1, 2010

Spring-Mass Systems

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].

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].

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)

Writing (1) and (2) in matrix form gives
(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 , so each piece must vanish separately ("separation of variables Eric Weisstein's World of Math"), yielding the coupled equations (3). The eigenmodes of the system follow from (3). Looking for a harmonic solution using the trial solution ,



(10)

(11)

This has solutions when the determinant Eric Weisstein's World of Math 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 , and the eigenfrequencies are

(19)
(20)

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Introductory Methods of Numerical Analysis