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.
The system of D. E.'s is

Computer Lab Work.

Exercise 1. Find the general solution to the system of D. E.'s

Plot the solution curves.

Put the D.E.'s in operator form and eliminate y to obtain a higher order D.E. for x, and find the roots of its characteristic equation.

The roots are pure complex, , so the general solution is formed as follows.

Next, construct the solutions x1[t,a1,a2] y1[t,a1,a2] involving the functions cos(t) and sin(t),
and the solutions x2[t,b1,b2] y2[t,b1,b2] involving the functions cos(2t) and sin(2t).
These are the two natural modes of oscillation of the spring mass system and they exhibit
the natural frequencies w1 = 1 and w2 = 2, respectively.

Plot the functions x1[t,1,0] and y1[t,1,0].
Since the mass on the right is free at the end of the spring, the function amplitude of y1 is larger than x1.

Plot the functions x2[t,1,0] and y2[t,1,0].
In this mode of oscillation the masses are moving in opposite directions and must have the same magnitude.

Assume that the equilibrium position along the horizontal axis is 2 and 6.
The two masses move in the same direction with the frequency w1 = 1,
as seen in the next graph, where time is along the vertical axis.

Assume that the equilibrium position along the horizontal axis is 2 and 6.
The two masses move in opposite directions with the frequency w2 = 2,
as seen in the next graph, where time is along the vertical axis.

More Background. Consider the system of two masses and three springs.
Visualize a wall on the left and to the right a spring , a mass, a spring, a mass, a spring and another wall.
The system of D. E.'s is

Exercise 2. Find the general solution to the system of D. E.'s

Plot the solution curves.

Put the D.E.'s in operator form and eliminate y to obtain a higher order D.E. for x, and find the roots of its characteristic equation.

The roots are pure complex, , so the general solution is formed as follows.

Next, construct the solutions x1[t,a1,a2] y1[t,a1,a2] involving the functions cos(t) and sin(t),
and the solutions x2[t,b1,b2] y2[t,b1,b2] involving the functions cos(3t) and sin(3t).
These are the two natural modes of oscillation of the spring mass system and they exhibit
the natural frequencies w1 = 1 and w2 = 3, respectively.

Plot the functions x1[t,1,0] and y1[t,1,0].
Since there is a wall on both sides of the spring mass system these two functions are the same.

Plot the functions x2[t,1,0] and y2[t,1,0].
In this mode of oscillation the masses are moving in opposite directions and must have the same magnitude.

Assume that the equilibrium position along the horizontal axis is 2 and 6.
The two masses move in the same direction with the frequency w1 = 1,
as seen in the next graph, where time is along the vertical axis.

Assume that the equilibrium position along the horizontal axis is 2 and 6.
The two masses move in opposite directions with the frequency w2 = 3,
as seen in the next graph, where time is along the vertical axis.

Solutions.

Return to the Differential Equations Project

Return to the Numerical Analysis Project

Return to the Complex Analysis Project

(c) John H. Mathews, 1998