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Here is a position (vertical) versus time (horizontal) movie for a 2 mass (filled black circle dots) 3 spring (connecting gray line segments) system confined between 2 walls (black lines, or floor/ceiling but no gravity). The equilibrium lengths of the 3 springs are chosen conveniently so the oscillations of the positions are kept separate. The two position variables show displacement from the equilibrium positions (green lines). Red shows x1, blue shows x2 , both measured upwards from the green lines.
An overview of 2 mass spring systems and the related files online here is given in 2mass-springsystems.htm.
This shows the case of spring constants and masses (k1, k2, k3) = (1,3/2,1), (m1, m2) = (1,1) of equal masses but a middle spring 50 percent stronger than the outer springs, leading to natural mode frequencies 1 (slow tandem mode) and 2 (fast accordian mode), so the motion is periodic with period 2 π . Details are given in figure8curve.mw and 2mass-3spring-figure8curves.mw, 2mass3spring.pdf (general), figure8curve1.pdf (specific).
Initial conditions are (x1(0), x2(0)) = (1,0) and (x1'(0), x2'(0)) = (0,1), where the first displacement in red from the lower green line is x1 and the second displacement in blue from the upper green line is x2 . The second mass is driven by a sinusoidal force of frequency 3, so that the combined motion of the two natural modes and the response mode is a periodic function with period 2 π (with individual periods of 1, 1/2 and 1/3 times this common period).
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The right/bottom video shows the corresponding individual modes for one period of the system. Eigenmodes are in blue, with the eigenmode part of the motion tracing out the blue curve, to which the green oscillation is superimposed to result in the red total solution curve.