Can a spring respectively its stiffness in a 2D-system be decomposed into its x- and y- components (stiffnesses)?

Assuming a spring located in an 2D coordinate system with positions p(x,y), length l and angle alpha, describing the orientation of the spring inside the system. Can the spring be represented by its xx, xy, yx and yy components?

Imagine an object which interacts with the spring not along its main direction, but along the global x-axis. Is it possible to model the interaction between spring and object by the decomposed xx-component of the spring?

Many thanks in advance

  • 1
    $\begingroup$ Forces can always be decomposed along different directions - not clear if that is what you are asking about? $\endgroup$
    – Floris
    Dec 27, 2015 at 0:30

1 Answer 1


If you are asking about the stiffness ( or spring constant $k$ ) , then obviously it cannot split into x and y components, because $k$ is a scalar constant. Suppose in your example, if the spring , at an angle $\alpha$ from x axis in $2D$ travels a distance $r$ then the displacement and force can be split into x and y component. That is, now the equation of such system is $$F=-kr$$
this can be split into $$F_x=-kx$$
and $$F_y=-ky$$ as $k$ is a scalar. You can connect these three equations using
$$F=\sqrt{F_x^2+F_y^2}\\ r=\sqrt{x^2+y^2}$$

  • $\begingroup$ In your example you assume a homogenous stiffness k for all directions. You apply the same k to a displacement along x and y direction. But actually, k is only known for displacements along the direction of r. $\endgroup$
    – Marcus
    Dec 27, 2015 at 23:53

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