Spring physics
Theory
In classical physics, a spring can be seen as a device that stores potential energy, specifically strain energy, by tensioning the bonds between the atoms of an elastic material "Elasticity (solid mechanics)").
Hooke's law of elasticity states that the extension of an elastic bar (its extended length minus its relaxed length) is linearly proportional to its tension "Strain (mechanical)"), the force applied to stretch it. Likewise, contraction (negative extension) is proportional to compression (negative tension).
This law actually only applies approximately, and only when the deformation (extension or contraction) is small compared to the total length of the bar. For deformations beyond the elastic limit, atomic bonds break or rearrange, and a spring may break, bend, or permanently deform. Many materials do not have a clearly defined yield stress, and Hooke's law cannot be meaningfully applied to these materials. Furthermore, for superelastic materials, the linear relationship between force and displacement is appropriate only in the low stress region.
Hooke's law is a mathematical consequence of the fact that the potential energy of the bar is minimum when its length is relaxed. Any infinitely differentiable function of a variable approximates a quadratic function when examined close enough to its minimum point, as can be seen by examining the terms of the Taylor series. Therefore, force, which is the derivative of energy with respect to displacement, approximates a linear function.
The force of a fully compressed spring takes the form.
Zero length springs
Zero length spring is a term for a specially designed coil spring that would exert zero force if it had zero length. If there were no restriction due to the finite wire diameter of such a coil spring, it would have zero length in the unstretched condition. That is, in a linear graph of the spring's force versus its length, the line would pass through the origin. Obviously, a coil spring cannot contract to zero length, because at some point the coils touch each other and the spring can no longer shorten. Zero length springs are made using a coil spring with built-in tension (twist is introduced into the wire as it is coiled during manufacturing, which works because a coil spring "unwinds" as it is stretched), so it could contract further, so the balance point of the spring, the point at which its restoring force is zero, occurs at a length of zero. In practice, zero-length springs are made by combining a negative-length spring, made with even more tension so that its balance point is "negative" length, with a piece of inelastic material of the appropriate length so that the zero-force point is at zero length.
A spring of zero length can be attached to a mass on an articulated arm in such a way that the force on the weight is almost exactly balanced by the vertical component of the spring force, whatever the position of the arm. This creates a horizontal "pendulum" with a very long period oscillation. These pendulums allow seismographs to detect the slower waves of earthquakes. The LaCoste" suspension with zero length springs is also used in gravimeters because it is very sensitive to changes in gravity. Door closing springs often have a length of approximately zero, so that they exert force even when the door is almost closed, so that they can keep it closed firmly.
strain energy
The simplest way to analyze a spring physically is through its global ideal model and under the assumption that it obeys Hooke's Law. The spring equation is thus established, where the force F exerted on it is related to the elongation/contraction or elongation "x" produced, as follows:
The strain energy or elastic potential energy associated with stretching or shortening a linear spring is given by the integration of work "Work (physics)") performed on each infinitesimal change in its length:
If the spring is not linear then the stiffness of the spring is dependent on its deformation and in that case we have a somewhat more general formula:
Differential equation and wave equation
We will now define an intrinsic constant of the spring independent of its length and thus establish the differential law constitutive of a spring. Multiplying by the total length, and calling the product or intrinsic k, we have:.
We will call the tension in a section of the spring located at a distance from one of its ends, which we will consider fixed and which we will take as the origin of coordinates, the constant of a small piece of spring of length at the same distance and the elongation of that small piece by virtue of the application of force. By the law of the complete dock:.
Taking the limit:.
which by the principle of superposition results:
If we also assume that both the section and the modulus of elasticity can vary with the distance from the origin, the equation becomes:
Which is the complete differential equation of the spring. If it is integrated for all x, the result is the value of the total unit elongation. Normally F (x) can be considered constant and equal to the total force applied. When F (x) is not constant and its inertia is included in the reasoning, we arrive at the one-dimensional wave equation that describes wave phenomena.
Let us assume, for simplicity, that both the section of the spring, its density (density being understood as the mass of a section of spring divided by the volume of the imaginary enveloping cylinder) and its modulus of elasticity are constant throughout it and that the spring is cylindrical. Let's call the displacement of a section of dock. Now let's take a differential spring section of length (dx). The mass of that portion will be given by:.
Applying Newton's second law to that section:
That is to say:.
On the other hand, it is easy to deduce that.
By introducing, therefore, this expression into the differential equation of the spring deduced above, we arrive at:
Deriving this expression with respect to x we obtain:.
Combining the temporal expression with the spatial expression, the general equation of a cylindrical spring of constant section, density and elasticity is finally deduced, which coincides exactly with the longitudinal wave equation:
From which the speed of propagation of disturbances in an ideal spring is deduced as:
In the case of a spring with a suspended mass,.
Whose solution is , that is, the mass performs a simple harmonic movement of amplitude and angular frequency .
Deriving and substituting:.
Simplifying:
This equation relates the natural frequency to the spring stiffness and the sprung mass.
For a spring of variable density, variable modulus of elasticity and variable envelope section, the generalized perturbation equation is as follows:
Solutions to the equation of waves in a spring
The general solution to the partial differential equation of the simplified spring of infinite length is described below. Given the initial conditions:
where , the D'Alembert function solution to the wave equation can be written as:.
Such a solution admits that F and G can be any kind of continuous functions and when .
For a spring of finite length L with its ends anchored, the problem becomes a boundary problem that can be solved by separation of variables with the Sturm-Liouville theory. Given initial conditions such as those previously described and boundary conditions with fixed ends. The initial conditions can be developed into a Fourier series as follows:.
Where the Fourier coefficients are obtained after integrating the functions f and g as follows:
for.
The solution to this problem is written as follows: