There are plenty of springs out there which have either a little PVC tube or a foam inside, when I first saw the tube I thought it was probably to prevent the metal spring from bending, but that wouldn't explain the foam that comes with others.
Is there an analysis for this materials within the springs?, in which situations shall I use this materials within the springs?
I know of at least one case where such material was added inside the spring to dampen it.
The spring will be resonant at some frequency. If there is enough component of that frequency in whatever is driving the spring, then it could oscillate or "ring". This can be undesirable because it will subject whatever it is connected to to forces at this frequency, the center can move substantially sideways, and it can cause audible noise.
Something squishy (like the foam) in the middle of the spring will dampen such oscillations, making any resonant ringing die down quickly if the spring happens to be excited at that frequency.
Something rigid (like the tube) is probably just to keep the spring from buckling or oscillating sideways.
To elaborate on Olin's answer, the foam (and also the tube if it is pneumatic) is likely used to add some damping to the mechanical resonance of the spring. The phenomenon of resonance in a spring-mass system is illustrated in the figure below. In that figure $\omega_0$ is the resonance frequency of the system which is determined by the spring constant and the sprung mass.
For frequencies well below the resonant frequency the amount of motion of is equal to the amount of the push. For frequencies well above the resonant frequency the amount of motion is much less than the amount of the push (this property is exploited in e.g. vehicle suspensions). But, for frequencies near the resonance the amount of motion can be significantly greater than the amount of the push. We have all experienced this when playing with a weight on a string or a glass of liquid; there is a certain frequency at which a very small input can create a very large motion of the mass/liquid.
Too much amplification at the resonance can literally tear the device apart (see e.g. the Tacoma Narrows bridge). This amount of amplification at the resonance is determined by the damping in the system. More damping means less amplification and vice versa. Damping is created in the system by inserting something which removes energy at the resonance. Foam is used in some of your examples because it is a very lossy material which efficiently converts mechanical motion into heat. Pneumatic tubes can be used (and often are in vehicle suspensions) to do work on a gas by forcing it through a small opening and thus, converting mechanical motion into heat. The tubes can also be used to restrict the motion of the spring to be only in the direction of the spring.
Depending on the mechanics of the design, the material is primarily there for damping. The design that compresses urethane (typically) foam uses the material's damping properties to convert the motion energy into heat. Other designs rely instead on friction to provide damping and similarly convert kinetic energy into heat. When I worked on designing suspension elements for washing machines, we used both methods.
Also, while a spring provides increased resistance with increased displacement, a damper provides increased resistance with increased velocity. Most elastomeric material provide both damping and stiffness.
