Mass-spring system

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A mass-spring system (or spring-mass system ) is on the one hand

• a concrete combination of mass- affected bodies and springs . The spring- mounted bodies can perform mechanical vibrations . Examples are the spring pendulum and the vehicle body suspended with springs . The expanded term spring-mass-damper system (or mass-spring-damper system) also refers to the decay of the vibration due to the damping . On the other hand it is too
• a basic model for every vibratory process. A combination of a mass and a spring is generally regarded as a second-order reference system (description with a second-order differential equation), although not all of the second-order structures that can be detected with it need to be oscillatory structures.

Spring-mass systems

Spring pendulum

→ Main article : Spring pendulum

The spring pendulum is a concrete mass-spring system that consists of a single mass, a single spring and optionally a single damping element ( ). ${\displaystyle \textstyle N=1}$

Railway technology

→ Main article : Mass-spring system (railway technology)

Resilient components can be inserted between the superstructure of railways and the subsurface (e.g. concrete tunnel floor) so that vibrations emanating from the vehicles do not spread into the environment and are not perceived there as structure-borne noise. Concrete masses are successive deck slabs or gravel troughs. They are mounted on concrete coil springs (metallic) or elastomer blocks or strips.

For this measure, which is generally used in technology, and its theoretical background for decoupling mechanical vibrations , the working term mass-spring system is explicitly used in railway technology.

Computer graphics

→ Main article : Mass-spring system (computer graphics)

For the two-dimensional pictorial representation of the variable elastic deformation of a body, it is simulated in a model made up of many finite (finitely small) masses, springs and damping elements. With the related method of finite elements i. d. Usually only the "static" elastic deformation is modeled, and the damping elements are omitted. Since the inertial mass is irrelevant here, the finite elements are i. d. Usually only elastic base bodies (e.g. bars).

Mathematical description

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The most general mathematical description of a mass-spring system with elements that each have freedom of movement is: ${\displaystyle \textstyle N}$ ${\displaystyle \textstyle D}$

${\displaystyle {\forall _{i=1}^{N}\forall _{d=1}^{D}\colon {\frac {d^{2}x_{i,d}}{dt^{2}}}=\sum _{k=0}^{D}F_{i,d,k}x_{i,k}+\sum _{k=0}^{D}D_{i,d,k}{\frac {dx_{i,k}}{dt}}+\sum _{j=1}^{N}\sum _{k=1}^{D}\sum _{l=1}^{D}F_{ij,kl}(x_{i,k}-x_{j,l})+\sum _{j=1}^{N}\sum _{k=1}^{D}\sum _{l=1}^{D}D_{ij,kl}{\frac {d}{dt}}(x_{i,k}-x_{j,l})+{\text{const}}}}$

Here, and count variable for the elements , and count variable for the degrees of freedom, and the spring constants of the deflection of element in the degree of freedom , the spring constants of the coupling between the element and upon displacement in the degree of freedom and , as well as the damping constant between the element and with respect to the relative velocities in the degree of freedom and . The constant term describes an optional existing external homogeneous field. In real systems, this description applies exactly to and approximately for small deflections . ${\displaystyle \textstyle i}$ ${\displaystyle \textstyle j}$ ${\displaystyle \textstyle d}$ ${\displaystyle \textstyle k}$ ${\displaystyle \textstyle l}$ ${\displaystyle \textstyle F_{i,dk}}$ ${\displaystyle \textstyle i}$ ${\displaystyle \textstyle k}$ ${\displaystyle \textstyle F_{ij,kl}}$ ${\displaystyle \textstyle i}$ ${\displaystyle \textstyle j}$ ${\displaystyle \textstyle k}$ ${\displaystyle \textstyle l}$ ${\displaystyle \textstyle D_{ij,kl}}$ ${\displaystyle \textstyle i}$ ${\displaystyle \textstyle j}$ ${\displaystyle \textstyle k}$ ${\displaystyle \textstyle l}$ ${\displaystyle \textstyle \mathrm {const} }$ ${\displaystyle \textstyle k=l=d}$ ${\displaystyle \textstyle k,l\neq d}$

Individual evidence

1. ↑ Mass-spring-damper system, description
2. ↑ ^{a b} Mass-damper-spring process basic model ( Memento from May 1st, 2015 in the Internet Archive )