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The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of
spring Spring(s) may refer to: Common uses * Spring (season), a season of the year * Spring (device), a mechanical device that stores energy * Spring (hydrology), a natural source of water * Spring (mathematics), a geometric surface in the shape of a h ...
s and dampers. This model is well-suited for modelling object with complex material properties such as
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
ity and
viscoelasticity In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly ...
. Packages such as
MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementatio ...
may be used to run
simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the ...
s of such models. As well as engineering simulation, these systems have applications in computer graphics and
computer animation Computer animation is the process used for digitally generating animations. The more general term computer-generated imagery (CGI) encompasses both static scenes (still images) and dynamic images (moving images), while computer animation refer ...
.


Derivation (Single Mass)

Deriving the equations of motion for this model is usually done by examining the sum of forces on the mass: :\Sigma F = -kx - c \dot x +F_ = m \ddot x By rearranging this equation, we can derive the standard form: :\ddot x + 2 \zeta \omega_n \dot x + \omega_n^2 x = u where \omega_n=\sqrt\frac; \quad \zeta = \frac; \quad u=\frac \omega_n is the undamped
natural frequency Natural frequency, also known as eigenfrequency, is the frequency at which a system tends to oscillate in the absence of any driving force. The motion pattern of a system oscillating at its natural frequency is called the normal mode (if all part ...
and \zeta is the
damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
.


See also

*
Numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...
* Soft body dynamics#Spring/mass models *
Finite element analysis The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...


References

Classical mechanics Mechanical vibrations {{Engineering-stub