Mass-spring-damper
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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) Spring, also known as springtime, is one of the four temperate seasons, succeeding winter and preceding summer. There are various technical definitions of spring, but local usage of ...
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 othe ...
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 linearl ...
. 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, implementa ...
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 ...
s of such models. As well as engineering simulation, these systems have applications in
computer graphics Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
and computer animation.


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 pa ...
and \zeta is the damping ratio.


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


References

Classical mechanics Mechanical vibrations {{Engineering-stub