In all these systems considered as examples so far, the properties of mass/inertia; springiness and dissipation are modeled as "lumped” into certain physical devices. For example the mass does not have any springiness nor dissipation. The spring does not have any mass and does not dissipate either. The mass-less dashpot is a pure viscous damper without any springiness. Thus these are idealized models of typical physical phenomena that are observed in vibrating system namely mass/inertia; restoring force, dissipative force etc. In a building subjected an earthquake, the whole building shakes and it is not possible to isolate which part of the building is a pure mass; which is a pure spring element and which is pure dashpot. These elements are all “distributed” within the complete system and need to be modeled as such.

However modeling as lumped elements eases up the modeling effort because the resulting equations are ordinary differential equations (ODEs), while the distributed parameter models are described by partial differential equations (PDEs). Since ODEs are lot easier to solve than PDEs, it has been common practice to develop simplified lumped parameter models first. Latest computational techniques such as finite element method convert given partial differential equations of distributed parameter models (such as a building etc) into ODEs before solving them on the computer.

A spring or a dashpot is mathematically represented by a force – deformation or force – velocity curve/equation. In general this curve can be straight line or a general curve as shown in Fig. 6.2.1. When the relationship is a straight line i.e., a linear spring or damper model is used, there is direct proportionality between the cause and the effect and the resulting governing equations of motion are also linear. Linear models are extensively used in most engineering analyses, at least as a first approximation in design. The linear equations (differential or algebraic) are lot easier to solve than non-linear equations and hence these models are commonly used eventhrough many times practical systems do exhibit non-linear behaviour. Nonlinear systems exhibit very interesting and sometime unexpected behavior (for example chaos) and do become important in special situations. In our course we will be limiting our attention to only linear systems.