Consider a linear system where (by definition) the response to a general excitation can be obtained by a superposition of simple excitation responses.

One of the simplest excitations is the delta function (or impulse function) which has the important property:

This property states that a general forcing function defined in the interval (t1 , t2) can be

expressed as the superposition (or integration) of many delta functions with magnitude positioned throughout the excitation time interval.

Hence, if we define our forcing function f(t) as equaling the sum of delta functions when inside the time interval t1 to t2,

and equaling zero otherwise, the displacement response x(t) of a linear SDOF system subjected to f(t) is then given by,

The function g(t) is the impulse response of the system. By definition, a system's impulse response is equal to x(t) when f(t) is just a single delta function,

Laplace Transform

When the response of a linear system is difficult to obtain in the time domain (for example, say the Convolution Integral did not permit a closed form solution), the Laplace transform can be used to transform the problem into the frequency domain. The Laplace Transform of h(t) is defined by,

Transforming a SDOF equation of motion converts an ODE into an algebraic expression which is typically much easier to solve. After obtaining a solution for the displacement X(s) in the frequency domain, the inverse Laplace Transform is used to find x(t), where the inverse

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transform is defined by,

Using Laplace transforms to solve a spring-mass vibration system is demonstrated in the Laplace transform example section.

Vibration analysis often makes use of the frequency domain method, especially in the field of control theory, since the method is straightforward and systematic. However, the inverse transform can be difficult to find for complex systems.

Multiple Degree-of-Freedom Example

Consider the 3 degree-of-freedom system,

There are 3 degrees of freedom in this problem since to fully characterize the system we must know the positions of the three masses (x1, x2, and x3).

Three free body diagrams are needed to form the equations of motion. However, it is also possible to form the coefficient matrices directly, since each parameter in a mass-dashpot-spring system has a very distinguishable role.

Equations of Motion from Free Body Diagrams

The equations of motion can be obtained from free body diagrams, based on the Newton's second law of motion, F = m*a.

The equations of motion can therefore be expressed as,

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