## Mechanical Vibrations

• The vibrations of mechanical systems can sometimes be analyzed in terms of a finite collection of masses, springs, and damping sources. This approach is referred to as lumped modeling and the resulting approximate system response is called a lumped characterization''.

• Figure 1 diagrams a system consisting of an ideal mass, spring, and damper connected in series'' (each element has the same velocity or displacement).

• An ideal mass (diagrammed in Fig. 2) is assumed to move on a friction-less surface or guide-rod and to be completely rigid.

• From Newton's Second Law:

• An ideal spring (diagrammed in Fig. 3) has no mass or internal damping and is characterized by a spring constant (or stiffness constant) k.

• By Hooke's Law: for x(0) = 0 (valid for small, non-distorting displacements)

• The spring's equilibrium position is given by x=0. A positive value of x produces a negative restoring force.

• The force necessary to overcome a mechanical dashpot or resistance (diagrammed in Fig. 4) is typically approximated as being proportional to velocity:

• In the series combination of an ideal mass, spring, and damper (Fig.  1), all of the forces must sum to zero (assuming no external force is applied). Thus, this system can be characterized by the equation:

• This second-order homogeneous differential equation has solutions of the form , where is a decay constant and is the characteristic (or natural) angular frequency of the system.

• A and are determined by the initial displacement and velocity.

• The natural frequency is lower than that of the mass-spring system ( ).