Page 55 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

52 Chapter 1 Fundamentals oF Vibration
1.7 spring elements

A spring is a type of mechanical link, which in most applications is assumed to have negli-
gible mass and damping. The most common type of spring is the helical-coil spring used in
retractable pens and pencils, staplers, and suspensions of freight trucks and other vehicles.
Several other types of springs can be identified in engineering applications. In fact, any
elastic or deformable body or member, such as a cable, bar, beam, shaft, or plate, can be
considered as a spring. A spring is commonly represented as shown in Fig. 1.18(a). If the
free length of the spring, with no forces acting, is denoted l, it undergoes a change in length
when an axial force is applied. For example, when a tensile force F is applied at its free end
2, the spring undergoes an elongation x as shown in Fig. 1.18(b) while a compressive force
F applied at the free end 2 causes a reduction in length x as shown in Fig. 1.18(c).
A spring is said to be linear if the elongation or reduction in length x is related to the
applied force F as
F = kx (1.1)

where k is a constant, known as the spring constant or spring stiffness or spring rate. The
spring constant k is always positive and denotes the force (positive or negative) required to
cause a unit deflection (elongation or reduction in length) in the spring. When the spring is
stretched (or compressed) under a tensile (or compressive) force F, according to Newton’s
third law of motion, a restoring force or reaction of magnitude -F1or +F2 is developed
opposite to the applied force. This restoring force tries to bring the stretched (or com-
pressed) spring back to its original unstretched or free length as shown in Fig. 1.18(b)
(or 1.18(c)). If we plot a graph between F and x, the result is a straight line according to
Eq. (1.1). The work done (U) in deforming a spring is stored as strain or potential energy
in the spring, and it is given by
1
U = kx 2 (1.2)
2

1 1 1

l x
l
F
l x
2
x
2
F x F

2

F
(a) (b) (c)

FiGure 1.18 Deformation of a spring.