upward gives
(2.8)
Figure 2.5 Movement of mass on a spring: (a) static deflection due to gravity and (b) oscillation due to initial displacement y0.
Thus the static deflection d of the mass is
The distance d is normally called the static deflection of the mass; we define a new displacement coordinate system, where Y = 0 is the location of the mass after the gravity force is allowed to compress the spring.
Suppose now we displace the mass a distance y from its equilibrium position and release it; then it will oscillate about this position. We will measure the deflection from the equilibrium position of the mass (see Figure 2.5b). Newton's law states that force is equal to mass × acceleration. Forces and deflections are again assumed positive upward, and thus
Let us assume a solution to Eq. (2.9) of the form y = A sin(ωt + φ). Then upon substitution into Eq. (2.9) we obtain
We see our solution satisfies Eq. (2.9) only if
The system vibrates with free vibration at an angular frequency ω rad/s. This frequency, ω, which is generally known as the natural angular frequency, depends only on the stiffness K and mass M. We normally signify this so‐called natural frequency with the subscript n. And so
and from Eq. (3.2)
The frequency, fn hertz, is known as the natural frequency of the mass on the spring. This result, Eq. (2.10), looks physically correct since if K increases (with M constant), fn increases. If M increases with K constant, fn decreases. These are the results we also find in practice.
Example 2.2
A machine of mass 600 kg is mounted on four springs of stiffness 2 × 105 N/m each. Determine the natural frequency of the system
Solution
We model the system as a hanging spring‐mass system (see Figure 2.5). Equation (2.9) governs the displacement of the machine from its static‐equilibrium position. Since we have four equal springs, the equivalent stiffness is 4 × 2 × 105 = 8 × 105 N/m. The natural frequency is then determined using Eq. (2.10) as
We have seen that a solution to Eq. (2.9) is y = A sin(ωt + ϕ) or the same as Eq. (2.3). Hence we know that any system that has a restoring force that is proportional to the displacement will have a displacement that is simple harmonic. This is an alternative definition to that given in Section 2.2 for simple harmonic motion.
b) Free Vibration – Damped
Many mechanical systems can be adequately described by the simple mass–spring system just discussed above. However, for some purposes it is necessary to include the effects of losses (sometimes called damping). This is normally done by including a viscous damper in the system (see Figure 2.6). See Refs. [8, 9] for further discussion on passive damping. With viscous or “coulomb” damping the friction or damping force Fd is assumed to be proportional to the velocity, dy/dt. If the constant of proportionality is R, then the damping force Fd on the mass is
(2.11)
and Eq. (2.9) becomes
(2.12)
or equivalently
where the dots represent single and double differentiation with respect to time.
Figure 2.6 Movement of damped simple system.
The solution of Eq. (2.13) is most conveniently found by assuming a solution of the form: y is the real part of A ejλt where A is a complex number and λ is an arbitrary constant to be determined. By substituting y = A ejλt into Eq. (2.13) and assuming that the damping constant R is small, R < (4MK)1/2 (which is true in most engineering applications), the solution is found that:
