1.1 One-Degree-of-Freedom Model
1.1.6 Undamped Natural Frequency: An Accurate Approximation
Because of the modest amounts of damping typical of most mechanical systems, the undamped model provides good answers for natural frequen- cies in most situations. Figure 1.5 shows that the natural frequency of the 1-DOF model is the frequency at which an excitation force produces maxi- mum vibration (i.e., a forced resonance) and is thus important. As shown in a subsequent topic of this chapter (Modal Decomposition), each natural mode of an undamped multi-DOF model is exactly equivalent to an undamped 1-DOF model. Therefore, the accurate approximation now shown for the 1-DOF model is usually applicable to the important modes of multi-DOF models.
The ratio (ς) of damping to critical damping (frequently referenced as a percentage, e.g., ς=0.1 is “10% damping”) is derivable as follows.
Shown with Equation 1.4, the defined condition for “critically damped”
is(c/2m)2=(k/m), which yields c=2√
km≡cc, the “critical damping.”
Therefore, the damping ratio, defined as ς≡c/cc, can be expressed as follows:
ς≡ c 2√
km (1.10)
With Equations 1.4 and 1.5, the following were defined: ωn= k/m (undamped natural frequency),α= −c/2m (real part of eigenvalue for an underdamped system), andωd=
ω2n−α2(damped natural frequency).
Using these expressions with Equation 1.10 for the damping ratio(ς)leads directly to the following formula for the damped natural frequency:
ωd=ωn
1−ς2 (1.11)
This well-known important formula clearly shows just how well the undamped natural frequency approximates the damped natural frequency for typical applications. For example, a generous damping estimate for most potentially resonant mechanical system modes is 10–20% of critical damp- ing (ς=0.1–0.2). Substituting the valuesς=0.1 and 0.2 into Equation 1.11 givesωd=0.995ωnfor 10% damping andωd=0.98ωnfor 20% damping, that is, 0.5% error and 2% error, respectively. For even smaller damping ratio values typical of many structures, the approximation just gets bet- ter. A fundamentally important and powerful dichotomy, applicable to the important modes of many mechanical and structural vibratory systems, becomes clear within the context of this accurate approximation: A natural frequency is only slightly lowered by the damping, but the peak vibration caused by an excitation force at the natural frequency is overwhelmingly lowered by the damping. Figure 1.5 clearly shows all this.
1.1.7 1-DOF Model as an Approximation
Equation 1.2 is an exact mathematical model for the system schematically illustrated in Figure 1.1. However, real-world vibratory systems do not look like this classic 1-DOF picture, but in many cases it adequately approxi- mates them for the purposes of engineering analyses. An appreciation for this is essential for one to make the connection between the mathematical models and the real devices, for whose analysis the models are employed.
One of many important examples is the concentrated mass (m) sup- ported at the free end of a uniform cantilever beam (length L, bending moment of inertia I, Young’s modulus E)as shown in Figure 1.6a. If the concentrated mass has considerably more mass than the beam, one may rea- sonably assume the beam to be massless, at least for the purpose of analyzing vibratory motions at the system’s lowest natural frequency transverse mode. One can thereby adequately approximate the fundamental mode by
L m xst (a)
m o
L
mg (b)
q
FIGURE 1.6 Two examples treated as linear 1-DOF models: (a) cantilever beam with a concentrated end mass and (b) simple pendulum.
a 1-DOF model. For small transverse static deflections(xst)at the free end of the cantilever beam resulting from a transverse static load(Fst)at its free end, the equivalent spring stiffness is obtained directly from the can- tilever beam’s static deflection formula. This leads directly to the equivalent 1-DOF undamped system equation of motion, from which its undamped natural frequency(ωn)is extracted, as follows:
xst= FstL3
3EI (beam deflection formula) and Fst≡kxst
∴k= Fst xst =3EI
L3 Then,
m¨x+ 3EI
L3
x=0, ∴ωn= k
m = 3EI
mL3 (1.12)
In this example, the primary approximation is that the beam is massless.
The secondary approximation is that the deflections are small enough so that simple linear beam theory provides a good approximation of beam deflection.
A second important example is illustrated in Figure 1.6b, the simple pla- nar pendulum having a mass(m)concentrated at the free end of a rigid link of negligible mass and length(L). The appropriate form of Newton’s Second Law for motion about the fixed pivot point of this model is M=Jθ,¨ where M is the sum of moments about the pivot point “o,” J (equal to mL2here) is the mass moment-of-inertia about the pivot point, andθis the single motion coordinate for this 1-DOF system. The instantaneous sum of moments about the pivot point “o” consists only of that from the grav- itational force mg on the concentrated mass, which is shown as follows (minus sign because M is always oppositeθ):
M= −mgL sinθ, ∴mL2θ¨+mgL sinθ=0
Dividing by mL2gives the following motion equation:
¨θ+g L
sinθ=0 (1.13)
This equation of motion is obviously nonlinear. However, for small motions (θ1) sinθ∼=θ; hence it can be linearized as an approximation, as follows:
θ¨+g L
θ=0, ∴ωn= g
L (1.14)
In this last example, the primary approximation is that the motion is small.
The secondary approximation is that the pendulum has all its mass concen- trated at its free end. Note that the stiffness or the restoring force effect in this model is not from a spring but from gravity. It is essential to make simplifying approximations in all vibration models, in order to have fea- sible engineering analyses. It is, however, also essential to understand the practical limitations of those approximations, to avoid producing analy- sis results that are highly inaccurate or, worse, do not even make physical sense.