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LECTURE NOTES
ON
MECHANICAL VIBRATION
AND
STRUCTURAL DYNAMICS
IV B. Tech I semester (JNTUH-R13)

MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS
UNIT-I
1 Introduction to Mechanical Vibrations
1.1
IMPORTANCE OF VIBRATIONS
Vibrations are oscillations in mechanical dynamic systems. Although any system can oscillate
when it is forced to do so externally, the term “vibration” in mechanical engineering is often
reserved for systems that can oscillate freely without applied forces. Sometimes these vibrations
cause minor or serious performance or safety problems in engineered systems. For instance, when
an aircraft wing vibrates excessively, passengers in the aircraft become uncomfortable especially
when the frequencies of vibration correspond to natural frequencies of the human body and
organs. In fact, it is well known that the resonant frequency of the human intestinal tract (approx.
4-8 Hz) should be avoided at all costs when designing high performance aircraft and reusable
launch vehicles because sustained exposure can cause serious internal trauma (Leatherwood and
Dempsey, 1976 NASA TN D-8188). If an aircraft wing vibrates at large amplitudes for an
extended period of time, the wing will eventually experience a fatigue failure of some kind,
which would potentially cause the aircraft to crash resulting in injuries and/or fatalities. Wing
vibrations of this type are usually associated with the wide variety of flutter phenomena brought
on by fluid-structure interactions. The most famous engineering disaster of all time was the
Tacoma Narrows Bridge disaster in 1940 (see Figure 1.1 below). It failed due to the same type of
self-excited vibration behavior that occurs in aircraft wings.
Figure 1.1: (left) View of Tacoma Narrows Bridge along deck; (right) view of torsional
vibration
In reading books and technical papers on vibration including the previous paragraph,
engineering students are usually left with the impression that all vibrations are detrimental
because most publicized work discusses vibration reduction in one form or another. But

MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS
vibrations can also be beneficial. For instance, many different types of mining operations rely on
sifting vibrations through which different sized particles are sorted using vibrations. In nature,
vibrations are also used by all kinds of different species in their daily lives. Orb web spiders, for
example, use vibrations in their webs to detect the presence of flies and other insects as they
struggle after being captured in the web for food. The reason that mechanical systems vibrate
freely is because energy is exchanged between the system’s inertial (masses) elements and elastic
(springs) elements. Free vibrations usually cease after a certain length of time because damping
elements in systems dissipate energy as it is converted back-and-forth between kinetic energy and
potential energy.
The role of mechanical vibration analysis should be to use mathematical tools for modeling
and predicting potential vibration problems and solutions, which are usually not obvious in
preliminary engineering designs. If problems can be predicted, then designs can be modified to
mitigate vibration problems before systems are manufactured. Vibrations can also be intentionally
introduced into designs to take advantage of benefits of relative mechanical motion and to
resonate systems (e.g., scanning microscopy). Unfortunately, knowledge of vibrations in
preliminary mechanical designs is rarely considered essential, so many vibration studies are
carried out only after systems are manufactured. In these cases, vibration problems must be
addressed using passive or active design modifications. Sometimes a design modification may be
as simple as a thickness change in a vibrating panel; added thickness tends to push the resonant
frequencies of a panel higher leading to less vibration in the operating frequency range. Design
modifications can also be as complicated as inserting magneto-rheological (MR) fluid dampers
into mechanical systems to take energy away from vibrating systems at specific times during their
motion. The point here is that design changes prior to manufacture are less expensive and more
effective than design modifications done later on.
1.2
Modeling issues
Modeling is usually 95% of the effort in real-world mechanical vibration problems; however, this
course will focus primarily on the derivation of equations of motion, free response and forced
response analysis, and approximate solution methods for vibrating systems. Figure 1.2 illustrates
one example of why modeling can be challenging in mechanical vibrating systems. A large crane
on a shipping dock is shown loading/unloading packages from a cargo ship. In one possible
vibration scenario, the cable might be idealized as massless and the crane idealized as rigid. In

MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS
this simple case, the package and crane both oscillate as rigid bodies; the package oscillates about
the end of the crane and the crane oscillates about its base point of rotation as the two exchange
energy. These vibrations would most likely correspond to relatively low frequencies and would
take place in addition to the gross dynamical motion of the crane and package. Two coupled
ordinary differential equations would be needed in this case to model the discrete, independent
motions of the crane and package.
This model might be sufficient in some cases, but what if the mass of the cable is comparable
to the mass of the package? In this case, the crane and package still behave like rigid bodies, but
the cable will probably vibrate either transversely or longitudinally as a continuous body along its
length. These higher frequency vibrations would require that both ordinary differential equations
for the crane and package and partial differential equations of the cable be used to model the
entire system. Furthermore, if the assumption of rigidity in the crane were also relaxed, then it too
would need to be modeled with partial differential equations. All of these complications would be
superimposed on top of the simple rigid body dynamics of the crane and package.
Figure 1.2: Crane for loading/unloading packages from cargo ship. Different regimes of
operation require different levels of sophistication in the mechanical vibration model.
We will have the opportunity to discuss modeling considerations throughout the course when
case studies of vibration phenomena are used to reinforce theoretical concepts and analysis
procedures. Before starting to analyze systems, we must be able to derive differential equations
1-4

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