Vibrations occur due to oscillations about an equilibrium point and can be periodic or random. The time for one full cycle is the period, the number of cycles per unit time is the frequency, and the maximum displacement from equilibrium is the amplitude. Vibrations can be free, forced, or damped. Free vibrations involve no external forcing, while forced vibrations are caused by an external periodic force. Damped vibrations dissipate energy through friction or resistance over time. Structures respond differently to short-period or long-period earthquake ground motions depending on their natural period. Models of single degree of freedom vibrating systems incorporate a mass, spring, and damper to formulate equations of motion.

1. DISASTERRESISTANTSTRUCTURES
TOPIC : VIBRATIONS
SUBMITTED BY –
NANDINI BUZAR BARUAH,
B. ARCH., SEMESTER – VII,
ROLL NO. – 191010015007,
GCAP, AZARA

2. VIBRATION
 Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point.
 The word comes from Latin word, vibrationem ("shaking, brandishing").
 The oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tyre
on a gravel road.
 Time interval required for a system to complete a full cycle of the motion is the period of the vibration.
 Number of cycles per unit time defines the frequency of the vibrations.
 Maximum displacement of the system from the equilibrium position is the amplitude of the vibration.

3. TYPESOFVIBRATION
 Free vibration is where there is no externally applied
vibrating forcing.
 In free vibration, energy will remain the same, and energy is
not added or removed from the body.
 The body keeps vibrating at the same amplitude.
 The solution to a free vibration is usually roughly sinusoidal.
 Examples: oscillations of object connected to a horizontal
spring, the sound produced by turning fork in a short
distance.
 There are three types of free vibrations : Longitudinal
vibrations, Transverse vibrations, Torsional vibrations.
FREEVIBRATION
The disc moves parallel to the axis.
The disc moves perpendicular to the axis.
The disc moves a circle about the axis.
TORSIONAL VIBRATIONS
TRANSVERSE VIBRATIONS
LONGITUDINAL VIBRATIONS

4. TYPESOFVIBRATION
 The tendency of one object to force another adjoining or interconnected object into vibrational motion is referred to as a
forced vibration.
 Vibrations of a body under the constant influence of an external periodic force acting on it are called the forced
vibrations.
 The external applied force is called the driving force.
 Amplitude of body oscillating under forced oscillations can be decreasing, constant or increasing depending on various
factors like difference in amount of driving force and resistive force, difference in frequency of driving force and actual
vibrations and difference in phase of driving force and actual vibrations.
 Examples of forced vibration are : Earthquake vibrations, wind forces which might cause resonance and collapse of
bridges, etc.
FORCEDVIBARTION
Earthquake vibrations Wind forces leading to rupture and collapse of bridges

5. TYPESOFVIBRATION
 When the energy of a vibrating system is gradually dissipated by
friction and other resistances, the vibrations are said to be damped
vibrations.
 The vibrations gradually reduce or change in frequency or intensity or
cease and the system rests in its equilibrium position. In order to
sustain the vibration an external force is required.
UNDAMPEDVIBRATION
DAMPEDVIBRATION
 If no energy is lost or dissipated in friction or other resistance during
oscillation, the vibration is known as undamped vibration.
 In undamped vibrations, no resistive force acts on the vibrating object.
 As the object oscillates, the energy in the object is continuously
transformed from kinetic energy to potential energy and back again,
and the sum of kinetic and potential energy remains a constant value.
 In practice, it’s extremely difficult to find undamped vibrations. For
instance, even an object vibrating in air would lose energy over time
due to air resistance.

6. TYPESOFVIBRATION
 A response spectrum is a plot of the peak or steady-state response (displacement, velocity or acceleration) of a
series of oscillators of varying natural frequency, that are forced into motion by the same base vibration or
shock. The resulting plot can then be used to pick off the response of any linear system, given its natural
frequency of oscillation. One such use is in assessing the peak response of buildings to earthquakes.
 The science of strong ground motion may use some values from the ground response spectrum (calculated from
recordings of surface ground motion from seismographs) for correlation with seismic damage.
 Response spectra are very useful tools of earthquake engineering for analyzing the performance of structures
and equipment in earthquakes, since many behave principally as simple oscillators (also known as single
degree of freedom systems). Thus, if you can find out the natural frequency of the structure, then the peak
response of the building can be estimated by reading the value from the ground response spectrum for the
appropriate frequency.
 In most building codes in seismic regions, this value forms the basis for calculating the forces that a structure
must be designed to resist (seismic analysis).
RESPONSESPECTRUM

7. LONG&SHORTPERIODSTRUCTURES
 When the ground shakes, the base of a building moves with the ground,
and the building swings back-and-forth. If the building were rigid, then
every point in it would move by the same amount as the ground. But,
most buildings are flexible, and different parts move back-and-forth by
different amounts.
 The ground shaking during an earthquake contains a mixture of many
sinusoidal waves of different frequencies, ranging from short to long
periods
 The time taken by the wave to complete one cycle of motion is called
period of the earthquake wave. In general, earthquake shaking of the
ground has waves whose periods vary in the range 0.03-33sec. Even
within this range, some earthquake waves are stronger than the others.
 Intensity of earthquake waves at a particular building location depends
on a number of factors, including the magnitude of the earthquake, the
epicentral distance, and the type of ground that the earthquake waves
traveled through before reaching the location of interest.
 In a typical city, there are buildings of many different sizes and shapes.
One way of categorizing them is by their fundamental natural period T.
The ground motion under these buildings varies across the city. If the
ground is shaken back-and-forth by earthquake waves that have short
periods, then short period buildings will have large response.
 Similarly, if the earthquake ground motion has long period waves, then
long period buildings will have larger response. Thus, depending on the
value of T of the buildings and on the characteristics of earthquake
ground motion (i.e., the periods and amplitude of the earthquake waves),
some buildings will be shaken more than the others.

8. DAMPEDFREEVIBARTIONSWITHVISCOUSDAMPNESS
 For a Free vibration (when there is no external force) of a single degree-of-freedom system with viscous damping can
be illustrated as following free body diagram,
 Damping that produces a damping vibration, proportional to the mass's velocity is referred to as viscous damp free
vibration.
 For an unforced damped Single Degree Of Freedom system, the general equation of motion becomes,
with the initial conditions,
 This equation of motion due to damped free vibration with viscous dampness.
 It is a second order, homogeneous, ordinary differential equation

9. DAMPEDFREEVIBRATIONOFASINGLEDEGREEFREEDOM
SYSTEM
 The simplest model to represent a single degree of
freedom system consisting of a rigid mass m
supported by a spring and dashpot.
 The motion of the mass m is specified by one co-
ordinate Z.
 Damping in this system is represented by the
dashpot, and the resulting damping force is
proportional to the velocity. The system is
subjected to an external time dependent force
F(t).
 According to De-Alembert’s principle, a body which is not in static equilibrium by virtue of some acceleration which it
posses, can be brought to static equilibrium by introducing on it an inertia force. This force acts through the centre of
gravity of the body in the direction opposite to that of acceleration. The equilibrium of mass m gives
MZ +CŻ + KZ = F (t) ---------- which is the equation of motion of the system called vibration.
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The formulation of governing equation of motion for an SDF system using the principle of virtual work
Spring-mass dashpot system Free-body diagram

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