mechanical vibration second chapter

1. Single degree of freedom system-
Free vibration
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2. Introduction
A system is said to undergo free vibration when it
oscillates only under an initial disturbance with no
external forces acting after the initial disturbance

3. Degree of Freedom
The number of degrees of freedom of a vibratory system is
the number of independent spatial coordinates necessary
to define its configuration. A is defined as the geometric
location of all the masses of the system. If the inter-
relationship of the masses is such that only one spatial
coordinate is required to define the configuration, the
system is said to possess one degree of freedom. A rigid
body in space requires six coordinates for its complete
identification, namely, three coordinates to define the
rectilinear positions and three to define the angular
rotations. Ordinarily, however, the masses in a system are
constrained to move only in a certain manner. Thus, the
constraints limit the of freedom to a much smaller number.

4. Frequency :-
No. of cycles per unit time or time taken to complete
one cycle.
Natural frequency :-
The frequency of free vibration when no external force
acts on the system after giving it an initial displacement
and body vibrates . These vibrations are called free
vibration and their frequency is called natural
frequency. Units :- Radian/sec or Hertz.

5. Equivalent systems
While we have discussed so far the vibration behavior of a
spring-mass system, in many practical situations we don't
readily find such simple spring-mass systems. Many a time, we
may find several springs and masses vibrating together and
then we will have several second order differential equations
to be solved simultaneously. In some special situations
however, we will be able to simplify the system by considering
equivalent stiffness and inertia. We may then still be able to
model the system as a simple single d.o.f spring-mass case.
When multiple springs are used in an application, they are
mainly found in two basic combinations.
• Series Combination
• Parallel Combination

6. Series Combination
•A typical spring mass system having springs in series
combination is shown above. The two springs can be
replaced by a equivalent spring having equivalent
stiffness equal to k. When springs are in series, they
experience the same force but under go different
deflections.
•For the two systems to be equivalent, the total static
deflection of the original and the equivalent system
must be the same.

7. Parallel combination
For the springs in parallel combination, the equivalent spring stiffness can
be found out as:
Each of the individual spring supports part of the load attached to it but
both the springs undergo same deflection.
Therefore the static deflection of the mass is,
Therefore if the springs are in parallel combination,
the equivalent spring stiffness is sum of individual
stiffnesses of each spring.

8. Newton’s method
Spring mass system in vertical position
Consider a spring mass system constrained to move in a
rectilinear manner along the axis of spring. Spring of constant
stiffness k which is fixed at one end carries a mass m at it’s
free end. The body is displaced from it’s equilibrium position
vertically downwards

13. Energy method
Energy methods are the methods which are based
on the conservation of energy. Assume the system
to be a conservative one. In a conservative system,
the total energy is constant. In a vibratory system
the energy is partly potential and partly kinetic.
The kinetic energy is because of velocity of mass
and potential energy is stored in the spring
because of it’s elastic deformation. According to
conservation law of energy, we know

15. Phase Plane method
The vibratory motion of a spring mass system with initial
conditions and has been obtained earlier give reference to
that section and is reproduced here
_____ (1)
We depict the vibratory motion in the form of a chart showing
displacement, x vs time, t. While this is one common way of
plotting the vibration response, we will now discuss another
very useful method of depicting the response viz., the
phase-plane plot.

16. Radius of the circle is the amplitude of oscillations and centre is at the origin.

17. Time is implicit in this plot and from this diagram, displacement
and velocity of motion are available from single point which
corresponds to a particular time instant. This is called the
phase-plane plot. The horizontal projection of the phase
trajectory on a time base gives the displacement-time plot of
the motion and similarly the vertical projection on time base
gives velocity-time plot of the motion.
The starting point (with finite displacement and velocity at time
t=0) is marked. After seconds, we reach where radians. There
are many other interesting forms of graphical representation of
dynamic response of a system. Since it is an undamped system,
when started with some initial conditions, if continues to move
forever. Staring point P1 is reached after every cycle (time
period).
If the system is damped, then the mass gradually dissipates
away energy and comes to rest.

18. THANK YOU