Engineering

1. TITLE – SINGLE DEGREE OF FREEDOM
NAME – AYAN DAS
ROLL – 35000720080
DEPT. – MECHANICAL ENGINEERING
SEMESTER – 7th
SUBJECT – MECHANICAL VIBRATION
SUBJECT CODE – PE - ME 702F

2. CONTENTS
 INTRODUCTION OF DEGREE OF FREEDOM
 SINGLE DEGREE OF FREEDOM SYSTEM
 DESCRIBE SINGLE DEGREE OF FREEDOM SYSTEM
 EQUATION OF MOTION FOR A SINGLE DEGREE OF FREEDOM SYSTEM
 SOLVING THE ODE FOR A SINGLE DEGREE OF FREEDOM SYSTEM
 VARIATIONS
 REAL-LIFE SINGLE DEGREE OF FREEDOM SYSTEMS
 EXAMPLES OF APPLYING SDOF SYSTEMS

4.  SINGLE DEGREE OF FREEDOM SYSTEMS
 When it comes to engineering, it is often ideal to simplify a system in a way that makes it
easier to work out any necessary calculations. This process of modeling a system is very
common in a variety of engineering fields, and quite possibly the simplest system that can
be modeled is a single degree of freedom (SDOF) system.
SDOF systems may be simple, but they have actual applicability to various engineering
problems, and if an engineer is able to understand how an SDOF system can be described and
modeled using an equation, it is not much of a leap to go from there to higher levels of
engineering systems

5.  SINGLE DEGREE OF FREEDOM SYSTEMS
 In the most general terms, an SDOF describes the motion of a system that is constrained
to only a single linear or angular direction. This means that the system will only move in
the x-, y-, or z-direction, or that the system will only rotate about the x-, y-, or z-axis.
To put this in perspective, an airplane moves in six degrees of freedom: it moves forward and
backward in the x-direction, right and left in the y-direction, up and down in the z-direction,
rolls about the x-axis, pitches about the y-axis, and yaws about the z-axis. But beginning
with a system that is constrained to moving in only one of these six ways provides a solid
foundation for subsequent modeling of systems that move in multiple ways.
The simplest SDOF system often describes a system that moves only linearly in the x- or y-
direction.

6.  DESCRIBING A SINGLE DEGREE OF FREEDOM SYSTEM
 An SDOF is often described using a damped spring-mass system in the x-direction. A mass is attached to a
spring and a damper, which are both fixed at the opposite end. The system starts with some initial
displacement and velocity, although usually, the initial velocity is zero.
After the mass is released, it will oscillate or vibrate in the x-direction as the spring is stretched and contracts.
The damper, however, will reduce the magnitude of this oscillation until the mass is no longer moving. In this
system, things like gravity and friction are often ignored, although both can be included.
The way the system moves can be described using an equation of motion which satisfies Newton’s Second Law.
This law is simplified to the following:
F = ma
where F is a force acting on some mass, m, causing some acceleration, a. This equation can be rearranged in
countless ways, including to describe the motion of an SDOF system. This equation is referred to as the equation
of motion of the system.

7.  Equation of motion for a Single Degree of Freedom System
A mass suspended by a spring is a single
degree of freedom system (SDOF)
A mass suspended by a spring is a single degree of freedom
system (SDOF).
An SDOF system can be described by a second-order, non-
homogenous, ordinary differential equation (ODE). This
equation can be simplified as follows:
f(t)=mẍ + cvẋ + kx
Where, m is the mass,
cv is the damping coefficient,
k is the spring constant,
x is the linear displacement,
ẋ is the linear velocity,
ẍ is the linear acceleration.

8. If things like gravity and friction are to be included, they can be incorporated into the values of cv and k.
To solve this ODE, it is necessary to specify the initial conditions, which are usually as follows:
x (t0) = x0
ẋ (t0) = vo
where t0 is the initial time, x0 is the initial displacement, and v0 is the initial velocity. As stated earlier, the initial
velocity is usually zero, but it is not completely uncommon to have some initial velocity of the system. The initial
displacement can be set to zero, or some other value, usually
depending on the engineering preference or the types of values that the engineer will need in subsequent
calculations.
By solving this ODE, a plot of the system as a function of the time can be generated, which will show the position
of the mass oscillating with the oscillations becoming less and less until the mass stops moving.

9.  Solving the ODE for a Single Degree of Freedom System
 There are many ways to solve the type of ODE that is used to describe an SDOF
system. With advancements in computers, several solvers, including online
examples, have been developed. In fact, most engineers are going to use some
type of computer software to solve the ODE that describes their SDOF system.
Whether the ODE is solved by hand or with a computer, the final solution will
provide the engineer with a numerical description of what the SDOF system is
doing. As discussed, for the simplest case of a damped spring-mass system, as time
varies, the linear x-position of the system will change, and these values can be
plotted against time.

10.  Variations
 Just because a general description of an SDOF is a damped spring-mass system
does not mean there are not variations. For example, the system may not be
damped – in which case the magnitude of the oscillations would never decrease.
Or the system may be driven – in which case the magnitude of the oscillations
would increase, rather than decrease.
For any system that only moves in a single degree of freedom, a corresponding model
can be developed to simplify that system, and from that model, an equation of motion
and necessary initial conditions can be specified to solve for the motion of the system.

11.  REAL-LIFE SINGLE DEGREE OF FREEDOM SYSTEMS
While it is great to have a general concept that describes an SDOF system, these mathematical models can be used to
describe real systems that have actual purposes.
 Example SDOF systems
One such example is an accelerometer, which is used to determine how much an object accelerates in a given direction. As
the object moves, the attached accelerometer measures and reports that acceleration. In this case, the equation of motion
previously introduced can be reformatted as follows:
Where,
is the acceleration of the object and
Using the above equation with the specified coefficients cv and k, the acceleration of the object can be worked out and plotted
in a fashion like plotting the response of the SDOF system discussed earlier.

12.  Examples of applying SDOF systems
 When it comes to applying SDOF systems to actual engineering problems, it must be
remembered that what has been designed is a model of the system. Using this model, an
engineer will be able to evaluate the motion that the system will undergo under various
conditions. With this information, it is possible to determine if the system will experience
any undesired behaviour.
From the model of an SDOF system, an actual working system can be built. For example,
accelerometers are very common in several applications, and they can all trace their
operation to a model of an SDOF system. Depending on how accurate the calculations
involved in the model are, referred to as the fidelity of the model, the model will be able to
mimic the actual system to varying degrees of acceptability.