2. Introduction
WHAT IS STRUCTURAL DYNAMICS ?
Dynamics concerned with the study of force and motion which are
time dependent
Methods for analyzing the stresses and deflection developed in any
given type of structure when it is subjected to an arbitrary dynamic
loading
Dynamic load is any load of which its magnitude, direction , and/or
position varies with time.
The structural response to a dynamic load , i.e., the resulting stresses
and deflection, is also time varying, or dynamic
Two basically different approaches are available for evaluating
structural response to dynamic loads:(based on how loading is
defined )
1. Deterministic Analysis and
2. Nondeterministic Analysis
3. Dynamic Analysis Approaches
Deterministic Analysis
The structural response i.e. displacement, acceleration ,velocity ,stress
etc., are completely known precisely as a function of time
Requires prefect control over all the variables that influence the
properties and loadings
Also known as prescribed dynamic loading
Nondeterministic Analysis
The time variation of vibration is not completely known
It provides only statistical information about the response statically
defined loading
Also known as random dynamic loading
4. TYPES OF PRESCRIBED LOADINGS
Classified in two categories , ‘ Periodic ‘ and ‘ Non-Periodic “
Periodic Loadings ;
loads which exhibit the same time variation successively for the large number of cycles.
The simplest form of periodic loading is a sinusoidal variation which termed as ‘simple
harmonic ‘
e.g. hydrodynamic pressures generated by a propeller at the stern of a ship or by
inertial effects in reciprocating machinery
5. Cont..
Non-Periodic Loadings
Loadings which doesn't exhibit the same time variation successively
It may be short duration (blast or explosion ) or long duration
impulsive loadings (earthquake)
6. Comparison of static loading and dynamic
loading
i. In static problem load is constant while in dynamic problem the load and its
responses varies with respect to time
ii. Static problem has only one response ,i.e. displacement but dynamic problem
has three responses ,such as displacement, velocity and acceleration
iii. Static problem only one solution whereas a dynamic problem has infinite
umber of solutions which are time dependent in nature
iv. In static problem response can be calculated by the principle of force or
static equilibrium whereas in dynamic problem the response depend not only
upon the load but also upon the inertia force which oppose the acceleration
7. Causes of dynamic effects
The most common types
a) Initial condition ; such as velocity and displacement produce dynamic effect in
the system e.g. the lift moving up and down suddenly stopped ,the cabin start to
vibrate
b) Applied force ; application of the external force e.g. bomb blast or wind force
on the building
c) Support motion ; the influence of support motion e.g. earthquake
8. Basic definitions
Mass ; dynamically ,it is the property that describe how an unrestricted body
resist the application of an external force (W/g) kgs
Stiffness ;force required to produce unite deformation or elastic property that
describe the level of resisting force that result when a body undergo a
change in length (N/m)
Natural period ;time required to complete one cycle of free vibration
(second)
Frequency ;number of cycles per unit time
Natural frequency ; the number of frequency of free vibration
Amplitude ; the maximum displacement or deformation of a vibrating system
from mean position
9. Basic definitions
Free vibration ; vibration which persists in structure after the force causing
the motion has been removed
Forced vibration ;the vibrating which maintained in a structure by steady
periodic force act on structure
Fundamental mode of vibration ;the fundamental mode of vibration od a
structure is the mode having the lowest natural frequency
Damping ;the resistance to the motion of vibrating body and the vibration
is called damped vibration(N/m/s)
Resonance ;when the frequency of the external force ids equal or much
with one of the natural frequency of the vibrating system, the amplitude of
the vibrating system become excessively large
10. Type of vibration
1. Fee and Forced vibration
Free vibration ; vibration which persists in a structure after
the force causing the motion has been removed
Forced vibration ;vibration maintained in a structure by
steady periodic force acting on the structure
2. Damped and undamped vibration
damped vibration ;when there is no damping element
Undamped vibration ;when there is damping element
3. Linear and Non-Linear vibration ;
4. Deterministic and random vibration
5. Longitudinal, transversal and torsional vibration
11. DEGREE OF FREDOM
Degree of Freedom is the number of coordinates necessary to specify the
position or geometry of mass point at any instant during its vibration
All real structures possess infinite number of dynamic degree of freedom.
Depending on the independent coordinates required to describe the motion
,systems divided into three
a) single degree of freedom system(SDOF system)
b) Multiple degree of freedom system(MDOF system)
c) Continuous system
12. Single degree of freedom system(SDOF system)
If a single coordinate is sufficient to define the position or geometry of the
mass of the system at any instant of time
13. Multiple degree of freedom system(MDOF system)
If more than one independent coordinate is required to completely specify
the position or geometry of different masses of the system at any instant of time
Continuous system (distributed system)
If the mass of a system may be considered to be distributed over its entire
length, in which the mass is considered to have infinite degrees of freedom
14. METHODS OF DISCRETIZATION
Lumped Mass Procedure
the distributed mass will be assumed as a
concentrated mass at discrete points
Generalized Displacements
based on the assumption that the deflected
shape of the structure can be expressed as the sum
of a series of specified displacement patterns ; these
patterns then become the displacement
coordinates of the structure
The Finite Element Concept
expressing the displacements of any given
structure in terms of a finite number of discrete
displacement coordinates, which combines certain
features of both the lumped mass and the
generalized coordinate procedures
15. FORMULATION OF THE EQUATIONS OF MOTION
The mathematical expressions defining the dynamic displacements are called the
equations of motion of the structure, and the solution of these equations of motion
provides the required displacement time histories
Direct Equilibration Using d'Alembert's Principle
The equations of motion of any dynamic system represent expressions of Newton's
second law of motion, which states that the rate of change of momentum of any
mass particle m is equal to the force acting on it
The concept that a mass develops an inertial force proportional to its acceleration
and opposing it is known as d'Alembert's principle.
16. Principle of Virtual Displacements
If a system which is in equilibrium under the action of a set of externally
applied forces is subjected to a virtual displacement, i.e., a displacement
pattern compatible with the system's constraints, the total work done by the
set of forces will be zero
Variational Approach
based on Hamilton's principle, makes no direct use of the inertial or
conservative forces acting in the system; the effects of these forces are
represented instead by variations of the kinetic and potential energies of the
system
