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1. Definitions: Dynamics vs Vibration ?
Def: Dynamics is the study relating the forces to
motion and the laws governing the motion are the
well-known Newton’s laws
 Dynamic load – any load of which the magnitude,
direction or position varies with time
Def: Vibration is an omnipresent type of dynamic
behaviour where the motion is actually an oscillation
about a certain equilibrium position
 Vibration - any motion that repeats itself after an
interval of time

2. Newton’s Law of Motion:
•First Law: A body continues to maintain its state of rest
or of uniform motion unless acted upon by an external
unbalanced force.
•Second Law: Momentum mv is the product of mass and
velocity. Force and momentum are vector quantities and
the resultant force is found from all the forces present by
vector addition. This law is often stated as, “F = ma: the
net force on an object is equal to the mass of the object
multiplied by its acceleration.”
•Third Law: To every action there is an equal and opposite
reaction.

3. Distinctive features of a dynamic analysis
Time-varying nature of the excitation (applied loads) and the
response (resulting displacements, internal forces, stresses,
strain, etc.)
 A dynamic problem does not have a single solution but a
succession of solutions corresponding to all times of interest in
the response history
 A dynamic analysis is more complex and computationally
intensive than a static analysis


Excitation
Response

4. Distinctive features of a dynamic analysis
Inertia forces when the loading is dynamically applied
 Inertia is the property of matter by which it remains at rest
or in motion at a constant speed along a straight line so long
as it is not acted by an external force
Translation motion, the measurement of inertia is the mass m
Rotational motion, the measurement of inertia is the mass moment
of inertia I0

M
V
F

F (t)
M (t)
V (t)
In e r tia fo rc es

5. Dynamic vs Static analysis
In reality, no loads that are applied to a structure are truly static
 Since all loads must be applied to a structure in some particular
sequence during a finite period of time, a time variation of the force is
inherently involved
When do we opt for dynamic analysis ?
 When forces change as a function of time ?
No, but when the nature of the force is such that causes
accelerations so significant that inertial forces can not be
neglected in the analysis
Static analysis – when the loading is such that the accelerations
caused by it can be neglected
The same load may be treated on one structure as dynamic
whereas on the other is static

6. 
Why not doing dynamic analysis always ?
Dynamic analysis is considerably more expensive than the
static analysis
More skills, knowledge, “feel” for the structural behaviour
under various types of dynamic loading are required in order
to deal with it both correctly and efficiently (a dynamic
analysis is much more computational than a static analysis)
The skill of the analyst is to make a judgement if a dynamic
analysis is necessary
Dynamic vs Static analysis

7. Dynamic vs Static analysis
 Situations in which dynamic loading must be
considered
response of bridges to moving vehicle
action of wind gusts, ocean waves, blast pressure upon a
structure
effect on a building whose foundation is subjected to
earthquake excitation
response of structures subjected to alternating forces
caused by oscillating machinery

8. Dynamic analysis procedure
Main steps of a dynamic investigation:
Identification of the physical problem (existing structure)
identifying and describing the physical structure or structural
component and the source of the dynamic loading
Definition of the mechanical (analytical) model
a set of simplifying assumptions (loading, boundary conditions, etc)
a set of drawings depicting the adopted analytical model
a list of design data – geometry, material properties, etc.
Definition of the mathematical model
a set of equations where the unknowns are the response sought
Having all this information available, investigation into dynamic
behaviour can start

9. Flow-chart of a typical dynamic analysis
P H Y S I C A L P R O B L E M
M E C H A N I C A L M O D E L
A s su m p tio n s o n :
* G e o m e try
* M a te rial la w s
* L o a d in g
* B o u n d a ry co n d itio n s
* E tc .
S O L U T I O N O F G O V E R N IN G
D I F F E R E N T I A L E Q U A T IO N S
* C o n tin u o u s m o d el:
P artial d iffe re n tia l e q u a tio n s
* D iscrete m o d e l:
O rd in a ry d iffe re n tia l e q u a tio n s
I N T E R P R E TA T IO N O F R E S U L T S R E F IN E A N A LY S I S
I M P R O V E M E C H A N I C A L M O D E L
C H A N G E O F P H Y S I C A L P R O B L E M
D E S I G N I M P R O V E M E N T S
S T R U C T U R A L O P T I M I Z AT IO N

10. Dynamic modelling of structures
Definition: Degree of freedom (DOF) – number of independent
geometrical coordinates required to completely specify the position
of all points on the structure at any instant of time
There are two types of geometrical coordinates:
Linear displacements (translations),

Angular displacements (rotations),
Three main procedure for the discretization of a structure:
Finite number of DOF – discrete parameter (lumped) model
Infinite number of DOF – distributed parameter (continuous) model
Combination of these two – finite element (FE) model
)(t
)(tx

11. Dynamic modelling of structures
Lumped mass model (discrete model)
The mass of the system is assumed to be concentrated
(localized/lumped) in various discrete points around the system
Single-degree-of-freedom (SDOF) system - the entire mass m of the
structure is localized at a single point
Multi-degree-of-freedom (MDOF) system - the mass m of the structure is
localized at many points around the system
m
x(t)
m 3
m 2
m 1
x 1(t)
x 2(t)
x 3(t)
a . b . c.

12. Dynamic modelling of structures
Lumped mass model (discrete model)
x (t)
x (t)
1
2
y (t) y (t) y (t)
1 2 3
m
1 m2 m3
x (t)
1
y (t)
2
(t)
y(t)

y(t)
x(t)m

13. Dynamic modelling of structures
Distribute model (continuous model)
mass is considered uniformly distributed throughout the system
in reality, structures have an infinite number of degrees of freedom
using the continuous model, a better accuracy of the results
can be achieved in a dynamic analysis than by the lumped mass model

x
y
x(y,t)
m(y)
a. b.

14. Comparison between static degree of freedom
and dynamic degree of freedom
From dynamic point of view, the system can have:
An infinity of DOF
(mass distributed)
6 DOF 3 DOF

15. Comparison between static degree of freedom
and dynamic degree of freedom
From dynamic point of view, the system can have:
1 DOF (SDOF)

16. Vibrations and classification of vibrations
Vibration is an omnipresent type of dynamic
behaviour where the motion is actually an oscillation
about a certain equilibrium position
Any motion that repeat itself after an interval of time
– vibration or oscillation
Vibration can be classified in several ways

17. Classification of vibrations
Free and forced vibration
Free vibration - the structure vibrates freely under the effect of the initial
conditions with no external excitations applied
Forced vibration - structure vibrates under the effect of external
excitation
Undamped and damped vibration
Undamped vibration – no energy is lost or dissipated in friction or
other resistance during oscillation
Damped vibration – any form of energy is lost during oscillation
x (t)
t
x (t)
t
Undamped vibration Damped vibration

18. Classification of vibrations
Periodic and nonperiodic vibration
Periodic vibration – repeats itself at equal time intervals called periods T .
The simplest form of periodic vibration is the simple harmonic vibration
T
T

19. Nonperiodic vibration – any other vibration that can not be
characterized as periodic

Periodic and nonperiodic vibration

20. Linear and nonlinear vibration
Linear vibration – all the components of a system (spring, mass and
damping) behave linearly. The principle of superposition is valid.
- eg. twice larger force will cause twice larger response
- mathematical techniques for solving linear systems are much more developed
than for the non-linear systems
Nonlinear vibration – any of the basic components behave
nonlinearly. The superposition principle is not valid.
Which is which?
 Linear-elastic
material
 Non-linear
material
 Non-linear
elastic material
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21. Deterministic and nondeterministic vibration
Deterministic vibration – the value (magnitude), point of application and
time variation of the loading are completely known
- eg. periodic vibration is a deterministic vibration
Nondeterministic (random) vibration – the time variation and other
characteristics of the load are not completely known but can be defined
only in a statistical sense
 Sources of dynamic loading
Environmental – wind load, wave load, earthquake load
Machine induced (in industrial installations) – rotating engines,
turbines, conveyer mechanisms, fans
Vehicular induced – road traffic, railway
Blast – explosive devices or accidental explosions

22. Components of a vibration system
k
c
x(t)
m
There are 3 key components of discrete systems:
Mass or inertia element
Spring element
Damping (dashpot) element
These interact with each other during the system’s motion
Therefore, it is very useful how each of the components
behave

23. Mass or inertia element
Mass relates force to acceleration
Mass is assumed to behave as rigid body (does not deform)
The 2
nd
Newton’s law relates forces to accelerations via
mass acting as a coefficient of proportionality
Inertia force “resisting” acceleration is developed and is
acting in the direction opposite to the external loading
Units N/(m/s^2) or kg
F (t)i
x(t)
..
slope = m
m
x(t)
..
F (t)=m x(t)i
..
.
F (t)i

24. Spring element
Spring relates force to displacement
Spring is assumed to have no mass and damping
An elastic force is developed whenever there is a relative
motion between the ends of the spring
k – spring constant or spring stiffness - Units N/m
Fe(t)
x(t)
slope = k

25. Spring element



n
i
ieq kk
1
Equivalent stiffness of
springs in series
Equivalent stiffness of
springs in parallel



n
i
i
eq
k
k
1
1

26. Spring element
Determine the equivalent spring stiffness of the following
dynamic systems
m
k1 k2
m
k1
k3
k2
21
kkk
eq

321
kkkk
eq

Spring are in parallel: Spring are in parallel:

27. Damping (dashpot) element
Dashpot relates force to velocity
Dashpot is assumed to have no mass and elasticity
A damping force is developed whenever there is a relative
velocity between the ends of the dashpot
c – viscous damping coefficient - Units Ns/m

28. Introduction
 Features of a dynamic analysis
 Dynamic analysis procedure
 Dynamic modelling of structures
 Vibrations
 Components of a vibration system
SUMMARY