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DYNAMICS OF STRUCTURES
ENGR. SUBHAN
PhD student, AIU
B.Sc. Civil, Suit Peshawar 2011
M.Sc. Structure INU Peshawar 2014
Former lecture & postgraduate coordinator, INU Peshawar

DYNAMICS OF SIMPLE STRUCTURES
INTRODUCTION TO BASIC DYNAMIC BEHAVIORS
• A simple structure is a structure that cab be idealized as a concentrated mass “m” supported by a
massless structure with stiffness “k” in the lateral direction .
• aa A single story Building
• A water Tank

• A simple structure which can be idealized as a
single-degree-of-freedom (SDF) System
• Figure shows a one story structure for which
most of the mass is concentrated at the roof
level and the roof is essentially rigid as
compared to the lateral-force resisting
system.

Some examples of simple structures which can
be idealized as single-degree-of-freedom (SDF)
systems.

IDEALIZED STRUCTURAL SYSTEM
• By this idealization, if the roof of a simple structure is displaced laterally by a distance and then released, the
idealized structure will oscillate around its initial equilibrium configuration:
The oscillation
with amplitude
The lateral displacement of roof as a function of time
The oscillation will continue with the same amplitude and
the idealized structure will never come to rest. This is an
unrealistic response because the actual structure will
oscillate with decreasing amplitude and will eventually come
to rest.

IDEALIZED STRUCTURAL SYSTEM
• To incorporate this feature into the idealized
structure, an energy dissipating mechanism is
required.
• Therefore, an energy absorbing element is
introduced in the idealized structure: the
viscous damping element (denoted by a
dashpot)
• This simple structure is sometimes called a
Single-Degree-of-Freedom Structure.
Many basic concepts of structure dynamics can be understood by studying
this simple structure.

EQUATION OF MOTION
• The motion of the idealized one-story structure caused by dynamic excitation is governed by an
ordinary differential equation, called the “equation of motion”
• Formulation of the equation is possibly the most important phase of the entire analysis procedure
(and sometime the most difficult phase).
• The equation can be determined using the following approaches:
(a) Direct Dynamic Equilibrium Approach
(b) Principle of Virtual Work (Energy Approach)

EQUATION OF MOTION
• The motion of the idealized one-story structure caused by dynamic excitation is governed by an
ordinary differential equation, called the “equation of motion”
• Formulation of the equation is possibly the most important phase of the entire analysis procedure
(and sometime the most difficult phase).
• The equation can be determined using the following approaches:
(a) Direct Dynamic Equilibrium Approach
(b) Principle of Virtual Work
(c) k (Energy Approach)

EQUATION OF MOTION
• The Direct Equilibrium using D’Alembert’s Principle will be employed in this lecture
Static Equilibrium Newton’s 2nd
Law D’Alembert’s Principle
+ + +
A particle mass is subjected to a system of dynamic force vectors and is the acceleration of the
particle mass m.

EQUATION OF MOTION
Newton’s 2nd
law states that, “The rate of change of momentum of any mass m is equal to the force
acting on it”.
+ =
D’ Alembert’s concept states that “ A mass develops an inertia force in proportion to its acceleration
and opposing it”
= -
Newton’s 2nd
law: + + All dynamic forces (Including the inertial force are in equilibrium):
Dynamic Equilibrium
This is very convenient concept in structure dynamics because it permits equations of motion to be
expressed as “equations of dynamic equilibrium”

EQUATION OF MOTION
Free body diagram
At any instantaneous time, the mass m is under the action of four types of
dynamic forces

1. External dynamic force:
2. Inertial force: = -
3. Elastic force: = -
Where k is the lateral stiffness of the two columns combined. The negative sign means that the forces are
always in the opposite direction the structural deformation ( this is to bring the structure back to its neutral
position)
a

By the application of D’Alembert’s principle, the sum of all four forces must be Zero.
+
The vector can be converted to scalar function by
=
=
a

Hence, the equation of motion in scalar form is
+ =
This is second-order linear (ordinary) differential equation.
a

Problem Statement
Given Data
a) The mass of the system ,
b) Applied dynamic load ,
c) Lateral stiffness of the system and
d) The damping coefficient of the system
a
Required Data
a) The displacement of the system
b) The other response quantities (e.g. the
response velocity , response acceleration ,
base shear, overturning moment etc.) can
be subsequently derived from

Problem Statement
Example
A 120-m-long concrete, box-girder bridge on four
supports-two abutments and two symmetrically located
bents-is shown. The cross-sectional area of the bridge
deck is The mass of the bridge is idealized as lumped at
the deck level; the unit mass of concrete is . The mass of
the bents may be neglected. Each bent consists of three
8-m-tall columns of circular cross section with Formulate
the equation of motion governing free vibration in the
longitudinal direction. The elastic modulus of concrete is
a

Problem Statement
• Solution:
For two bents the total stiffness of:
The equation governing the longitudinal displacement is:

Multi-Degree-of-Freedom Structure
• The example (idealized one-story) structure
described earlier is single-degree-of-freedom
system because its motion can be completely
described by only one scalar function-.
• A 3-story building is a three-degree-of-
freedom system because at least 3 response
functions
A three-degree-of-freedom system

Equation of Motion of One-story Building Subjected to Earthquake
• Consider a case when an SDF system is subjected to a lateral ground displacement .
• This represents a simplified earthquake excitation (i.e. the ground motion is assumed to be a one-
dimensional lateral motion).

Equation of Motion of One-story Building Subjected to Earthquake
Let’s denote the ground displacement, ground velocity and ground acceleration as
The total displacement at the roof is defined by
There are three dynamic forces acting on the roof mass:
1. Elastic force:
2. Damping force

Each of these forces is a function of “relative” motion, and not the absolute (or total) motion.
However the mass undergoes and acceleration of
Therefore,
3. Inertial force
Applying D’ Alembert’s dynamic equilibrium to this case, we get,

Equation of Motion of One-story Building subjected to Earthquake
The deformation of the structure due to ground acceleration is identical to the deformation of the
structure if its base were stationary and if it were subjected to an external force

Kinemetric Altus K2 Strong Motion Accelerograph System
Applications
• Structural monitoring arrays
• Dense arrays, two and three
dimensional
• Aftershock study arrays
• Local, regional and seismic
networks and arrays

High Dynamic Range Strong Motion Accelerograph

Ground Acceleration Recordings

Solution of Equation of Motion

Free Vibration Response of SDF Systems
• Free vibration response: the motion of an SDF system with the applied force set equal to zero
• Free vibration response in mathematical terms is the mathematical solution of the following
homogeneous differential equation
--------------------Equation (1)

A Quick Review of Basic Mathematical Concepts
• Solution Form
Consider a first-order differential equation
By separation of variables,
Integrate both sides
Where c is an arbitrary constant.
• By applying exponential operation,
Where is an arbitrary constant.
It can be shown that the solutions of higher order
differential equations are also this exponential form.

• Superposition
• If a solution of a homogeneous linear differential
equation is multiplied by a constant, the resulting
function is also a solution.
• The sum of two solutions is also a solution.
• Proof:
Let and be independent solutions of governing
differential equation of and SDF system, such that
.
• Substituting into the left-hand side of
equation-1, we get
=
Hence is also a solution of the equation of
motion.
In similar manner, by a direct substitution of
into the left-hand side of Eq(1),
It can be shown that is also a solution of the
equation of motion.

• Initial Value Conditions
Consider as a general solution of the governing
equating of motion. Since the constants and can
have any value, the general solution can represent
different solutions.
Usually initial conditions are known and we are
seeking for specific Solution that satisfies these
initial conditions.
Example of initial Conditions:
are the initial displacements and initial velocity of
the SDF system.
.
Two conditions are needed because there are
two unknown arbitrary constants to be
specified.
, , , all are known. Therefore, and can be
determined.

Free Vibration Response of SDF Systems
• Now consider the equation governing the free vibration of an SDF system:
Assuming that the solution of Eq(1) is in the exponential form:
Where G and s are constants.
Substituting this solution into the equation of motion (Eq(1)),
to have a non-zero solution , the term must be zero,

Case 1: Undamped Free Vibration Response
In this case,
Introducing the notation
The equation (4) becomes,
Which has two solutions,
----------(6)
Where
Hence the general solution of is
Where and are arbitrary constants.

Since there are two arbitrary constants, two initial conditions need to specified, i.e.
Therefore,
-------------(8)

• Taylor Series of (expand around )
•
•
•
• Taylor Series of
Similarly Taylor Series of
Therefore,
This is called Euler’s Equation.

Introducing the Euler’s equations:
----------(9)
And the expressions for and (Eq(8)) into the solution (Eq(7)), we obtain
It is easy to verify that this equation is the solution of governing equation of motion by direct
substitution.

Amplitude

The structure vibrates in simple harmonic motion (oscillation)
Amplitude -------------(11)
--------------------(12)

• The oscillation does not decay because the structure is undamped. The period of oscillation is the
time required for one cycle of free oscillation. For undamped structure,
Where is the natural circular frequency,
is the natural (cyclic) frequency (cycle/sec, Hz), and
is the natural period (sec)
• This term “natural” is used to qualify each of the above quantities to emphasize the fact that
these are “natural properties” of the structure.
• These properties are independent of the initial conditions.

Case 2: Damped Free Vibration Response
• In this case i.e. damping is present in the structure.
• The solutions of for this case are:
----------------------(14)
The characteristics of “ depends upon the sign of the term
Case 2(a): The equation will have distinct real roots, if
Case 2(b): The equation will have complex conjugate roots, if
Case 2(c): The equation will have real double roots, if

Case 2 (b): Underdamped Systems
• Let’s define : critical damping:
• Let’s define ----------------------(15)
• Hence, in underdamped systems,
• Rewriting the solution in terms of , we get
------------(16)
Where ----------------(17)

Then the general solution of is
)
)---------------(18)
When the initial conditions of and are introduced, the constants and can be evaluated, and after
using Euler’s equations we finally obtain
-------------(19)

The response in the previous equation can also be presented as
--------------(20)
-------------------eq(21a, b)
equation (20) says that the underdamped system in its free vibration stage will oscillate with circular
frequency and with exponentially decreasing amplitude.

0 1 2 3 4 5 6 7 8 9
-1.5
-1
-0.5
0
0.5
1
1.5
Series1
Series3
Series5
Series7
Time
Displacement

Case 2 (c): Critically damped Systems
• In this case and This will yield,
The general solution of the governing equation of motion will be of the form
(
The second term must contain because the two roots of the quadratic equation are identical.

Critically Damped System Underdamped System

• Using initial value conditions and , the constants and can be determined as follows:
The general solution will be
No Oscillation. critical damping just eliminated them

Case 2 (a): Overdamped Systems
The response of an Over-Critically-Damped System is similar to the motion of a critically-
damped system.
• Not encountered in practice.
• No oscillation.

Effects of Damping on Free Vibration
• In most structures the critical damping ratio and hence and .
• The rate of amplitude decay depends on .
0 0.2 0.4 0.6 0.8 1 1.2
0
0.2
0.4
0.6
0.8
1
1.2
Damped Natural Frequancey/Undamped Natural Frequency=ωD/ω
Damping
Ratio
ξ
Range of
Damping for
most of the
structures

Effects of Damping on Free Vibration
0 2 4 6 8 10 12 14
-1.5
-1
-0.5
0
0.5
1
1.5
t/T
u(t)/u0
0%, 1%, 2% and 5% Damping

Damping in Structures
• In Seismic of most structures is used.
• For tall structures subjected strong wind, we
generally assume
• For Single cables, assume

Decay of Free Vibration Response
Measured displacement response from a free-vibration test

-----------------(20) is called logarithmic decrement

• If and this gives an approximate equation:
----
--------(22)
To determine the number of cycles elapsed for a
50% reduction in displacement amplitude, we
obtain the following relation
----------------(23)
𝛿≃
2 𝜋𝜉
𝛿=
2 𝜋𝜉
√1− 𝜉
2

----------------(23)
The plot of this relation is shown in figure
0 0.05 0.1 0.15 0.2
0
2
4
6
8
10
12
Damping ratio x
J50%

Free Vibration Tests
• It is not possible to analytically the damping ratio for practical structures.
• It is achieved through free vibration tests.
• The Natural period can also be determined from the free vibration record by measuring the time
required to complete one cycle of vibration.

Numerical Problems
• Example-01: Determine the natural vibration period and damping ratio of the plexiglass frame
model shown in chapter-1 from the acceleration of its free vibration.
Solution:
Peak Time(sec) Peak,
1 1.11 0.915
11 3.884 0.076

Numerical Problems
• Example-02: A free vibration test is conducted on an empty elevated water tank such
as the one shown in figure. A cable attached to the tank applies a lateral force of 80
and pulls the tank horizontally by 5cm. The cable is suddenly cut and the resulting free
vibration is recorded. At the end of four complete cycles, the time is 2.0 sec and the
amplitude is 2.5 cm. from these data compute the following: (a) damping ration; (b)
natural period of undamped vibration: ( c) stiffness; (d) weight; (e) damping coefficient;
and (f) number of cycles required for the displacement amplitude to decrease to 0.5
cm.
• Solution: (a)
(b)
( c)
(d) ;
(e)
(f)

Numerical Problems
• Example-03: The weight of water required to fill the tank of the previous
example is Determine the natural vibration period and damping ratio of the
structure with the tank full.
• Solution:

Energy in Free Vibration
• The energy input to an SDF system by imparting to it the initial displacement
• At any instant of time the total energy in a freely vibrating system is made up of two parts, kinetic energy of
the mass and potential energy equal to the strain energy equal to the strain energy of deformation in the
spring:
Substituting from response of free vibration
Summing up the above two equations we get
+ Proved
The total energy is independent of time and equal to the input energy

Energy in Free Vibration
• For System with viscous damping, the kinetic energy and potential energy could be determined by
substituting and its derivative into
• The total energy will now be a decreasing function of time because of energy dissipated in viscous damping,
which over the time duration is
All the input energy will eventually get dissipated in viscous damping; as the dissipated energy tends to the
input energy.

Coulomb-Damped Free Vibration
• Equation governing the motion from right to
left is
For which the solution is
--------(1)
For motion of the mass from left to right, the
governing equation is
For which the solution is
--------(2)
• Initial value conditions:
Putting in eq(1) we get:
Substituting back in eq(1)
At
Putting in eq(2) we get:

• At time the motion reverses and is described by the below
equation at this time
2
• The time taken for each half-cycle is and duration of full
cycle the natural period of vibration, is
• The natural period of a system with Coulomb damping is the
same as for the system without damping. In contrast, viscous
damping had the effect of lengthening the natural period.
• In each cycle of motion, the amplitude is reduced by ; that is,
the displacements and at successive maxima are related by

Example-01: A small building consists of four
steel frames, each with a friction device,
supporting a reinforced-concrete slab, as
shown. The normal force across each of the
spring-loaded friction pads is adjusted to equal
2.5% of the slab weight. A record of the building
motion in free vibration along the x-axis is also
shown. Determine the effective coefficient of
friction.

• Solution: Determining .
• .
• .
• Determining coefficient of friction

RESPONSE TO HARMONIC AND PERIODIC EXCITATIONS

RESPONSE TO HARMONIC EXCITATION
Harmonic Force
• A simple structure subjected to a harmonic
loading,
• .

RESPONSE TO HARMONIC EXCITATION
In mathematics, the response is the solution of the
following linear non-homogeneous differential equation:
The solution must also satisfy the prescribed initial
conditions:

• Solution form:
A general solution of linear nonhomogeneous differential equation is the sum of a general solution
of the corresponding homogeneous differential equation and a particular solution
Where
And

is the specific response generated by the form of external force function (in this case external force function is
harmonic and is also harmonic) does not need to satisfy the initial conditions.
Introducing the general solution into the governing equation of motion, we obtain

Response to Harmonic Loading
(Undamped Systems,
Homogeneous (complementary) Solution
From the previous section, we have already
obtained as

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