Lecture 1: Structural dynamics

Reference Books:

• Anil K. Chopra, (2006), Structural Dynamics and its Application in Earthquake Engineering, 2nd Edition, John Wiley and Sons, New York, NY, USA.

• Clough-Penzien, 1975-2006, Dynamics of Structures, 3rd Ed., Mcgraw-Hill.

• Mario Paz, (1996), Structural Dynamics, Theory and Computation, 1st Indian Edition, CBS Publishers and Distributed, New Delhi.

Introduction

System analysis

Input System Output

• Engineering System

• Civil System: Structural System, Environmental System

• Mechanical System

• Electrical System

• Economic System

• Social System

Classification of Structural analysis

• Static Analysis: response analysis of structures under static loads

• Dynamic Analysis: response analysis of structures under dynamic loads

• Deterministic Analysis: loads are prescribed (known)

• Random Analysis: loads are known only in a statistical sense

• Linear Analysis: • Nonlinear Analysis: material non-linearity,

geometric non-linearity

Degree of freedom (DOF):

• Minimum Number of Coordinates (Number of Independent Coordinates) to define the Positions of a Body

• Number of Variables

• The Coordinates General Coordinates

• SDOF: Single Degree Of Freedom, DOF = 1

• MDOF: Multi-Degrees Of Freedom, DOF ≥ 2

m m

Generalised single degree of freedom

Car wheel modeled as single and double degree of freedom systems

Introduction to Structural Dynamics

Dynamic Loading:

• Varying loads changing in position, magnitude and direction with respect to time.

Dynamics:

• It is the branch of structural engineering in which the response of structures to time varying loads is considered.

• The response of structure to dynamic loading is also dynamic.

Rigid and deformable response:When a load is applied on a body and no (or negligible) strains or deformations are allowed to occur, then it is called a rigid body response and the opposite is deformable response.

Examples of rigid body response:• Flight of an aircraft *• Motion of earth or a planet• Movement of machinery• Punching of a pile into the ground ** may also be considered deformable response

Examples of deformable response:• Structural frames supporting manufactured objects like; buildings,

bridges, off shore structures, automobiles, ships, aircrafts, etc.

Oscillatory motion: Dynamic response of structures involving deformations is usually oscillatory.

Importance of vibration (dynamics) Analysis: Current engineering analysis tools have made the structures light weight with high strength materials and lower damping properties (energy dissipating mechanisms). This results in intense vibration response. Thus dynamic analysis has become more important for these modern structures.

In earthquake prone areas structures have to withstand violent shaking of the ground to resist damaging effects of earthquakes. Dynamic analysis needs to be carried out for these structures.

Example of light weight structure

Nature of exciting forces:

• Classification according to nature of variation with time:

– Periodic

– Non-Periodic

– Random

• Classification as being function of time:

– Deterministic: being specified as a definite fn. of time

– Nondeterministic: being known only in a statistical sense

Examples

• Rotating machinery

• Wind loading

• Bomb blast

• Earthquake

• Impact

At 4 micro seconds

At 7 micro seconds

Impact on a steel bar

Methods of discretization

Lumped mass procedure:

Lumped mass idealization of simple beam

Generalized displacements:

Sine series representation of simple beam deflection

Finite element concept

Typical finite-element beam coordinates

Equations of motion

• Newton’s second law of motion

• Direct equilibration using D'Alembert'sprinciple (dynamic equilibrium)

• Principle of virtual work

• Hamilton’s principle

fI = mü, fD = cú, fS = ku

Newton’s second law of motion:

p(t) - fD - fS = mass x acceleration

p(t) - fD - fS = mü

p(t) - cú - ku = mü

mü + cú + ku = p(t)

For linearly elastic system

Direct equilibration using D'Alembert's principle:

There is an inertial force fD acting

opposite to direction of motion

Sum of all forces is equal to zero

+ p(t) - fD - fS - fI = 0

mü + cú + ku = p(t)

For linearly elastic system

Method of virtual work:

-fS·δu- fI·δu- fD·δu + p(t) ·δu = 0

For linearly elastic system

[-fS - fI - fD + p(t)]·δu = 0

mü + cú + ku = p(t)