lecture 1 structural dynamics
TRANSCRIPT
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)