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### CE 3202 STRUCTURAL ENGINEERING

Module Reader: Mr. Noor M S Hasan

Civil Structural Dynamics broadly covers:

- Blast and ImpactEngineering

- VibrationEngineering

- Building vibration due to external ground borne vibration
- Typically dealt via vibration isolation of the whole building or of the machinery
- Vibration due to human-induced excitation

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

- 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.

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

Response

Excitation

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

- 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

- 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

- 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

- 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

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

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

Dynamic modelling of structures

- Lumped mass model (discrete model)

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

Dynamic modelling of structures

- Finite element (FE) model

- Combination of the discrete and continuous model
- The structure is divided into elements, which are connected at discrete points called nodes
- The nodes are allowed to displace in a prescribed manner to represent the motion of the structure
- The sum of the displacements (translations and rotations) represents the total number of DOF for the system
- The mass of the system is concentrated within each element

Comparison between static degree of freedom

and dynamic degree of freedom

- The number of dynamic coordinates can be maximum equal to the number of static degrees of freedom of the system

6 static DOF

(3 for each node)

3 static DOF

(axial def neglected)

An infinity of static DOF

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

Comparison between static degree of freedom

and dynamic degree of freedom

- From dynamic point of view, the system can have:

1 DOF (SDOF)

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

- 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

Undamped vibration

Damped vibration

- 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

Periodic and nonperiodic vibration

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

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

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

Components of a vibration system

- 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

- Mass relates force to acceleration
- Mass is assumed to behave as rigid body (does not deform)
- The 2nd 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

- 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

- Equivalent stiffness of springs in series

- Equivalent stiffness of springs in parallel

- Determine the equivalent spring stiffness of the following dynamic systems

Spring are in parallel:

Spring are in parallel:

- 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

Introduction

Features of a dynamic analysis

Dynamic analysis procedure

Dynamic modelling of structures

Vibrations

Components of a vibration system

SUMMARY

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