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chapter 1

structural dynamics introduction

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chapter 1

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  1. STRUCTURAL DYNAMICS Seye Nigussie

  2. Introduction WHAT IS STRUCTURAL DYNAMICS ? • Dynamics concerned with the study of force and motion which are time dependent • Dynamic load is any load of which its magnitude, direction , and/or position varies with time. • The structural response to a dynamic load , i.e., the resulting stresses and deflection, is also time varying, or dynamic Fundamental Objective of Structural Dynamics The primary purpose of the course is to present Methods for analyzing the stresses and deflection developed in any given type of structure when it is subjected to an arbitrary dynamic loading • Two basically different approaches are available for evaluating structural re­sponse to dynamic loads:(based on how loading is defined ) 1. Deterministic Analysis and 2. Nondeterministic Analysis By Seyfe N.

  3. Dynamic Analysis Approaches Deterministic Analysis • The structural response i.e. displacement, acceleration ,velocity ,stress etc., are completely known precisely as a function of time • Requires prefect control over all the variables that influence the properties and loadings • Also known as prescribed dynamic loading Nondeterministic Analysis • The time variation of vibration is not completely known • It provides only statistical information about the response statically defined loading • Also known as random dynamic loading By Seyfe N.

  4. TYPES OF PRESCRIBED LOADINGS • Classified in two categories , ‘ Periodic ‘ and ‘ Non-Periodic “ Periodic Loadings ; • loads which exhibit the same time variation successively for the large number of cycles. • The simplest form of periodic loading is a sinusoidal variation which termed as ‘simple harmonic ‘ • e.g. hydrodynamic pressures generated by a propeller at the stern of a ship or by inertial effects in reciprocating machinery By Seyfe N.

  5. Cont.. Non-Periodic Loadings • Loadings which doesn't exhibit the same time variation successively • It may be short duration (blast or explosion ) or long duration impulsive loadings (earthquake) By Seyfe N.

  6. Comparison of static loading and dynamic loading • In static problem load is constant while in dynamic problem the load and its responses varies with respect to time • Static problem has only one response ,i.e. displacement but dynamic problem has three responses ,such as displacement, velocity and acceleration • Static problem only one solution whereas a dynamic problem has infinite number of solutions which are time dependent in nature • In static problem response can be calculated by the principle of force or static equilibrium whereas in dynamic problem the response depend not only upon the load but also upon the inertia force which oppose the acceleration By Seyfe N.

  7. Causes of dynamic effects • The most common types • Initial condition ; such as velocity and displacement produce dynamic effect in the system e.g. the lift moving up and down suddenly stopped ,the cabin start to vibrate • Applied force ; application of the external force e.g. bomb blast or wind force on the building • Support motion ; the influence of support motion e.g. earthquake By Seyfe N.

  8. Basic definitions • Mass ; dynamically ,it is the property that describe how an unrestricted body resist the application of an external force (W/g) kgs • Stiffness ;force required to produce unite deformation or elastic property that describe the level of resisting force that result when a body undergo a change in length (N/m) • Natural period ;time required to complete one cycle of free vibration (second) • Frequency ;number of cycles per unit time f=1/T • Natural frequency ; the number of frequency of free vibration • Amplitude ; the maximum displacement or deformation of a vibrating system from mean position By Seyfe N.

  9. Basic definitions • Free vibration ; vibration which persists in structure after the force causing the motion has been removed • Forced vibration ;the vibrating which maintained in a structure by steady periodic force act on structure • Fundamental mode of vibration ;the fundamental mode of vibration of a structure is the mode having the lowest natural frequency • Damping ;the resistance to the motion of vibrating body and the vibration is called damped vibration(N/m/s) • Resonance ;when the frequency of the external force is equal with one of the natural frequency of the vibrating system, the amplitude of the vibrating system become excessively large By Seyfe N.

  10. Type of vibration • Free and Forced vibration Free vibration ; vibration which persists in a structure after the force causing the motion has been removed Forced vibration ;vibration maintained in a structure by steady periodic force acting on the structure 2. Damped and undamped vibration damped vibration ;when there is no damping element Undamped vibration ;when there is damping element 3. Linear and Non-Linear vibration ; 4. Deterministic and random vibration 5. Longitudinal, transversal and torsional vibration By Seyfe N.

  11. Response of the system • If dynamic force is applied on the structure ,it produce displacement, velocity, acceleration and also develops stress, strain ,reaction etc. • A motion due to initial condition is generally known as free vibration and when the motion is due to applied force ,it is known as forced response By Seyfe N.

  12. DEGREE OF FREDOM Degree of Freedom is the number of coordinates necessary to specify the position or geometry of mass point at any instant during its vibration • All real structures possess infinite number of dynamic degree of freedom. • Depending on the independent coordinates required to describe the motion ,systems divided into three a) single degree of freedom system(SDOF system) b) Multiple degree of freedom system(MDOF system) c) Continuous system By Seyfe N.

  13. Single degree of freedom system(SDOF system) If a single coordinate is sufficient to define the position or geometry of the mass of the system at any instant of time By Seyfe N.

  14. Multiple degree of freedom system(MDOF system) If more than one independent coordinate is required to completely specify the position or geometry of different masses of the system at any instant of time Continuous system (distributed system) If the mass of a system may be considered to be distributed over its entire length, in which the mass is considered to have infinite degrees of freedom By Seyfe N.

  15. Simple Harmonic Motion (SHM) • Harmonic motion is one of the form of periodic motion • Can be represented in terms of circular sine and cosine function SHM 1.periodic 2.when displaced from the fixed point or the mean position, a restoring force acting on the particle to bring it to mean position 3.The restoring force on the particle is directly proportional to its displacement Consider the harmonic motion of type x =A sin ( + Ф ) Where x - is the displacement A - is the amplitude - is the frequency Ф – is the phase angle By Seyfe N.

  16. SHM • Displacement x =A sin ( + Ф) • The velocity and the acceleration are = velocity = = A cos ( + Ф) = acceleration = = -Asin ( + Ф ) For the maximum value of displacement , sin ( + Ф) =1 , To get the maximum value of velocity ,cos( + Ф ) =1 , To get the maximum value of acceleration , sin ( + Ф ) =1 , At mean position –velocity is max, acceleration is zero, KE is max, PE is zero At position of vibration – velocity is zero ,acceleration is maximum, KE is zero , PE is max By Seyfe N.

  17. Representation of harmonic motion in complex form vector form of x = x +yi makes an angle θ with X axis X = A cosθ +i A sin θ = A Where A is the modular or the absolute value of vector X θ = Velocity = = iAω = = iω since θ = Acceleration = =A = -A =-x Phase –is difference between tow SHM indicates how much the two motions are out of steps with each other or how much angle or how much time one is ahead of the other By Seyfe N.

  18. Consequence of vibration Vibration of structures is undesirable for a number of reasons , • Overstressing and collapse of structure, • Cracking and other damage requiring repair, • Damage to safety related equipment’, • Reduce performance of equipment or delicate apparatus, • Adverse human response, • Fatigue fracture By Seyfe N.

  19. Vibration control in the design of structures • The three steps in design of structures for dynamic loading • Identifying the dynamic loading • Analyze the response of the structure • Check the performance of the structure against specified criteria to ensure that there is no adverse consequence of the dynamic load But the dynamic analysis should not be left until the end, some method must be used to minimize its effect, such as; • Careful detailing of expansion joints, provide smooth surfaces • Increase the cross-section of floors • Provide sufficient ductility • Special vibration absorbing devices or vibration isolation methods • etc By Seyfe N.

  20. Example 1.1. A harmonic motion has a time period of 0.2 s and an amplitude of 0.4 cm Find the maximum velocity acceleration and velocity. 1.2. A harmonic motion has a maximum velocity of 6 m/s and it has a frequency of 12 cps. determine its amplitude ,its period and its maximum acceleration. By Seyfe N.

  21. The end of chapter one By Seyfe N.

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