11Introduction to dynamics

Abstract: The effects of vibration are discussed. Different types of dynamicloads that are encountered in nature are listed. The differences between staticand dynamic problem are illustrated. Modelling of structures subjected todynamic loads is discussed.

Key words: periodic motion, stochastic, natural frequency, finite elementmethod, resonance.

1.1 Introduction

This text is concerned with the analysis of structures subjected to dynamicloads. Dynamics in this context means time varying. Both application ofload and removal of load necessarily vary with time. Hence, the internalstresses and the resulting deflections are also time dependent or dynamic innature.

In the real world, no loads that are applied to a structure are truly static.All bodies possessing mass and elasticity are capable of vibration. Thusmost engineering machines and structures experience vibration to some degreeand their design generally requires consideration of oscillatory behaviour.

The effects of vibration are very common in our daily life. We live on thebeating of our hearts. Planetary motion is also another example of vibration.These motions are called periodic motions (periodic motion is a motion thatrepeats itself regularly after a certain interval of time). This interval of timeis known as the period of the system or motion. In general, vibration hasboth good and bad effects. In civil engineering, the good effect of vibrationis harnessed by the compaction of fresh concrete. The bad effects of vibrationson a structure are those produced by natural forces such as wind gusts, andearthquakes and by mechanical forces on a bridge.

Oscillatory systems can be broadly characterized as linear or nonlinear. Ingeneral, for linear systems, the principle of superposition is valid, and themathematical techniques available are well developed. Techniques for theanalysis of nonlinear systems are less well known and difficult to apply,since changes in stiffness and damping characteristics occur during nonlinearinelastic response.

In general, vibration can be classified as free vibration and forced vibration.Free vibration takes place when a system oscillates under the actions offorces inherent in the system itself and external forces are absent. The system

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under free vibration will vibrate at one or more of its natural frequencies,which are properties of the dynamics system.

Vibration that takes place under the excitation of external forces is calledforced vibration. If the frequency of excitation coincides with one of thenatural frequencies of the system, a condition of resonance occurs, resultingin large oscillations in structures, which ultimately results in the failure ofthe structure.

1.2 Different types of dynamic loads

Dynamic loads may be classified as deterministic and non-deterministic.If the magnitude, point of application of the load and the variation of the loadwith respect to time are known, the loading is said to be deterministic and theanalysis of a system to such loads is defined as deterministic analysis. On theother hand, if the variation of load with respect to time is not known, theloading is referred to as random or stochastic loading and the correspondinganalysis is termed as non-deterministic analysis.

Dynamic loads may also be classified as periodic and non-periodic loadings.When a loading repeats itself at equal time intervals then it is called periodicloading. A single form of periodic loading is either a sine or cosine functionas shown in Fig. 1.1a. A vibration induced due to rotating mass is a periodicmotion. This type of periodic loading is called simple harmonic motion asshown in Fig. 1.1a. The type of loading shown in Fig. 1.1b is a periodicloading but non-harmonic. Later on we will see that most periodic loads canbe represented by summing sufficient number of harmonic terms in a Fourierseries. Any loading which does not come under the category of periodicloading is termed as non-periodic. Blast loading shown in Fig. 1.1c andearthquake ground motion as shown in Fig. 1.1d are the examples of non-periodic loads.

1.3 Difference between dynamic and staticproblems

In two aspects, a dynamic problem is different from a static problem. Thefirst and most obvious difference is the time-varying load and the responseis also time varying. This needs analysis over a specific interval of time.Hence dynamic analysis is complex and computationally extensive andexpensive compared with static analysis. The other difference between dynamicand static problems is the major occurrence of inertia forces when the loadingis dynamically applied. Consider a water tank as shown in Fig. 1.2a subjectedto load F at the top. The resulting deflection, shear force and bending momentcan be calculated on the basis of static structural analysis principles. On theother hand, if the time-varying load F(t) is applied at the top, the structure is

Introduction to dynamics 3

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1.1 Different types of dynamic loads: (a) simple harmonic; (b) non-harmonic (periodic); (c) non-periodic (short duration); (d) non-periodic (long duration).

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set to motion or vibration and experiences accelerations. According to Newtonssecond law, inertia force is proportional to acceleration. Inertia forces areproportional to the mass and they develop in the structure that resists theseaccelerations. Depending on the contribution made by inertia force to shearand bending moment will determine whether dynamic analysis is warranted.

1.4 Methodology

A physical problem based on certain assumptions is idealized into a mechanicalmodel by correctly defining in terms of geometry, kinematics and loadingand boundary conditions. The governing differential equations are solved toobtain the dynamic response and the results are interpreted in a meaningfulmanner. For complex structures, it may be necessary to refine the analysis byconsidering a more detailed mechanical model. In a mechanical model, infinitedegrees of freedom are converted to finite degrees of freedom, and for eachdegree of freedom, exhibiting the structure, there exists a natural frequencyat which the structure vibrates in a particular mode of vibration. A mechanicalmodel can also be categorized as either continuous or discrete. Consider achimney as shown in Fig. 1.3a. In a real structure, the structure manifestsdistributed mass and stiffness characteristics along the height. The continuousmodel is shown in Fig. 1.3b. Hence the mathematical continuous modelincorporates distributed mass and distributed stiffness to arrive at the responseof the system. The displacement v is a function of space and time. On theother hand, we can place the whole mass at the top of the chimney as shownFig. 1.3c and can consider single-degree-of-freedom (SDOF) in which entiremass m of the structure is localized (lumped) at the top of the structure hasconstant stiffness k. The independent displacement V(t) is a function of timealone. The lumped mass representation shown in Fig. 1.3d is a three-degrees-

F F(t)

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1.2 Water tank subjected to static and dynamic loads: (a) static load;(b) dynamic load.

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of-freedom system in which each localized mass m1, m2, m3 has its owndisplacements V1(t), V2(t), V3(t) respectively. Practical dynamic analysis oflarge complicated multiple-degrees-of-freedom (MDOF) is generallyaccomplished by computer-implemented numerical analysis techniques suchas the finite element method.

1.5 Types of vibration

Vibration can occur in a structure by imposing initial conditions, whichgenerally manifest themselves as energy input. If the initial input is impartingvelocity to the system, kinetic energy is produced. If the displacement isimparted to the system, potential energy is produced. If the structural vibrationoccurs in the absence of external loads, it is termed free vibration. Freevibration usually occurs at the fundamental natural frequency. Owing todamping present in the system, the free vibration eventually dampens out. Ifthe vibration takes place under the excitation of external force, it is calledforced vibration. If the source of vibration is periodic, the resulting vibrationmay constitute both steady state and transient. Steady state response transpiresat the frequency of excitation. The transient response is due to the initialenergy stored in the structure. A transient response may also occur if thestructure is subjected to blast loads.

X

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1.3 Different mechanical models of a chimney: (a) physicalrepresentation; (b) continuum method; (c) SDOF discrete method;(d) 3-DOF discrete method.

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1.6 Further reading

Biggs J M (1964) Introduction to Structural Dynamics, McGraw-Hill, New York.Chopra A K (2002) Dynamics of Structures Theory and applications to earthquake

engineering, Eastern Economy Edition, Prentice Hall of India, New Delhi.Clough R W and Penzien J (1974) Dynamics of Structures, McGraw-Hill, New York.DenHartog J P (1956) Mechanical Vibrations, 4th ed., McGraw-Hill, New York.Paz M (1980) Structural Dynamics, Theory and Computation, Van Nostrand Reinhold,

New York.Rao S S (2003) Mechanical Vibrations, 4th ed., Prentice Hall, Inc., Englewood Clifps,

NT.Thompson W T (1981) Theory of Vibration with Applications, 2nd ed., Prentice Hall,

Englewood Cliffs, NJ.Wilson E L (2002) Three Dimensional Static and Dynamic Analysis of Structures, Computers

and Structures, Inc., Berkeley, CA.