In this chapter, we discuss harmonic oscillation in systems with only one degree of freedom. Determine the ratio of successive amplitude if the amount of damping is a double b halve exercise. The motion takes the form of a nonoscillatory or oscillatory decay. In particular, the response of a single degree offreedom system subjected to a base excitation or direct excitation such as that shown in figure 2. We analyzed vibration of several conservative systems in the preceding section. Undamped systems and systems having viscous damping and structural damping are included. Then, newtons second law of motion for the translational part of motion is given by. Dynamics of simple oscillators single degree of freedom systems cee 541. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Free vibration of single degree of freedom sdof chapter 2 2. Mechanical vibration palm solutions manual the solutions. Vibration of single degree of freedom systems in this chapter, some of the basic concepts of vibration analysis for single degree of freedom sdof discrete parameter systems will be introduced. A singledegreeoffreedom system consists of a mass of 20 kg and a spring of stiffness 4,000 nm. Chapter iii harmonic excitation of singledegreeoffreedom.
View notes chapter 2 free vibration of single degree of freedom from mae 3400 at delaware technical community college. The term free vibration is used to indicate that there is no external force causing the motion. Free vibration means that no time varying external forces act on the system. Free vibration of singledegree of freedom systems systems are said to undergo free vibration when they oscillate about their static equilibrium position when. If the mass m is displaced from its equilibrium position and then allowed to vibrate free from further external forces, it is said to have free vibration. Chapter 03 free vibration of single degree of freedom systems contents introduction undamped free vibrations in. State the necessary assumptions to reduce this problem to a one degree offreedom oscillator. Single degree of freedom sdof system m k ft ut figure 1. For a single time history record, the period is t and the bandwidth b is the reciprocal so that the bt product is unity, which is equal to 2 statistical degrees of freedom from the definition in equation 6. Recall that a system is conservative if energy is conserved, i. Consider an undamped system with two degrees of freedom as shown in figure 6. In each case, we found that if the system was set in motion, it continued to move indefinitely.
Equivalent singledegreeoffreedom system and free vibration 7 vc f1 c f2 f3 1 2 3 x y. Free vibration of singledegree of freedom systems systems are said to undergo free vibration when they oscillate about their static equilibrium position when displaced from those positions and then released. Introduction a system is said to undergo free vibration when it oscillates only under an initial disturbance with no external forces acting after the initial disturbance 3. Unit 6 vibrations of two degree of freedom systems. The motion is primarily the result of initial conditions, such as an initial displacement of the mass element of the system from an equilibrium position andor an initial velocity. Free vibration of single degree of freedom sdof chapter 2 introduction a. Undamped sdof system its acceleration and opposing its motion. Free vibration of single degree of freedom systems. Me 563 mechanical vibrations fall 2010 15 of motion that adequately describe the systems. This chapter introduces some of the basic concepts of vibration analysis for multiple degree of freedom mdof discrete parameter systems, since there are many significant differences to single degree of freedom sdof systems. Chapter 2 free vibration of single degree of freedom. Vibrations in free and forced single degree of freedom. Describes free vibration, the ode, natural frequency, and natural period.
Example of overhead water tank that can be modeled as sdof system 1. Simple vibration problems with matlab and some help. Unit 22 mit opencourseware free online course materials. Free vibration of single degree of freedom sdof chapter 2 introduction a system is said to undergo free vibration when it oscillates only under an initial disturbance with no external forces acting after the initial disturbance. The amplitudes of successive cycles are found to be 50, 45, 40, 35. The mathematical models that govern the free vibration of single degree of freedom systems can be described in terms of homogeneous secondorder ordinary differential equations that contain displacement, velocity, and acceleration terms. In this chapter the free vibration of undamped and damped single degree of freedom systems is discussed.
Single degree offreedom systems are treated at length in chapters 3 to 6. Start of chapter 2 for dynamics, noise and vibration module code ufmeaw203 at uwe bristol. Chapter 2 free vibration of single degree of freedom 1. Dynamics of simple oscillators single degree of freedom.
Forced vibration of singledegreeoffreedom sdof systems. This video is an introduction to undamped free vibration of single degree of freedom systems. Thus, first deal wit h free vibration do this by again setting forces to zero. Analysis, measurement, design, and control of a single degree offreedom system often abbreviated sdof is discussed. Chapter 9 multi degree offreedom systems equations of motion, problem statement, and solution methods twostory shear building a shear building is the building whose floor systems are rigid in flexure and several factors are neglected, for example, axial deformation of beams and columns. Chapter 2 free vibration of single degree of freedom systems chapter 3 harmonically excited vibration chapter 4 vibration under general forcing conditions chapter 5 two degree of freedom systems chapter 6 multidegree of freedom systems. Information included in this chapter, as a part of the second year subject mechanics 1. Blake introduction this chapter presents the theory of free and forced steadystate vibration of single degree offreedom systems. The solution of this problem for single degree of freedom systems has been obtained by lewis and by ellington and mccallion for mechanical vibrations and by hok for the equivalent electrical case. Let x c and y c be x and y coordinates of the center of mass c with respect to the. Gavin fall, 2018 this document describes free and forced dynamic responses of simple oscillators somtimes called single degree of freedom sdof systems. In such cases, the oscillation is said to be free damped vibration.
The term discrete or sometimes lumped parameter implies that the system is a combination of discrete rigid masses or components. Determination of natural frequencies and mode shapes. A free body analysis of this system in the framework of newtons second law, as performed in chapter 2 of the textbook, results in. Equivalent single degree offreedom system and free vibration 7 vc f1 c f2 f3 1 2 3 x y. First, we will explain what is meant by the title of this section. Introduction to undamped free vibration of sdof 12. Rao 5th ed the ratio of successive amplitudes of a viscously damped singledegreeoffreedom system is found to be 18. The equation of motion for the free vibration of an undamped single degree of freedom system can be rewritten as. Determine the nature and magnitude of the damping force and the frequency of the damped vibration.
Structural dynamics department of civil and environmental engineering duke university henri p. Response of single degree offreedom systems to initial conditions. Chapter 2 sdof undamped oscillation the simplest form of vibration that we can study is the single degree of freedom system without damping or external forcing. Inertia force which work to eliminate the acceleration of. The concepts developed in this chapter constitute an. We discuss linearity in more detail, arguing that it is the generic situation for small. A given time history is thus worth 2 degrees of freedoms, which is poor accuracy per chi.
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