The term discrete or sometimes lumped parameter implies that the system is a combination of discrete rigid masses or components. The concepts developed in this chapter constitute an. Equivalent single degree offreedom system and free vibration 7 vc f1 c f2 f3 1 2 3 x y. A singledegreeoffreedom system consists of a mass of 20 kg and a spring of stiffness 4,000 nm. Free vibration of singledegree of freedom systems systems are said to undergo free vibration when they oscillate about their static equilibrium position when. 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. Recall that a system is conservative if energy is conserved, i. 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. A given time history is thus worth 2 degrees of freedoms, which is poor accuracy per chi. Rao 5th ed the ratio of successive amplitudes of a viscously damped singledegreeoffreedom system is found to be 18. Dynamics of simple oscillators single degree of freedom. Example of overhead water tank that can be modeled as sdof system 1. Unit 22 mit opencourseware free online course materials.
Determine the nature and magnitude of the damping force and the frequency of the damped vibration. Me 563 mechanical vibrations fall 2010 15 of motion that adequately describe the systems. Blake introduction this chapter presents the theory of free and forced steadystate vibration of single degree offreedom systems. Describes free vibration, the ode, natural frequency, and natural period. Dynamics of simple oscillators single degree of freedom systems cee 541. First, we will explain what is meant by the title of this section. Free vibration of single degree of freedom sdof chapter 2 2.
Undamped systems and systems having viscous damping and structural damping are included. 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. Determination of natural frequencies and mode shapes. 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. Single degree of freedom sdof system m k ft ut figure 1. 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. 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. Chapter iii harmonic excitation of singledegreeoffreedom. Gavin fall, 2018 this document describes free and forced dynamic responses of simple oscillators somtimes called single degree of freedom sdof systems. Free vibration of single degree of freedom systems. Analysis, measurement, design, and control of a single degree offreedom system often abbreviated sdof is discussed. Consider an undamped system with two degrees of freedom as shown in figure 6.
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 each case, we found that if the system was set in motion, it continued to move indefinitely. 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. Information included in this chapter, as a part of the second year subject mechanics 1. Inertia force which work to eliminate the acceleration of. Unit 6 vibrations of two degree of freedom systems. Mechanical vibration palm solutions manual the solutions. We analyzed vibration of several conservative systems in the preceding section. State the necessary assumptions to reduce this problem to a one degree offreedom oscillator. Let x c and y c be x and y coordinates of the center of mass c with respect to the. Simple vibration problems with matlab and some help.
Structural dynamics department of civil and environmental engineering duke university henri p. In this chapter, we discuss harmonic oscillation in systems with only one degree of freedom. Start of chapter 2 for dynamics, noise and vibration module code ufmeaw203 at uwe bristol. We discuss linearity in more detail, arguing that it is the generic situation for small. The equation of motion for the free vibration of an undamped single degree of freedom system can be rewritten as. 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. A free body analysis of this system in the framework of newtons second law, as performed in chapter 2 of the textbook, results in. 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. Then, newtons second law of motion for the translational part of motion is given by. In this chapter the free vibration of undamped and damped single degree of freedom systems is discussed. This video is an introduction to undamped free vibration of single degree of freedom systems. Vibrations in free and forced single degree of freedom.
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. View notes chapter 2 free vibration of single degree of freedom from mae 3400 at delaware technical community college. In such cases, the oscillation is said to be free damped vibration. Undamped sdof system its acceleration and opposing its motion. Determine the ratio of successive amplitude if the amount of damping is a double b halve exercise. The amplitudes of successive cycles are found to be 50, 45, 40, 35. Introduction to undamped free vibration of sdof 12.
The motion takes the form of a nonoscillatory or oscillatory decay. Single degree offreedom systems are treated at length in chapters 3 to 6. 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. 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. Response of single degree offreedom systems to initial conditions. The term free vibration is used to indicate that there is no external force causing the motion.
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