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Note for Mechanical Vibrations and Structural Dynamics - MVSD By JNTU Heroes

  • Mechanical Vibrations and Structural Dynamics - MVSD
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MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS this simple case, the package and crane both oscillate as rigid bodies; the package oscillates about the end of the crane and the crane oscillates about its base point of rotation as the two exchange energy. These vibrations would most likely correspond to relatively low frequencies and would take place in addition to the gross dynamical motion of the crane and package. Two coupled ordinary differential equations would be needed in this case to model the discrete, independent motions of the crane and package. This model might be sufficient in some cases, but what if the mass of the cable is comparable to the mass of the package? In this case, the crane and package still behave like rigid bodies, but the cable will probably vibrate either transversely or longitudinally as a continuous body along its length. These higher frequency vibrations would require that both ordinary differential equations for the crane and package and partial differential equations of the cable be used to model the entire system. Furthermore, if the assumption of rigidity in the crane were also relaxed, then it too would need to be modeled with partial differential equations. All of these complications would be superimposed on top of the simple rigid body dynamics of the crane and package. Figure 1.2: Crane for loading/unloading packages from cargo ship. Different regimes of operation require different levels of sophistication in the mechanical vibration model. We will have the opportunity to discuss modeling considerations throughout the course when case studies of vibration phenomena are used to reinforce theoretical concepts and analysis procedures. Before starting to analyze systems, we must be able to derive differential equations 1-4

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MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS of motion that adequately describe the systems. There are many different methods for doing this; these are discussed in Chapter 2. 1.3 Linear superposition as a “working” principle We cannot discuss everything in this course. In particular, there is not sufficient time to present linear and nonlinear methods of vibration analysis. Therefore, the course will primarily focus on linear vibrating systems and linear approaches to analysis. Only certain special characteristics of nonlinear systems will be introduced during the semester. Because the decision has been made to talk primarily about linear systems, the principle of superposition will hold in every problem that is discussed. Instead of stating this principle at the beginning of the course, and referencing it when it is needed in proofs and derivations, we will view it more as a “working” principle. In other words, linear superposition will guide us in our analysis of free and forced linear vibrations. When we begin to analyze vibrations, we will look to the principle of superposition to help us move forward in our analyses. Recall that a mathematical operator, L[], which obeys the principle of linear superposition by definition satisfies the following two expressions: ( 1.1 ) and ( 1.2 ) where L is said to operate on the two different functions, x and y, and a and b are constants. Eq. ( 1.1 ) is the principle of homogeneity and Eq. ( 1.2 ) is the principle of additivity. These two expressions may seem trivial or obvious, but they will in fact be extraordinarily useful later in the course. The important point to remember is that linear systems, which are governed by linear operators, L[], are equal to the sum of their parts. Although this statement is profound and may even be fruit for philosophical discussions, the motivation for putting linear vibration into the context of linear superposition here is that it makes vibration analysis in free and forced systems much easier to develop and understand. More will be said about superposition in Chapter 3. 1-5

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MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS 1.4 Review of kinematics and generalized coordinate descriptions This section will review some of the fundamental techniques in particle kinematics. Note that this is only a review so no attempt is being made to cover everything here. Student should take this opportunity to refresh their memories of undergraduate courses in mechanics. Generalized coordinates are the basis for our kinematic description of vibrating bodies. Generalized coordinates are usually either position variables (e.g., x, y, z, and r), angular variables (e.g., φ, θ, and α), or a combination thereof (e.g., rcosφ). We must choose our set of generalized variables in each problem to adequately describe the position and orientation of all bodies in the mechanical system of interest. Note that the position and orientation are both important because both of these coordinates are associated with kinetic and potential energy storage. The minimum number of coordinates required is equal to the degrees-of-freedom (DOFs). Sometimes the number of DOFs is not obvious. For example, Figure 1.3 illustrates a pendulum-cart system that has many translating and rotating inertias. The question is: How many generalized coordinates are required to locate and orient all of the inertias? Figure 1.3: Mass-spring-damper-pendulum cart system In order to locate and orient every body, it appears as if one coordinate is needed for M1, one coordinate for M2, one coordinate for M3, two coordinates for M4, and two coordinates for M5: a total of seven coordinates are needed. But all of these coordinates are not actually required because there are some constraints between them. First, the translation of M3 is equal to the 1-6

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MECHANICAL VIBRATIONS AND STRUCTURAL DYNAMICS motion of the center-of-mass of M4. Second, the translation of M4 is proportional to its rotation, x4=R4θ4. Third, the translation of M5 is equal to the translation of M2 plus a component due to the rotation of M5. With a total of three constraints, the number of DOFs is reduced as follows: Simply put, this statement implies that there should be four differential equations of motion for this system. The first step in any analytical vibrations problem should be to compute the number of DOFs. As an aside, the number of DOFs (i.e., ½ the order of the system) must also always be estimated prior to applying experimental vibration techniques. There are many different common sets of generalized coordinates in use for mechanical vibration analysis. If a position vector, r, of a particle (or center-of-mass) is written in terms of its associated generalized (possibly) curvilinear coordinates, qi , as follows: ( 1.3 ) then the differential tangent vector, dr, is given by, ( 1.4 ) where the generalized coordinate unit vectors, ek, are given in parenthesis in the second of these equations and are chosen to be orthogonal in most vibration problems, and the hkk are scale factors associated with the generalized coordinate differentials, dqk. For example, the cylindrical coordinate system in terms of the Cartesian coordinate unit vectors is given by, 1-7

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