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**Contents:**

The perspective to keep in focus is that the same steps that analysts must jump through to solve this simple spring-mass system are the same steps needed to solve the real problem using the computer. The only difference between the FEA approach and the hand solutions are usually the complexity of the geometry. Table 1, lists a comparison of summary results when an FEA model idealization of Fig. Table 1 shows that for flexible systems gravity effects cannot be neglected otherwise FEA models for resonant frequency calculations will not be correct. Notes: 1. When the critical buckling load is reached or exceeded then the frequency becomes zero.

Hand Calculations contains the detailed calculations and summary arithmetic. Raymond J. Roark and Warren C. William Jones , Dr. Robert E. Ensure proper resonant vibration analysis with sample problems and calculations. Upgrade enables better use of test-based data from design and simulation to validation and certification. Software includes the ability to use regular hexahedra instead of unstructured tetrahedra in regions away This issue informs readers about the teamwork involved in a variety of multiphysics projects.

Lateral Vibration of Prismatical Bars Forced Vibration of a Beam with Supported Ends Vibration of Bridges Effect of Axial Forces on Lateral Vibrations Vibration of Beams on Elastic Foundation Ritz Method Vibration of Bars of Variable Cross Section Vibration of Turbine Blades.

19. Introduction to Mechanical Vibration

Vibration of Hulls of Ships Lateral Impact of Bars Vibration of a Circular Ring ' Vibration of Membranes Vibration of Plates General 2. Frequency Measuring Instruments 3. The Measurement of Amplitudes 4. Seismic Vibrographs 5. Torsiograph 6.

Torsion Meters 7. Free Harmonic Vibrations. If an elastic system, such as a loaded beam, a twisted shaft or a deformed spring, be disturbed from its position of equilibrium by an impact or by the sudden application and removal of an additional force, the elastic forces of the member in the disturbed posi- tion will no longer be in equilibrium with the loading, and vibrations will ensue.

Generally an elastic system can perform vibrations of different modes. For instance, a string or a beam while vibrating may assume the different shapes depending on the number of nodes subdividing the length of the member. In the simplest cases the configuration of the vibrating system can be determined by one quantity only. Such systems are called systems having one degree of freedom. Let us consider the case shown in Fig.

If the arrangement be such that only vertical displacements of the weight W are possible and the mass of the spring be small in compari- son with that of the weight W, the system can be considered as having one degree of freedom. The configuration will be determined completely by the vertical displacement of the weight. By an impulse or a sudden application and removal of an external force vibrations of the system can be produced.

Such vibrations which are maintained by the elastic force in the spring alone are called free or natural vibrations. An analytical expression for these FIG.

Let k denote the load necessary to produce a unit extension of the spring. This quantity is called spring constant. If the load is measured in pounds and extension in inches the spring constant will be obtained in Ibs. The static deflection of the spring under the action of the weight W will be If, for instance, the body in its vibrating motion takes a position above the position of equilib- rium and such that a compressivc force in the spring is produced the expres- sion a becomes negative, and both terms on the right side of eq.

Thus in this case the force in the spring is added to the gravity force as it should be. Its magnitude, from eq. If the statical deflec- tion 8st is determined theoretically or experimentally the period r can be calculated from eq. The number of cycles per unit time, say per second, is called the fre- quency of vibration.

In order to determine the constants of integration Ci and C2, the initial conditions must be considered. P 7 It is seen that in this case the vibration consists of two parts; a vibration which is proportional to cos pt and depends on the initial displacement of the system and another which is proportional to sin pt and depends on the FIG. Each of these parts can be represented graphically, as shown in Figs.

The total displacement x of the oscillating weight W at any instant t is obtained by adding together the ordinates of the two curves, Fig. Another method of representing vibrations is by means of rotating vectors.

Imagine a vector OA, Fig. This velocity is called circular frequency of vibration. The projection OA of the vector on the x axis is equal to xo cos pt and represents the first term of expression 7. The same result will be obtained if, instead of vectors A and OB, we consider the vector C, equal to the geometrical sum of the previous two vectors, and take the projection of this vector on the x axis. The magni- tude of this vector, from Fig. The result of this addition is a simple harmonic motion, proportional to cos pt a , which is represented by Fig.

In such a case it may be said that the vibration, represented by the curve, Fig.

Then, from eq. The amplitude of vibration, from eq. Solve the previous problem assuming that instead of a vertical wire a helical spring is used for suspension of the load W. In what proportion will the frequency of vibration be changed if the spring has 10 coils, the other characteristics of the spring remaining the same? A load W is supported by a beam of length l t Fig.

Determine the spring constant and the frequency FIG. The statical deflection of the beam under load is -c 2 Here c is the distance of the load from the left end of the beam and El the flexural rigidity of the beam in the vertical plane. It is assumed that this plane contains one of the two principal axes of the cross section of the beam, so that vertical loads produce only vertical deflections. The effect of the mass of the beam on the frequency of vibration will be discussed later, see Art.

A load W is vertically suspended on two springs as shown in Fig. Determine the resultant spring constant and the frequency of vertical vibration of the load if the Determine the frequency of vibration of the load W if it is suspended on two equal springs as shown in Fig. Solution: In the case shown in Fig.

Substituting dgt in eq. Determine the period of horizontal vibrations of the frame, shown in Fig. The mass of the frame should be neglected in this calculation. We begin with a statical problem and determine the horizontal deflection 6 of the frame which a horizontal force H acting at the point of application of the load W will produce.

Neglecting deformations due to tension and compression in the members FIG. Assuming that the load W in Fig. The weight of the cable is to be neglected. Due to the velocity v the cage will not stop suddenly and will vibrate on the cable. Counting time from the instant of the accident, the displace- ment of the cage from the position of equilibrium at that instant is zero and its velocity is v.

From eq. It is seen that due to the sudden stoppage of the upper end of the cable the stress in the cable increased in this case about five times. The maximum dynamical deflection is 5.

Thus the It is seen that by introducing the spring a considerable reduction in the maximum stress is obtained. Torsional Vibration. The equation of motion in the case of a body rotating with respect to an immovable axis states that the moment of inertia of the body with respect to this axis multiplied with the angular acceleration is equal to the moment of the external forces acting on the body with respect to the axis of rota- tion. Proceeding as in the previous article we conclude from eq.

Substituting this in eqs.

Call number: Copyrightdate: /00/ Digitalpublicationdate: /04/ Identifier: vibrationproblemmbp. Identifier-ark. The Fifth Edition of this classic work retains the most useful portions of Timoshenkos book on vibration theory and introduces powerful, modern computational.

In general if we have a shaft consisting of portions with different diam- eters we can, without changing the spring constant of the shaft, replace any Such a case is of practical importance since an arrangement of this kind may be encountered very often in machine design. A propeller shaft with the propeller on one end and the engine on the other is an example of this kind. This cross section is called the nodal cross section, and its position will be found from the condition that both por- tions of the shaft, to the right and to the left of the nodal cross section, must have thejsame period of vibra- tion, since otherwise the condition that the masses at the ends always are rotating in opposite directions will not be fulfilled.

Applying eq. These quantities, as seen from eq. Frahm, V. At the initial instant the moment of momentum of the two discs with respect to the axis of the shaft is zero and must remain zero since the moment of external forces with respect to the same axis is zero friction forces are neglected. The equality to zero of moment of momen- tum requires that both masses rotate in opposite directions. The mass of the shaft is neglected in our present discussion and its effect on the period of vibration will be considered later, see Art.

It can be seen eq. Determine the frequency of torsional vibration of a shaft with two circular discs of uniform thickness at the ends, Fig.

From eqs. In what proportion will the frequency of vibration of the shaft considered in the previous problem increase if along a length of 64 in. The length of 64 in. Since the frequency of vibration is inversely proportional to the square root of the length of the shaft see eq.

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