mechanical_vibration,_unit_2_ppt,_krishna_R [1].pptx

MECHANICALVIBRATION

1. UNIVERSITY OF VISVESVARAYA COLLEGE OF ENGINEERING K.R. Circle, Bengaluru - 560001 Department of Mechanical Engineering Presentation on Unit – 2 Presented by Name : Krishna R USN : P25UV23T069006 Branch : Machine Design , 2 nd Semester Subject : Advanced Vibrations and Acoustics Under the Guidance of Dr. H C CHITTAPPA Department of Mechanical Engineering

2. TOPICS Vibration of Membranes Introduction to Vibration Control Vibration Isolation Theory Vibration Isolation for Harmonic Excitation for Different Types of Foundations

3. VIBRATION OF MEMBRANES A membrane is a plate that is subjected to tension and has negligible bending resistance.
Thus a membrane bears the same relationship to a plate as a string bears to a beam. A drumhead is an example of a membrane. Vibration of Membranes can be classified into two types, They are : Equation of Motion Initial and Boundary Conditions

4. Equation of Motion : To derive the equation of motion of a membrane, consider the membrane to be bounded by a plane curve S in the xy -plane, as shown in Fig. 8.21. Let f (x , y, t) denote the pressure loading acting in the z direction and P the intensity of tension at a point that is equal to the product of the tensile stress and the thickness of the membrane. The magnitude of P is usually constant throughout the membrane, as in a drumhead. If we consider an elemental area dx dy , forces of magnitude P dx and P dy act on the sides parallel to the y- and x-axes, respectively, as shown in Fig. 8.21. The net forces acting along the z direction due to these forces are

5. The pressure force along the z direction is f(x, y, t) dx dy , and the inertia force is Where is the mass per unit area. The equation of motion for the forced transverse vibration of the membrane can be obtained as If the external force Eq. (8.137) gives the free-vibration equation

6. Where Equations (8.137) and (8.138) can be expressed as and Where

7. is the Laplacian operator. Initial and Boundary Conditions : Since the equation of motion, Eq. (8.137) or (8.138), involves second-order partial derivatives with respect to each of t, x, and y, we need to specify two initial conditions and four boundary conditions to find a unique solution of the problem. Usually , the displacement and velocity of the membrane at t = 0 are specified as w (x, y) and w˙ (x, y). Hence the initial conditions are given by

8. w(x, y, 0) = w (x, y) The boundary conditions are of the following types: 1. If the membrane is fixed at any point (x1, y1)on a segment of the boundary, we have w(x 1 , y 1 , t) = 0, t ≥ 0 2. If the membrane is free to deflect transversely (in the z direction) at a different point (x 2 , y 2 ) of the boundary, then the force component in the z direction must be zero. Thus

9. P ∂w/∂n (x2, y2, t) = 0, t ≥ 0 (8.151) Where ∂w/∂n represents the derivative of w with respect to a direction n normal to the boundary at the point (x 2 , y 2 ).

10. INTRODUCTION TO VIBRATION CONTROL In many practical situations, it is possible to reduce but not eliminate the dynamic forces that cause vibrations. Several methods can be used to control vibrations. Among them, the following are important: Controlling the natural frequencies of the system and avoiding resonance under external excitations. Preventing excessive response of the system, even at resonance, by introducing a damping or energy-dissipating mechanism. Reducing the transmission of the excitation forces from one part of the machine to another by the use of vibration isolators. Reducing the response of the system by the addition of an auxiliary mass neutralizer or vibration absorber .

11. VIBRATION ISOLATION THEORY Vibration isolation is a procedure by which the undesirable effects of vibration are reduced. Basically , it involves the insertion of a resilient member (or isolator) between the vibrating mass (or equipment or payload) and the source of vibration so that a reduction in the dynamic response of the system is achieved under specified conditions of vibration excitation. An isolation system is said to be active or passive depending on whether or not external power is required for the isolator to perform its function. A passive isolator consists of a resilient member (stiffness) and an energy dissipater (damping).

12. Examples of passive isolators include metal springs, cork, felt, pneumatic springs, and elastomer (rubber) springs. Figure 9.17 shows typical spring and pneumatic mounts that can be used as passive isolators.

13. Fig . 9.18 illustrates the use of passive isolators in the mounting of a high-speed punch press.

14. VIBRATION ISOLATION FOR HARMONIC EXCITATION FOR DIFFERENT TYPES OF FOUNDATIONS Vibration Isolation System with Rigid Foundation : When a machine is bolted directly to a rigid foundation or floor, the foundation will be subjected to a harmonic load due to the unbalance in the machine in addition to the static load due to the weight of the machine. Hence an elastic or resilient member is placed between the machine and the rigid foundation to reduce the force transmitted to the foundation. The system can then be idealized as a single-degree-of-freedom system, as shown in Fig. 9.20(a). The resilient member is assumed to have both elasticity and damping and is modeled as a spring k and a dashpot c, as shown in Fig. 9.20(b).

15. It is assumed that the operation of the machine gives rise to a harmonically varying force F(t) = F cos ωt . The equation of motion of the machine (of mass m) is given by

16. Since the transient solution dies out after some time, only the steady-state solution will be left. The steady-state solution of Eq. (9.88) is given by (see Eq. (3.25)) The force transmitted to the foundation through the spring and the dashpot, F t (t), is given by

17. The magnitude of the total transmitted force (F T ) is given by The transmissibility or transmission ratio of the isolator ( T f ) is defined as the ratio of the magnitude of the force transmitted to that of the exciting force:

18. where r = ω/ ω n is the frequency ratio. The variation of T f with the frequency ratio r = ω/ ω n is shown in Fig. 9.21. In order to achieve isolation, the force transmitted to the foundation needs to be less than the excitation force. It can be seen, from Fig. 9.21, that the forcing frequency has to be greater than √2 times the natural frequency of the system in order to achieve isolation of vibration. For small values of damping ratio ζ and for frequency ratio r > 1, the force transmissibility, given by Eq. (9.94), can be approximated as

19. Vibration Isolation System with Flexible Foundation : In many practical situations, the structure or foundation to which the isolator is connected moves when the machine mounted on the isolator operates. For example, in the case of a turbine supported on the hull of a ship or an aircraft engine mounted on the wing of an airplane, the area surrounding the point of support also moves with the isolator. In such cases, the system can be represented as having two degrees of freedom. In Fig. 9.28, and denote the masses of the machine and the supporting structure that moves with the isolator, respectively. The isolator is represented by a spring k, and the damping is disregarded for the sake of simplicity.

20. The equations of motion of the masses and are

21. By assuming a harmonic solution of the form Eqs . (9.108) gives The natural frequencies of the system are given by the roots of the equation The roots of Eq. (9.110) are given by

22. The value ω 1 = 0 corresponds to rigid-body motion, since the system is unconstrained. In the steady state, the amplitudes of m 1 and m 2 are governed by Eq. (9.109), whose solution yields The force transmitted to the supporting structure (F t ) is given by the amplitude of m 2 x¨ 2 :

23. The transmissibility of the isolator ( T f ) is given by

24. Where ω 2 is the natural frequency of the system given by Eq. (9.111). Equation (9.114) shows, as in the case of an isolator on a rigid base, that the force transmitted to the foundation becomes less as the natural frequency of the system ω 2 is reduced. 3. Vibration Isolation System with Partially Flexible Foundation : Figure 9.29 shows a more realistic situation in which the base of the isolator, instead of being completely rigid or completely flexible, is partially flexible. We can define the mechanical impedance of the base structure, Z(ω), as the force at frequency ω required to produce a unit displacement of the base.

25. The equations of motion are given by

26. By substituting the harmonic solution Into Eqs. (9.115) and (9.116), X 1 and X 2 can be obtained as in the previous case: The amplitude of the force transmitted is given by

27. and the transmissibility of the isolator by In practice, the mechanical impedance Z(ω) depends on the nature of the base structure. It can be found experimentally by measuring the displacement produced by a vibrator that applies a harmonic force on the base structure. In some cases for example, if an isolator is resting on a concrete raft on soil the mechanical impedance at any frequency ω can be found in terms of the spring-mass-dashpot model of the soil.

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