Equilibrium, Energy and Rayleigh’s Method

In this slides you will gain the knowledge of different types of methods for vibrating systems and his derivations.

1. GANDHINAGAR INSTITUTE OF TECHNOLOGY MECHANICAL DEPARTMENT DYNAMICS OF MACHINE

2. CONTENT :- Introduction Determination of Natural frequency Types of Method Equilibrium Method Energy Method Rayleigh’s Method

3. INTRODUCTION :- If the external forces is removed after giving an initial displacement to the system, then the system vibrates on its own due to internal elastic forces. Such vibrations are known as free vibration . T h ere is no external artificial resistance to the vibrations then such vibrations are known as undamped free vibratio n. In most of the free vibration there is always certain amount of damping associated with the system. However damping is very small, for all practical purpose it can be neglected and the vibrations considered as undamped vibration.

4. DETERMINATION OF NATURAL FREQUENCY :- The natural frequency of any body or a system is depends upon the geometrical parameters and mass properties of the body. There are various methods to obtained the equation of vibrating system, Equilibrium method Energy method Rayleigh’s method

5. EQUILIBRIUM METHOD :- According to D’Alembert’s Principal, a body or a system which is not in static equilibrium due to acceleration it possess, can be brought to static equilibrium by introducing the inertia force on it. This Principal is used for developing the equation of the motion for vibrating the equation of the motion for vibrating system which if further used to find the natural frequency of the vibrating system.

6. Consider a spring-mass system as shown in fig ure ;

7. A Spring has a negligible mass. The forces acting on the masses are :- Inertia Force, Spring Force, Gravitational Force, According to D’Alembert’s Principal, Inertia Force + External Force] = 0 - = 0 + = 0  

8. Comparing the above equation with the fundamental equation of the simple harmonic equation. We get, + = 0 The Natural Frequency is, = , = The Time Period equation is, =  

9. ENERGY METHOD :- According to law of conservation of energy, the energy can neither be created nor be destroyed but it can be transfer from the one from of energy to another form of energy. In free damped vibration, no energy is transferred to the system or from the system, therefore total mechanical energy remains constant. The potential energy due to Gravitational potential energy Strain energy At equilibrium position the kinetic energy is maximum and the potential energy is zero and vice versa .

10. According to law of energy conservation, Total energy = Constant KE + PE = Constant Differentiating equation, = 0 Kinetic Energy (KE) = Potential Energy (PE) = Substituting equations we get, + = 0  

11. Comparing it with fundamental equation of Simple Harmonic Equation, we get, = , =  

12. RAYLEIGH’S METHOD This is the extension of the energy method, which is developed by the Lord Rayleigh. Total energy = constant + = + The subscripts 1 & 2 denotes the two different positions. Let subscripts 1 will denotes the mean position where the potential energy is zero. And subscripts 2 will denotes the extreme position where kinetic energy is zero. The above equation will be, =  

13. But at mean position the kinetic energy is maximum and at extreme position the potential energy is maximum. = According to Lord Rayleigh’s the maximum kinetic energy which is at the mean position is equal to maximum potential energy which is the extreme position. Let’s Body is moving with simple harmonic motion, therefore the displacement of the body is given by, Differentiating above equation, = = = (t = 0, at mean position)  

14. Maximum kinetic energy at mean position, = = Maximum potential energy at extreme position, PE = Comparing both equations, = The Natural Frequency is, = , = The Time Period equation is, =  

15. THANK YOU…