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Introduction to Mechanical Vibration Introduction to Mechanical Vibration lec4

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manojkumarg1990

Introduction to Mechanical Vibration

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FUNDAMENTAL OF VIBRATIONS VIBRATION ANALYSIS PROCEDURE DR ANIL KUMAR DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING 1
CONTENTS VIBRATION ANALYSIS OF A PHYSICAL SYSTEM STEPS OF VIBRATION ANALYSIS ENERGY STORING (SPRING OR STIFFNESS) ELEMENT ENERGY DISSIPATING (DAMPING) ELEMENT INERTIA (MASS) ELEMENT 2
VIBRATION ANALYSIS OF A PHYSICAL SYSTEM A vibratory system is a dynamic one as the variables such as the excitations (inputs) and responses (outputs) are time dependent To understand the behaviour of a physical system we model it mathematically Usually, we make a discrete model representing it in terms of its basic elements Mass, Stiffness and Damping 3
STEPS OF VIBRATION ANALYSIS 1. Mathematical modelling 2. Derivation of the governing equations 3. Solution of the equations 4. Interpretation of the results 4
1. MATHEMATICAL MODELLING Represent all the important features of the system Include enough details to allow describing the system in terms of equations The mathematical model may be linear or nonlinear, depending on the behavior of the systems components Linear models permit quick solutions and are simple to handle However, nonlinear models sometimes reveal certain characteristics of the system that cannot be predicted using linear models Sometimes the mathematical model is gradually improved to obtain more accurate results 5
MATHEMATICAL MODELLING In this approach, first a very crude or elementary model is used to get a quick insight into the overall behavior of the system. Subsequently, the model is refined by including more components and/or details so that the behavior of the system can be observed more closely. 6 Modelling of a forging hammer
2. DERIVATION OF GOVERNING EQUATIONS Use the principles of dynamics and derive the equations that describe the vibration of the system Draw the free-body diagrams of all the masses involved The free-body diagram of a mass can be obtained by isolating the mass and indicating all externally applied forces, the reactive forces, and the inertia forces The equations of motion of a vibrating system are usually in the form of a set of ordinary differential equations for a discrete system and partial differential equations for a continuous system. Newtons second law of motion, DAlembert s principle, and the principle of conservation of energy 7
3. SOLUTION OF THE GOVERNING EQUATIONS Solve the equations to find the response of the vibrating system standard methods of solving differential equations, Laplace transform methods, matrix methods, and numerical methods Numerical methods involving computers can be used to solve the equations 8
4. INTERPRETATION OF THE RESULTS The solution of the governing equations gives the displacements, velocities, and accelerations of the various masses of the vibrating system These results must be interpreted with a clear view of the purpose of the analysis and the possible design implications of the results 9
MATHEMATICAL MODEL OF A MOTORCYCLE WITH A RIDER Develop a sequence of three mathematical models of the system for investigating vibration in the vertical direction. Consider the elasticity of the tires, elasticity and damping of the struts (in the vertical direction), masses of the wheels, and elasticity, damping, and mass of the rider. 10
11 keq: Stiffness of tire, strut and rider ceq: Damping of struts and rider meq: mass of wheels, vehicle body and rider
ENERGY STORING (SPRING OR STIFFNESS) ELEMENT A spring is a type of mechanical link, which in most applications is assumed to have negligible mass and damping In fact, any elastic or deformable body or member, such as a cable, bar, beam, shaft or plate, can be considered as a spring A spring is said to be linear if the elongation or reduction in length x is related to the applied force F as, F=kx k : spring constant or spring stiffness or spring rate The spring constant k is always positive and denotes the force (positive or negative) required to cause a unit deflection (elongation or reduction in length) in the spring 12
ENERGY STORED IN SPRING ELEMENTS The work done (U) in deforming a spring is stored as strain or potential energy in the spring, and it is given by, U= kx2/2 13
EXAMPLES Stiffness of a rod Stiffness of a cantilever beam 14
SPRING COMBINATIONS Case 1: Springs in series Case 2: Springs in parallel 15 = 1 + 2 1 = 1 1 + 1 2
ENERGY DISSIPATING (DAMPING) ELEMENT In many practical systems, the vibrational energy is gradually converted to heat or sound. Due to the reduction in the energy, the response, such as the displacement of the system, gradually decreases. The mechanism by which the vibrational energy is gradually converted into heat or sound is known as damping. A damper is assumed to have neither mass nor elasticity, and damping force exists only if there is relative velocity between the two ends of the damper. 16
VISCOUS DAMPING Viscous damping is caused by such energy losses as occur in liquid lubrication between moving parts or in a fluid forced through a small opening by a piston, as in automobile shock absorbers. The viscous-damping force is directly proportional to the relative velocity between the two ends of the damping device. 17 F = C v F= damping force C= damping constant v= relative velocity
COMBINATION OF DAMPERS Case 1: Dampers in parallel Case 2: Dampers in series 18 = 1 + 2 1 = 1 1 + 1 2
MASS OR INERTIA ELEMENT The mass or inertia element is assumed to be a rigid body; it can gain or lose kinetic energy whenever the velocity of the body changes. From Newtons second law of motion, the product of the mass and its acceleration is equal to the force applied to the mass. Work is equal to the force multiplied by the displacement in the direction of the force, and the work done on a mass is stored in the form of the masss kinetic energy. 19

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Introduction to Mechanical Vibration Introduction to Mechanical Vibration lec4

  • 1. FUNDAMENTAL OF VIBRATIONS VIBRATION ANALYSIS PROCEDURE DR ANIL KUMAR DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING 1
  • 2. CONTENTS VIBRATION ANALYSIS OF A PHYSICAL SYSTEM STEPS OF VIBRATION ANALYSIS ENERGY STORING (SPRING OR STIFFNESS) ELEMENT ENERGY DISSIPATING (DAMPING) ELEMENT INERTIA (MASS) ELEMENT 2
  • 3. VIBRATION ANALYSIS OF A PHYSICAL SYSTEM A vibratory system is a dynamic one as the variables such as the excitations (inputs) and responses (outputs) are time dependent To understand the behaviour of a physical system we model it mathematically Usually, we make a discrete model representing it in terms of its basic elements Mass, Stiffness and Damping 3
  • 4. STEPS OF VIBRATION ANALYSIS 1. Mathematical modelling 2. Derivation of the governing equations 3. Solution of the equations 4. Interpretation of the results 4
  • 5. 1. MATHEMATICAL MODELLING Represent all the important features of the system Include enough details to allow describing the system in terms of equations The mathematical model may be linear or nonlinear, depending on the behavior of the systems components Linear models permit quick solutions and are simple to handle However, nonlinear models sometimes reveal certain characteristics of the system that cannot be predicted using linear models Sometimes the mathematical model is gradually improved to obtain more accurate results 5
  • 6. MATHEMATICAL MODELLING In this approach, first a very crude or elementary model is used to get a quick insight into the overall behavior of the system. Subsequently, the model is refined by including more components and/or details so that the behavior of the system can be observed more closely. 6 Modelling of a forging hammer
  • 7. 2. DERIVATION OF GOVERNING EQUATIONS Use the principles of dynamics and derive the equations that describe the vibration of the system Draw the free-body diagrams of all the masses involved The free-body diagram of a mass can be obtained by isolating the mass and indicating all externally applied forces, the reactive forces, and the inertia forces The equations of motion of a vibrating system are usually in the form of a set of ordinary differential equations for a discrete system and partial differential equations for a continuous system. Newtons second law of motion, DAlembert s principle, and the principle of conservation of energy 7
  • 8. 3. SOLUTION OF THE GOVERNING EQUATIONS Solve the equations to find the response of the vibrating system standard methods of solving differential equations, Laplace transform methods, matrix methods, and numerical methods Numerical methods involving computers can be used to solve the equations 8
  • 9. 4. INTERPRETATION OF THE RESULTS The solution of the governing equations gives the displacements, velocities, and accelerations of the various masses of the vibrating system These results must be interpreted with a clear view of the purpose of the analysis and the possible design implications of the results 9
  • 10. MATHEMATICAL MODEL OF A MOTORCYCLE WITH A RIDER Develop a sequence of three mathematical models of the system for investigating vibration in the vertical direction. Consider the elasticity of the tires, elasticity and damping of the struts (in the vertical direction), masses of the wheels, and elasticity, damping, and mass of the rider. 10
  • 11. 11 keq: Stiffness of tire, strut and rider ceq: Damping of struts and rider meq: mass of wheels, vehicle body and rider
  • 12. ENERGY STORING (SPRING OR STIFFNESS) ELEMENT A spring is a type of mechanical link, which in most applications is assumed to have negligible mass and damping In fact, any elastic or deformable body or member, such as a cable, bar, beam, shaft or plate, can be considered as a spring A spring is said to be linear if the elongation or reduction in length x is related to the applied force F as, F=kx k : spring constant or spring stiffness or spring rate The spring constant k is always positive and denotes the force (positive or negative) required to cause a unit deflection (elongation or reduction in length) in the spring 12
  • 13. ENERGY STORED IN SPRING ELEMENTS The work done (U) in deforming a spring is stored as strain or potential energy in the spring, and it is given by, U= kx2/2 13
  • 14. EXAMPLES Stiffness of a rod Stiffness of a cantilever beam 14
  • 15. SPRING COMBINATIONS Case 1: Springs in series Case 2: Springs in parallel 15 = 1 + 2 1 = 1 1 + 1 2
  • 16. ENERGY DISSIPATING (DAMPING) ELEMENT In many practical systems, the vibrational energy is gradually converted to heat or sound. Due to the reduction in the energy, the response, such as the displacement of the system, gradually decreases. The mechanism by which the vibrational energy is gradually converted into heat or sound is known as damping. A damper is assumed to have neither mass nor elasticity, and damping force exists only if there is relative velocity between the two ends of the damper. 16
  • 17. VISCOUS DAMPING Viscous damping is caused by such energy losses as occur in liquid lubrication between moving parts or in a fluid forced through a small opening by a piston, as in automobile shock absorbers. The viscous-damping force is directly proportional to the relative velocity between the two ends of the damping device. 17 F = C v F= damping force C= damping constant v= relative velocity
  • 18. COMBINATION OF DAMPERS Case 1: Dampers in parallel Case 2: Dampers in series 18 = 1 + 2 1 = 1 1 + 1 2
  • 19. MASS OR INERTIA ELEMENT The mass or inertia element is assumed to be a rigid body; it can gain or lose kinetic energy whenever the velocity of the body changes. From Newtons second law of motion, the product of the mass and its acceleration is equal to the force applied to the mass. Work is equal to the force multiplied by the displacement in the direction of the force, and the work done on a mass is stored in the form of the masss kinetic energy. 19