Simple mass spring system^JDamped vibration.pptx

Simple mass spring system

1. DIFFERENTIAL EQUATION PROJECT-3 SIMPLE MASS SPRING SYSTEM, DAMPED VIBRATION SYSTEM BTECH(CSE) SECTION-E PRESENTED BY: NAME: REG NO. SARBAJEET MUDULI 230301120242 DIBYA SANKET PRADHAN 230301120243 SOUMYA RANJAN PARIDA 230301120244 BONELA PRAVEEN KUMAR 230301120245 SUBHAM MAHAPATRA 230301120246

2. CONTENTS Spring mass system. Vibration. Damped Vibration

3. WHAT IS SPRING MASS SYSTEM ? IT IS SPRING SYSTEM IN WHICH A BLOCK IS ATTACHED AT THE FREE END OF THE SPRING. Let us Consider a mass ‘M’ and Spring having spring constant ‘K’. The spring is suspended from a wall by one end. The mass is then suspended from the free end of a spring. By forming this arrangement, we can obtain a spring- mass system. K M Its application involves calculating the time period of an object which is in a simple harmonic motion.

4. SPRING MASS SYSTEM EQUATION The time period (T) of a spring-mass system is given by, Where , ‘M’ denotes mass, and ‘K’ denotes spring constant and is a measure of the stiffness of the spring. The more stiff a spring is , the more force is exerted by the spring when it is displaced by a certain amount. The spring constant is measured Newton’s/meter.

5. A simple horizontal spring mass system consists of a spring attached to a wall on one side and a mass on the other. The spring mass system in the given figure is in its relaxed state. And the mass is at its equilibrium position.

6. If the mass is pulled to the right ,the string will be stretched and exert a force on the mass directed back towards the left. If the mass is pushed to the left, the string will be stretched and exert a force on the mass directed back towards the right. when the mass is displaced, the spring will always exert a restoring force i.e F = -kx (THIS EQUATION IS HOOKE’S LAW) F = force acting on the mass in ‘N’ . k =spring constant in N/m . x = displacement in ‘M’.

7. DIFFERENTIAL EQUATION (SHM - HORIZONTAL MOTION) From Newton 2 nd law , we have F= Ma From Hooke’s law we have restoration force F=-KX (i) Acceleration ‘a’ is the 2 nd derivative of distance x. (ii) BY COMBINING EQUATION. I AND II WE GET

8. WHAT IS VIBRATION? Vibration is the periodic back and fourth motion of the particles of an elastic body or medium. Example - A weight suspended from a spring is the best example of a free vibration. DAMPED VIBRATION Damped vibrations are periodic vibrations with a continuously diminishing amplitude in the presence of a resistive force. The resistive force is usually the frictional force acting in the direction opposite to vibration. Example - Friction in a mass spring system causes damping.

9. DAMPING COEFFICIENT When a damped oscillator is subject to a damping force which is linearly dependent upon the velocity, such as viscous damping, the oscillation will have exponential decay terms which depend upon a damping coefficient. If the damping force is of the form BY DIFFERENTIAL Based on Newton's 2nd Law: Total force applied to a body= motion of the body F= ma

10. From Hooke's law We have restoration force F r = -kx.... (1) And Damping force F d = - c dx/ dt ..... (2) We know that acceleration (a) is the 2nd derivative of distance i.e ………. (3) By combining equation (1), (2) and (3) We get :-