Topic of computational methods for mechanical engineering. Information about spring mass system. Mathematical modelling of spring mass system. free mass spring system. Damped vibration. Forced damped system. Free oscillation.

1. SPRING MASS SYSTEM
PREPARED BY:-
BHAVESH PANCHAL

2. INDEX :
1. FREE OSCILLATIONS
2. DAMPED OSCILLATIONS
3. FORCED OSCILLATION
1. UNDAMPED FORCED OSCILLATIONS
2. DAMPED FORCED OSCILLATIONS

3. FREE OSCILLATIONS :

4. By Newton’s law
Mass X Acceleration = Force
𝑚𝑦′′ = Force
𝑦′′
=
𝑑2𝑦
𝑑𝑥2
Where,
y(t) = Displacement of the body and t is time
We take,
Downward direction is POSITIVE Direction
Upward direction is NEGATIVE Direction

5. When we attach the mass to spring, spring stretches by an amount x.
This causes an upward force in the spring.
By Hooke’s law,
F = -kx
Where, k is called spring constant.
The extension x is such that F balances the weight W=mg of body.
Consequently,
F + W = -kx + mg = 0
This force will not affect the motion. Spring and body are against the Static equilibrium of the system.
From the position y = 0 we pull the body downward. This further stretches the spring by some amount y > 0.
By Hooke’s law upward force F in the spring
F = -ky

8. DAMPED OSCILLATIONS :

11. THREE CASES OF SYSTEM :

12. CASE 1 : OVERDAMPING

13. CASE 2 : CRITICAL DAMPING

14. CASE 3 : UNDERDAMPING