This topic is about Free Oscillation. Spring-Mass system is an application of Simple Harmonic Motion (SHM). This topic is Depend on the Ordinary Differential Equation.

1. INFORMATION AND TECHNOLOGY Branch Code : 016
Advanced Engineering Mathematics Subject code : 2130002
Presentation on
Modeling-Free oscillation(Mass spring system)
By Divya S. Modi

2. Spring- Mass System
 A mass m attached to a spring of spring
constant k exhibits simple harmonic motion in closed
space. The equation shows that the
period of oscillation is independent of both the
amplitude and gravitational acceleration. The above
equation is also valid in the case when a constant force
is being applied on the mass, i.e. a constant force can
not change the period of oscillation.
What is spring-Mass System ?

3. INDEX
Setting up the Model
Undamped System
Damped System
References

4. 1. Setting up the Model
What is a spring-mass system and why it is important?
(Hooke’s Law)
W = Gravitational force
Fs = Spring Force
g = Gravitational acceleration
k = Spring constant

5. 1. Setting up the Model
Dynamic problem : What is motion of the mass when acted by an external
force or is initially displaced?

6. 1. Setting up the Model
Forces acting on the mass
Net Force acting on the mass

7. 1. Setting up the Model
Newton’s Second Law of Motion
 the acceleration of an object due to an applied force is in the direction of the
force and given by:
For our spring-mass system

8. 2. Undamped System
My’’(t) + cy’(t) + ky(t) = F(t)
no damping no external force
Particular Solution
General Solution

9. 2. Undamped System

10. 2. Undamped System

11. 3. Damped System
no external force
Assume an exponential solution
Then
and substituting in equation above, we have
(characteristic equation)

12. 3. Damped System
Solutions to characteristic equation:
overdamped
critically damped
underdamped

13. 3. Damped System
The most interesting case is underdamping, i.e:

14. 3. Damped System

15. 4. References

16. Thank you