spring–mass system
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The document is a presentation on modeling free oscillation in a mass-spring system, focusing on the principles of simple harmonic motion and the effects of external forces. It outlines the set up of the model, discussing both undamped and damped systems in relation to Hooke's law and Newton’s second law of motion. The presentation also touches on the significance of the spring-mass system in dynamic problems and includes references for further reading.
spring–mass system
- 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
- 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