Advanced Engineering
Mathematics I
Chapter 2. Second-Order ODEs
2.4 Modeling of Free Oscillations of a Mass-Spring System
2.4 Modeling of Free Oscillations of a Mass-Spring System
▪ ODE of the Undamped System
• An ordinary coil spring that resists extension as well as compression
• Direction and sign
• Downward force: positive
• Upward force: negative
Hooke’s law
k: spring constant
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2.4 Modeling of Free Oscillations of a Mass-Spring System
▪ ODE of the Undamped System
k: spring constant
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2.4 Modeling of Free Oscillations of a Mass-Spring System
▪ ODE of the Undamped System
k: spring constant
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2.4 Example 1: Harmonic oscillation of an undamped system
▪
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2.4 Modeling of Free Oscillations of a Mass-Spring System
▪ ODE of the Damped System
c: damping constant
k: spring constant
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2.4 Modeling of Free Oscillations of a Mass-Spring System
▪ Case I. Overdamping
c: damping constant
k: spring constant
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2.4 Modeling of Free Oscillations of a Mass-Spring System
▪ Case II. Critical Damping
c: damping constant
k: spring constant
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2.4 Modeling of Free Oscillations of a Mass-Spring System
▪ Case III. Underdamping
c: damping constant
k: spring constant
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2.4 Example 2: The three cases of damped motion
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