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Math / Differential Equations and Their Applications: An Introduction to Applied Mathematics 3 / Chapter 4.7 / Problem 11

Differential Equations and Their Applications: An Introduction to Applied Mathematics | 3rd Edition | ISBN: 9780387908069 | Authors: M. Braun

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The equation of motion of a spring-mass system with damping (see Section 2.6) is mi'+ ci

Chapter 4, Problem 11

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QUESTION:

The equation of motion of a spring-mass system with damping (see Section 2.6) is mi'+ ci + kz =O, where m, c, and k are positive numbers. Convert this equation to a system of first-order equations for x = z, y = i, and draw the phase portrait of this system. Distinguish the overdamped, critically damped, and underdamped cases.

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QUESTION:

The equation of motion of a spring-mass system with damping (see Section 2.6) is mi'+ ci + kz =O, where m, c, and k are positive numbers. Convert this equation to a system of first-order equations for x = z, y = i, and draw the phase portrait of this system. Distinguish the overdamped, critically damped, and underdamped cases.

ANSWER:

Step 1 of 4

m = mass of the body

k = stiffness of spring

c = damping coefficient

x = displacement of the body from the equilibrium position

= velocity of the body

= acceleration of the body

The system is displaced in downward direction through a distance x from the equilibrium position as shown in the figure. The force acting on the body or mass in displaced position are

i) Inertial force mx (upward)

ii) Damping force cx(upward)

iii) Spring force kx (upward)

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