Download Section 5.1 - damped motion

Section 5.1 ­ damped motion
MTH 225 Differential Equations
Notes 5.1 Linear Models: Initial­Value Problems ­ Part II
March 23, 2011
P. Seeburger
Part II: Spring/Mass Systems: Free Damped Motion
In mechanics, friction and other damping forces acting on a body are typically proportional to some power of its velocity. Here we will assume that the damping force is proportional to dx/dt.
So, taking the equation we developed earlier in this section and adding this damping force, we get the equation:
Where β is a positive damping constant. This term is negative because the damping force acts in a direction _________________________ the motion.
Dividing by the mass m, we obtain the following DE of the free damped motion:
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Section 5.1 ­ damped motion
March 23, 2011
Case 1: λ2 ­− ω2 < 0 (β2 < 4mk or _______________) In this case the system is said to be
_______________________ because the damping constant β is small when compared to mass m and the spring constant k. The roots of the characteristic equation are complex. They are:
The corresponding solution is:
Typical graphs will appear similar to the following example.
The motion _______________________________ but as t → ∞, the amplitudes of vibration → _____.
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Section 5.1 ­ damped motion
March 23, 2011
Case 2: λ2 ­− ω2 > 0 (β2 > 4mk or _______________) In this case the system is said to be
_______________________ because the damping constant β is large when compared to mass m and the spring constant k. The corresponding solution is:
This equation represents a smooth and ____________________________ motion. Below are three possible graphs of x(t).
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Section 5.1 ­ damped motion
March 23, 2011
Case 3: λ2 ­− ω2 = 0 (β2 = 4mk or _______________) In this case the system is said to be
________________________ _________________ because even the slightest increase in (β) the damping force would cause the motion to oscillate. The corresponding solution is:
Note that this equation tells us that the mass can pass through the equilibrium position at most ________.
Typical graphs of critically damped systems appear very similar to those of overdamped systems. Here are two examples:
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Section 5.1 ­ damped motion
March 23, 2011
Ex. 1: a. A mass weighing 64 pounds stretches a spring 5.12 feet. Assuming a damping force that is 10 times the velocity of the mass, determine the equation of the motion, if the mass is initially released from a point 1 foot below the equilibrium position with a upward velocity of 5 ft/s.
b. Determine when the object passes through the equilibrium position (if at all).
c. Determine when the object reaches a maximum displacement above the equilibrium position.
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Section 5.1 ­ damped motion
March 23, 2011
Ex. 2: An object with a mass of 4 slugs is attached to a 5­ft spring stretching it to a length of 9 ft. If the object is initially released from rest at a point 1.5 feet above the equilibrium position, find the equation of motion if it is also known that the surrounding medium resists the motion of the object with a resistance equal to twice the instantaneous velocity of the object.
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Section 5.1 ­ damped motion
March 23, 2011
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Section 5.1 ­ damped motion
March 23, 2011
Similar to what was done for undamped motion, we can express any undamped solution using a single sine function as follows:
The coefficient is sometimes called the ________________ ______________________ of the vibrations. Although this function is not truly periodic:
The quasi period of this function is:
(The interval between successive maxima of x(t))
And the quasi frequency is:
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