Download Section 5.1 Free Undamped Motion

Section 5.1 Free Undamped Motion
Mar. 21­22, 2011
Spring­Mass Demo ­ Kilner
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MTH 225 Differential Equations
Notes 5.1 Linear Models: Initial­Value Problems
­ Part I
P. Seeburger
In this section we will study two related applications of second­order linear DEs:
Spring/Mass Systems
and
LRC Series Circuits
There are three types of Spring/Mass Systems we will consider:
1. Free ____________________________________ Motion
2. Free ____________________________________ Motion
3. ___________________________ Motion
Part I: Spring/Mass Systems: Free Undamped Motion
By Hooke’s Law, the restoring force exerted by a spring when it is stretched or compressed is proportional to the distance x that it is stretched or compressed. That is:
In this context, the constant of proportionality k is called the __________________ constant.
Ex. If a mass weighing 20 pounds stretches a spring 2 feet, then find k.
When a mass is attached to a spring, it will stretch the spring a distance s feet
where it reaches an equilibrium position. At this point, the weight W is
balanced by the restoring force ks. Remember that weight is defined
by W = mg.
Then: If the spring is then displaced by a distance x from its equilibrium position, the restoring force of the spring will be:
If I release the mass at this position x units below the equilibrium position, what will the restoring force cause to happen next? Will x change as time passes?
So we see that x is ­­­­­­­­­­­­­______________________________________________. In fact, the velocity of the mass can be represented by: , and the acceleration of the mass is:
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Section 5.1 Free Undamped Motion
Mar. 21­22, 2011
Assuming no other forces are present in the system except gravity and the spring
itself (free motion), we can use Newton’s 2nd Law ( F = ma ) to write:
Note: This textbook considers displacements below the equilibrium position to be positive
and those above the equilibrium position to be negative.
Dividing our DE for this spring motion through by m, we obtain:
This equation describes simple __________________________ motion or free undamped motion.
The initial conditions would be x(0) = x0 (the initial ______________________________ of the mass) and x′(0) = v0 (the initial _____________________________ of the mass). If x(0) = 2 ft and x′(0) = 0 ft/s, the mass was released from rest (v0 = 0) at a position 2 ft below the equilibrium position.
If x(0) = ­1 ft and x′(0) = ½ ft/s, the mass was released from a position 1 ft above the equilibrium position with an initial velocity of ½ ft/s.
Solving the second form of the DE above:
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Section 5.1 Free Undamped Motion
Mar. 21­22, 2011
The period of the motion described by this curve is:
(the number of seconds for the mass to complete one cycle of motion)
The frequency of the motion is:
(the number of cycles completed each second)
The number ω = is called the circular frequency
of the system.
(Circular frequency is measured in radians per second.)
Note: Before we get to the examples, it is useful to remember that
possible units of mass are slugs (lbs * s2/ft), kilograms, or grams
and g = 32 ft/s2, 9.8 m/s2, or 980 cm/s2, depending on the units
of the problem.
Examples: a. If a rock weighs 8 lbs, the force due to gravity is: and its mass is:
b. If a rock has a mass of 25 kg, its mass is 25 kg, but the force due to gravity is:
c. If a weight has a mass of 45 grams, its mass is 45 grams, but the force due to gravity is:
Ex. 1: A mass weighing 2 pounds stretches a spring 6 inches. At t = 0, the mass is released from a point 8 inches below the equilibrium position with an upward velocity of 4/3 ft/s. Determine its equation of motion.
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Section 5.1 Free Undamped Motion
Mar. 21­22, 2011
Note that the amplitude of the motion is not clear from the equation above. This is the case whenever c1 and c2 are both not zero. For this reason, it is helpful to rewrite the equation of motion
in a simpler form using only one of these trig functions.
Ex. 1 (continued): So the equation of motion for Ex. 1 above can be rewritten as:
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Attachments
SpringMass.nbp