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Finding a cosine function that models damped harmonic motion

Find and graph a function of the form y = a e c t cos ( ω t ) that models the information given.

  1. a = 20 , c = 0.05 , p = 4
  2. a = 2 , c = 1.5 , f = 3

Substitute the given values into the model. Recall that period is 2 π ω and frequency is ω 2 π .

  1. y = 20 e 0.05 t cos ( π 2 t ) . See [link] .
    Graph of f(t) = 20(e^(-.05t))cos(pi/2 * t), which begins with a high amplitude and slowly decreases.
  2. y = 2 e 1.5 t cos ( 6 π t ) . See [link] .
    Graph of f(t) = 2(e^(-1.5t))cos(6pi * t), which begins with a small amplitude and quickly decreases to almost a straight line.
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The following equation represents a damped harmonic motion model:   f ( t ) = 5 e 6 t cos ( 4 t ) Find the initial displacement, the damping constant, and the frequency.

initial displacement =6, damping constant = -6, frequency = 2 π

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Finding a sine function that models damped harmonic motion

Find and graph a function of the form y = a e c t sin ( ω t ) that models the information given.

  1. a = 7 , c = 10 , p = π 6
  2. a = 0.3 , c = 0.2 , f = 20

Calculate the value of ω and substitute the known values into the model.

  1. As period is 2 π ω , we have
        π 6 = 2 π ω   ω π = 6 ( 2 π )     ω = 12

    The damping factor is given as 10 and the amplitude is 7. Thus, the model is y = 7 e 10 t sin ( 12 t ) . See [link] .

    Graph of f(t) = 7(e^(-10t))sin(12t), which spikes close to t=0 and quickly becomes almost a straight line.
  2. As frequency is ω 2 π , we have
       20 = ω 2 π 40 π = ω

    The damping factor is given as 0.2 and the amplitude is 0.3. The model is y = 0.3 e 0.2 t sin ( 40 π t ) . See [link] .

    Graph of f(t) = .3e^(-.2t)sin(40pi*t), which has a small amplitude but quickly decreases to the appearance of a straight line. The frequency is so high that, in this scaling, the function looks like a solid shape. The zoom in cut out of the graph shows the actual sinusoidal image of the function.
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Write the equation for damped harmonic motion given a = 10 , c = 0.5 , and p = 2.

y = 10 e 0.5 t cos ( π t )

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Modeling the oscillation of a spring

A spring measuring 10 inches in natural length is compressed by 5 inches and released. It oscillates once every 3 seconds, and its amplitude decreases by 30% every second. Find an equation that models the position of the spring t seconds after being released.

The amplitude begins at 5 in. and deceases 30% each second. Because the spring is initially compressed, we will write A as a negative value. We can write the amplitude portion of the function as

A ( t ) = 5 ( 1 0.30 ) t

We put ( 1 0.30 ) t in the form e c t as follows:

0.7 = e c     c = ln .7     c = 0.357

Now let’s address the period. The spring cycles through its positions every 3 seconds, this is the period, and we can use the formula to find omega.

3 = 2 π ω ω = 2 π 3

The natural length of 10 inches is the midline. We will use the cosine function, since the spring starts out at its maximum displacement. This portion of the equation is represented as

y = cos ( 2 π 3 t ) + 10

Finally, we put both functions together. Our the model for the position of the spring at t seconds is given as

y = 5 e 0.357 t cos ( 2 π 3 t ) + 10

See the graph in [link] .

Graph of the function y = -5e^(-.35t)cos(2pi/3 t) + 10 from 0 to 24. It starts out as waves with a high amplitude and decreases to almost a straight line very quickly.
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A mass suspended from a spring is raised a distance of 5 cm above its resting position. The mass is released at time t = 0 and allowed to oscillate. After 1 3 second, it is observed that the mass returns to its highest position. Find a function to model this motion relative to its initial resting position.

y = 5 cos ( 6 π t )

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Finding the value of the damping constant c According to the given criteria

A guitar string is plucked and vibrates in damped harmonic motion. The string is pulled and displaced 2 cm from its resting position. After 3 seconds, the displacement of the string measures 1 cm. Find the damping constant.

The displacement factor represents the amplitude and is determined by the coefficient a e c t in the model for damped harmonic motion. The damping constant is included in the term e c t . It is known that after 3 seconds, the local maximum measures one-half of its original value. Therefore, we have the equation

a e c ( t + 3 ) = 1 2 a e c t

Use algebra and the laws of exponents to solve for c .

  a e c ( t + 3 ) = 1 2 a e c t e c t e 3 c = 1 2 e c t Divide out  a . e 3 c = 1 2   Divide out  e c t . e 3 c = 2 Take reciprocals .

Then use the laws of logarithms.

e 3 c = 2   3 c = ln 2     c = ln 2 3

The damping constant is ln 2 3 .

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Questions & Answers

what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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