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Harmonic motion

The kind of motion exhibited by a simple pendulum is often referred to as periodic motion in general, and simple harmonic motion in particular.

Other devices that exhibit periodic motion may exhibit more complex harmonic motion. For example, Figure 2 illustrates a more complex form of harmonic motion.

Figure 2. Complex harmonic motion.
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As you can see, the positive excursions are greater than the negative excursions in Figure 2 . In addition, the negative excursions exhibit some ripple that is not exhibited by the positive excursions.

A complex sinusoidal function:

Figure 2 is a plot of the following Java expression:

Math.cos(2*pi*x/50) + Math.sin(2*pi*x/50)+ Math.sin(2*pi*x/25);

Sinusoids

Figure 1 shows a curve whose shape is commonly referred to as a sinusoid. Eliminating the syntax required by the Java programming language, the expressionused to produce the data plotted in Figure 1 was:

f(x) = cos(2*pi*x/50) + sin(2*pi*x/50)

As you can see, this function is composed of two components, one cosine component and one sine component.

Separate the cosine and sine components

Figure 3 shows separate plots of the two components that were added together to produce the plot in Figure 1 .

Figure 3. Separate cosine and sine functions.
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Plotting separate sine and cosine functions

The black curve in Figure 3 shows the cosine function produced by evaluating and plotting the following equation:

f(x) = cos(2*pi*x/50)

The blue curve shows the sine function produced by evaluating and plotting the following equation:

g(x) = sin(2*pi*x/50)

The point-by-point sum of these two curves would produce the sinusoidal curve shown in Figure 1 .

Time varying functions

Many processes produce functions whose values vary over time. For example, the temperature in your office is a function that varies continuously with time.

(The temperature in your office is probably not a periodic function because the temperature probably doesn't repeat its values on any periodicbasis).

Sampled data

If you were to measure and record the temperature in your office once each minute, you could plot the recorded values in the manner shown in Figure 1 through Figure 3 . You could plot the temperature along the vertical axis against time (or sample number) along the horizontal axis.

A common name for sampled data of this type is time series. That name reflects the fact that your data is a recording of the temperaturevalues for a series of measurements that occur over time. (I will have quite a lot more to say about time series in future modules.)

The period of the sine and cosine functions

Figure 4 is very similar to Figure 3 . However, there is one major difference. In Figure 3 , the repetition period of the sine and cosine functions is the same. In other words, the shape of the sine function is the same as the shape of thecosine function. They are simply shifted relative to one another along the horizontal axis.

Each of the curves in Figure 3 has the same period, where the period is the horizontal distance from one peak to the next.

Questions & Answers

Ayele, K., 2003. Introductory Economics, 3rd ed., Addis Ababa.
Widad Reply
can you send the book attached ?
Ariel
?
Ariel
What is economics
Widad Reply
the study of how humans make choices under conditions of scarcity
AI-Robot
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn Reply
U(x,y) = (x×y)1/2 find mu of x for y
Desalegn
what is ecnomics
Jan Reply
this is the study of how the society manages it's scarce resources
Belonwu
what is macroeconomic
John Reply
macroeconomic is the branch of economics which studies actions, scale, activities and behaviour of the aggregate economy as a whole.
husaini
etc
husaini
difference between firm and industry
husaini Reply
what's the difference between a firm and an industry
Abdul
firm is the unit which transform inputs to output where as industry contain combination of firms with similar production 😅😅
Abdulraufu
Suppose the demand function that a firm faces shifted from Qd  120 3P to Qd  90  3P and the supply function has shifted from QS  20  2P to QS 10  2P . a) Find the effect of this change on price and quantity. b) Which of the changes in demand and supply is higher?
Toofiq Reply
explain standard reason why economic is a science
innocent Reply
factors influencing supply
Petrus Reply
what is economic.
Milan Reply
scares means__________________ends resources. unlimited
Jan
economics is a science that studies human behaviour as a relationship b/w ends and scares means which have alternative uses
Jan
calculate the profit maximizing for demand and supply
Zarshad Reply
Why qualify 28 supplies
Milan
what are explicit costs
Nomsa Reply
out-of-pocket costs for a firm, for example, payments for wages and salaries, rent, or materials
AI-Robot
concepts of supply in microeconomics
David Reply
economic overview notes
Amahle Reply
identify a demand and a supply curve
Salome Reply
i don't know
Parul
there's a difference
Aryan
Demand curve shows that how supply and others conditions affect on demand of a particular thing and what percent demand increase whith increase of supply of goods
Israr
Hi Sir please how do u calculate Cross elastic demand and income elastic demand?
Abari
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Source:  OpenStax, Digital signal processing - dsp. OpenStax CNX. Jan 06, 2016 Download for free at https://legacy.cnx.org/content/col11642/1.38
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