Chapter4-HarmonicAnalysis

a
ds s 0 cos t dt
(4.2)
dv s 0 2 sin t s 2 dt
(4.3)
In which
a 0 s 0 2
v---velocity; v0---velocity amplitude or maximum velocity; a---acceleration a0---acceleration amplitude or maximum acceleration.
In which A,An, and Bn---harmonic coefficients; n---harmonic orders, integers from 1 to .
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y C sin( x 1 )
Case II: For y =A cos x - B sin x, y C cos( x 2 )
Main topics:
Chapter 4 The Analog Measurand: Time-Dependent Characteristics
Harmonic signals Frequency Spectrum Fourier Analysis Fourier Series for continuous-time periodic signals Fourier Series for discrete-time periodic signals
or
A general mathematical statement:
f (t ) A (4.5) A n cos n t B n sin n t ) 2 n 1 A The static component 2 The fundamental harmonic A1 cos t B 1 sin t A 2 cos 2 t B 2 sin 2 t ......
(a)
15 10 5 0 -5
-20 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(d)
20 15 10 5
y 2 10 sin t 4 sin 2t
11
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
--the crank angle --the circular
frequency (rad/s); f—the frequency (Hz)
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The piston displacement
s s 0 sin t
It shows that the yoke-piston combination moves with simple harmonic motion.
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Crank
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15 10 5 0
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The effects of the variables
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4.2 Simple Harmonic Relations (简谐关系）
Simple Harmonic function ：A function is called to be simple harmonic in terms of a variable (often it is time t ) when its second derivative is proportional to the function but of opposite sign. （当一个函数对某一变量的二阶偏 导数与该函数成正比，但符号相反时，则称之为简谐 函数。） Most quantities that are time functions may be expressed harmonically.
（1） pendulum motions of small amplitude;(小幅度的钟摆） （2） a mass on a bean; (一个位于梁上的质量块） （3）a weight suspended by a rubber band.(一个用橡皮筋 悬挂起来的 重物）
v 0 s 0
The yoke-piston combination. The piston displacement slider block
(rad);
The phase angle when they both have the same frequency , but do not oscillate together, the time relation (lag or advance) between their motions may be expressed by an angle referred to as the phase angle .
or In which
The 2nd harmonic …….
y C sin( - x 1 )
C
1
A2 B2
1 tan
10
A B
2 tan
1
B A
Positive acute angles
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6 6
The acceleration a is proportional to the displacement s but is of opposite sign.
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Another example: A spring-mass system In phase: (P98) If the mass and piston have the same frequencies and simultaneously reach corresponding extremes of displacement. (move up and down in perfect synchronization（同步）)
4.1 Introduction
All measurands have time-related characteristics. The nature of any change is often fully as important as that of any discrete amplitude. The time-related measurands can be classified as: 1.Static 2.Dynamic a. Steady-state periodic (稳态周期） b. Non-repetitive or transient (非重复 或 瞬态） i. Signal-pulse or aperiodic （单脉冲 或 非周期） ii. Continuing or random (连续 或 随机）
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4.4 Complex Relations
Most complex dynamic-mechanical signals may be expressed as a combination of simple harmonic components.
Conversion is made to write the Eq.(4.5) in terms of either the sine or the cosine alone. Case I: For y =A cos x + B sin x, y C cos( x 2 )
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4.3 The Significance of Circular Frequency
Circular frequency （圆频率）(rad/s); Cyclic relations （周期关系） Scotch-yoke （苏格兰轭，挡车轭）mechanism • The crank turns at radians per second. • =t=2ft
In which
(4.1)
By differentiation
and Also，
v
s---instantaneous displacement from equilibrium(平稳); s0---amplitude, or maximum displacement from equilibrium; ---circular frequency （圆频率）(rad/s), and t---any time interval measured from the instant when t=0 s. For example：
15
-5 -10 -15 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(c)
20 15
10 5 0 -5
10 5
y1 10 sin t 2 sin 2t
0 -5 -10
y 4 10 sin t 8 sin 2t
-10 -15 0
-15
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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Simple harmonic motion:
s s 0 sin t
amplitudes C, harmonic orders n, and phase angles
y3 10 sin t 6 sin 2t
Figure 4.2 Examples of two component waveforms with second-harmonic component of various relative amplitudes
For example: in mechanical engineering: displacement - time; in electrical engineering: alternating-current voltage - time.
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