Modul II
VECTORIAL REPRESENTATION OF HARMONIC MOTIONS
The motion sustained by the sinusoidal excitation is called the steady -state
vibration or the steady-state response. Hence the steady-state response must be at the
excitation frequency regardless of the initial conditions or the natural frequency of the
system. It will be shown in Chap. 2 that the steady-state response is described by the
particular integral and the transient motion by the complementary function of the
differential equation of the system.
Resonance occurs when the excitation frequency is equal to the natural
frequency of the system. No energy input is needed to maintain the vibrations of an
undamped system at its natural frequency. Thus, any energy input will be used to build
up the amplitude of the vibration, and the amplitude at resonance of an undamped
system will increase without limit.
where k2 is a constant. When the acceleration of a particle with rectilinear motion is
always proportional to its displacement from a fixed point on the path and is directed
towards the fixed point, the particle is said to have simple harmonic motion. It can be
shown that the solution of Eq. (1-7) has the form of a sine and a cosine function with
circular frequency equal to ω.
It is convenient to represent a harmonic motion by means of a rotating vector X of
constant magnitude X at a constant angular velocity ω. In Fig. 1-7, the displacement of
P from the center O along the x axis
(a) Vectorial representation
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(b) Harmonic motions FIG. 1-7. Harmonic motions represented by rotating vector.
In complex variables, the length of a vector is called the absolute value or modulus, and
the phase angle is called the argument or amplitude. The length of the vector in this
discussion is the amplitude of the harmonic motion. To avoid confusion, we shall use
magnitude to denote the length of the vector.
is OP = x(t)=X cosωt. This is the projection of the rotating vector X on the
diameter along the x axis. Similarly, the projection of X on the y axis is OQ= y(t) = Xsin
ωt. Naming the x axis as the "real" axis and the y axis as the "imaginary" one, the
rotating vector X is represented by the equation:
X = X cos ωt + jX sin ωt = Xeωt (1-9)
where X is the length of the vector or its magnitude and j = v (-1) is called the imaginary
unit.
If a harmonic function is given as x(t) = X cos ωt, it can be expressed as x(t) =
Re[Xejωt], here the symbol Re denotes the real part of the function Xeiait. Similarly, the
function y(t) = Xsin ωt can be expressed as y(t} = Im[Xelωt!], where the symbol Im
denotes the imaginary part of Xejωt. It should be remembered that a harmonic motion is a
reciprocating motion. Its representation by means of a rotating vector is only a
convenience. This enables the exponential function ejωtto be used in equations involving
harmonic functions. The use of complex functions and complex numbers greatly
simplifies the mathematical manipulations of this type of equations. In reality, all physical
quantities, whether they are displacement, velocity, acceleration, or force, must be real
quantities.
The differentiation of a harmonic function can be carried out in its vectorial form.
The differentiation of a vector X is
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………(1.10)
Thus, each differentiation is equivalent to the multiplication of the vector by jω.
Since X is the magnitude of the vector X, ω is real, and |j| = 1, each differentiation
changes the magnitude by a factor of ωt. Since the multiplication of a vector by |j| is
equivalent to advancing it by a phase angle of 90°, each differentiation also advances a
vector by 90°.
If a given harmonic displacement is x(t) = X cos ωt , the relations between the
displacement and its velocity and acceleration are
…………(1.11)
These relations are identical to those shown in Eqs. (1-4) to (1-6). The representation of
displacement, velocity, and acceleration by rotating vectors is illustrated in Fig. 1-8.
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Since the given displacement x(t) is a cosine function, or along the real axis, the velocity
and acceleration must be along the real axis. Hence the real parts of the respective
vectors give the physical quantities at the given time t.
Harmonic functions can be added graphically be means of vector addition. The vectors
X1 and X2 representing the motions Xl cos ωt and X2 cos(ωt + α), respectively, are
added graphically as shown in Fig. l-9(a). The resultant vector X has a magnitude
X  ( X 1  X 2 cos  ) 2  ( X 2 sin  ) 2
And a phase angle
  tan 1
sin 
X 1  X 2 cos 
with respect to X1. Since the original motions are given along the real axis, the sum of
the harmonic motions is Re[X] = X cos(ωt + α). The addition operation can readily be
extended to include the subtraction operation.
Since both X1 and X2 are rotating with the same angular velocity ωt, only the
relative phase angle of the vectors is of interest. It is convenient to assign arbitrarily ωt
=0 as a datum of measurement of phase angles. The vector X 1 ,X2, and their sum X are
plotted in this manner in Fig. 1-9(b). Note that the vector X2 can be expressed as
………(1.12)
The quantity X2 = X2ejωt is a complex number and is called the complex amplitude
or phasor of the vector X2. Similarly, X= Xejωt in Fig. 1-9(b) is the phasor of the vector X.
Harmonic functions can be added algebraically by means of vector addition. Using the
same functions x: = X1 cos ωt and X2 = X2 cos(ωt+ α), their vector sum is
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Since the given harmonic motions are along the real axis, their sum is
In representing harmonic motions by rotating vectors, it is often necessary to
determine the product of complex numbers. The product can be found by expressing the
complex numbers in the exponential form. For example, the product of the complex
numbers A and B is
……….(1.13)
where A = a  a and B = b  b are the magnitudes of the numbers and α =tan-1
a2/a1 and β = tan-1 b2/bl are their phase angles. Equation (1-13) indicates that
2
1
2
2
2
1
2
2
……(1.14)
AB = (phase of A) + (phase of B)…….. (1-15)
Obviously, the multiplication operation can be generalized to include the division
operation.
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Dr. Ir. Abdul Hamid M.Eng. GETARAN MEKANIS
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Example 2-1.Manipulation of Complex Numbers
The last two examples indicate that the multiplication of a vector by j advances the
vector –clockwise by a phase angle of 900 and a division by j retards it by 900.
1.6 Units
Since there will be a change from the English engineering (customary) to the
International System of Units (SI), the two systems of units will co-exist for some years.
The student and the practicing engineer will need to know both systems. We shall briefly
discuss the English units and then the SI units in some detail in this section.
Newton's law of motion may be expressed as
Force = (mass)(acceleration)…… (1-16)
Dimensional homogeneity of the equation is obtained when the force is in pounds Ity, the
mass in slugs, and the acceleration in ft/sec2. This is the English ft-lbpsec system in
which the mass has the unit of Ibf-sec2/ft. A body falling under the influence of gravitation
has an acceleration of g ft/sec2, where g=32.2 is the gravitational acceleration. Hence
one pound-mass lbm exerts one pound-force lb; under the gravitational pull of the earth.
In other words, 1 lbm weighs one pound on a spring scale. If a mass is given in pounds,
lbm or weight, it must be divided by g to obtain dimensional homogeneity in Eq. (1-16).
The in.-lbf-sec system is generally used in the study of vibrations. The
gravitational acceleration is (32.2)(12) = 386 in./sec2. Hence the weight is divided by 386
in order to obtain dimensional homogeneity in Eq. (1-16). We assume that the
gravitational acceleration is constant unless otherwise stated. In the derivation of
equations, the mass m is assumed to have the proper units.
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The International System of Units (SI) is the modernized version of the metric
system. SI consists of (1) seven well-defined base units, (2) derived units, and (3)
supplementary units.
The base units are regarded as dimensionally independent. Those of interest in
this study are the meter, the kilogram, and the second. The meter m is the unit of length.
It is defined in terms of the wave-length of a krypton-85 lamp as "the length equal to 1
650 763.73 wave-lengths in vacuum of the radiation corresponding to the transition
between the levels 2p10 and 5d5 of the krypton-86 atom." The kilogram kg is the unit
of mass. The standard is a cylinder of platinum-indium, called the International Standard,
kept in a vault at Sevres, France. The second s is the unit of time. It is defined in terms
of the frequency of atomic resonators. "The second is the duration of 9 192631 770
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Dr. Ir. Abdul Hamid M.Eng. GETARAN MEKANIS
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periods of the radiation corresponding to the transition between the two hyperfine levels
of the ground state of the cesium-133 atom."*
The derived units are formed from the base units according to the algebraic relations
linking the corresponding quantities. Several derived units are given special names and
symbols. The supplementary units form a third class of SI units. Examples of derived
and supplementary units are shown in Tables l-2(a) and (&), respectively.
The common prefixes for multiples and submultiples of SI units are shown in Table 1-3.
Examples of conversion from the English to the SI units are given in Table 1-4. Note that
a common error in conversion is to become ensnared in too many decimal places. The
result of a computation cannot have any more significant numbers than that in the
original data.
For uniformity in the use of SI units, the recommendations* are: (1) In numbers, a
period (dot) is used only to separate the integral part of numbers from the decimal part.
Numbers are divided into groups of three to facilitate reading. For example, in defining
the meter above, we have ". . . equal to 1 650 763.73 wave-lengths . . ." (2) The type
used for symbols is illustrated in Table 1-2. The lower case roman type is generally
used. If the symbol is derived from a proper name, capital roman type is
used for the first letter. The symbols are not followed by a period. (3) The product of
units is denoted by a dot, such as N • m shown in Table 1-2. The dot may be omitted if
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there is no risk of confusion with another unit symbol, such as Nm but not mN. (4) The
division of units may be indicated by a solidus (/), a horizontal line, or a negative power.
For example, velocity in Table 1-2 can be expressed as m/s, —, or m • s. The
solidus must not be repeated on the same line unless ambiguity is avoided by
parentheses. For example, acceleration may be expressed as m/s2 or m • s-2 but not
m/s/s. (5) The prefix symbols illustrated in Table 1-3 are used without spacing between
the prefix symbol and the unit symbol, such as in mm. Compound prefixes formed by the
use of two or more SI prefixes are not used.
1.7 Bentuk Exponential
Dari Gamb.5,vektor
A
dpt ditulis:
A  x  jy .....(1-17).
Gamb.5.
,dari
pers.(1)
didpt :
1
Disini j 
x  A cos(t   )
y  A sin( t   )
yg bila disubstitusikan ke pers.(10) didpt:
A  Acos(t   )  j sin( t   ) ...(1-18).
Dari hubungan : e
j
 cos   j sin  ,maka
A  Ae j (t  )  Ae j e jt  A0 e jt ....(1-19).
disini A0  Ae
j
disebut amplitudo komplek .Dlm bentuk bilangan komplek adalah:
x  A0 e jt ....(13)
Contoh.2-2
Hitunglah amplitudo
6e
j (t  )
2
dan.8e
maksimum
dari
gabungan
antara
dua
vektor
yaitu
j (t  )
6
Jawab.
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Dr. Ir. Abdul Hamid M.Eng. GETARAN MEKANIS
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Ambil t=0,sehingga ωt=0.Dan kalau diambil komponen datar dan tegak lurus yaitudari
pers.(13) bahwa bagian riel:
x  A cos(t   ) dan bagian imaginer:
y  A sin( t   )
Dari ωt=0,maka α=π/2 dan α=π/6,maka
x  6 cos  / 2  8 cos  / 6  8 cos 30 0  6.93
y  6 sin  / 2  8 sin  / 6  6  8 sin 30 0  10
Resultante: R= 10 2  6.932  12.2 ,dan besar sudut α:
Tanα=10/6.93=1.443,maka
0
  55 013'  55 13
360 0
Gabungan dari dua vektor tsb :
j (t  / 2 )
j (t  / 6 )
 8e
6e
 2  0.962  rad
 12.2e j (t  0.962)
Soal-soal latihan 2:
2-1.Tunjukkanlah bahwa komplek 5+j2 adalah bentuk dari Aej
2-2.Hitunglah:
j / 6
j / 2
5e
 7e
2-3. Sketch the motion described by each of the following equations:
(a) x = 5e~2rsin(10irt + 7r/4)
(b) x = 5<T2'sinC
for 0<rsl.0s. 1-18 Express the following complex numbers in the exponential form Ae'°.
(a) 1+jJl (e) 3/(V3-j)
(b) -2 (I) (V3+j)(3
(c) 3/(V3-j) (g)
(d) 5j (h)
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