MTE314, IME 414: Mechanical Vibrations

Lecture 5
Assoc. Prof. Mohamed G. Alkalla
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 Details of the Course
Course lecturer: Assoc. Prof.\ M. G. Alkalla
Lecture Theatre: Red Hall
Email address: mohamed.gouda@ejust.edu.eg,
Lecture schedule: Sunday 12:30 – 2:00
Office hours: Monday – Wednesday (10:00 – 2:00)
◦ B7 G38

Grading system:
➢Semester Work: MTE (30 marks), IME (20 marks)
➢Midterm Exam: 30 marks
➢Lab: MTE (MTE315), IME (10 marks)
➢Final Exam: 40 marks
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 Boycott x

Assoc. Prof. Mohamed G. Alkalla

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 Contents

1. Fundamentals of Vibration
2. Free Vibration of Single-Degree-of-Freedom Systems
3. Harmonically Excited Vibration
4. Vibration Under General Forcing Conditions
5. Two-Degree-of-Freedom Systems
6. Multi-degree-of-Freedom Systems
7. Determination of natural Frequencies and Mode Shapes
8. Continuous Systems
9. Vibration Control
10. Vibration Measurement and Applications

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF.

MOHAMED G. ALKALLA
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Harmonically Excited
Vibration

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF.

MOHAMED G. ALKALLA
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Harmonically Excited Vibration

A mechanical or structural system is said to undergo forced vibration whenever

external energy is supplied to the system during vibration
External energy can be supplied through either an applied force or an imposed
displacement excitation

Applied force
or
Displacement

Nonharmonic Random in
Harmonic Nonperiodic
but periodic nature

The response of a dynamic system to suddenly applied nonperiodic excitations is called

transient response
EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 6
MOHAMED G. ALKALLA
Harmonically Excited Vibration

The dynamic response of a single-degree-of-freedom system under harmonic excitations of

the form 𝐹(𝑡) = 𝐹0 𝑒 𝑖(𝜔𝑡+𝜙) or 𝐹(𝑡) = 𝐹0 cos(𝜔𝑡 + 𝜙) or 𝐹(𝑡) = 𝐹0 sin(𝜔𝑡 + 𝜙), where
𝑭𝟎 is the amplitude, 𝝎 is the frequency of excitation, and 𝝓 is the phase angle of the
harmonic excitation.

The value of 𝜙 depends on the value of 𝐹 𝑡 at 𝑡 = 0 and is usually taken as zero

Under a harmonic excitation, the response of the system will also be harmonic.

If the frequency of excitation coincides with the natural frequency of the system, the
response will be very large, the phenomenon is called resonance.

Examples of harmonically excited vibrations:

1. Vibration produced by an unbalanced rotating machine,
2. Oscillations of a tall chimney due to vortex shedding in a steady wind,
3. Vertical motion of an automobile on a sinusoidal road surface

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 7

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 8

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of an undamped system under harmonic force

The homogeneous solution of this equation is given by

where 𝜔𝑛 is the natural frequency of the system. Because the exciting force F(t) is
harmonic, the particular solution 𝒙𝒑 (𝒕) is also harmonic and has the same frequency
𝜔. Thus, we assume a solution in the form:
where X is the maximum
amplitude of 𝑥𝑝 (𝑡)
𝑥ሶ 𝑝 = −𝜔𝑋 sin 𝜔𝑡
Substitute both 𝑥𝑝 and 𝑥ሷ 𝑝 in the
𝑥ሷ 𝑝 = −𝜔2 𝑋 cos 𝜔𝑡
1st eqn.
EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 9
MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of an undamped system under harmonic force

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 10

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of an undamped system under harmonic force

𝑭
where 𝜹𝒔𝒕 = 𝟎ൗ𝒌 denotes the deflection of the mass under a force F0 and is sometimes
called static deflection. Thus, the total solution becomes:

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 11

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of an undamped system under harmonic force

General displacement
equation

The quantity 𝑋ൗ𝛿𝑠𝑡 represents the ratio of the dynamic

to the static amplitude of motion and is called the
magnification factor, amplification factor, or
amplitude ratio
frequency ratio r

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 12

MOHAMED G. ALKALLA
Harmonically
Excited Vibration Resonance

Equation of motion
1 . Response of an
undamped system under
harmonic force

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 13

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion

1 . Response of an
undamped system under
harmonic force

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 14

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of an undamped system under harmonic force

F(t), x(t) Pump

Kplate

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 15

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of an undamped system under harmonic force

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 16

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of an undamped system under harmonic force

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 17

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of an undamped system under harmonic force

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 18

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of a damped system under harmonic force

The particular solution of this equation is also expected to be harmonic; we assume it in

the form:
X is amplitude of the response
ϕ is phase angle of the response

Using the trigonometric relations

Then,

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 19

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of a damped system under harmonic force

Then, By squaring both eqns

and summing them

By using second equation

By inserting the expressions of X and 𝜙 into the particular equation in the previous
slide, , we obtain the particular solution of damped
system under harmonic force,

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 20

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of a damped system under harmonic force

Dividing both the numerator and

denominator of above equation by k
and making the following substitutions

we obtain:
𝑴 (𝒓, 𝝃) =

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 21

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of a damped system under harmonic force

𝒇(𝒓, 𝝃)

As the quantity M = 𝑋ൗ𝛿 is called the magnification factor, the variations M and 𝜙 with
𝑠𝑡
the frequency ratio r and the damping ratio ζ are shown in the following figure

EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. 22

MOHAMED G. ALKALLA
Harmonically Excited Vibration
Equation of motion
1 . Response of a damped system under harmonic force

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Harmonically Excited Vibration
Equation of motion
1 . Response of a damped system under harmonic force
The following characteristics of the magnification factor (M) can be noted from Fig. 3.11(a)

1. For an undamped system (𝜁 = 0), the magnification factor equation reduces to the
undamped one and M →∞ as r →1.

2. Any amount of damping (𝜁 > 0) reduces the magnification factor (M) for all values of
the forcing frequency.
3. For any specified value of r, a higher value of damping reduces the value of M.
4. In the degenerate case of a constant force (when r = 0), the value of M = 1.
5. The reduction in M in the presence of damping is very significant at or near resonance.
6. The amplitude of forced vibration becomes smaller with increasing values of the forcing
frequency (i.e., M → 0 as r → ∞).

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Harmonically Excited Vibration
Equation of motion
1 . Response of a damped system under harmonic force

It can be used for the experimental

determination of the measure of
damping present in the system. In
a vibration test, if the maximum
amplitude of the response (X)max is
measured, the damping ratio of the
system can be found
9. For (𝜁 > 1/√2), the graph of M monotonically decreases with increasing values of r.

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Harmonically Excited Vibration
Equation of motion
1 . Response of a damped system under harmonic force

The following characteristics of the phase angle can be observed from Fig. 3.11(b):
1. For an undamped system (𝜁 = 0), the phase angle equation shows that ϕ is 0 for 0 < 𝑟
< 1 and 180° for 𝑟 > 1. This implies that the excitation and response are in phase for 0
< 𝑟 < 1 and out of phase for 𝑟 > 1 when 𝜁 = 0.

2. For 𝜁 > 0 and 0 < 𝑟 < 1 , the phase angle is given by 0 < 𝜙 < 90°, implying that the
response lags the excitation.
3. For 𝜁 > 0 and 𝑟 > 1, the phase angle is given by 90° < 𝜙 < 180°, implying that the
response leads the excitation.
4. For 𝜁 > 0 and r = 1, the phase angle is given by 𝜙 = 90°, implying that the phase
difference between the excitation and the response is 90°.
5. For 𝜁 > 0 and large values of r, the phase angle approaches 180°, implying that the
response and the excitation are out of phase.

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Harmonically Excited Vibration
Equation of motion
Total response damped system under harmonic force Underdamped system

The complete solution is given by 𝑥 𝑡 = 𝑥ℎ 𝑡 + 𝑥𝑝 𝑡 , where 𝑥ℎ 𝑡 was given in the

last lecture (damped free vibration). Thus, for an underdamped system, we have

Where, 𝑋0 and 𝜙0 can be determined from the initial conditions of the system. For
𝑥 𝑡 = 0 = 𝑥0 and 𝑥ሶ 𝑡 = 0 = 𝑥ሶ 0 , then

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Harmonically Excited Vibration
Equation of motion
Total response damped system under harmonic force Underdamped system

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Harmonically Excited Vibration
Equation of motion
Total response damped system under harmonic force Underdamped system

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EGYPT-JAPAN UNIVERSITY OF SCIENCE AND TECHNOLOGY - ASSOC. PROF. MOHAMED G. ALKALLA 31