Mechanical Vibrations
(MED 343)
Lecture 5: Forced vibration of SDOF systems
(Harmonically Excited Vibration)
Dr. Mahmoud Atef
Department of Mechanical Design and Production
Engineering
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Physical model of SDoF system
The basic components of a mechanical system are:
1- Inertia components store kinetic energy
2- Stiffness components store potential energy
3- Damping components dissipate energy
4- Energy sources provide energy to the system
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Forced vibration system
• Forced vibration occurs when external energy is supplied to the system
during vibration.
• External energy can be supplied through :
1- Force excitation.
2- Displacement excitation (Base excitation).
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Forced vibration system
• External Excitation can vary in nature:
1- Harmonic excitation:
• A regular, repeating force or displacement, often sinusoidal.
• The system’s response to such excitation is known as the harmonic response.
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Forced vibration system
• External Excitation can vary in nature:
2- Nonharmonic but periodic excitation:
Repetitive, non-sinusoidal forces or displacements that recur at regular
intervals.
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Forced vibration system
• External Excitation can vary in nature:
3- Nonperiodic excitation:
Irregular, one-time or intermittent forces or displacements.
F(t) F(t) F(t)
t t t
Step force Ramp force impulse force
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Forced vibration system
• External Excitation can vary in nature:
Nonperiodic excitations vary in duration and can be classified based on their
impact on the system:
➢ Long Duration Excitation: A sustained, continuous excitation that acts over
an extended period, significantly influencing the system’s behaviour through
prolonged energy input.
➢ Short Duration Excitation (Shock or Impulse): A brief, intense excitation
with a duration much shorter than the system’s natural period, delivering a
sudden energy surge and causing a rapid transient response
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Forced vibration system
• External Excitation can vary in nature:
3- Random excitation
• It refers to external forces or displacements that vary unpredictably over time,
lacking any regular or periodic pattern.
• Random excitation is commonly encountered in real-world situations where
forces arise from complex, uncontrollable sources.
Real examples of random excitation are:
➢ Earthquake ground motion
➢ Wind turbulence
➢ Vehicle vibrations
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Harmonically Excited Vibration
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Force excited vibration system
• Harmonic Excitation in Engineering: it commonly arises from rotational
unbalance in machinery, producing a sinusoidal force or displacement.
• Importance for Analysis: Understanding harmonic response helps predict system
behaviour under more complex excitations, aiding in resonance and stability
assessments.
• Harmonic excitation may be in the form of a force or displacement of some point
in the system.
𝑡 = 𝐹𝑜 sin(𝜔𝑡)
𝑌 𝑡 = 𝑌𝑜 sin(𝜔𝑡)
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Force excitated Mass-spring- damper system
Equation of motion Vibration response
c k
k.x
m
m.x
𝑥(𝑡) = 𝑥ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑜𝑢𝑠 + 𝑥𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟
𝐹(𝑡) = 𝐹𝑜 sin 𝜔 𝑡
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Force excited Mass-spring- damper system
Vibration response 𝑥(𝑡) = 𝑥ℎ𝑜𝑚𝑜𝑔𝑒𝑛𝑜𝑢𝑠 + 𝑥𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟
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Force excited Mass-spring- damper system
P𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 Solution (steady state response)
x(t ) = X sin(t − ) m.x + cx + kx = Fo sin t
x (t ) = X cos(t − ) = X sin(t − + / 2)
x(t ) = − 2 X sin(t − )
−𝑚𝜔2 𝑋 sin( 𝜔𝑡 − 𝜑) + 𝑐𝑋𝜔 sin( 𝜔𝑡 − 𝜑 + 𝜋/2) + 𝑘𝑋 sin( 𝜔𝑡 − 𝜑) − 𝐹𝑜 sin 𝜔 𝑡 = 0
m 2 X sin(t − ) − cX sin(t − + / 2) − kX sin(t − ) + Fo sin t = 0
kX Fo sin t
cX X sin(t − )
t
m 2 X
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Force excited Mass-spring- damper system
P𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 Solution (steady state response)
F 2 = (cX ) 2 + (kX − m 2 X ) 2
Fo
X=
(k − m 2 ) 2 + (c ) 2
kX Fo sin t
cX X sin(t − )
t
m 2 X
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Force excited Mass-spring- damper system
P𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 Solution (steady state response)
Non- dimensional form
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Force excited Mass-spring- damper system
P𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 Solution (steady-state response)
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Force excited Mass-spring- damper system
P𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 Solution (steady-state response)
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Force excited Mass-spring- damper system
General response of forced vibration
Standard form
Steady state response
Transient response
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Quality Factor and Bandwidth
Quality Factor
• It is the value of the amplitude ratio at
resonance frequency.
• It is a measure of a system's resonant
sharpness.
• A high Q factor means low energy loss and
a narrow, sharp resonance peak.
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Quality Factor and Bandwidth
Bandwidth
• It is the range of frequencies over which the
system's response remains significant, typically
around the resonant frequency.
• Bandwidth is the frequency range where the
amplitude falls to 70.7% of the peak value.
• High Q has narrow bandwidth, and low Q has
wider bandwidth.
• Q and bandwidth determine the frequency
response and damping characteristics of a
system
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Example
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Example
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Example
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