APPLIED MECHANICS

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APPLIED MECHANICS
Slovak University of Technology Faculty of
Material Science and Technology in Trnava

• Lecture 05

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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The system is excited by a harmonic force of the
  form

where F0 - amplitude of the forced vibration,
w - the forced angular frequencies.
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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The equation of motion

• The solution of equation

• The particular solution xp

• The constant Cp is determined for

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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The solution

resp.

• The constants A and B (C and j) are determined
  from the initial conditions

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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The constants

• The derivative with respect to time

• The solution is

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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The displacement is a combined motion of two
  vibrations
• one with the natural frequency w0,
• one with the forced frequency w
• The resultant is a nonharmonic vibration

• The amplitude is

where
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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• Resonance - excitating frequency w is equal to
  the natural angular frequency w0 - the resonance
  phenomenon appears.

Curve of resonance
Diagram of resonance phenomenon
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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
CENTRIFUGAL EXCITING FORCE

• Unbalance in rotating machines is a common source
  of vibration excitation. Frequently, the excited
  harmonic force came from an unbalanced mass that
  is in a rotating motion that generates a
  centrifugal force

m0 is an unbalanced mass connected to the mass m1
with a massless crank of lengths r, the mass m0
rotates with a constant angular frequency w.
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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
CENTRIFUGAL EXCITING FORCE

• The amplitude of the combined vibration

where m m1 m0.

• The magnification factor

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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
CENTRIFUGAL EXCITING FORCE
Variation of the magnification factor
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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
ARBITRARY EXCITING FORCE

• The general case of exciting force is an
  arbitrary function of time

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SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION -
ARBITRARY EXCITING FORCE

• The differential equation of motion

• The vibration in this case is described

where t is presented in Figure A, B are
constants.
The integral in equation is called the Duhamel
integral.
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SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The mechanical model

• The equation of motion

The following notation is used
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SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The equation of motion becomes

• Case 1

or

• The characteristic equation

with the roots

• The general solution of differential equation

x1 - solution of the differential homogenous
equation, x2 - particular solution of the
differential nonhomogeneous equation
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SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The solution of the free damped system

• The solution of the forced (excited) vibration

• D1, D2 are determined by the identification
  method.

• Solution of the forced vibration is introduced
  into equation of motion

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SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The linear system of algebraic equation

• D1, D2 are obtained

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SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The forced vibration x2

or

• The motion of the system

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SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The amplitude of forced vibration

• The magnification factor and phase delay

- damping ratio
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SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• The graphic of the vibration

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SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION -
HARMONIC EXCITING FORCE

• Resonance

A-F characteristics
Phase delay