Forced Vibration

6 Key excerpts on "Forced Vibration"

• 

  eBook - ePub

  Vibration Control and Actuation of Large-Scale Systems

  • Hamid Reza Karimi(Author)
  • 2020(Publication Date)
  • Academic Press

    (Publisher)

  Vibration is defined as an oscillation around an equilibrium position. The etymology “vibration” comes from the Latin word “vibrationem.” Oscillations of a vibration can be periodic or random. Vibrations can be desirable in some cases such as musical instruments, etc. However, vibration is an undesired phenomenon in motion control. Undesired vibrations can create noise, waste of energy, wear off mechanical components, and damage structures. Therefore, vibration suppression is a widely studied area by the researches from the early days.

  Performance deterioration of motion control systems such as velocity controllers, position controllers, and force controllers is a critical issue caused by vibrations. Therefore, control engineers have developed various methods to suppress vibrations in motion control systems [1 –3 ].

  Force controllers are not popular due to the inherent drawbacks of traditional force measuring transducers. Attaching traditional force sensors to measure forces can alter the system parameters such as inertia, mass, and transfer function [4] . Furthermore, force sensing is limited to the position where the sensor is attached. Force sensors have a narrow bandwidth. Therefore, a limited range of exerted forces can be identified [5] . Hence, vibration analysis of the force controllers is not a heavily studied area in the control literature. Since torque and force proportional to each other, torque can be controlled by controlling the force and vice versa.

  Consider Fig. 3.1 . It represents an actuator, which is controlled by a motor, exerting a constant force on a stationary object. An external vibration torque τ v is applied on the actuator axis. There is a positional vibration at the end of the actuator due to vibration torque τ v . Eq. (3.1) can be derived from Newton's third law.

  F r

  =

  F m

  +

  F v

    (3.1)

  Fig. 3.1 Force actuator model.

  where

  • F r : reaction force
  • F v : force on the object due to the vibration torque τ v
  • F m : force on the object due to the motor torque τ m

  F

  m

  is a constant in a traditional force controller. Therefore, according to Eq. (3.1) , reaction force F r on the object fluctuates with the vibration force F v . In other words, object can feel the external vibrations. Assuming the torque vibration τ v is a sine wave and the motor torque τ m is a constant, F r , F m , and F v fluctuations are illustrated in Figs. 3.2 –3.4

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  eBook - ePub

  Polymers for Vibration Damping Applications

  • Bikash C. Chakraborty, Debdatta Ratna(Authors)
  • 2020(Publication Date)
  • Elsevier

    (Publisher)

  When an external dynamic force is applied to an object for a period greater than its own time period of oscillation, the vibration is termed as Forced Vibration. The object vibrates at the same frequency as is imposed by the external force. Running machinery (such as motor, engine, pump, centrifuge, etc.) is subjected to Forced Vibration.

  When the frequency of an external dynamic force coincides with the natural frequency of the body, there will be a very large amplitude of displacement and the phenomenon is termed as resonance. It is also the frequency, at which the potential energy of the object is totally converted to kinetic energy. Therefore, at resonance, the vibration amplitude attains a maximum value. There can be several modes of resonances, which are higher harmonics of the first natural frequency. In each mode, the intensity peaks are observed. System resonance for machinery with very large amplitude may cause severe damage or catastrophic failure. Thus, the study of the vibration response of a structure with respect to time and frequency is very important to take measures to avoid such damages or failures.

  1.4 Random vibration

  If the magnitude of external dynamic force at a given time is known, then the vibration is deterministic vibration. When the magnitudes cannot be determined at a given time, then it is random vibration. In a random vibration scenario, a cluster of vibration intensities and frequencies would exist. Therefore, whatever be the complexity of a random vibration signature, it can be assumed as a sum of many pure sine waves of different amplitudes with corresponding harmonic frequencies such as

  f

  t

  =

  u 0

  +

  u 1

  sin

  ωt −

  φ 1

  +

  u 2

  sin

  2 ωt −

  φ 2

  +

  u 3

  sin

  3 ωt −

  φ 3

  + ⋯ +

  u n

  sin

  nωt −

  φ n

  Generally, the recording of a random vibration spectrum is done with respect to time (time domain). Fourier Transform is applied to find the individual intensities in the frequency scale after selection of a time domain window of the random signal. The example of random vibration is wind velocity, earthquakes, etc. If large data is available, a statistical analysis may be done to determine the different magnitudes and frequencies of the random vibration [10 , 11]

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  eBook - ePub

  Mechanical Vibrations and Condition Monitoring

  • Juan Carlos A. Jauregui Correa, Alejandro A. Lozano Guzman(Authors)
  • 2020(Publication Date)
  • Academic Press

    (Publisher)

  c can also be obtained. However, it's important to remark that the value of the damping coefficient of a system only depends on the damper while with the damping factor, it is determined by the parameters of the mechanical system.

  Forced Vibration

  Every machine is subjected to external excitations, which are related to the operating conditions and the mechanical configuration of each component. For conditioning monitoring systems, this characteristic boosts the prediction of failures at a component level. Thus, it is important to deeply understand the response of a mechanical system to a forced excitation. The simplest system is a one-degree-of-freedom mass-spring system (a simple linear pendulum of a single-degree-of-freedom oscillator has the same dynamic response). The most illustrative case is when the excitation force is represented as a simple harmonic excitation. The response of a one-degree-of-freedom system to a harmonic excitation is presented next. An essential part of the study of mechanical vibrations is the knowledge of the response of the system to an external excitation. In the case of the machinery, the main source of this external excitation comes from the power supply to the machine through the motors used for its operation. This means that, once the machine starts up, Forced Vibrations will occur. However, the total elimination of these vibrations is impossible because the very operation of the engines in industrial conditions is subject to the variation of their components, tolerances, mismatches, imbalances, variations in the power supply, and wear of parts; in other words, countless causes. The purpose, therefore, is to keep these vibrations at tolerable levels. With vibration monitoring for predictive maintenance, the intention is to identify the source of any change in the tolerable levels of Forced Vibration before they exceed the reference levels of normal operation.

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  eBook - ePub

  Vibrations and Waves

  • A.P. French(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  4 Forced Vibrations and resonance THE PRECEDING CHAPTER was concerned entirely with the free vibrations of various types of physical systems. We shall now turn to the remarkable phenomena, of profound importance throughout physics, that occur when such a system—a physical oscillator—is subjected to a periodic driving force by an external agency. The key word is “resonance.” Everybody has at least a qualitative familiarity with this phenomenon, and probably the most striking feature of a driven oscillator is the way in which a periodic force of a fixed size produces very different results depending on its frequency. In particular, if the driving frequency is made close to the natural frequency, then (as anyone who has pushed a swing knows) the amplitude of oscillation can be made very large by repeated applications of a quite small force. This is the phenomenon of resonance. A force of about the same size at frequencies well above or well below the resonant frequency is much less effective; the amplitude produced by it remains quite small. To judge by the quotation at the beginning of this chapter, the phenomenon has been recognized for a very long time. 1 It is typical of this type of motion that the driven system is compelled to accept whatever repetition frequency the driving force has ; its tendency to vibrate at its own natural frequency may be in evidence at first, but ultimately gives way to the external influence. 1 As Alexander Wood remarks in his book Acoustics (Blackie & Son, London, 1940): “It seems difficult to believe that legislation should be designed to cover a situation that had never arisen.” The example does seem rather bizarre, however, and H

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  eBook - ePub

  Fundamentals of Metal Machining and Machine Tools

  • Winston A. Knight, Geoffrey Boothroyd(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  3 ].

  9.2 Forced Vibrations

  A machine structure that is subjected to a periodic force will vibrate at the forcing frequency. Several basic results can be illustrated by first considering a single-degree-of-freedom system.

  9.2.1 Single-Degree-of-Freedom System

  For the one-degree-of-freedom system shown in Figure 9.1 , the equation of motion is

  where

  x	=	displacement
  t	=	time
  me	=	equivalent mass
  cd	=	damping force per unit velocity (viscous damping constant)
  Se	=	restoring force per unit displacement (spring stiffness)
  Fmax	=	peak value of the external harmonic force
  ω f	=	angular frequency of the external harmonic force

  m e

  d 2

  x

  d

  t 2

  +

  c d

  d x

  d

  t ′

  +

  S e

  x =

  F  max 

   cos 

  ω f

  t ′

  (9.1)

  FIGURE 9.1 Model of one-degree-freedom mass-spring system with viscous damping.

  The steady-state vibration of this system is given by where

  F0	=	Fmax /me
  ω n	=	the natural angular frequency
  ξ	=	the damping ratio

  x =

  F 0

   cos  (

  ω f

  t ′

  −

  ϕ f

  )

  4

  ξ 2

  ω f 2

  −

  (

  ω n 2

  −

  ω f 2

  )

  2

  (9.2)

  Equation 9.2 represents a motion of angular frequency ay at an amplitude given by:

  a 0

  =

  F 0

  4

  ξ 2

  ω f 2

  −

  (

  ω n 2

  −

  ω f 2

  )

  2

  (9.3)

  and lagging the disturbing force by a phase angle φ

  f

  where

  ϕ f

  =

  tan

  − 1

  2 ξ

  ω n

  (

  ω n 2

  −

  ω f 2

  )

  (9.4)

  Resonance occurs when ωn equals ωn and the amplitude at resonance is F0 /2ξω

  n

  . These results are shown in Figure 9.2

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  eBook - ePub

  Structural Dynamics of Earthquake Engineering

  Theory and Application Using Mathematica and Matlab

  • S Rajasekaran(Author)
  • 2009(Publication Date)
  • Woodhead Publishing

    (Publisher)

  4 Forced Vibration (harmonic force) of single-degree-of-freedom systems in relation to structural dynamics during earthquakes Abstract In this chapter, Forced Vibration of single-degree-of-freedom (SDOF) systems (both undamped and under-damped) due to harmonic force is considered. Governing equations are derived and the displacement response is determined using Wilson’s recurrence formula. Vibration excitation due to imbalance in rotating machines is discussed. Equations for transmissibility are derived for force and displacement isolation. The underlying principle of vibration-measuring instruments is illustrated. Key words resonance transient steady state magnification factor beating transmissibility seismometer accelerometer 4.1 Forced Vibration without damping In many important vibration problems encountered in engineering work, the exciting force is applied periodically during the motion. These are called Forced Vibrations. The most common periodic force is a harmonic force of time such as P = P 0 sin ω t 4.1 where P 0 is a constant, ω is the forcing frequency and t is the time. The motion is analysed using Fig. 4.1. m x ¨ + kx = P 0 sin ωt 4.2 4.1 Spring-mass system subjected to harmonic force. The general solution of Eq. 4.2 (non-homogeneous second order differential equation) consists of two parts x = x c + x p where x c = complementary solution, and x p = particular solution. The complementary solution is obtained by setting right hand side as zero. m x ¨ c + k x c = 0 4.3 x c = c 1 sin ω n t + c 2 cos ω n t 4.4 where ω n = k / m and c 1 and c 2 are arbitrary constants. Assume x p = A sin wt and. substituting m x ¨ p + k x p = P 0 sin ωt 4.5 − ω 2 mA + kA sin ωt = P 0 sin ωt 4.6 A = P 0 k 1 − ω 2 m / k = P 0 k 1 − ω 2 / ω n 2 4.7 Since β = ω / ω n, A = P 0 k 1 − β 2 = P 0 / k 1 − β 2 4.8 If[

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