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Basic Concepts of Vibration
Vibration: mechanical oscillations about an equilibrium point.
or any motion that repeats itself after an interval of time is called vibration or oscillation. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.
The swinging of a pendulum and the motion of a plucked string are typical examples of vibration. The study of vibration deals with the study of oscillatory motions of bodies and the forces associated with them.
Vibratory System consists of:
1) Spring or elasticity
2) Mass or inertia
3) Damper
Involves transfer of potential energy to kinetic energy and vice versa
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Why study vibration?
• Vibrations can lead to excessive deflections and failure on the machines and structures
• To reduce vibration through proper design of machines and their mountings
• To utilize profitably in several consumer and industrial applications (quartz oscillator for computers)
• To improve the efficiency of certain machining, casting, forging & welding processes
• To stimulate earthquakes for geological research and conduct studies in design of nuclear reactors
Imbalance in the gas or diesel engines
• Blade and disk vibrations in turbines
• Noise and vibration of the hard-disks in your computers
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• Vibration testing for electronic packaging to conform International standard for quality control (QC)
• Safety engine: machine vibration causes parts loose from the body
Degree of Freedom (D.O.F.) = the number of independent co-ordinates required to describe the motion of a system.
or Minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time
Examples of single degree-of-freedom systems:
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Examples of two degree-of-freedom systems:
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Desirable “Good” (useful) vibration
o General industries – crushers, jackhammer, concrete compactor, etc.
o Medical and health – electric massage, high frequency vibration probe for heart disease treatment
o Motion of a tuning fork
o Cone of a loudspeaker
Undesirable “Bad” (unwanted) vibration
o Vibration is, wasting energy and creating unwanted sound -- noise.
o Poor ride comfort in vehicle due to road irregularities
o Sea sickness when traveling on ships, boats, etc.
o Earthquakes
o Fatigue failures in machine and structures
o Vibrational motions of engines, electric motors, or any mechanical device in operation Such vibrations can be caused by imbalances in the rotating parts
o Uneven friction
o Meshing of gear teeth
o Careful designs minimize unwanted vibrations. Sound and vibration are closely related. Sounds, or ―pressure waves‖, are generated by vibrating structures (e.g. vocal cords); these pressure waves can also induce the vibration of structures (e.g. ear drum). Hence, when trying to reduce noise it is often a problem in trying to reduce vibration
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Classifying Vibrating Systems
Lumped parameter models
Distributed parameter models
Describe a vibrating system as a ―concentrated‖ quantity (i.e. body mass, stiffness and/or damping).
When mass, stiffness and/or damping cannot be described as concentrated at a specified point.
Solve for the response using Ordinary Differential equations
Solve using partial differential equations
Rigid body vibration
Elastic body vibration
Vibration measurement terminology
(Some useful quantities)
For a complete cycle of sine wave, ( )
Where A is the amplitude of the displacement
ω =Excitation frequency rad/sec
Peak value indicates the maximum response of a vibrating part.
A=peak value
Max or peak value of displacement
Max or peak value of velocity
Max or peak value of acceleration ̈
Average value Indicates a steady or static value (somewhat like the DC level of an electrical current) and it is defined as
̅ ∫ ( ) ( )
Mean square value Square of the displacement generally is associated with the energy of the vibration for which the mean square value is a measure and is defined as
̅ 〈 〉 ∫ ( )
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Root mean square value (rms) is the square root of the mean square value.
√ ̅ [〈 〉]
Decibel (dB) it is a unit of the relative measurement of the vibration and noise. It is defined in terms of a power ratio:
( ⁄)
Where p is the power, since power is proportion to square of amplitude of vibrations
Octave: The octave is used for the relative measurement of the frequency. If two frequencies have ratio 2:1, the frequency span is one octave.
( )
Types of vibration
Free vibration
Free vibration takes place when a system oscillates under the action of forces inherent in the system itself due to initial disturbance, and when external impressed (applied) forces are absent
or (Free vibration occurs when a mechanical system is set off (provide) with an initial input (conditions) and then allowed the system to vibrate freely (respond).
or A system is left to vibrate on its own after an initial disturbance and no external force acts on the system
For examples pulling a child back on a swing and then letting go or hitting a tuning fork and letting it ring and simple pendulum
The system under free vibration will then vibrate at one or more of its "natural frequencies"(which are properties of the dynamic system established by its mass and stiffness distribution) and damp down to zero
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Forced vibrations
The vibration that takes place under the excitation of external forces
or Forced vibration a system that is subjected to a repeating external force. E.g. oscillation arises from diesel engines
If excitation is harmonic, the system is forced to vibrate at excitation frequency.
If the frequency of excitation (external force) coincides with one of the natural frequencies of the system, a condition of resonance is encountered and dangerously large oscillations may result, which results in failure of major structures, i.e., bridges, buildings, or airplane wings etc.
Thus calculation of natural frequencies is of major importance in the study of vibrations.
Because of friction & other resistances vibrating systems are subjected to damping to some degree due to dissipation of energy.
Damping has very little effect on natural frequency of the system, and hence the calculations for natural frequencies are generally made on the basis of no damping.
Damping is of great importance in limiting the amplitude of oscillation at resonance.
Forced vibration continuous forcing the system by an oscillating force or motion
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Damping vibrating systems are all subjected to damping to some degree because energy is dissipated by friction or other resistance
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Damped system any energy is lost or dissipated in friction or other resistance during oscillations
Undamped system
No energy is lost or dissipated in friction or other resistance during oscillations
Types of Damping
o Air Damping
o Coulomb dry Friction
o Fluid Friction
o Internal damping
o Magnetic damping
Seismic instruments Vibratory systems consisting from the support or the base and the mass with spring attached, the base is attached to the body whose motion is to be measured .the relative motion between the mass and the base recorded by a rotating drum or some other devices inside the instrument will indicated the motion of the body.
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Types Seismic instruments
o Vibrometer used to measure displacement
o Accelerometer used to measure acceleration
o Seismographs measures earthquake vibration
o Torsiograph are used to record torsional vibration
Modelling
A process of developing equations or sets of equations to describe motions of physical systems
Linear Vibration When all basic components of a vibratory system, i.e. the spring, the mass and the damper behave linearly
Nonlinear Vibration: If any of the components behave nonlinearly
Oscillating Motions
The study of vibrations is concerned with the oscillating motion of elastic bodies and the force associated with them.
All bodies possessing mass and elasticity are capable of vibrations.
Most engineering machines and structures experience vibrations to some degree and their design generally requires consideration of their oscillatory motions.
Oscillatory systems can be broadly characterized as linear or nonlinear.
Oscillatory motion repeat itself regularly for example pendulum of a wall clock
Oscillatory motion display irregularity for example earthquake
Periodic Motion
This motion repeats itself at equal (constant) interval of time T.
Examples sine or cosine functions sin (ωt) and cos(ωt)
ω = radian frequency (rad/sec)
, where f is frequency (Hz)
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Frequency It is defined reciprocal of time period.
The condition of the periodic motion is ( ) ( )
Period of Oscillatory
The time taken for one repetition is called period.
or Period = time between two adjacent peaks or valleys;
Time delay (𝛕) the time required for a wave to travel between two points in space.
Non-periodic motion: Any motion whose characteristics don't repeat at regular intervals
Deterministic Vibration: If the value or magnitude of the excitation (force or motion) acting on a vibratory system is known at any given time
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Nondeterministic or random Vibration: When the value of the excitation at a given time cannot be predicted
Examples of deterministic and random excitation:
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Simple Harmonic Motion SHM ( ) ( ) ( ) ̇( ) ( ) ( ) ̈( ) ( ) ( )
Simplest form of periodic motion is harmonic motion and it is called simple harmonic motion (SHM). It can be expressed as
Where A is the amplitude of motion, t is the time instant and T is the period of motion, ψ is the phase angle. Phase shift is any change that occurs in the phase of one quantity, or in the phase difference between two or more quantities
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Example A harmonic motion has amplitude of 0.20 cm and a period of 0.15 sec. Determine the maximum velocity and acceleration.
Solution:
The frequency is given as
Maximum velocity is given as ̇
Maximum acceleration is given as ̈ ( )
Example A harmonic motion has a frequency of 10 cps (cps = cycles per second=Hz) and its maximum velocity is 4.57 m/s. Determine its amplitude, its period, and its maximum acceleration.
Solution: ( ) ̈ ( )
Torsional spring stiffness
T = Kθ , T = torque θ = twist angle
K= Torsional spring stiffness K=N.m/rad
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Types of Spring Element
1. Linear spring element 2.Torsional spring element
Force-Deflection Relation of Linear spring element
L = undeformed length (free length)
x = deflection due to applied load, k = slope of force/deflection
Torsional spring Where; T = torque θ = twist angle
Spring stiffness depends on:
Material properties (E, G, υ)
Mechanical properties (A, L, d, w, v, h)
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Equivalent spring constant, eq for springs in parallel and series
1-Parallel Arrangement
Assume same deflection, x for both spring elements
During static deflection condition, perform, Σ Σ
2-Series Arrangement
During static deflection / equilibrium, the deflection, x of both springs are not the same unless k1 = k2, but both springs experience the same force.
Therefore we have, Σ Σ
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Damping elements
1. Force generated between solid surface and fluid (viscous damping). Linear relationship of force and velocity. E.g.: Automobile shock absorber.
2. Force generated between 2 solid surfaces (frictional damping). Nonlinear characteristics. E.g.: Magnetorheological (MR) damper, Electrorheological (ER) damper, building rubber damper.
Viscous damper
Linear model where damping force is proportional to relative velocity ̇
= damping force;
c = damping coefficient;
̇ relative velocity across damper
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Equivalent damping constant, Ceq for dampers in parallel and series
1-Parallel arrangement
1. Force applied such that the velocities across damper are the same
2. During static deflection condition, perform, Σ
3. For ―n‖ dampers;
Σ
2-Series arrangement
1. During static deflection / equilibrium, the deflection, both dampers experience same force but with different relative velocities across the dampers.
2. Therefore the velocity difference is given by;
3. For ―n‖ dampers;
Σ Σ
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Example model the system shown in Fig by a block attached to a single spring of an equivalent stiffness.
Solution Replace the parallel combinations by springs with equivalent stiffness springs as shown in Fig. (a)
Left springs in series as shown in Fig. (b)
Right springs in series as shown in Fig. (b)
Springs in the two sides of the block are parallel as shown in Fig. (c)
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Example A reciprocating engine is mounted on a foundation as shown in Fig. The unbalanced forces and moments developed in the engine are transmitted to the frame and the foundation. An elastic pad is placed between the engine and the foundation block to reduce the transmission of vibration. Develop two mathematical models of the system using a gradual refinement of the modeling process.
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Example Figure shows a human body and a restraint system at the time of an automobile collision Suggest a simple mathematical model by considering the elasticity, mass, and damping of the seat, human body, and the restraints for a vibration analysis of the system.
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Single degree of freedom (SDOF) Undamped Free Vibration Systems
Translational System
Solving using Newton’s Law of Motion Σ ̈
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Free Body Diagram: Static Case
δst: Static deflection due to the weight of the mass acting on the spring
Note: For small angles sinθ ≈ tanθ ≈ θ , cosθ ≈ 1
Free Body Diagram: Dynamic Equilibrium
Free Body Diagram: Dynamic case
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Apply Newton’s 2nd Law Σ ̈ ( ) ̈
Equation of motion (EOM) ̈
2nd order differential equation homogenous linear constant coefficients
Forms of solution ( ) ( ) ( ) ( ) ( )
Where phase angle
Assume ( ) ̇( ) ̈( ) ( )
For non-trivial solution
where equation called Characteristic Equation (c/cs eqn)
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√ √ ( ) ( ) √ √
where and are arbitrary constants to be determined from the initial conditions.
Recall Euler’s identity: ( ) ( ) ( ) . √ √ / . √ √ / ( ) ( ) √ ( ) √ ( ) √ √ ( )
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√ ( ⁄) ( ) ( ⁄ )
Rotational System- a pendulum system
̈ Σ ̈ ( ) ̈ √
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Example
A circular disc of moment of inertia J is attached to the lower end of an elastic vertical shaft If the mass of the shaft is small. and the shaft has torsional stiffness K, derive the differential equation of motion for the free torsional vibration of the disc. What is the natural frequency?
The Energy method ̇ ( ) ( ̈ ̇ ̇) ̇
̈ √ ⁄
K =Torsional stiffness (N.m/rad) and = angle of Twist (rad)
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SDOF Damped Free Vibration Systems
Viscous Damping Element (Dashpot)
Damping Force is Linear and Proportional to Velocity.
Case study: given an initial condition, determine the resulting motion
Maintain Dynamic Equilibrium
Apply Newton’s 2nd Law Σ ̈
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Σ ̈ ( ) ̇ ̈
Equation of motion ̈ ̇
Equation called characteristic (C⁄ Cs) equation.
2nd order Differential equation homogenous linear constant coefficients
Form of solution: ( ) ( ) ( )
assume ( ) ̇( ) ̈( )
substitute in Equation of motion ( )
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For a non-trivial solution
1st method √( )
Critical damping unit of CC = N.s/m ( ) √ √
damping ratio or damping factor (dimensionless)
2nd method √ ( )
A1 and A2 are determined from initial conditions.
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For critical damping Consider a case when,
Solving for c: √
Define √ ⁄ natural frequency , damping ratio √(
) * √
+ No damping -The system oscillates at its natural resonant frequency (ωo)
Critical damping (Transition between oscillatory & Non-oscillatory motion)
Over damping (Non-oscillatory motion)
Under damping (Oscillatory motion)
Case 1: ζ=1 critically damped (Real equal roots)
∴ ( ) ( ) ∴ ( ) ( )
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Fig.3.5 ( ) ̇( ) ( ) * ( ) ( ̇( ) ( )) +
Case 2: ζ>1 Over- damped [Real unequal (distinct) roots]
( ) ∴ ( ) ( √
) ( √
)
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Case 3: ζ< 1 under damped (Complex conjugate roots)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) √ ( ) | | √( ) ( )
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Damping ratio (for many structural materials)
Logarithmic Decrement, δ
A method to experimentally measure damping ratio in under damped systems. In practice, damping coefficient is difficult to estimate. To measure mass and stiffness only require static test. However, in estimating damping requires a dynamic test. Example: Estimating damping of an aircraft wings.
By definition, logarithmic decrement can be calculated as the natural log of the ratio of 2 successive amplitudes.
Vibration Isolation
In many industrial applications, one may find the vibrating machine transmit forces to ground which in turn vibrate the neighboring machines. So in that
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contest it is necessary to calculate how much force is transmitted to ground from the machine or from the ground to the machine.
In order to reduce the amount of force transmitted to the foundation due to the vibration of machinery ,machines are isolated from the foundation by mounting them on springs And dampers.
Transmissibility ( )
The amplitude Ratio of the max transmitted force to the floor (foundation) to the max excitation force (impressed-input force) √ ( ) √( ( ) ) ( ) √ ( ) √( ) ( ) √
Problem A precision grinding machine is supported on an isolator that has a stiffness of 1 MN/m and a viscous damping constant of 1 kN-s/m. The floor on which the machine is mounted is subjected to a harmonic disturbance due to the operation of an unbalanced engine in the vicinity of the grinding machine. Find the maximum acceptable displacement amplitude of the floor if the resulting amplitude of vibration of the grinding level is to be restricted to 10-6 m. Assume that the grinding machine and the wheel are a rigid body of weight 5000 N.
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Problem A reciprocating engine of mass m1 is mounted on a fixed-fixed
beam of length l, width a, thickness t, and Young’s modulus E, as shown in
Fig. A spring mass system (k2, m2) is suspended from the beam as indicated
in the figure. Find the relation between m2 and k2 that leads to no steady-state
vibration of the beam when a harmonic force, F1 (t) = F0 cosωt, is developed
in the engine during its operation.
** Note: The spring-mass system (k2, m2) added to make the amplitude of the
first mass zero is known as ―vibration absorber.‖
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Dynamic Vibration Absorber
It is a simply a single-degree of freedom system, usually in the form of a simply spring-mass system. When added to another single-degree of freedom system as an auxiliary system, it will transform the whole system into 2DOf with two natural frequencies of vibration. One of the natural frequencies is set above the excitation frequency while the other is set below it, so that the main mass of the entire system will have very small amplitude of vibration instead of very large amplitude under the given excitation.
̈ ( )
̈ ( ) ̈ ̈ ( )
( )
( ) EOM #1
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( ) EOM #2 * ( )+, - , - | | ( )( )
∵ , ( )( ) ( )( )
Using Crammer rule | | ( ) | |
For dynamic absorber ( ) ( )
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Example Imagine that you are a Vibrations Engineer working for Thi-Qar
University power plant company, where you are investigating the properties of a foundation that will be used to support an electric motor of weight mg =500 N. Your boss wants you to identify the following for the foundation:
a) The nature of damping provided by the foundation.
b) The damped and undamped natural frequencies of the motor/foundation combination.
c) The foundation stiffness and damping.
Thus you perform a free vibration tests whereby the motor—supported by the particular foundation—is released from rest with an initial displacement x0 = 8 mm and the subsequent response is measured. The response is shown in Figure.
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Whirling of rotating shafts
Rotating shafts tends to bow out at a certain speed and whirl in a complicated manner. Whirling is defined as the rotation of the plane made by the bent shaft and the line of the centre of the bearing. It occurs due to a number of factors, some of which may include (i) eccentricity, (ii) unbalanced mass, (iii) gyroscopic forces, (iv) fluid friction in bearing, viscous (hysteresis) damping.
Consider a shaft AB on which a disc is mounted at S. G is the mass center of the disc, which is at a distance e from S. As the mass center of the disc is not on the shaft center, when the shaft rotates, it will be subjected to a centrifugal force. This force will try to bend the shaft. Now the neutral axis of the shaft, which is represented by line ASB, is different from the line joining the bearing centers AOB. The rotation of the plane containing the line joining bearing centers and the bend shaft (in this case it is AOBSA) is called the whirling of the shaft.
Whirling of rotating shafts rotation: of the plane made by the bent shaft and the line of centers of the bearings
Cause: mass unbalance, hysteresis damping in the shaft, gyroscopic forces, fluid friction in the bearings, etc.
Note: Whirling and shaft rotation are not necessarily synchronous (they are not necessarily in the same direction or of the same magnitude).
S=contact point between disc and shaft
G = mass center of the disc
e=distance between S and G
O= center of line joining the bearing centers of shaft
r=distance between S and O
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Considering unit vectors i , j , k, the acceleration of point G can be given by [( ̈ ̇ ) ( )] [( ̈ ̇ ̇) ( )]
Assume: the restoring force of the shaft
Assuming a viscous damping acting at = ̇ ̇
The equation of motion in radial direction [i-direction] [ ̈ ̇ ( )] ̇
The equation of motion in angular direction- [j-direction] [ ̈ ̇ ̇ ( )] ̇
Rearranging ̈ ̇ [ ̇ ] ( ) ̈ ( ̇) ̇ ( )
These 2 equations represent the general case for whirling of rotating shafts.
We refer to whirl as being synchronous if: ̇ ̈ ̈ ̇
Integrating ̇ gives: where is the phase angle between e and r and is constant.
Equation of motion: [ ] ( )
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( ) ( ) ( ) ( ) ( ) ( )
Fig.5.35 ( ) ( ) * + √( ) ( ) √( ) ( ) √( ) ( ) ( ) √( ( ) ) ( ) [( ) ( ) ] ⁄ ( ) √
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Note: If (resonance) then and the system is restrained only by damping.
The eccentricity line e = SG leads the displacement line r = OS by phase angle ϕ which depends on the amount of damping and the rotation speed ratio When the rotational speed equals to the natural frequency or critical speed, the amplitude is restrained by damping only. From equation (iv) at very high speed ω≫ωn, ϕ 180oand the center of mass G tends to approach the fixed point O and the shaft center S rotates about it in a circle of radius e.
