Vibration Analysis of A Cantilevered Beam

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    i

    Table of ContentsTable of Figures ................................................................................................................................... ii

    Table of Graphs ................................................................................................................................... ii

    A. Literature Review ........................................................................................................................ 1

    A.1 - Vibration TransducersDetecting & Measuring Vibrations ..................................................... 1

    A.1.1 - Velocity Pickups ................................................................................................................ 1

    A.1.2 - Acceleration Pickups ......................................................................................................... 2

    A.1.3 - Displacement Probes ........................................................................................................ 3

    A.2 - Use of Frequency Spectral Analysis & Frequency Response function of a System ..................... 4

    A.3 - Vibrations Induced by Rotary Machines ................................................................................... 6

    A.3.1 - Force Induced Vibration.................................................................................................... 6

    A.3.2 - Structural Vibration ........................................................................................................ 10

    A.4 - Distinguishing between Force-Induced and Structural Vibrations .......................................... 11

    A.4.1 - Coast-down/ Run-up Test (Internal Excitation) ................................................................ 11

    A.4.2 - Bump Test (External Excitation) ...................................................................................... 11

    A.4.3 - Shaker Test ..................................................................................................................... 11

    A.5 - Cepstrum and Envelope Analyses .......................................................................................... 12

    A.5.1 - Cepstrum Analysis (aka Cepstrum Alansys) ..................................................................... 13

    B. Fault Diagnosis .......................................................................................................................... 14

    C. Theoretical Design ..................................................................................................................... 18

    C.1 - Pre-Isolation .......................................................................................................................... 19

    C.1.1 - Static Equations .............................................................................................................. 19

    C.1.2 - Finding Equivalent Mass of System ................................................................................. 20

    C.1.3 - Finding Equivalent Stiffness of Cantilever Beam .............................................................. 20

    C.1.4 - Finding Un-damped Natural Frequency of System ........................................................... 22

    C.1.5 - Forced Vibrations of a Cantilever Beam due to Unbalance .............................................. 23

    C.1.6 - Calculating Maximum Bending Stress .............................................................................. 29

    C.2 - Post-Isolation ........................................................................................................................ 30

    C.2.1 - Finding Equivalent Stiffness of New Structure ................................................................. 31

    C.2.2 - Finding Equivalent Mass of New Structure ...................................................................... 31

    C.2.3 - Finding Equivalent Damping Coefficient of New Structure............................................... 32

    C.2.4 - Finding Un-Damped Natural Frequency of New Structure ............................................... 32

    C.2.5 - Finding Damping Ratio of New Structure......................................................................... 32

    C.2.6 - Transmissibility (Vibration Isolation) ............................................................................... 33

    C.3 - Natural Frequencies of Unloaded Cantilever Beam ................................................................ 36

    C.4 - Natural Frequencies of Loaded Cantilever Beam .................................................................... 40

    Group Effort ...................................................................................................................................... 43

    References ........................................................................................................................................ 44

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    Table of FiguresFigure 1 - Magnet-in-Coil (left) & Coil-in-Magnet (right) type transducers. [1] ...................................... 1

    Figure 2 - Diagram showing a basic setup of a 3-axis accelerometer. [13] ............................................ 2Figure 3 - General method adopted to obtain the Velocity Spectrum [2] ............................................. 4

    Figure 4 - Application of the Fourier transforms [2, p. 178] .................................................................. 5

    Figure 5 - The relationship between Time, Frequency and Amplitude [3] ............................................. 5

    Figure 6 - Mass eccentricity causing a rotating imbalance [5] ............................................................... 6

    Figure 7 - FFT analysis of an unbalance defect [4] ................................................................................ 6

    Figure 8 - Types of shaft misalignments [6] .......................................................................................... 8

    Figure 9 - FFT analysis for Parallel and Angular shaft misalignment [4] ................................................. 8

    Figure 10 - FFT analysis of roller bearing just before failure [4] ............................................................ 9

    Figure 11 - FFT analysis of loose journal bearings [4] ........................................................................... 9

    Figure 12 - Synchronous or DC motor vibrations due to electrical problems [4] (FLElectrical LineFrequency) ........................................................................................................................................ 10

    Figure 13 - Causes and effects of Structural Vibration [7] ................................................................... 10

    Figure 14 - Carrier & Envelope Frequency [8]..................................................................................... 12

    Figure 15 - Second Modulation; Carrier and Envelope Signal ............................................................. 13

    Figure 16- Schematic of motor-fan mechanical system ..................................................................... 14

    Figure 17 - The Frequency spectrum of a gearbox.............................................................................. 15

    Figure 18 - Setup used for initial calculation ...................................................................................... 18

    Figure 19 - Simplified setup ............................................................................................................... 19

    Figure 20 - Illustration of masses ....................................................................................................... 20

    Figure 21 - Equivalent kinematic diagram .......................................................................................... 23

    Figure 22 - Illustration of rotating imbalance ..................................................................................... 24

    Figure 23 - Illustration of the isolated structure ................................................................................. 30

    Figure 24 - Kinematic representation of the isolated structure .......................................................... 33

    Figure 25 - The deflected cantilever beam, under its own weight ...................................................... 36

    Figure 26 - The deflected cantilever beam, under its own weight and those of the edge loads .......... 40

    Table of GraphsGraph 1 - Vibration amplitude frequency spectrum of motor fan system ........................................... 14

    Graph 2 - Plot of the frequency response function for different values of

    ....................................... 28

    Graph 3 - Plot of the phase shift against the frequency ratio for different values of ........................ 28Graph 4 - Magnitude of transmitted vibrations against frequency ratio, for different values of ....... 35Graph 5 - Magnification factor against excitation frequency for different natural frequencies of the

    unloaded system ............................................................................................................................... 39

    Graph 6 - Plot of the magnification factor against the excitation frequency for different values of the

    natural frequency of the loaded structure ......................................................................................... 42

    http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987447http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987464http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987473http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987473http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987473http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987474http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987474http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987474http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987476http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987476http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987476http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987476http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987474http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987473http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987464http://c/Users/Racer/Dropbox/Year%203%20-%20Semester%202/Vibrations%20Analysis/Vibrations%20Assignment/FINAL/Final%20Report.docx%23_Toc356987447
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    A.Literature ReviewA.1 - Vibration TransducersDetecting & Measuring Vibrations

    A transducer is a device that converts energy from one form to another. In the case of

    vibration transducers, these convert the kinetic energy from the vibrating object into an electric

    signal, which can then in turn be processed in order to accurately analyse the nature of the

    vibrations. There are three main types of vibration transducers, each suited to a different

    measurement requirement, these being:

    - Velocity Pickups- Accelerometers- Proximity Sensors

    A.1.1 - Velocity Pickups

    Velocity pickups work on the principle of electromagnetism, that is, when a conductor is

    moved inside a magnetic field, a voltage is generated across that conductor. These types of sensors

    come in two main types, Coil-in-Magnet & Magnet-in-Coil. The difference between the two is in

    which element is kept stationary and which is allowed to vibrate.

    Figure 1 - Magnet-in-Coil (left) & Coil-in-Magnet (right) type transducers. [1]

    Due to the fact that the signal is self-generated (meaning that no external power supply is

    required to operate the sensor), these types of transducers are relatively cheap. They are also very

    easy to mount onto machinery, however they do have some limitations. As shown in Figure 1, the

    moving element of the sensor is constrained to move in only one axis, this means that one sensor can

    measure vibrations in only one axis. Moreover, cross-axis vibrations can also prove to be damaging to

    these transducers, so care must be taken before installing them on machinery.

    Velocity pickups are usually used to measure vibrations in the range of between 10 Hz to 1

    kHz, and while having a narrow frequency response, they give a relatively strong output within said

    range. This serves to ameliorate its immunity to electrical noise.

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    A.1.2 - Acceleration Pickups

    Also known as accelerometers, these transducers work on the phenomenon of

    piezoelectricity. In essence, the mechanism consists of a small mass encased inside a piezoelectric

    crystal (Figure 2). When the sensor is subjected to vibrations the mass is given an acceleration,causing it to impinge on the crystal. The force of the mass on the crystal causes deformation of the

    crystals lattice, leading to the generation of a charge. The charge generated, measured in [Pico

    Coulombs per g] is proportional to the vibratory force experienced by the accelerometer.

    The accelerometer is then accompanied by an amplifier, which can be either internal or

    external, that converts the output charge to a proportional voltage output.

    As mentioned above, two types of accelerometers exist. Those with an internal amplifier

    circuit, called Currentor Voltage-mode sensors. These types of sensors are limited in that they can be

    used in a restricted range of temperatures due to the internal circuitry. To cater for this limitation,

    sensors with external charge amplifiers, called Charge-mode accelerometers, are used. The charge

    signal is drawn from the accelerometer via two wires and amplified to give the final output signal.

    Accelerometers may be mounted onto machinery in a variety of ways. However the most

    ideal way, ideal meaning that it has the most secure attachment and the widest possible frequency

    response range, is by direct stud-mounting. In this way, the sensor is screwed onto the machine so

    that vibrations are transmitted directly.

    Other mounting methods include adhesive mounting, magnetic mounting and handheld

    probes. Each of these introduces its own disadvantages, but they all tend to narrow down the

    frequency response range for the sensor.

    The typical frequency response for an accelerometer that measures machine vibrations will

    lie in the range of 2 Hz10 kHz, notably larger than that for a velocity pickup.

    One pitfall for these piezoelectric sensors is that, although they are not subject to fatigue as

    they have no moving components, they cannot be recalibrated. This means that if the sensor has

    suffered damage due to inappropriate operating temperatures or misuse, it cannot be brought back

    into spec.

    Figure 2 - Diagram showing a basic setup of a 3-axis accelerometer. [13]

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    A.1.3 - Displacement Probes

    Also known as Eddy current (or inductive) transducers, they are used mainly on rotary

    machinery. The working principle if this sensor is as follows. The probe emits an electromagnetic

    field, with a frequency in the range of 2 MHz, which interacts with the shaft surface. This causes eddycurrents to be generated on the shafts surface and these same eddy currents are read back via

    sensing circuitry. Given that the shaft is made of a homogeneous material, its surface is adequately

    polished and that the input field has a constant frequency and magnitude, the variation of the output

    signal will describe the vibratory motion of the shaft.

    The probes are generally mounted at right angles to the shaft in one of three different configurations.

    i) They can be internally mounted, meaning that the probe is mounted directly inside thejournal bearing housing via a special bracket. The advantages are that the probe has an

    unconstrained view of the shaft surface. However the probe is inaccessible while the

    machine is in operation.

    ii) They can be mounted using an external adaptor that fits onto the bearings. The probeitself is still located inside the bearing. This allows for access to the probe while in

    operation while still keeping an unconstrained view of the shaft. However the probe

    would have to be lengthened which may give rise to resonance problems.

    iii) The final configuration is that of an external mounting. This should be only a last resortfor when there is absolutely no way of mounting the probe internally. Even though it is

    the least expensive method, the readings will not be as accurate as it may be influenced

    by the electrical & mechanical run-out of the shaft itself.

    For this probe to work, the shaft material needs to be a conductor and moreover, the probe

    has to be calibrated depending on what type of conductor the shaft is made of, as different materialsgive different responses to inductive fields.

    The gap between the probe tip and the shaft surface is set depending upon the operating

    voltage of the probe itself. However for a 12 V supply, the gap is usually around 1.5mm. [1]

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    A.2 - Use of Frequency Spectral Analysis & Frequency Response function of a System

    Various methods are available to analyse frequencies and as described above, each method

    results in different frequency spectra which affects the ease of analysis. The techniques used to

    collect data vary from one analyst to another depending on the level of detail required by theresearcher. One common method is the overall level data in which the summation of the vibration

    amplitude over a wide range of frequencies is measured; hence a single value for the overall vibration

    magnitude is obtained. This method is very cheap and easy to do unlike the high frequency

    diagnostics, such as shock pulse and ultrasonic energy, which are more expensive and complex to

    carry out. The user is able to detect early signs of bearing wear, however they suffer to detect lower

    frequency machinery defects such as misalignment. The most commonly used method to analyse the

    frequency response of a system, is the narrow band technique.

    The narrow band techniqueis carried out by researchers to detect the causes of vibrations

    such as imbalance, misalignment, clearance issues and resonance, so as to eliminate or minimise the

    effect of these causes as much as possible. In reality, the researchers produce an acceleration signalwhich is received by an instrument and covert it to a velocity signal. The velocity signal can either be

    displayed as a velocity wave form (in the time domain), or as velocity spectrum (in the frequency

    domain). The velocity spectrum is obtained by applying the Fast Fourier Transform method on the

    velocity waveform. The Fast Fourier Transform technique is a mathematical operation that extracts

    the frequency information from a time domain signal and converts it to the frequency domain.

    Figure 3 - General method adopted to obtain the Velocity Spectrum [2]

    For the frequency spectral analysis, the measured vibrations are transformed form the timedomain into discrete frequency components after applying the Discrete Fourier transform technique.

    To change from the frequency domain to the time domain, the inverse Fourier transform is

    applied. This is further explained inFigure 4.

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    Figure 4 - Application of the Fourier transforms [2, p. 178]

    All input signals can be represented as a group of cosine waves. Each cosine wave has acertain amplitude and phase shift. In both the frequency domain graph and the time domain graph,

    the height of the peak represents the amplitude of the signal (Figure 5). In the frequency domain

    analysis, all running speeds may be analysed whereas in the time domain analysis, only the actual

    running speed can be analysed.

    Figure 5 - The relationship between Time, Frequency and Amplitude [3]

    Distortions are observed when converting from one domain to another; hence theresearcher reduces noise by using either a low-pass digital filter or by multiplying the signal to a

    smooth curve, also known as the Hamming window. After multiplying the original signal to the

    Hamming window, the researcher can easily identify the required data which are the frequency,

    the amplitude and the phase.

    Once the signal is changed from the time domain to the frequency domain using FFT, the

    researcher may interpret the peaks obtained by calculating the shafts rotating speed and the

    frequencies that are being transmitted to all components in the system. Different types of

    defects have different harmonic patterns, frequency and amplitude expectations. The last step

    involves checking how severe the fault is and what can be done to eliminate or minimise the

    probability of this fault to occur. Once these values are obtained, the system is continuously

    monitored to make sure that the overall machine is in good condition.

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    A.3 - Vibrations Induced by Rotary Machines

    Vibrations induced by rotating machinery may be subdivided into two main categories; force-

    induced vibrations and structural vibrations. Force-induced vibrations occur in the presence of an

    excitation source, which is necessary for vibrations to be initiated and sustained. Examples ofexcitation sources include mass unbalance, shaft misalignment and loose journal bearings. On the

    other hand, structural vibrations or self-excited vibrations are initiated and sustained without the

    presence of a forcing phenomenon.

    Vibration in machines such as electric motors, rotary pumps and compressors, may indicate

    deterioration of the equipment. Failure-Mode Analysis is generally carried out to define the failure

    mode present and hence identify which component is degrading. Machine vibration signatures,

    including both Fast Fourier Transforms (FFT) and time traces, are essential for failure-mode analysis.

    A.3.1 - Force Induced Vibration

    As previously outlined, these vibrations may be induced by faults within several componentssuch as bearings, gears, shafts and electric motors (Table 1). The most common causes of failure may

    be identified by obtaining the relationship between the frequencies of induced vibrations to the

    frequency of the rotating shaft within the machine-train. The table overleaf is a vibration

    troubleshooting chart that pinpoints some of the most common failure modes.

    A.3.1.1-RotatingUnbalance

    Vibration due to the unbalance of a rotor is the most common fault present in rotary

    machines, and is also the easiest to detect and amend. The International Standards Organisation (ISO)

    defines unbalance as: That condition, which exists in a rotor when vibratory, force or motion is

    imparted to its bearings as a result of centrifugal forces [4]. A rotating unbalance occurs due to the

    presence of an uneven distribution of mass about the rotating axis of a rotor (Figure 6). This may becaused by manufacturing defects such as machining errors or maintenance issues including corrosion

    or deformation. The effects of imbalance greatly increase as the rotor speed increases, generating

    higher amplitude vibrations while severely reducing bearing life.

    Figure 6 - Mass eccentricity causing a rotating imbalance [5]

    For all types of unbalance, a predominant

    1 rpm frequency of vibration will be observed as

    shown on the FFT spectrum in Figure 7. The

    vibration amplitude at the 1 rpm frequency is

    generally always present and varies proportionally

    to the square of the rotational speed.

    Figure 7 - FFT analysis of an

    unbalance defect [4]

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    Table 1 - Vibration Troubleshooting Chart

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    A.3.1.2-MisalignmentinShafts

    Shaft misalignment may be subdivided into two: angular misalignment and parallel

    misalignment (Figure 8). Angular

    misalignment occurs when two shaftsmeet at an angle whereas parallel

    misalignment occurs when two parallel

    shafts are at an offset. The latter is also

    known as Offset Misalignment. Both

    misalignments may originate from

    assembly or develop over time due to

    thermal expansion or improper

    reassembly after maintenance. The

    resulting vibration may be radial, axial or

    both.

    Figure 8 - Types of shaft misalignments [6]

    Angular misalignments produce axial vibrations at the 1 rpm frequency while parallel

    misalignments generate 2 rpm vibrations in the radial direction. Since pure angular or parallel

    misalignments are rare, there will typically be high axial or radial vibrations at 1, 2 or 3 rpm as

    shown inFigure 9.Such results may also indicate faults in couplings.

    Figure 9 - FFT analysis for Parallel and Angular shaft misalignment [4]

    A.3.1.3-FaultsinBearings

    Faults in a roller bearing may occur in any of its four separate components namely; inner and

    outer races, cage and rolling elements. Faults in bearings cause high-frequency vibrations, which

    amplify the severity of wear. This results in a continuously changing vibration pattern. Faults on

    rolling elements or raceways are the most evident on an FFT spectrum (Figure 10).

    When a bearing starts to wear, minute pits are developed on the raceways. As rolling

    elements pass over these raceways, natural frequencies that predominantly occur in the 30120

    kcpm range are developed. At a later stage the minute pits present continue to grow into larger pits,

    until they merge together, spalling the passing rolling elements. By this time, the bearing is severely

    damaged and is vibrating excessively creating a lot of noise. The FFT spectrum of a bearing at this

    instance is shown inFigure 10.Bearing failure may occur due to inadequate lubrication, installation,

    age or excessive load caused by misalignment, rotating unbalance or a bent shaft.

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    Figure 10 - FFT analysis of roller bearing just before failure [4]

    (BPFO - Ball-Pass Outer-Race, BPFI - Ball-Pass Inner-Race)

    A.3.1.4-LooseJournalBearingsLoose components such as journal bearings

    create vibrations which may cause destructive

    damage, wear and fatigue in equipment mounts and

    other components.

    Journal bearings with high clearances usually

    display a series of running speed harmonics, which

    can be up to 10 or 20 rpm (Figure 11). Higher

    vibration amplitudes are generally induced with the

    presence of unbalance or misalignment.

    Figure 11 - FFT analysis of loose journal bearings [4]

    A.3.1.5-ElectricMotorVibration

    Electrical machines such as motors, generators and alternators may generate mechanically or

    electrically induced vibrations. Faults present within the electric motor such as a broken rotor bar and

    open windings of the rotor or stator, induce electrical vibrations. This results from unequal magnetic

    forces acting on the rotor or the stator.

    Due to the aforementioned electrical problems, a vibratory response of 1 rpm is produced,

    which will appear similar to a rotating unbalance. A technique used to differentiate between the two

    is to keep the analyser capturing the FFT spectrum in the livemode and turning off the power.

    Different vibratory responses exist for AC and DC motors. Loose stator coils in synchronous

    motors will generate fairly high vibrations due to the alternating forces present in the stator. These

    alternating forces are produced by the rotating magnetic field, which is being generated by the stator

    coils. The electric motor will vibrate at the coil pass frequency (CPF) which will be surrounded by 1

    rpm sidebands, as seen on the FFT spectrum inFigure 12.

    On the other hand, DC motor defects generate high vibration amplitudes at the SCR firing

    frequency (6FL) and harmonics (Figure 12). Broken field windings, loose connections and bad SCRs

    may all produce these vibrations.

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    Figure 12 - Synchronous or DC motor vibrations due to electrical problems [4] (FL Electrical Line Frequency)

    A.3.2 - Structural Vibration

    Structural vibration occurring in most machines is undesirable due to unpleasant motions,

    noise and dynamic stresses as well as reduction in performance resulting from energy losses.

    The frequency and amplitude of structural vibrations are dependent on the excitation applied

    and the structures response to that particular excitation. A variance in the applied excitation or the

    structure characteristics would stimulate different vibrations. External sources such as cross winds,

    waves, currents, ground vibration and earthquakes are the applied excitation forces, which may be

    periodic in time or random in nature. The structures response to the applied excitation is dependent

    on the location of the excitation and on the structures natural frequency, which is directly related to

    the stiffness and damping coefficient. Structural vibrations may cause impairment of function or

    failure of the structure, corrosion and environmental noise.Figure 13 outline the various causes andeffects of structural vibrations.

    Figure 13 - Causes and effects of Structural Vibration [7]

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    A.4 - Distinguishing between Force-Induced and Structural Vibrations

    A common way to identify resonances empirically is by means of run-up and run-down test. In

    these tests the machine is operated from standstill to maximum speed, across its range of operating speeds,

    and back down, while measuring the vibration it exhibits. The natural frequencies that the machine exhibits,either of the machine itself, or the supporting structure, may cause amplitude enhancement of forcing

    frequencies that can severely reduce component life and adversely affect operating parameters. When

    testing, if the natural frequencies are reached, discrete peaks will pop-up in the graph which are a clear

    indication that resonance exists at that point. There are three methods that can be used to understand and

    examine vibration in a structure or a machine.

    A.4.1 - Coast-down/ Run-up Test (Internal Excitation)

    This is a real time test as it records and analysis how the vibrations happening inside the machine

    vary with different operating speeds of the machine. After the test data is recorded, the engineer analysing it

    will establish critical and resonant speeds of the machine. The results can be presented in a variety of formats

    (such as: Band RMS v.s. RPM, waterfall plots, colour spectrograms) which enable the user to understand and

    characterize the critical or resonant speeds. Discrete peaks in the magnitude that are accompanied by a phase

    shift at the same frequency normally show that a natural frequency exists there. This means that the machine

    operator should avoid operating the machine at those frequencies or else the dynamics should be modified to

    adapt the resonance effect.

    A.4.2 - Bump Test (External Excitation)

    This test produces the same results as the run-up/ run-down tests but in a fairly crude way

    because it does not provide meaningful amplitude values (relative comparison only) as the input is not

    measured directly. In this method an external force has to be applied as the machine is switched off duringthe testing. It requires the engineer to apply a controlled bump or hit to the machine structure. When

    impacted, the structure produces a frequency band of excitation components and when these frequencies

    coincide with the structural natural frequencies, resonant conditions are present which result in higher than

    normal vibration levels. These high vibrations levels should be noted so that the machines normal operating

    conditions will be well away to avoid resonant responses that can cause very high and destructive vibration

    levels.

    A.4.3 - Shaker Test

    This test is accomplished by connecting a testing shaker to the structure. Different types of shakersare used according to the type of frequency. For high frequencies, an electro-dynamic shaker is used and for

    low frequencies a servo-hydraulic is used. This equipment is driven by a source signal from the analyser, which

    is used to excite the structure. The excitation is then transmitted to a load cell attached to the structure which

    provides a measure of the input force. The signal analyser then measures the input and output response

    simultaneously and plots magnitude and phase versus frequency which identifies the resonance.

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    A.5 - Cepstrum and Envelope Analyses

    These techniques are usually used in order to help identify faults which occur in rotating

    machinery. These methods are useful as they help to form a distinction between the oscillatory

    frequencies which occur during normal operation of the apparatus, and those frequencies whichoccur when damage is present in the system.

    In the example of a roller bearing fault, an impact is produced each time the components

    come into contact with the deformed area. This impact is analogous to an impulse in a vibrating

    system, and causes a vibration to occur, followed by a series of oscillations of regressing amplitude

    until the energy from the impulse is dissipated. These impacts superimpose upon the normal

    vibration signal and an amplitude modulation signal is developed.

    Envelope analysis consists of tracing the frequency of the crests of the occurring sinusoidal

    oscillations of the structure and forming an envelope.[8] The Envelope is formed by an amplitude

    demodulation of the frequency obtained. The demodulation is simply a band pass filtering of thecarrier frequency, which reveals the envelope. InFigure 14 the envelope is clearly shown as the red

    line. [9]

    Figure 14 - Carrier & Envelope Frequency [8]

    The carrier frequency explains the vibration which is transferred from the faulty part to the frameof the structure being tested. This occurs due to the effect of resonance of some of the fault

    frequencies, as they reach a factor of the natural frequency of the structure.

    This carrier acts as the Primary Modulation of the fault signal. There is then the second modulation

    which forms the envelope. This is the actual aforementioned band pass filtering which forms the

    envelope frequency. This envelope frequency directly shows the frequency of the fault as it occurs in

    the machinery.Figure 15 shows the stated fault frequencies defined by the envelope signal; one for a

    fault in the inner race of a bearing and one for a fault on the outer race. [9]

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    Figure 15 - Second Modulation; Carrier and Envelope Signal

    A.5.1 - Cepstrum Analysis (aka Cepstrum Alansys)

    The development of Cepstrum analysis was done in 1963; the aim being to distinguish

    between ground vibrations caused by earthquakes and those caused by nuclear explosions. The

    System was then adapted for audio signal processing, where speech components could be identified

    and distinguished from all other sounds within a recording [10]. The Cepstrum is preferred over

    Spectrum because it senses and clearly defines any periodicities in the wave being processed, by

    displaying an amplified signal at the repeated frequency. Periodicities in mechanical or

    electromechanical components are usually caused by defects in gears and bearings and thus they can

    easily be identified. Cepstrum analysis is also beneficial because the periodic signals cannot becancelled out by any harmonics or sidebands, however in spectrum analysis this may occur. [10]

    Mathematically, the Cepstrum is defined as the inverse Fourier transform of the logarithm of

    the Autospectrum, and it is in the (lag) time domain. It can be considered as the Spectrum of the

    Autospectrum. The logarithm attenuates the lower levels of the Autospectrum, meaning that the

    Cepstrum would display these lower levels while the Autospectrum would not. Thus a fault which

    creates a low amplitude of vibration is easier to identify using Cepstrum analysis.

    In relation to the Autospectrum, the Cepstrum ( in the time domain can be equated as: = ()

    Where is the autospectrum in the frequency domain, and is the inverse Fourier transform.To show the references between the terms used in

    spectral and cepstral analysis and to distinguish them for the

    different types of analysis, the following terms (Table 2) are

    used to describe functions of the analysis. [9] [11]

    Table 2 - Jargon used in Cepstrum analysis

    Spectrum Cepstrum

    Frequency Quefrency

    Harmonics Rahmonics

    Low-pass filter Short-pass Lifter

    High-pass filter Long-pass Lifter

    Phase Saphe

    Magnitude Gamnitude

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    B.Fault DiagnosisThe task given for this part was to identify the causes of vibrations occurring in the machinery

    setup shown inFigure 16. The system is described to be a three bladed extractor fan being driven by

    a 50Hz Ac electric motor. The electric motor runs at 1200rpm, and drives the fan via a 2 gearreduction system, having a 20 tooth gear driving an 80 tooth one. Thus the fan rotates at the speed

    of the motor, i.e. at 300 rpm. The shafts of the 2 gears rotated on journal bearings, which we

    assumed to be mounted on pillow-blocks.

    Figure 16- Schematic of motor-fan mechanical system

    The vibration frequency spectrum derived from an accelerometer reading is shown inGraph 1

    and the peaks to be investigated were marked.

    Graph 1 - Vibration amplitude frequency spectrum of motor fan system

    Initially we converted all given values to frequency to be able to compare these values with

    the peaks shown in Figure 2. It was stated that the motor is rotating at 1200 rpm, hence this value

    was changed to frequency to determine the frequency transmitted to gear 1 using the equation

    below. = 2

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    Our motor rotates at 1200 rpm, i.e. 1200 2 per 60 seconds, which is equal to 20 2radians/s20 2 = 2 = Hence, the frequencyof the rotations of the motor and of the driver gear is 20. Since the

    reduction is 20 teeth to 80 teeth, the driven gear rotates at 14 the frequency of the driver gear. =204 =

    Hence, the frequency of rotation of the driven gear, the second shaft and the fan is equal to 5.A three-bladed fan is attached to the second shaft hence; the blade pass frequency was

    calculated to check whether vibration peaks would occur at this particular frequency. = . = 3 5 = A gearbox experiences ongoing rotation of the gears which causes both normal low-

    frequency harmonics and high frequency harmonics to occur due to the gear teeth and bearing

    impacts. The spectrum of a gear box is as shown inFigure 17.

    Figure 17 - The Frequency spectrum of a gearboxThe GMF (gear mesh frequency) is the product of the number of teeth of a pinion or a gear,

    and its respective running speed.

    = . = 20 20 = Some common types of defects that are commonly found in gears are gear tooth wear, gear

    tooth load, gear eccentricity and backlash, gear misalignment, broken or cracked gear tooth andhunting gear tooth problems. In this case, the phase factor was not given hence we couldnt check

    whether there are going to be any vibration due to the hunting gear tooth problem.

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    As explained in detail inA.3 - Vibrations Induced by Rotary Machines,all components in the

    system may contribute to the occurrence of vibrations. After analysing all the frequency spectra

    graphs of each component, we used the values calculated above to identify which faults might

    contribute to the occurrence of the frequency peaks shown inGraph 1.Our deductions are shown in

    Table 3 overleaf.

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    Characteristic/FaultPeaks in Hz

    5 15 20 40 50 100 300 400 800

    Shaft 1

    Mass Unbalance

    Bent Shaft

    Motor

    Eccentric rotor

    Stator Eccentricity, Shorted

    Laminations & Loose Iron

    Eccentric air gap

    Shorted Rotor / Cracked Rotor bars

    Phasing Problems

    Looseness in Winding slots, Iron, End

    Turns or Connections

    Mechanical Looseness

    Internal Assembly Looseness

    Looseness of system to base plate and

    bearings

    Structure Looseness

    Misalignment

    Angular Misalignment

    Driver gear- 20 tooth Gear

    Tooth load

    Gear tooth wear

    Gear Misalignment

    Cracked or broken gear tooth

    Gear Eccentricity

    Shaft 2Mass Unbalance

    Bent Shaft

    Eccentric Fan

    Overhung load

    80 Tooth Gear

    Tooth load

    Gear tooth wear

    Gear Misalignment

    Cracked or broken gear tooth

    Gear Eccentricity

    Mechanical Looseness

    Internal Assembly Looseness

    Looseness of system to base plate and

    bearings

    Structure Looseness

    Misalignment

    Angular Misalignment

    Three bladed fan

    Blade pass and vane pass vibrations

    Flow Turbulence

    Eccentric Blade

    Table 3 - Cause analysis of recorded vibration

    [15][16]

    [17]

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    C.Theoretical DesignFor continuous system vibration, the forced vibration of a cantilever beam was considered, as shown

    inFigure 18.

    Figure 18 - Setup used for initial calculation

    Where is the mass per unit length of the cantilever beam is the applied harmonic force due to unbalanceis the length of the cantilever beam

    is the mass of the motor, which is providing the applied harmonic force

    is the mass of the accelerometeris the equivalent stiffness of the cantilever beamis the equivalent damping coefficient of the damperFor analysis purposes, the distributed mass of the cantilever beam was replaced by a concentrated

    load at the edge of the cantilever by using the equation,

    = 0.2427

    Eqn. 1

    Where is the mass of the beam as a concentrated load in is the distributed mass of the cantilever beam given in /is the length of the cantilever beam

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    Hence the system may now be adapted to the one inFigure 19,

    C.1 - Pre-Isolation

    C.1.1 - Static Equations

    : = 0Eqn. 2

    : = 0 = + + Eqn. 3 : + + = 0

    = + + Eqn. 4

    Figure 19 - Simplified setup

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    C.1.2 - Finding Equivalent Mass of System

    Figure 20 - Illustration of masses = + + Eqn. 5

    C.1.3 - Finding Equivalent Stiffness of Cantilever Beam

    A static analysis of the structure was done in order to derive the equivalent stiffness of the cantilever

    beam. Sectioning the cantilever beam,

    For 0 < < , : + = 0Substituting for and fromEqn. 3 andEqn. 4 yields, = + + + + UsingEqn. 5,

    = Given that, = Where is Youngs Modulus of the cantilever beamis the area moment of inertiais the deflection of the cantilever beam

    is the bending moment

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    = = 2 +

    Eqn. 6

    = 2 6 + + Eqn. 7

    Applying boundary conditions,

    For = 0, = 0For = 0, = 0

    Substituting boundary conditions intoEqn. 6 andEqn. 7 and solving for integration constants,

    = 0 and = 0HenceEqn. 7 reduces to,

    = 2 6 Eqn. 8

    Solving for maximum deflection, which occurs at = , = 2 6 = 3 = 3

    Eqn. 9

    Given that,

    =

    Then,

    = 3 Eqn. 10

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    C.1.4 - Finding Un-damped Natural Frequency of System

    =

    Substituting for the equivalent stiffness and equivalent mass of the system fromEqn. 5 andEqn. 10,

    = 3 + + Eqn. 11

    Therefore, the un-damped natural frequency of vibration may be derived from the formula,

    = 2

    Substituting,

    = 12 3 + + Eqn. 12

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    C.1.5 - Forced Vibrations of a Cantilever Beam due to Unbalance

    Figure 21 - Equivalent kinematic diagram

    Where is the applied harmonic force due to unbalanceis the length of the cantilever beamis the mass of the beam as a concentrated load in

    is the mass of the motor, which is providing the applied harmonic force

    is the mass of the accelerometeris the equivalent stiffness of the cantilever beamis the equivalent damping coefficient of the damperis the mass equivalent of the system

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    From the above diagram, the equation of motion is given by = Eqn. 13

    The applied harmonic force , is due to a rotating unbalance. As shown in the diagram below, twoforces are experienced by the rotating mass; a centripetal force and a tangential force.

    Figure 22 - Illustration of rotating imbalance

    Where

    is the unbalanced massis the eccentricityis the angular acceleration of the unbalanced massis the angular velocity of the unbalanced mass or motoris the mass of the motor, including the unbalanced massHowever, the unbalance is rotating at constant velocity,

    = 0Thus, the applied harmonic force is only due to the centripetal force, = sin

    Eqn. 14

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    SubstitutingEqn. 14 intoEqn. 13, sin = + + = sin + + = sin Eqn. 15

    The equation of motion may also be expressed as, + + 2 = sinEqn. 16

    ComparingEqn. 16 withEqn. 15,

    = Eqn. 17

    Which have been defined previously.

    2= = 2Eqn. 18

    Where the damping ratio = 0 for an undamped system. = =

    Eqn. 19

    When enteringEqn. 17 into the derived equation.

    Given that, sin= Eqn. 20

    And =

    =

    =

    Eqn. 21

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    Where = Eqn. 22

    Hence fromEqn. 21,

    = Eqn. 23 = Eqn. 24

    SubstitutingEqn. 20, Eqn. 21,Eqn. 23 andEqn. 24 intoEqn. 16, + + 2 = (+ + 2)= Dividing by

    throughout,

    1 + 2 = = 1 +2

    Using the conjugate,

    = 1 +2 1 2 1 2

    = 1 2 1

    +2

    Eqn. 25

    FromEqn. 22, = Therefore,

    = 1 2

    1

    + 2 =

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    Hence to solve for the dynamic amplitude,, the magnitude ofis found

    = 1 +2

    1 +2

    = 1 + 2

    Eqn. 26

    SubstitutingEqn. 19 intoEqn. 26,

    =

    1 + 2

    Eqn. 27

    And the phase angle is given by,

    = tan [ 2 1 ]Eqn. 28

    Where = 0or 180for an undamped system since the damping ratio, = 0.Therefore, the magnification factor (M) or amplification ratio is defined as,

    = 11 +2

    Eqn. 29

    The magnification factor and the phase-shift were plotted against the frequency ratio, fordifferent values of . (Graph 2, Graph 3)

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    0.000

    20.000

    40.000

    60.000

    80.000

    100.000

    120.000

    140.000

    160.000

    180.000

    200.000

    0 0.5 1 1.5 2 2.5 3 3.5

    Phase

    Difference

    W/Wn

    Phase Difference

    0

    0.1

    0.2

    0.3

    0.5

    0.75

    1

    0.000

    1.000

    2.000

    3.000

    4.000

    5.000

    6.000

    0 0.5 1 1.5 2 2.5 3 3.5

    M-

    Magnification

    W/Wn - Frequency Ratio

    Frequency Response Function

    0

    0.1

    0.2

    0.3

    0.5

    0.75

    1

    Graph 2 - Plot of the frequency response function for different values of

    Graph 3 - Plot of the phase shift against the frequency ratio for different values of

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    C.2.1 - Finding Equivalent Stiffness of New Structure

    The equivalent stiffness of the new structure is given by,1

    = 1

    + 1

    Eqn. 34

    Where

    is the equivalent stiffness of the cantilever beam =3 is the equivalent stiffness of all six isolators = 6 Substituting for

    and

    intoEqn. 34,

    1 = 16+ 3 1 =3 + 663 = 183 + 6Eqn. 35

    C.2.2 - Finding Equivalent Mass of New Structure

    The equivalent mass of the new structure is given by,

    = + Eqn. 36

    Where is the mass equivalent of the pre-isolation system = + + is the added mass in new structure = Substituting for and intoEqn. 36, = + + +

    Eqn. 37

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    C.2.3 - Finding Equivalent Damping Coefficient of New Structure

    For the pre-isolation structure,

    2=

    = 2Where=

    Also, the equivalent damping coefficient of all four isolators is defined as, = 6Hence, the equivalent damping coefficient of the new structure is given by,

    1 = 1+ 1 Eqn. 38

    Substituting and intoEqn. 38, 1 = 12+ 16 = 122+ 6

    Eqn. 39

    C.2.4 - Finding Un-Damped Natural Frequency of New Structure

    FromEqn. 35 andEqn. 37,the new natural frequency of the structure is given by,

    = = 18( + + + )3 + 6Eqn. 40

    C.2.5 - Finding Damping Ratio of New Structure

    From the equation of motion, it is defined that,2= = 2Eqn. 41

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    C.2.6 - Transmissibility (Vibration Isolation)

    Figure 24 - Kinematic representation of the isolated structure

    Where

    is the mass of the frame

    is the mass equivalent of the full systemis equivalent damping coefficient of all six isolatorsis the equivalent stiffness of all six isolatorsis the force transmitted by the vibrating structure to ground

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    FromEqn. 27,the dynamic amplitude of vibration transmitted is given by,

    =

    1

    + (2 )

    Eqn. 42

    And fromEqn. 21 the steady-state solution is defined as,= Eqn. 43 = Eqn. 44

    The force transmitted to ground is given by, = + Eqn. 45

    SubstitutingEqn. 43 andEqn. 44 intoEqn. 45 = + || = +

    || = 1 + = || 1 +

    Eqn. 46

    SubstitutingEqn. 46 intoEqn. 42,

    ||= 1 + 1 +(2 )

    Eqn. 47

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    But,

    =

    = 2

    Substituting intoEqn. 47,

    ||= 1 + (2 )1 +(2 )

    Hence the transmissibility, which is the amplitude ratio of the transmitted force to the impressed

    force, is defined as,

    TR = || = 1 +(2 )

    1 + (2 )

    Eqn. 48

    Graph 4 - Magnitude of transmitted vibrations against frequency ratio, for different values of

    .

    0.000

    2.000

    4.000

    6.000

    8.000

    10.000

    12.000

    14.000

    16.000

    18.000

    20.000

    0 0.5 1 1.5 2 2.5

    Transmissibility-TR

    Frequency Ratio -

    Transmissibility

    0

    0.010

    0.020

    0.029

    0.039

    0.057

    0.091

    0.129

    0.163

    sqrt2

    /

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    C.3 - Natural Frequencies of Unloaded Cantilever Beam

    The exact solution for distributed loads was used to determine the multiple harmonics of the

    unloaded cantilever beam, as shown hereunder.

    Figure 25 - The deflected cantilever beam, under its own weight

    Where is the mass per unit length of the cantilever beamis the length of the cantilever beamis an arbitrary length section of the cantilever beam

    is the deflection of a point distant

    from the origin at a given instant

    From beam deflection theory, = Where is Youngs Modulus of the cantilever beamis the area moment of inertia

    is the bending moment

    = Eqn. 49

    Hence the shear force is defined as,

    = = Eqn. 50

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    And the rate of loading is given by,

    = = Eqn. 51

    Given that the inertia loading at the point which is distant x from the origin,

    = Eqn. 52

    Which acts in the opposite direction to the acceleration.

    SubstitutingEqn. 52 intoEqn. 51,

    =

    + = 0 Eqn. 53Assuming simple harmonic motion, = Where is the circular frequency of the natural vibrations of the beam. Considering the beam inthe maximum deflected position,Eqn. 53 is reduced to,

    +

    = 0

    Eqn. 54

    Where

    = Eqn. 55

    The solution toEqn. 54 is defined as,

    = + + +

    Eqn. 56 = + + Eqn. 57 = + + Eqn. 58 = + + + Eqn. 59

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    Applying the boundary conditions,

    For = 0, = 0, = 0For = , = 0, = 0

    Substituting boundary conditions intoEqn. 56 toEqn. 59,the solution below is obtained [12], = 1Eqn. 60

    Solving this frequency equation for the constant, , for different values of 1,2,3,4 = 1.8751041 = 4.6940911 = 7.8547574

    = 10.9955407

    Eqn. 61

    Therefore, by substitutingEqn. 61 intoEqn. 55 for different values of , the multiple naturalfrequencies of the cantilever beam are given by,

    = 1.875 = 4.694 = 7.855 = 10.996

    Eqn. 62

    Overleaf,Graph 5 shows how the magnification factor varies with the excitation frequency for

    different values of

    .

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    0.000

    1.000

    2.000

    3.000

    4.000

    5.000

    6.000

    0.000 1000.000 2000.000 3000.000 4000.000 5000.000 6000.000 7000.000

    Magnification/M

    Excitation frequency, omega/ rad/s

    Magnification v.s. Omega for different natural frequencies - Unloaded

    Beam

    1

    2

    3

    4

    Graph 5 - Magnification factor against excitation frequency for different natural frequencies of the unloaded system

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    C.4 - Natural Frequencies of Loaded Cantilever Beam

    Figure 26 - The deflected cantilever beam, under its own weight and those of the edge loads

    The Durnkerleys empirical method was used to calculate the multiple natural frequencies of the

    loaded cantilever beam as shown below, 1 = 1+ 1 Eqn. 63

    Where

    is the natural frequency of the loaded cantilever beam

    =4 Eqn. 64

    For each natural frequency for = 1to 4.is the frequency of vibration of the both concentrated loads acting alone on the beam

    =4

    Eqn. 65

    Where

    = Where = + And fromEqn. 10,

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    = 3 Therefore,

    =3Eqn. 66

    Substituting for , = 34

    Eqn. 67

    is a natural frequency of the unloaded cantilever beam. Hence from the exact solution above,

    =4 Eqn. 68

    For each natural frequency for = 1to 4.SubstitutingEqn. 64,Eqn. 67 andEqn. 68 intoEqn. 63,

    4 =43 + 4 1 =3 + 1 1 = + 33 = 3 + 3

    Eqn. 69

    For each natural frequency for = 1to 4.Overleaf, Graph 6 shows how the magnification factor varies with the excitation frequency fordifferent values of . It can be seen that once the masses have been added the higher ordernatural frequencies tend to converge.

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    Graph 6 - Plot of the magnification factor against the excitation frequency for different values of the natural frequency of the loaded structure

    0.000

    1.000

    2.000

    3.000

    4.000

    5.000

    6.000

    0.000 100.000 200.000 300.000 400.000 500.000 600.000

    Magnification/M

    W/ rad/s

    Magnification v.s. Omega for different natural frequencies - Loaded Beam

    1

    2

    3

    4

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    Group Effort

    Agius, Maria Kristina _______________________________

    (389492M)

    Borg, Joseph Alexander _______________________________

    (487092M)

    Buhagiar, Liana _______________________________

    (557992M)

    Farrugia, Matthew _______________________________

    (122190M)

    Zammit, Nathan John _______________________________

    (9993M)

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    Page | 44

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