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    Tugas

    Vibration Conceptand MethodsbyJumaddil Hair (P22022013401)

    Ratnawati (P22022013403)

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    Preface

    Every spinning rotor has some vibration, at least a once-per-revolutionfrequency component, because it is of course impossible to make anyrotor perfectly mass balanced. Experience has provided guidelines forquantifying approximate comfortable safe upper limits for allowablevibration levels on virtually all types of rotating machinery. That suchlimits are crucial to machine durability, reliability, and life is rarely

    disputed. The mechanics of rotating machinery vibration is an interesting subject

    with considerable technical depth and breadth. Many industries relyheavily on reliable trouble-free operation of rotating machinery, e.g.,power generation; petrochemical process; manufacturing; land, sea,and air transportation; heating and air conditioning; aerospace;computer disk drives; textiles; home appliances; and various military

    systems. Even with the best of design practices and most effective methods of

    avoidance, many rotor vibration causes are so subtle and pervasive thatincidents of excessive vibration in need of solutions continue to occur.Thus, a major task for the vibrations engineer is diagnosis andcorrection. To that end, this book comprises four sequential parts.

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    The One Degree of Freedom

    Model

    Mass Spring Damper Model is the starting point to

    understanding mechanical vibration

    The fundamental physical law governing all vibration

    phenomena is Newtons second law

    F = m.a

    F = m.a yields its differential equation of motion, as follows.

    m + + = ()

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    The One Degree of Freedom Model

    Assumption of Linearity

    In the model of Eq. (2), as in most vibration analysis models,

    spring and damper connection forces are assumed to be linear

    with (proportional to) their respective driving parameters, i.e.,

    displacement (x) across the spring and velocity (x) across thedamper.

    These forces are therefore related to their respective driving

    parameters by proportionality factors, stiffness k for the spring

    and c for the damper.

    Linearity is a simplifying assumption that permeates most

    vibration analyses because the equations of motionare then

    made linear,even though real systems are

    never completely linear.

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    The One Degree of Freedom

    Model Unforced System

    The solution for the motion of the unforced one-degree-of-freedom systemis important in

    its own right but specifically important in laying the groundwork to studyself-excited

    instability rotor vibrations.If the system is considered to be unforced,then(t) = 0 and Eq.

    (2) becomes the following.

    m + + =0

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    The One Degree of Freedom

    Model

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    The well-known x(t) time signals for these three solution categories

    are illustrated in Fig. 2 along with the undampedsystem (i.e., c0). In

    most mechanical systems, the important vibration characteristics are

    contained in modes with so-called underdampedroots, as is certainly

    the case for rotor dynamical systems. The general expression for the

    motion of the unforced underdampedsystem is commonly

    expressed in any one of the following four forms

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    The One Degree of Freedom

    Model Stady-State Sinusoidally Forced System

    If the system is dynamically stable (c 0), i.e., the natural mode is

    positively damped as illustrated in Fig. 2, then long-term vibration

    can persist only as the result of some long-term forcing mechanism

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    The One Degree of Freedom

    ModelIf the system is dynamically stable (c > 0), i.e., the natural mode is

    positively damped as illustrated in Fig. 2, then long-term vibration can

    persist only as the result of some long-term forcing mechanism. In

    rotating machinery, the one longterm forcing mechanism that is always

    present is the residual mass unbalance distribution in the rotor, and

    that can never be completely eliminated. Rotor mass unbalances are

    modeled by equivalent forces fixed in the rotor, in other words, a group

    of rotor-synchronous rotating loads each with a specified magnitude

    andphase angle locating it relative to a common angular reference point

    (key phaser) fixed on the rotor

    Without preempting the subsequent treatment in this book of theimportant topic of rotor unbalance,suffice it to say that there is a

    considerable similarity between the unbalance-driven vibration of a

    rotor and the steady-state response of the 1-DOF system described by

    Eq. (2) with (t) = F0. sin(t + ).

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    The One Degree of Freedom

    ModelEquation (2) then becomes the following

    m + + = F0. sin(t + )

    F0. = force magnitude

    = force phase angle

    = forcing frequency

    Rotating machinery designers and troubleshooters are concerned with longterm-exposure vibration levels, because of material fatigue considerations, and are

    concerned with maximum peak vibration amplitudes passing through forcedresonances within the operating zones. It is therefore only the steady-state solution,

    such as of Eq. (6), that is most commonly extracted. Because this system is linear,

    only the frequency(s) in (t)will be present in the steady-state (particular) solution.Thus the solution of Eq. (6) can be expressed in any of the following four

    steady-state solution forms, with phase angle as given for Eq. (5) for each torepresent the same signal

    .

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    The One Degree of Freedom

    Model Undamped Natural Frequency : An Accurate Approximation

    The ratio () of damping to critical damping(frequently referenced as a

    percentage; e.g., =0.1 is 10% damping) is derivable as follows. Shown with

    Eq. (4), the defined condition for critically damped is c = 2 k/m= cc, thecritical damping. Therefore, the damping ratio,defined as = cc, can be

    expressed as follows.

    = 2 k/m

    So

    n = k/m(undamped natural frequency)

    = -c/2m (real part of eigenvalue for underdamped system)

    d = n (damped natural frequency)

    So

    d = n 1 Formula damped natural frequency with ratio

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    The One Degree of Freedom

    Model

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    The One Degree of Freedom

    Model The One Degree of Freedom Model as a Approximation

    Equation (2) is an exact mathematical model for the system schematically illustratedin Fig. 1. However, real-worldvibratory systems do not look like this classic 1-DOFpicture, but in many cases it adequately approximates them for the purposes ofengineering analyses. An appreciation for this is essential for one to make theconnectionbetween the mathematical models and the real devices that the modelsare employed to analyze.

    One of many important examples is the concentrated mass (m) supported at thefree end of a uniform cantilever beam (length L,bending moment of inertia I, Youngsmodulus E) as shown in Fig. 5a. If the concentrated masshas considerably more massthan the beam, one may reasonably assume the beam to be massless,at least for thepurpose of analyzing vibratory motions at the systems lowest natural frequencytransverse mode. One can thereby adequately approximate the fundamentalmodeby a 1-DOF model. For smalltransverse static deflections (xst) at the free end of

    the cantilever beam resulting from a transverse static load (Fst) at its free end, theequivalent spring stiffness is obtained directly from the cantilever beams staticdeflection formula. This leads directly to the equivalent 1-DOF undamped-systemequation of motion, from which its undamped natural frequency(n) is extracted, asfollows.

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    The One Degree of Freedom

    ModelThis leads directly to the equivalent 1-DOF undamped-system equation of motion, from

    which its undamped natural frequency(n) is extracted, as follows.

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    The instantaneous sum of moments about pivot point o consists only of

    that from the gravitational force mgon the concentrated mass, as follows

    (minus

    sign because Mis always opposite ()

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    Multi Degree of Freedom Models Two Degree of Freedom Models

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    Multi Degree of Freedom

    Models

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    The Lagrange equations are derived directly from Fmaand therefore embody thesame physical principle. Their derivation can be found in virtually any modern

    second-level text on dynamics or vibrations, and they are expressible as follows

    Multi Degree of Freedom

    Models

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    Then The Kinetic Energy :

    Then becomes

    Multi Degree of Freedom

    Models

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    Multi Degree of Freedom Models Matrix Bandwidth and Zeros

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    Multi Degree of Freedom

    Models Standard Rotor Vibration Analyses

    Achieving good models for rotor vibration analyses of many single-spantwobearing rotors may require models with as many as 100 DOFs. For amultispan rotor model of a complete large steam-powered turbogenerator,models of 300 to 500 DOFs are typically deemed necessary to characterizethe system accurately.

    Obtaining the important vibration characteristics of a machine or structurefrom large DOF models is not nearly as daunting as one might initially think,because of the following axiom. Rarely is it necessary in engineeringvibration analyses to solve the models governing equations of motion intheir totality.For example, lateral rotor vibration analysesgenerally entail no

    more than the following threecategories.1. Natural frequencies (damped or undamped) and corresponding mode

    shapes

    2. Self-excited vibration threshold speeds, frequencies, and mode shapes

    3. Vibration over full speed range due to specified rotor mass unbalances