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    Assignment I - Vibrations of Physical Systems

    Uploaded by

    nhc2009
    This document outlines the requirements for Assignment 1 on vibrations of physical systems. It contains 4 questions covering various topics in vibrations including: [1] analyzing the resultant amplitude and phase of multiple simple harmonic motions with different amplitudes and phases; [2] analyzing beat periods and Lissajous figures for combined vibrations; [3] deriving and solving the differential equation of motion for a damped harmonic oscillator; and [4] analyzing forced oscillations including steady state solutions and resonance conditions. The document provides marking criteria and instructs students to show their working clearly and attempt all questions.

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    Assignment I - Vibrations of Physical Systems

    Uploaded by

    nhc2009
    260 views2 pages
    This document outlines the requirements for Assignment 1 on vibrations of physical systems. It contains 4 questions covering various topics in vibrations including: [1] analyzing the resultant amplitude and phase of multiple simple harmonic motions with different amplitudes and phases; [2] analyzing beat periods and Lissajous figures for combined vibrations; [3] deriving and solving the differential equation of motion for a damped harmonic oscillator; and [4] analyzing forced oscillations including steady state solutions and resonance conditions. The document provides marking criteria and instructs students to show their working clearly and attempt all questions.

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    This document outlines the requirements for Assignment 1 on vibrations of physical systems. It contains 4 questions covering various topics in vibrations including: [1] analyzing the resultant amplitude and phase of multiple simple harmonic motions with different amplitudes and phases; [2] analyzing beat periods and Lissajous figures for combined vibrations; [3] deriving and solving the differential equation of motion for a damped harmonic oscillator; and [4] analyzing forced oscillations including steady state solutions and resonance conditions. The document provides marking criteria and instructs students to show their working clearly and attempt all questions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    260 views2 pages

    Assignment I - Vibrations of Physical Systems

    Uploaded by

    nhc2009
    This document outlines the requirements for Assignment 1 on vibrations of physical systems. It contains 4 questions covering various topics in vibrations including: [1] analyzing the resultant amplitude and phase of multiple simple harmonic motions with different amplitudes and phases; [2] analyzing beat periods and Lissajous figures for combined vibrations; [3] deriving and solving the differential equation of motion for a damped harmonic oscillator; and [4] analyzing forced oscillations including steady state solutions and resonance conditions. The document provides marking criteria and instructs students to show their working clearly and attempt all questions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as DOCX, PDF, TXT or read online from Scribd
    Download as docx, pdf, or txt
    of 2

    PLEDGE NOT TO PLAGIARIZE

    Assignment I – Vibrations of Physical Systems


    SUBMISSION DATE: 15.04.11

    Steps that you should adhere to while answering the questions:

    1. Write a paragraph (should include a diagram or two) that analyzes the question and sets
    forth the method of attack.
    2. Set up the equations explaining what each equation stands for.
    3. All intermediate computations should be clearly and neatly presented.

    Marking Criteria:
    1. Analysis - 50%
    2. Equations – 10%
    3. Intermediate Computations – 30%
    4. Correct Answer – 10%

    Attempt all questions.

    1. (a) [10 Marks] A particle is simultaneously subjected to three simple harmonic motions, all
    of the same frequency and in the x -direction. If the amplitudes are 0.25, 0.20, and 0.15 mm,
    respectively, and the phase difference between the first and second is 45°, and between the
    second and the third is 30°, find the amplitude of the resultant displacement and its phase
    relative to the first (0.25 mm amplitude) component.
    (b) [5 Marks] Two vibrations along the same line are described by the equations
    y 1= A cos 10 πt
    y 2= A cos 12 πt
    Find the beat period, and draw a careful sketch of the resultant displacement over one beat
    period.
    (c) [5 Marks] Two vibrations are at right angles to one another are described by the
    equations
    x=10 cos ( 5 πt )
    y=10 cos (10 πt + π /3 )
    Construct the Lissajous figure of the combined motion.

    2. [5 Marks] Show that a two-body simple harmonic oscillator oscillates with the same time-
    period and frequency as a one-body oscillator of mass 𝜇 (called the reduced mass) where
    m1 m2
    μ=
    m 1 +m 2
    [The quantities m1∧m2 are the masses of the two−body system .]
    3. An object of mass 0.2 kg is hung from a spring whose spring constant is 80 N /m . The object
    is subject to a resistive force given by – bv , where v is its velocity in metres per second.
    (a) [10 Marks] Set up the differential equation of motion for damped oscillations of the
    system. If the damped frequency is √ 3/2 of the undamped frequency, what is the value
    of the constant b ?
    (b) [5 Marks] What is the Q of the system, and by what factor is the amplitude of the
    oscillation reduced after 10 complete cycles?

    4. (a) [5 Marks] Consider how to solve the steady-state motion of a forced oscillator if the
    driving force is of the form F 0 sin ω d t instead of F 0 cos ωd t .
    (b) A simple pendulum has a length ( l ) of 1m . In free vibration the amplitude of its swing falls
    off by a factor e in 50 swings. The pendulum is set into forced vibration by moving its point
    of suspension in SHM with amplitude of 1m m.
    (i) [5 Marks] Show that if the horizontal displacement of the pendulum bob is x , and the
    horizontal displacement of the support isξ , the equation of motion of the bob for small
    oscillation is
    d2 x dx g g
    2
    + γ + x= ξ
    dt dt l l
    Solve this equation for steady-state motion, if ξ=ξ0 cos ω d t . (Put ω 2=g/ l.)
    (ii) [5 Marks] At exact resonance, what is the amplitude of the motion of the pendulum
    bob?
    (iii) [5 Marks] At what angular frequencies is the amplitude half of its resonant value?

    ________________________________________________________________________

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