Differential Equations and Matrices

    Differential Equations and Matrices

    This is an individual coursework.
    Percentage of module mark: 30
    Intended Learning Outcomes:
    1.
    Ability to formulate and analyse mathematical model
    s of engineering problems
    2.
    Ability to apply mathematical techniques and modern
    mathematical software to
    solving engineering problems, in particular: soluti
    on of ordinary differential equations
    using analytical techniques and Matlab
    Hand Out: 28th October 2013
    Hand in: 10th January 2014
    Question 1: 30 marks
    Question 2: 35 marks
    Question 3: 35 marks
    Total: 100 marks
    All students have been assigned a unique number
    N
    which is used for setting parameter
    values in all questions.
    Please look up your number in the table provided wi
    th this assignment!
    REMEMBER: coursework hand-in now has a hard deadlin
    e. Be late, get zero
    (ABsent / failure to submit). Plan to hand in at le
    ast a day early. Better still, a week.
    For full marks, all working must be shown.
    Where boxes are provided, please fit all the workin
    g and the answers in the boxes.
    You are allowed to use Matlab unless specified othe
    rwise explicitly. You are not allowed to
    use software not covered in the module (e.g. Maple
    or Simulink)
    Please submit an electronic version via Moodle. Onl
    y two files up to 20 Mb can be uploaded
    to Moodle, multiple files can be combined in a ZIP
    archive if necessary. It is however
    recommended to only submit one file.
    208MAE, coursework 1 (2013/2014)
    Page 2 of 20
    1 Beam bending
    Problem:
    A uniformly thick beam of length
    L
    with square cross-section with width
    a
    has a distributed
    force,
    ?
    ?
    ?
    ?
    ?
    ?
    +

    =

    x
    e
    L
    x
    Q
    x
    q
    2
    2
    sin
    1
    )(
    p
    N/m,
    applied
    0
    =
    x
    to
    L
    x
    =
    (
    Q
    is a constant). A point force
    P
    is applied at
    L
    x
    =
    .
    P
    x
    y
    x
    = 0
    q
    x L
    =
    Find expressions for shear and moment
    analytically
    (not using Matlab).
    Sketch distributions of shear forces and moment alo
    ng the beam length (using Matlab)
    Parameters:
    Values of
    L, EI, Q
    and
    P
    for each student are calculated as follows (here
    N
    is the number
    assigned to you in the table):
    N
    L
    *
    01.0
    1
    +
    =
    (m)
    600
    /)
    300(*
    50
    N
    P

    =
    (kN)
    300/
    5
    N
    e
    Q
    =
    (kN/m)
    5
    10
    6
    ×
    =
    EI
    (Pa m
    4
    )
    Marking scheme:
    Finding correct expressions for shear and moment
    ANALYTICALLY
    (not using Matlab,
    please show your working)
    (20 marks)
    Plots of shear force and moment
    (10 marks)
    208MAE, coursework 1 (2013/2014)
    Page 3 of 20
    Solution:
    1. Correct parameters for my student number
    N
    =
    L
    =
    P
    =
    Q
    =
    EI
    =
    2. Shear
    Equation for shear:
    General solution of shear equation (please put your
    working here):
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