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    Mathematical Model

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    ardius ebenezer
    The document discusses calculating the system response of a damped spring-mass system using Laplace transforms. It provides the steps to derive the transfer function of the system in terms of its poles and zeros. The transfer function is then used to find the time domain response by taking the inverse Laplace transform. Some examples are provided to illustrate calculating the system response for different mass, spring, and damping values, as well as homework problems involving similar damped spring-mass systems.

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    Mathematical Model

    Uploaded by

    ardius ebenezer
    15 views18 pages
    The document discusses calculating the system response of a damped spring-mass system using Laplace transforms. It provides the steps to derive the transfer function of the system in terms of its poles and zeros. The transfer function is then used to find the time domain response by taking the inverse Laplace transform. Some examples are provided to illustrate calculating the system response for different mass, spring, and damping values, as well as homework problems involving similar damped spring-mass systems.

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    The document discusses calculating the system response of a damped spring-mass system using Laplace transforms. It provides the steps to derive the transfer function of the system in terms of its poles and zeros. The transfer function is then used to find the time domain response by taking the inverse Laplace transform. Some examples are provided to illustrate calculating the system response for different mass, spring, and damping values, as well as homework problems involving similar damped spring-mass systems.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    15 views18 pages

    Mathematical Model

    Uploaded by

    ardius ebenezer
    The document discusses calculating the system response of a damped spring-mass system using Laplace transforms. It provides the steps to derive the transfer function of the system in terms of its poles and zeros. The transfer function is then used to find the time domain response by taking the inverse Laplace transform. Some examples are provided to illustrate calculating the system response for different mass, spring, and damping values, as well as homework problems involving similar damped spring-mass systems.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    of 18

    Mathematical Model of System

    TKI3I3
    Exercise

    The closed-loop transfer function T(s) = Y(s) / R(s) is…


    Problem
    How to calculate total force..?
    damper
    k b
    From the second Newton’s law,
    we can obtain :

    f (t )  Mg (t )  bv(t )  kx(t )
    x(t)
    or equals…
    f(t)
    2
    d x(t ) dx(t )
    f (t )  M 2
    b  kx(t )
    dt dt
    Problem
    How to find the response of
    damper
    the system, y(t)..?
    k b if M = 10 kg, k = 20 N/m, and b = 30 Ns/m

    x(t)

    f(t)
    System Response
    To find the system response, use Laplace transform

    d k f (t ) k 1  k 2  ( k 1) 
    k
     s k
    F ( s )  s f ( 0 )  s f ' ( 0 )  ...  f ( 0 )
    dt
    2
    from previous eq. f (t )  M
    d x ( t ) dx (t )
    2
    b  kx(t )
    dt dt
    so, we get a new equation:
     2  dy  
     

    M  s Y ( s )  sy (0 )  (0 )   b sY ( s )  y (0 )  kY ( s )  R ( s )
    dt
     
    System Response
    Set the initial condition
     dy
    f (t )  0, and y (0 )  y0 , and 0
    dt t 0-
    so, we simplify the previous equation:
    Ms 2Y ( s )  Msy0  bsY ( s )  by0  kY ( s )  0

    Solving for Y(s), we obtain:


    ( Ms  b) y0
    Y (s) 
    Ms  bs  k
    2
    System Response
    Substitute M, k, and b with its value, and set the initial
    position y0 = 1, we obtain:
    10s  30
    Y (s) 
    10s  30s  20
    2

    then simplify the equation:

    s3 s3 p( s)
    Y (s)  2  
    s  3s  2 ( s  1)( s  2) q ( s )
    Pole and Zero
    The roots of the numerator p(s) is called zeros of the
    system
    The roots of the denumerator q(s) is called poles of
    the system

    So, from the previous equation:

    s3
    Y (s) 
    ( s  1)( s  2)
    The zero is –3, meanwhile the poles are –1 and –2
    Residue
    When we expand the equation in a partial fraction
    expansion, we obtain:
    k1 k2
    Y (s)  
    ( s  1) ( s  2)
    where k1 and k2 are the coefficients of the expansion, and
    called residue
    Coefficients ki are evaluated by multiplying through by
    denominator factor corresponding to ki and setting s
    equal to the root.
    Residue
    Evaluating k1 and k2, we obtain:

    ( s  s1 ) p ( s ) ( s  1)( s  3)
    k1   2
    q( s) s  s1
    ( s  1)( s  2) s  1

    ( s  s2 ) p ( s ) ( s  2)( s  3)
    k2    1
    q( s) s  s2
    ( s  1)( s  2) s  2
    Residue
    Then the equation will be:

    2 1
    Y (s)  
    ( s  1) ( s  2)
    To find the y(t), use inverse Laplace transform
    Residue
    From the Laplace transform table, we find

     at 1
    e 
    sa
    1

    so, we can obtain


    0.8

    0.6

    y (t )  2e  t  e 2t

    y(t)
    0.4

    0.2

    0
    0 1 2 3 4 5 6 7 8
    t
    Any Question…?

    “No man really becomes


    a fool until he stops
    asking question”

    – Charles P. Steinmetz –
    Homework
    Find the system response y(t)

    if M = 10 kg, k = 60 N/m, and b = 50 Ns/m,


    k b meanwhile f(t) = 0, initial velocity y’(t) = 0,
    and initial position y0 = 2

    x(t)

    f(t)
    Homework

    Find the transfer function between the speed of the truck (VT) and the speed
    of the cart (VC) if mass of cart m = 1.000 kg is attached to a truck using a
    spring of stiffness k = 20.000 N/m, and a damper of constant b = 200 Ns/m.

    Answer choices :
    Homework

    Find the transfer function X2(s)/F(s) while both masses slide on a frictionless
    surface, and k = 1 N/m.
    X 2 (s) 1 X 2 (s) 1
    Answer choices : a.  2 2 c.  2
    F ( s ) s ( s  2) F ( s ) s ( s  2)

    b. X 2 (s) 1 d. X 2 (s) 1
     
    F ( s ) s ( s 2  2) F ( s ) s ( s  2)
    Homework Soal UTS Tahun 2015/16

    Determine G(s) and H(s) of


    the block diagram in Fig.(b)
    that are equivalent to the
    block diagram in Fig.(a)
    To be continued…

    つづく…

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