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Math Help - How to calculate min acceleration between two points of different velocities

  1. #1
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    How to calculate min acceleration between two points of different velocities

    Hi all.

    Start Point A at a velocity of 100mm/sec
    Finish Point B at a velocity of 40 mm/sec

    Distance to travel = 300mm
    Time to do it in = 2 seconds

    Now obviously we have to apply an excessive acceleration to a "Point C" (between A and B) to achieve the 300mm movement within the 2s time.

    So we have an acceleration Point A to Point C
    Then a deceleration from Point C to Point B

    Point C will be the peak velocity to achieve the distance in the time constraints

    Can anyone point me in the direction to solve the minimum value of acceleration, such that acceleration = deceleration

    Cheers
    Last edited by calltronics; August 6th 2012 at 01:31 PM. Reason: more explaination
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    Re: How to calculate min acceleration between two points of different velocities

    Start Point A at a velocity of 100mm/sec
    Finish Point B at a velocity of 40 mm/sec

    Distance to travel = 300mm
    Time to do it in = 2 seconds
    given those constraints ...

    v_0 = 100 \, mm/s

    v_f = 40 \, mm/s

    \Delta t = 2 \, s

    would require a constant acceleration of a = -30 \, mm/s^2

    however, the distance traveled under that constant acceleration would be \Delta x = 140 \, mm, much less than the given 300 \, mm of travel ... any other given information?
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    Re: How to calculate min acceleration between two points of different velocities

    The issue is that you have over-constrained the problem. As skeeter points out if you want the deceleration to take 2 seconds it only requires 140 mm of distance to affect the change of velocity that you want. Alternatively if you want to use the full 300 mm of distance then you could use the formula a = (vf^2 - vi^2)/2d to get a = -14 mm/s^2, but this requires time of 4.3 seconds to make the change in velocity.
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    Re: How to calculate min acceleration between two points of different velocities

    sorry further explanation added to question
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    Re: How to calculate min acceleration between two points of different velocities

    Quote Originally Posted by calltronics View Post
    Hi all.

    Start Point A at a velocity of 100mm/sec
    Finish Point B at a velocity of 40 mm/sec

    Distance to travel = 300mm
    Time to do it in = 2 seconds

    Now obviously we have to apply an excessive acceleration to a "Point C" (between A and B) to achieve the 300mm movement within the 2s time.

    So we have an acceleration Point A to Point C
    Then a deceleration from Point C to Point B

    Point C will be the peak velocity to achieve the distance in the time constraints
    "peak velocity" ? ... if you mean a velocity > 100 mm/s ... do you understand this can't happen?

    maybe you should post the entire problem as it was given to you.
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    Re: How to calculate min acceleration between two points of different velocities

    I am not very good at this am I?

    Travelling in a straight line between Point A and Point B

    Starting at Point A at an initial velocity of 100 mm/s I have to get to Point B which is at final velocity of 40mm/s.
    The movement has to be achieved in 2 seconds and over the distance of 300mm

    To achieve these distance and time constraints I have to perform an acceleration/deceleration along the straight line path between Point A and Point B: -
    Accelerate from Point A to a point (C) somewhere between point A and Point B
    Then decelerate from this intermediate point(C) to Point B

    Such that: -
    the distance travelled is 300mm
    the time taken is 2 seconds
    and acceleration = deceleration (and at a minimum value)

    Point (C) will have a velocity greater than Point A thereby basically increasing the distance travelled in the time available.


    Is this any better?
    Last edited by calltronics; August 6th 2012 at 02:47 PM. Reason: addition
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    Re: How to calculate min acceleration between two points of different velocities

    one more question ...

    is the acceleration from 100 mm/s to vmax at C constant and equal in magnitude to the deceleration from vmax at C to 40 mm/s ?

    in other words, is the velocity vs, time graph piece-wise linear ?
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    Re: How to calculate min acceleration between two points of different velocities

    Yes.
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    Re: How to calculate min acceleration between two points of different velocities

    If we let t_1 = the time under acceleration (and hence 2 seconds - t1 = time under deceleration), and let d_1 = the distance travelled under acceleration (and hence  0.3m -d_1 = d_2 = distance travelled under deceleration) then you can set up two equations in  a and  t_1 as follows. First consider the velocity at point c; under constant acceleration starting with velocity  v_a and then under constant deceleration to end with velocity  v_b you have:

    v_c = v_a + at_1 = v_b + a(T-t_1) where v_c = peak velocity somewhere in the middle, T = total time (2 seconds)

    So the first equation is

    (1) a = \frac {v_b-v_a} {2t_1-T}

    Now consider the distances travelled under acceleration from a to c and then under delleration from c to b. The total distance in the acceleration phase is d_1 = v_a t_1 + \frac 1 2 a t_1^2. The distance travelled in the deceleration phase is d_2 = v_c(T-t_1) - \frac 1 2 a (T-t_1)^2. Add these together, replace  v_c with  v_a + a t_1 and set equal to total distance D (which is 0.3m), and after some manipulation you get your second equation:

    (2) a = \frac {D-v_aT} {2t_1T - t_1^2-\frac 1 2 T^2}

    Set equations (1) and (2) equal to each other and you get a quadratic in t_1:

     (v_a - v_b)t_1^2 + 2(v_b T - D)t_1 + (TD - \frac 1 2 v_aT^2 - \frac 1 2 v_bT^2) = 0

    Apply the quadratic equation to solve, and you find that a = 0.818665m/s^2. This yields point c at 0.149m, and v_c = 0.235m/s.
    Last edited by ebaines; August 7th 2012 at 05:50 AM.
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    Re: How to calculate min acceleration between two points of different velocities

    I saw it this way ...

    let t = time interval of acceleration

    2-t = time interval of deceleration

    a = \frac{v_c - 100}{t}

    -a = \frac{40 - v_c}{2-t}

    solving this system for v_c ...

    v_c = \frac{10(7t-10)}{t-1}


    total displacement ...

    \frac{1}{2}(100+v_c)t + \frac{1}{2}(v_c+40)(2-t) = 300

    simplifying ...

    v_c = 260 - 30t

    substituting for v_c ...

    \frac{10(7t-10)}{t-1} = 260 - 30t

    simplifying ...

    3t^2 - 22t + 16 = 0

    t \approx 0.819 \, s

    v_c \approx 235.44 \, mm/s

    ... same result as ebaines
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    Re: How to calculate min acceleration between two points of different velocities

    Wow!!!!!!!!!
    I am so so pleased.
    I got to nearly where ebains was but it took me all morning.
    Trying to go back 30 years to my school maths took some doing.
    Absolutely great job.
    Thank you very very much both of you - amazing.
    This is one part of a really complex machine movement.
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    Re: How to calculate min acceleration between two points of different velocities

    I am forever in your debt, but to push your kindness.......
    If I now want the minimum time (T) and t1
    With a fixed maximum acceleration is it easy to resolve?
    Same parameters (but with variable time) and a max acceleration = 0.7m/s2 Max Accel
    Last edited by calltronics; August 8th 2012 at 12:50 PM. Reason: clarification
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    Re: How to calculate min acceleration between two points of different velocities

    Quote Originally Posted by calltronics View Post
    If I now want the minimum time (T) and t1
    With a fixed maximum acceleration is it easy to resolve?
    Same parameters (but with variable time) and a max acceleration = 0.7m/s2 Max Accel
    You will have to clarify - is the total time (T) still 2 seconds? And covering 300 mm with the same initial and final velocities as before? Because if so it can't be done, as we've already shown that a = 0.82 m/s^2.
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    Re: How to calculate min acceleration between two points of different velocities

    The total time is now the variable.
    The acceleration is fixed at 0.7m/s^2
    Same type of solution Passing through "Point C".
    Solved for either t1 or t2 as a minimum total time.
    Last edited by calltronics; August 8th 2012 at 01:42 PM.
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    Re: How to calculate min acceleration between two points of different velocities

    Quote Originally Posted by skeeter View Post
    I saw it this way ...

    let t = time interval of acceleration

    2-t = time interval of deceleration

    a = \frac{v_c - 100}{t}

    -a = \frac{40 - v_c}{2-t}

    solving this system for v_c ...

    I am lost on how you did this bit......
    Can you please explain how you got to the next point?


    v_c = \frac{10(7t-10)}{t-1}


    total displacement ...

    \frac{1}{2}(100+v_c)t + \frac{1}{2}(v_c+40)(2-t) = 300

    simplifying ...

    v_c = 260 - 30t

    substituting for v_c ...

    \frac{10(7t-10)}{t-1} = 260 - 30t

    simplifying ...

    3t^2 - 22t + 16 = 0

    t \approx 0.819 \, s

    v_c \approx 235.44 \, mm/s

    ... same result as ebaines
    Please see comments in your answer.
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