ME 358 Mechanism Analysis Straight Line Mechanism

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ME 358 Mechanism Analysis Straight Line Mechanism
11/22/04
Tony Frego
J. Cole Schalk
Pat Michels Darkness
Nate Dogg Williams
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Abstract
The problem given to us was to convert rotary
motion into straight line motion. In order to
properly solve this problem, we researched
straight line mechanisms, and selected the design
which best fit our application. Once the general
design was selected, we used the Fourbar program
to determine the lengths of the links to be used,
and Working Model to draw it out. We then used
our knowledge of dynamics to analyze the
mechanism and determine the required angular
velocity for the motor. From here we compared the
results of the dynamics work with the results of
the graphical analysis, analytical analysis, and
Working Model. The dynamics work resulted in an
angular velocity of .856 radians per second for
the motor, and this result was confirmed by two
of the other three methods.
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Introduction

• The goal of this project is to design a mechanism
  using four-bar linkages that will transform the
  rotary motion of a motor into a vertical motion
  without the need for sliders or a guide.

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Requirements

• The mechanism must conform to the following
  requirements
• The output must travel 9 inches vertically,
  reaching but not exceeding a speed of 10 inches
  per second
• The mechanism should be designed using a minimal
  material area.
• The mechanism must consist of one or more
  four-bar linkages.
• The motor must be located within the box outlined
  below

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The Mechanism
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Mechanism Specifications

• First Four-bar
• Link a 6 inches
• Link b 20.04 inches
• Link c 14.25 inches
• Ground 17.498 inches
• Second Four-bar
• Link a 33.75 inches
• Link b 13.5 inches
• Link c 33.75 inches
• Ground 27 inches
• Total Area of material used in links
• 165.788 square inches
• Total Area of rectangular package, with a 6 inch
  margin on all sides
• 3002.4 square inches

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Analysis

• Methods used to analyze the mechanism
• We first broke the mechanism into two separate
  four-bar mechanisms.
• We used Fourbar, the four-bar program from the
  Design of Machinery CD to determine the link
  lengths required for the output four-bar.
• We used four different methods to analyze the
  mechanism
• Working Model
• Dynamic Analysis
• Analytical Analysis
• Graphical Analysis

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Working Model
This graph shows the velocity of the output
versus the position of the output path. This
graph was generated by the program working model.
(This is only an approximation)
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The Mechanism
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Dynamic Analysis

• Equations used
• V r x w
• VB VA VB/A
• Dynamics Analysis We used the knowledge that we
  learned in Dynamics to solve for the angular
  velocities that we needed. In Working Model, we
  used an arbitrary motor speed to determine where
  the maximum velocity of the output would be.
  Then we set the velocity of output equal to 10
  in/s. From there we solved for the angular
  velocity of the motor. We then verified this
  velocity with Working Model.
• In the next few slides they will show you the
  work that was done in order to calculate the
  angular velocity of the motor.

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Dynamic Analysis
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Dynamic Analysis
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Dynamic Analysis
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Analytical Analysis

• Analytical Analysis For this we used the
  equations in the Design of Machinery book and
  Microsoft Excel to develop the necessary
  information for each four-bar mechanism. However,
  since the book equations were developed assuming
  that q1 0, it was necessary for us to changer
  our point of view on the mechanism so as to make
  q1 0.
• From these equations we plotted the velocity of
  the output vs. the position of the motor every 5
  degrees. We also made several other graphs that
  display various angular velocities vs. position.

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Analytical Analysis

• Formulas used
• Position Analysis Formulas
• K1 d/a
• K2 d/c
• K3 (a2 b2 c2 d2)/(2ac)
• K4 d/b
• K5 (c2 d2 a2 b2)/(2ab)
• A Cosq2 - K1 - K2Cosq2 K3
• B -2Sinq2
• C K1 (K2 1)Cosq2 K3
• D Cosq2 K1K4Cosq2 K5
• E -2Sinq2
• F K1 (K4 -1)Cosq2 K5
• q4 2tan-1(-E /- (E2 4DF)1/2)/ (2D)
• q3 2tan-1(-B /- (B2 4AC)1/2)/ (2A)
• - open
• crossed

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Analytical Analysis

• Velocity Analysis Formulas
• w3 aw2Sin(q4 - q2)/(bSin(q3 - q4)
• w4 aw2Sin(q2 - q3)/(cSin(q4 - q3)
• V r x w

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Analytical Analysis

• Acceleration Analysis Formulas
• a3 (CD AF)/(AE BD)
• a4 (CE BF)/(AE BD)
• A cSinq4
• B bSinq3
• C aa2Sinq2 aw22Cosq2 bw32Cosq3 cw32Cosq4
• D cCosq4
• E bCosq3
• F aa2Cosq2 aw22Sinq2 bw32Sinq3 cw32Sinq4
• AB AA AB/A
• AN rw2
• AT ra

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Analytical Analysis
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Analytical Analysis
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Analytical Analysis
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Analytical Analysis
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Analytical Analysis
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Analytical Analysis
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Graphical Analysis

• Equations used
• V r x w
• AB AA AB/A
• AN rw2
• AT ra
• Graphical Analysis For this we used Autocad to
  draw three different positions of our mechanism.
  From there we did graphical velocity and
  acceleration analysis on Autocad. We used Autocad
  because it generates far more accurate results
  compared to drawing it out by hand.

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Graphical Analysis
Position 1
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Graphical Analysis
Position 2
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Graphical Analysis
Position 3
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Graphical Analysis
Velocity 1
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Graphical Analysis
Velocity 2
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Graphical Analysis
Velocity 3
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Graphical Analysis
Acceleration 1
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Graphical Analysis
Acceleration 2
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Graphical Analysis
Acceleration 3
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Results Dynamic Method

• Using Working Model, we determined the location
  of the maximum velocity of the output link would
  be at the bottom of the 9 inches.
• Once we had this location, we solved for the
  angular velocities required to generate an output
  of 10 inches per second at this location.
• The resulting angular velocity of the motor for
  this location is .8561322 radians per second.

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Results Working Model

• The output velocity reached a maximum of
  approximately 10 in/s on the 9 inch output path.
• Once we got this value we treated it as merely an
  approximation because of some fundamental flaws.
  These flaws stemmed from the fact that we
  couldnt precisely determine the locations of the
  motor and pins in the program. The velocity of
  the output exceeds 10 in/s, but not on the 9 inch
  required output path.

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Result Analytical Analysis

• We used Microsoft Excel to analytically analyze
  the position, velocity, and acceleration of the
  mechanism for every 5 degrees of motor rotation.
• The position and angular velocity results seem to
  be accurate, and were verified using the
  graphical approach
• The analytical velocity of the center of the
  output link seems to have a problem. The graph
  of the velocity of this point follows a curve
  that seems accurate, but the actual value
  oscillates above and below this curve. We do not
  know why this is occurring.
• The analytical acceleration analysis is also
  questionable. Using the analytical formulas, the
  denominator nearly equals zero for the second
  four-bar in several locations, causing the values
  to asymptote. This also happens once in the
  first four-bar.

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Results Graphical Analysis

• Using Autocad we drew 3 different locations from
  which we would analyze the velocity and the
  acceleration of the output.
• We first had to assume that the angular velocity
  of the motor was what we calculated from the
  Dynamics Method.
• Using this method we determined our velocities at
  these locations to be consistent with our values
  from the Analytical Method.
• We also calculated our acceleration at these
  three locations to range from 5.6 in/s2 to 966
  in/s2. Leading us to believe that there might be
  a problem.

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Comparison of Methods

• To compare the results, we will accept the
  dynamically calculated value as the standard.
  This was the only method that allowed us to
  calculate a motor velocity exactly. It was also
  the simplest method used, and the method we are
  most confident in. The value calculated for the
  angular velocity of the motor was .8561322
  radians per second.
• The Working Model results agreed that this result
  was close to what it should be.
• The analytical method disagreed with our dynamic
  answer. Using trial and error, the analytical
  method yielded an angular velocity of roughly .6
  radians per second for the motor. Though we are
  confident in our analytical solutions for
  position and angular velocity, we are not
  confident in the analytical solution for the
  velocity of the output along the 9 inches, and
  therefore believe the dynamic answer to be more
  accurate.
• The results of the graphical analysis also tend
  to agree with the dynamics method, or rather do
  not disprove the result.

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Conclusions

• In conclusion, we were able to design a mechanism
  to convert the angular motion of a motor into a
  vertical output. The velocity requirement was
  more difficult to achieve, but we were able to
  calculate an angular velocity for the motor, and
  verify it with two out of three methods.

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Recommendations

• There is a definite problem with our analytical
  calculation of the velocity of the center of the
  output link. We also believe there is a problem
  with the analytical acceleration analysis, as
  their graphs do not follow any trend (not that we
  are qualified to judge the acceleration results).
  
• Therefore, our recommendation would be to use the
  dynamics method to solve for the velocities of
  more points along the output to confirm our
  calculated solution, and to reattempt to solve
  the velocity of the output and the accelerations
  analytically.

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Resources

• Working Model 2D
• Fourbar
• Design of Machinery (Third Edition)
• Engineering Mechanics Dynamics (Third Edition)
• Autocad 2000
• Microsoft Excel
• Microsoft PowerPoint
• Microsoft Word