four_bar_kinematic_analysis_of_a_mechanical_bird_device.pdf

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Four Bar Kinematic Analysis of a Mechanical Bird Device

Evan A. Kontras

University of Missouri

Department of Mechanical and Aerospace Engineering

November 2008

Many important and practical machines utilize a four bar mechanism. Though simple,

having only three links and a fixed reference or ground, the four bar mechanism has a wide

variety of uses in engineering. Having only one degree of freedom, a four bar mechanism can be

operated with a single input, yet a wide range of operations can be accomplished with different

categories of the four bar. A common utilization of this mechanism is in transforming rotational

motion into translational or oscillatory motion. In designing a flap wing style flying vehicle, the

four bar mechanism is ideal.

With many electric motors providing rotation, and the need to have a quickly oscillatinglink, the crank rocker category four bar has been selected for the vehicle design. A motor with

6000 rpm capability was chosen. It will be assumed that the optimal wing speed is 30 flaps per

second, therefore gears will have to be used to reduce the motor speed. This can be done using

the simple relationship,

(1)

where r is the radius of each respective gear, and the angular velocity. A 3 mm gear on the

motor hub in mesh with a 10 mm gear will result in 30 oscillations per second of L4, as shown in

Fig. 1 below, which is a simplified schematic of the vehicle mechanism and structure.

Figure 1. Crank Rocker Mechanism.


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For simplicity the 10 mm gear will be represented in the four bar mechanism as L2, a

strait line. A small, light design is desirable, therefore the other links were decided accordingly,

taking into consideration the need for a moderate range of oscillation for link L4. This link is

particularly important because it is where the wing of the vehicle connects to input motion. With

link dimensions known, the position, velocity, and acceleration of a point on the four bar

mechanism can be determined using the complex number method, utilizing

d

dt + (2)

dt + + (3)

where z, , and are the position, velocity, and acceleration respectively, the angular position, the angular velocity, and the angular acceleration. It should be noted that the terms in order

from left to right in Eq. (3) represent the radial, coriolis, tangential, and normal components of

acceleration. Using the complex number method is very helpful in understanding the kinematics

of the four bar mechanism, but it can become quite tedious to repeat the calculations for many

values of. However, using a spread sheet or similar computational program, multiple values of

can be quickly and easily calculated. Also, the use of computer programs can provide for a

quick means of plotting data as well. Using Microsoft Excel, the position, velocity, and

acceleration of a point at the midspan of the wing was determined for crank angles between 0

and 360. As the crank 360 the wing completes one flap. The plots showing how each

parameter changes as the crank rotates are shown in the figures below.

Figure 2. Displacement of Wing Center from Original Position.

-300

-250

-200

-150

-100

-50

0

50

0 50 100 150 200 250 300 350 400

Displacement(mm)

Crank Angle (Degrees)


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As expected with oscillating motion, the point at the center of the wing ends at the same

point that it started. From crank angles between approximately 100 and 275, the wing is

moving downward for more than 20 cm generating lift. The midpoint of the wing, which is

initially at rest, reaches a peak velocity of 164 mm/s. Again, the velocity curve ends at the same

value it begins, as shown in Fig. 3. The angular velocity of the wing is the same as that of link L4

from Fig. 1.

Figure 3. Angular Velocity of Wing.

Finding the acceleration is more involved, though other important vehicle criteria can bedetermined once the components of acceleration are found. The low points on the angular

acceleration curve correspond to points when the wing is changing direction. The maximum

velocity occurs at the minimum acceleration as seen from Fig. 4.

-200

-150

-100

-50

0

50

100

150

200

0 50 100 150 200 250 300 350 400

AngularVe

locity(rad/s)



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Figure 4. Angular Acceleration of Wing.

It is at this point, approximately 175 of crank rotation, where the wing has experienced the

longest time under acceleration and is traveling at 164 mm/s. After the maximum velocity, the

acceleration changes direction, eventually stopping the wing and changing its direction of motion

as well.

Considerable forces can be developed along the wing due to the abrupt changes in both

velocity and acceleration. A lift force is produced by each wing, proportional to the square of the

angular velocity. For each wing, the lift force has been determined to be 60% of the total vehicle

weight. Unlike the other links of the four bar mechanism analyzed for this vehicle, the wing itself

is assumed to have mass. Each wing is one quarter of the total mass of the vehicle, with theentire mass being concentrated at the wing midpoint for simplicity. Inertial forces are produced

as the mass of the wing rotates. Knowing the angular acceleration of the wing and the lift force

generated,Newtons second law can be used to solve for the inertial force created by the

accelerated mass of the wing. Balancing these forces is the torque on the other end of the link. As

the angular acceleration of the wing changes, so too does the torque, as seen in Fig. 5.

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0 50 100 150 200 250 300 350 400

AngularAcceleration(rad/s^2)



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Figure 5. Driving Torque for One Wing.

Torque must originate at the motor, actually driving both wings. With a combined mass of halfthe vehicle, it is important that the motor have enough power to accelerate the wings as needed,

without any extra gearing for mechanical advantage. Knowing the combined torque on each

wing, the necessary toque the motor must output can be determined. Fig. 6 shows the energy

required to move the wings during operation, which can provide information on how much

power the motor must output.

Figure 6. Energy for Complete Cycle.

The motor must be able to reliably output the maximum energy, approximately

400,000 kgf-mm. This is around 0.0146 hp, a power requirement many electric motors are

capable of satisfying. Weight is one of the most important parameters for any flying vehicle, so

the lightest motor available to produce 0.0146 hp is desirable. The weight of each structural

-100000

-50000

0

50000

100000

0 50 100 150 200 250 300 350 400

Torque

(N-mm)


-300000

-200000

-100000

0

100000

200000

300000

400000

500000

0 50 100 150 200 250 300 350 400Energy(kgf-mm)



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component should also be considered. Due to the torque and inertial forces on the wing, a

relatively high strength material will be used. The maximum torque is 75,000 -, which is

equal to the force on the wing multiplied by the distance from the frame, which is half the

wingspan. At this distance, a force of approximately 400 N is developed. Once a cross sectional

area for the wing is determined, the stress can be found, and a proper material selected. The

flapping wing flying vehicle design will use 3 mm diameter wing supports, with each connecting

link, L2-L4 being 3x5 mm rectangles. The highest stresses are developed where the wing support

meets the larger area connectors, with a maximum value of approximately 56 MPa. Plastics such

as PVC and Nylon will yield under such stresses, so a more high strength material will be

necessary. Most steels have a yield strength between 200 and 600 MPa, but unfortunately they

are too heavy to be suitable for the flying vehicle. Aluminum, with a density nearly one third of

steel, has an equally high strength, around 400 MPa. Due to its high strength to weight ratio and

abundance, aluminum was selected as construction material for the vehicle. A simplified model

of the four bar mechanism and aluminum frame for the flapping wing vehicle is shown below in

Fig. 7.

Figure 7. Basic Frame and Mechanism.

To save weight, only wing supports will be used for the vehicle, with the wing itselfbeing made of a thin light weight fabric or plastic. The rear inside edge of each wing skin will

connect to a rocker at the rear of the frame to control direction during flight. Also, a large

adjustable tail section at the rear of the frame will control the altitude. Both direction control

devices are actuated with linear servo motors. The entire vehicle will be powered with

lightweight, rechargeable lithium batteries. A completed prototype is shown in Fig. 8, 9,

10,and11.


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Figure 8. Front View.

Figure 9. Arial View.


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Figure 10. Altitude Control.

Figure 10. Directional Control.

Animations, SolidWorks assembly files, and other perspective pictures are included on

disk.


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OMEGA 2= 628.3 rad/s

AB

C

D

NOTES:

-Units are in millimeters

10.0000

20.0000

40.0000


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