By Nasser M. Abbasi, Donny Kuettel III and Paul Frisch.
This report outlines a simple passive vibration isolation system design for use in the first class cabin of a Boeing 757-200 airplane with the goal of reducing the vibrations felt by the passengers in the first class cabin. This was done by simulation in order to select suitable design parameters that produced an acceptable absolute acceleration time history compared the rest of the airplane during a turbulent flight.
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mass of first class cabin |
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spring constant |
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critical damping constant |
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ratio of external load frequency to the natural frequency of first class
cabin |
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ratio of external load |
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Transmissibility. The ratio of cabin absolute displacement to base absolute displacement |
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Natural frequency of first class cabin |
| Fundamental frequency of the external load frequency. |
EOM | Equation Of Motion |
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damping constant for damper under first class cabin |
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the complex amplitude of the term associated with the |
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the complex amplitude of the term associated with the |
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the complex amplitude of the term associated with the |
Table 1. Description of mathematical notations used in report
Reducing the vibration effect felt by the passengers in the first class cabin was based on reducing
the transmissibility ratio
Figure 1. Mechanical model view of vibration isolation system in place.
The absolute acceleration of the first class cabin,
Assuming the mass of cabin is
The transfer function between
To compare the absolute acceleration of the first class cabin with the rest of the airplane, the
absolute acceleration,
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Table 3. Final values of design parameters
Figure 2 below shows the result using the above parameters
Figure 2. First class cabin absolute acceleration compared to rest of airplane.
We see from figure 2 that the absolute acceleration of the first class cabin has much less variation
and is much smoother than the absolute acceleration of the rest of the airplane. From this we can
see that the first class passengers experience a much more comfortable flight than the rest of the
airplane. In addition, the transmissibility plot was found to be acceptable since
Figure 3. Transmissibility plot of first class cabin.
In addition to producing a smooth absolute acceleration time history, the goal was also to insure
that
Figure 4. Animation of vibration isolation during flight.
The force shown in Figure 4. below the airplane is the numerical value of
The vibration dampening system proposed for the first class cabin is a simple spring dashpot system that utilizes the additive properties of springs and dashpots to dampen the vibration of the first class cabin in the Boeing 757-200.
Figure 5. Schematic diagram of vibration isolation system in place
The design of our passive vibration isolation system is simple and effective with a minimal costs. It starts by defining the area that represents the first class cabin, which is at the front of the plane right behind the cockpit.
The cabin spans the entire inside width of the airplane body, which is
The next step in our design is to define the area that will actually be part of the vibration
isolation system. We cannot use the whole floor of the first class cabin because the
rounded body of the airplane would not allow the floor to travel up and down rendering
our whole system ineffective. To solve this problem we started at the center of the
plane’s cross section and went out
To begin the actual design, additional support must be given to the aluminum floor of
the cabin. The use of
The key component of the vibration dampening system is the use of carbon fiber leaf springs. We
chose carbon fiber leaf springs in place of steel for several reasons. They provide a softer ride at a
lower noise level and excellent stability due to better damping characteristics than steel. Placed
in series, the use of
The dashpots needed for our design,
Our design for this passive vibration isolation system works whether the first class cabin is full, empty, or half way in-between. The system works best when the cabin is fully loaded with passengers, and has almost identical results with no passengers on board. Even though the results are slightly diminished with fewer passengers, the system still creates a noticeably smoother flight.
The total cost of our vibration isolation system is around $
The damping effects of the system could be improved if weight were added to the cabin. However the additional cost of the added weight over the lifetime of the plane would outweigh the benefits for the passengers. If however some heavy components of the plane were to be attached to the first class cabin, the system could be redesigned for an even better ride. This would require further investigation into the balance of the plane, flight dynamics and a deeper knowledge of the various components of the plane so it falls out of the scope of this project.
weight
This table shows the design values based on weight
Description of item | Mass (kg) |
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Miscellaneous weight | |
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Weight of First Class Cabin before vibration isolation system | |
Weight of First Class Cabin after vibration isolation system | |
Weight of First Class Cabin with maximum passengers | |
Table 3. Mass of items used in design calculations
The total mass
Leaf springs and spring K value
The most important aspect of picking a
Figure 6. absolute acceleration of first class cabin compared to rest of airplane
To keep the weight of our vibration isolation system as small as possible we opted to use
carbon fiber leaf springs. The
Figure 7. Leaf spring design used in vibration isolation system
Since our springs are in parallel, the
After finding the materials we needed, the following describes how we calculated the total cost of our vibration isolation system.
The simulation program was a GUI program written in Matlab version 2013a, which made it easier to determine the parameters to use for the design. The following is a screen shot of the program. The program can be downloaded from the project web site
Figure 6. Simulation Matlab program used for obtaining the design parameters.
The first step is to load the Matlab .mat file which contains the acceleration time history. Then
one can use the sliders to adjust the system parameters and see the effect on the absolute
acceleration of the first class cabin. Computation was done in the FFT domain using
the functions
Assuming the mass of cabinet is
The time history of the turbulent acceleration
Substituting back into Eq ?? and simplifying, the magnitude of the absolute displacement of the
first class cabin relative to absolute displacement of airplane is found to be
Where
Where
But
The magnitude of the absolute displacement of first class cabinet relative to absolute
displacement of the airplane is
http://www.boeing.com/boeing/commercial/757family/index.page
The following zip file contains the current version of Matlab software to use to design the vibration isolation system.
1absolute position of the first class cabin was computed from the absolute acceleration of the cabin in the frequency domain. Hence the average value was not used due to the division by zero problem with this method. We do not have another method to find absolute position from absolute acceleration (unless we use more advanced numerical integration method in time domain, which is beyond the scope of this course)