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.
| \(M\) | mass of first class cabin |
| \(k\) | spring constant |
| \(\zeta \) | critical damping constant |
| \(r\) | ratio of external load frequency to the natural frequency of first class cabin \(\frac {\varpi }{\omega _{natural}}\) |
| \(r_{n}\) | ratio of external load \(n^{th}\) harmonic frequency to the natural frequency of first class cabin \(\frac {\varpi _{n}}{\omega _{natural}}\) |
| \(T_{r}\) | Transmissibility. The ratio of cabin absolute displacement to base absolute displacement |
| \(\omega _{nat}\) | Natural frequency of first class cabin |
| \(\omega _{1}\) | Fundamental frequency of the external load frequency. |
| EOM | Equation Of Motion |
| \(c\) | damping constant for damper under first class cabin |
| \(Z_{n}^{acc}\) | the complex amplitude of the term associated with the \(n^{th}\) harmonic of the frequency \(z^{\prime \prime }\relax (t) \) |
| \(Z_{n}^{disp}\) | the complex amplitude of the term associated with the \(n^{th}\) harmonic of the displacement \(z\relax (t) \) |
| \(Y_{n}\) | the complex amplitude of the term associated with the \(n^{th}\) harmonic of the displacement of \(y\relax (t) \) |
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 \(\left (T_{r}\right ) \) of the absolute acceleration of the airplane to that of the first class cabin. A passive vibration isolation system was used for its ease of implementaion and its low cost. The model is based on figure 1 below
Figure 1. Mechanical model view of vibration isolation system in place.
The absolute acceleration of the first class cabin, \(y^{\prime \prime }\left ( t\right ) \), was calculated with the vibration isolation system in place and then compared to the absolute acceleration, \(z^{\prime \prime }\relax (t) \), of the rest of the airplane. The goal was to produce a smooth absolute acceleration time history when compared to the rest of the airplane. This was done by adjusting \(M,\zeta \) and \(K\) and running a simulation of the motion of the plane with our vibration isolation system in place. A plot of \(T_{r}\) vs. \(r\) was also used to insure that the maximum \(T_{r}\) remained small as the frequency ratio \(r\) was increased.
Assuming the mass of cabin is \(M,\) which includes the live load (passengers), then applying Newton’s laws the the first class cabin results in the equation of motion
\begin {align*} my^{\prime \prime }+c\left (y^{\prime }-z^{\prime }\right ) +k\left (y-z\right ) & =0\\ my^{\prime \prime }+cy^{\prime }+ky & =cz^{\prime }+kz \end {align*}
The transfer function between \(y\relax (t) \) and \(z\relax (t) \) in the frequency domain can now be derived (Appendix contains complete derivation) resulting in
\[ T\relax (r) =\left \vert \frac {Y_{n}}{Z_{n}^{disp}}\right \vert =\frac {\sqrt {1+\left (2\zeta r_{n}\right ) ^{2}}}{\sqrt {\left (1-r_{n}^{2}\right ) ^{2}+\left (2\zeta r_{n}\right ) ^{2}}}\]
To compare the absolute acceleration of the first class cabin with the rest of the airplane, the absolute acceleration, \(y^{\prime \prime }\relax (t) \), is now found from \(Y_{n}\). Since \(y\relax (t) =\operatorname {Re}\left \{ Y_{n}e^{i\varpi _{n}t}\right \} \) then \(y^{\prime \prime }\left ( t\right ) =\operatorname {Re}\left \{ -\varpi _{n}^{2}Y_{n}e^{i\varpi _{n}t}\right \} \).
\(z^{\prime \prime }\relax (t) \) (given) and \(y^{\prime \prime }\left ( t\right ) \) (computed) are now plotted on the same plot in order to compare the effect of our vibration isolation system to the comfort of the first class passengers. The final design parameters used are (Appendix 5.1)
| \(M\) Mass of first class cabin (dead+live) | \(\allowbreak 3050\) kg |
| \(\zeta \) | \(0.7\) |
| \(k\) | \(9700\) N/M |
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 \(T_{r}\) decreases with increasing \(r\)
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 \(T_{r}\) decreased as \(r\) increased. This implies that at higher external acceleration relative to the natural frequency, our vibration isolation system remained effective. The simulation program generated a mechanical view showing the absolute position of the first class cabin, with an offset, and the absolute position of the airplane during the flight as shown in figure 4 below.
Figure 4. Animation of vibration isolation during flight.
The force shown in Figure 4. below the airplane is the numerical value of \(-Mz^{\prime \prime }\) where \(z^{\prime \prime }\) is the absolute acceleration of the airplane and \(M\) is the total mass of the first class cabin.
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 \(3.53\) m (\(11.58\) ft), and then extends down the body of the plane roughly \(3.35\) m (\(11\) ft) giving the first class cabin a total area of \(3.53\times 3.35\) m\(^{2}\) (\(11.58\times 11\) ft\(^{2}\)).
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 \(1.524\) m (\(5\) ft) in either direction giving a total area of the platform used in our vibration isolation system \(3.048\times 3.35\) m\(^{2}\) (\(10\times 11\) ft\(^{2}\)) as seen above in figure \(5\).
To begin the actual design, additional support must be given to the aluminum floor of the cabin. The use of \(6061\) T\(6\) Aluminum I-beams (specifications are given in appendix \(5\)) spanning the width of the platform provides the needed support. In addition the I-beams provide a sturdy surface for the spring and dashpot system to contact the cabin floor.
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 \(5\) carbon fiber leaf springs provides the spring constant required (\(9700\) N/m) and a low increase in weight.
The dashpots needed for our design, \(2K325\) Dashpots, can be purchased from many manufactures. When added in parallel they provided the necessary damping coefficient of \(7425\) N*s/m needed when the first class cabin is full and \(5800\) N/s*m when the cabin is empty.
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 $\(16500\) (appendix \(5\)). The cost of the aluminum support beams, dashpots and carbon fiber leaf springs make up the majority of the material cost totaling only about $\(3000\). The majority of the total cost comes from the additional weight of the system and the resulting price of fuel used during the planes lifetime. The additional weight results in an expected cost of about $\(13500\) over the lifetime of the plane.
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) |
| \(15\) Chairs \(@\) \(100\) kg/chair | \(1500\) |
| \(10\) ft by \(11\) ft aluminum flooring | \(200\) |
| \(5\) Aluminum I-beams \(@\) \(20\) kg/beam | \(100\) |
| \(5\) Carbon Fiber leaf springs @ \(5\) kg/spring | \(25\) |
| Miscellaneous weight | \(25\) |
| \(15\) Passengers @ \(80\) kg | \(1200\) |
| Weight of First Class Cabin before vibration isolation system | \(1700\) |
| Weight of First Class Cabin after vibration isolation system | \(1850\) |
| Weight of First Class Cabin with maximum passengers | \(3050\) |
Table 3. Mass of items used in design calculations
The total mass \(M\) has the value of \(3050\) kg. For our \(\zeta \) value we choose the value \(0.7\) as it worked well in simulations to provide a smooth ride for the passengers while still keeping \(T_{r}\) small.
Leaf springs and spring K value
The most important aspect of picking a \(k\) value is the total allowed clearance the first class cabin floor has to move. The first class cabin’s floor has a displacement relative to the body of the aircraft and if that gets too large the floor will make contact with the body of the airplane. The lower the \(k\) value we choose, the larger the displacement of the first class cabin relative to the body of the airplane will become. The maximum travel distance of the first class cabin is around \(20\) cm (\(7.87\) in) and we can use this value to pick an appropriate \(k\) value. A \(k\) value around \(10000\) N/m keeps the first class cabin floor within this tolerance. The following plot shows the absolute acceleration of the cabin vs. the rest of the airplane during the turbulent flight 1 .
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 \(k\) value of any leaf spring system can be calculated by the equation
\[ k=\frac {8Enbt^3}{3L^3} \]
Figure 7. Leaf spring design used in vibration isolation system
Since our springs are in parallel, the \(k\) values add to give a total equivalent \(k\). We took the \(k\) value selected (\(10000\) N/m) and divided it by \(5\) giving us an individual \(k\) value of \(2000\) N/m. Using the following dimensions for the leaf spring resulted in a \(k\) value of \(1940\) N/m for a total \(k\) value of \(9700\) N/m.
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 \(fft\_easy()\) and \(ifft\_easy()\) in the class web site. The absolute displacement was found from the absolute acceleration in the frequency domain. Due to the problem of division by zero for the first component in the frequency vector, this was set to zero before using \(ifft\_easy().\)
Assuming the mass of cabinet is \(M\) which includes passengers weight, by applying Newton’s laws the EOM for the first class cabin is\begin {align} my^{\prime \prime }+c\left (y^{\prime }-z^{\prime }\right ) +k\left (y-z\right ) & =0\nonumber \\ my^{\prime \prime }+cy^{\prime }+ky & =cz^{\prime }+kz \label {eq:1} \end {align}
The time history of the turbulent acceleration \(z^{\prime \prime }\left ( t\right ) \) was given to us in the matlab mat file. Therefore in the frequency domain, and assuming the time history represents one period we can write\[ z^{\prime \prime }=\operatorname {Re}\left \{ Z_{n}^{acc}e^{i\left (\omega _{1}n\right ) t}\right \} \]
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\[ \left \vert \frac {Y_{n}}{Z_{n}^{disp}}\right \vert =\frac {\sqrt {1+\left (2\zeta r_{n}\right ) ^{2}}}{\sqrt {\left (1-r_{n}^{2}\right ) ^{2}+\left (2\zeta r_{n}\right ) ^{2}}}\]
Where \(Z_{n}^{acc}\) is the complex amplitude of the \(n^{th}\) harmonic component in the acceleration data. Letting \(\omega _{1}n\equiv \varpi _{n}\) then in the frequency domain Eq ?? becomes\begin {align*} \operatorname {Re}\left \{ \left (-m\varpi _{n}^{2}+i\varpi _{n}c+k\right ) Y_{n}e^{i\varpi _{n}t}\right \} & =\operatorname {Re}\left \{ \left ( c\frac {Z_{n}^{acc}}{i\varpi _{n}}+k\frac {Z_{n}^{acc}}{-\varpi _{n}^{2}}\right ) e^{i\varpi _{n}t}\right \} \\ Y_{n} & =\left (\frac {\frac {c}{i\varpi _{n}}-\frac {k}{\varpi _{n}^{2}}}{-m\varpi _{n}^{2}+i\varpi _{n}c+k}\right ) Z_{n}^{acc}\\ & =-\frac {Z_{n}^{acc}}{\varpi _{n}^{2}}\frac {1+i2\zeta r_{n}}{\left ( 1-r_{n}^{2}\right ) +2i\zeta r_{n}} \end {align*}
Where \[ r_{n}=\frac {\varpi _{n}}{\omega _{natural}}\]
But \(-\frac {Z_{n}^{acc}}{\varpi _{n}^{2}}\)is the absolute displacement of the airplane, say \(Z_{n}^{disp}\), hence the transfer function between the absolute displacement of first class cabin and the absolute displacement of the airplane is\[ Y_{n}=\frac {1+i2\zeta r_{n}}{\left (1-r_{n}^{2}\right ) +2i\zeta r_{n}}Z_{n}^{disp}\]
The magnitude of the absolute displacement of first class cabinet relative to absolute displacement of the airplane is\[ \left \vert \frac {Y_{n}}{Z_{n}^{disp}}\right \vert =\frac {\sqrt {1+\left (2\zeta r_{n}\right ) ^{2}}}{\sqrt {\left (1-r_{n}^{2}\right ) ^{2}+\left (2\zeta r_{n}\right ) ^{2}}}\]
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)