Model a passenger aircraft as spring-mass-damper system and find suitable values of spring constant and damping ratio for the system.
Theory:
An aircraft can be taken as a lumped mass attached to a spring and a damper. If the aircraft has a mass `m`, net spring constant of suspension system be `k` and net damping coefficients of the damper system is `C`, the damping ratio can be defined as:
`\xi = \frac(C) (C_c)`
`C` = Actual damping
`C_c` = Critical damping
`C_c` can be defined as:
`C_c = 2\sqrt(km)`
Thus in terms of `\xi`, `C` can be written as:
`C = 2\xi\sqrt(km)`
The benefit of using this conversion is that `\xi` is a better known term in damping systems rather than `C`.
Thus differential equation of the aircraft system can be written as:
`m\ddotu + 2\xi\sqrt(km)\dotu + ku = 0`
Maximum landing weight of a passenger aircraft is around 260,000 kg and maximum take off weight is around 360,000 kg.
Assuming worst case scenario when aircraft needs to land right after take off, we have taken 360,000 kg as the mass for all calculations.
Approximate selection of spring constant value:
A typical race car weighs 1200 kg and net spring constant of the suspension system is around 3600 N/m.
Scaling this up for the aircraft, we can select spring constant values up to 900,000 N/m.
Approximate selection of damping ratio value:
Datasheets suggest damping ratio value of 0.3 for auto shock absorbers.
Initial conditions of a plane landing:
In case of standard landing, the vertical speed of aircraft is 1 m/s.
Initially, just before impact, the spring will be in relaxed state and thus initial displacement will be equal to undamped amplitude for the above case. Solving the differential equation with zero damping and initial condition `\dotu_0 = 1`, we get maximum displacement to be around 0.6 m.
Thus initial conditions can be selected as `u_0 = 0.6` and `\dotu_0 = -1`.
Negative initial velocity means the mass will be pushed in opposite direction of the initial displacement, at the time of the impact.
Results:
1) For `\xi = 0.3`
2) For `\xi = 0.6`
3) For `\xi = 0.9`
4) For `\xi = 0.999`
5) For `\xi = 1.2`
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