Determine the amplitude ration:

A trailer weighs 1 tonne while fully loaded and 250 kg while empty. The stiffness of suspension spring is equal to 350 kN/m. The damping factor is equal to 0.5 while  fully loaded. Determine the amplitude ration while it is fully loaded & empty. The speed of trailer is equal to 100 km/hr on a road with sinusoidal track along 5 m wavelength.

Solution

 It is given that:

Mass of fully loaded trailer = 1000 kg

Mass of empty trailer = 250 kg

Velocity 'v' = 100 km/hr = 100/3.6  = 27.78 m/s

Wavelength = 5 m

Stiffness of spring = 350 kN/m

The sinusoidal function of road surface

 = y₀  (sin 2π vt)/5

here      v = 27.78 m/s

While fully loaded

∴             r =  ω /ω_n = 34.9 /18.7 = 1.865

Amplitude ratio = Transmissibility

= 2.12/3.10 = 0.68

While trailer is empty

Here the value of damping factor will be different because critical damping shall be different now but damping coefficient of damper shall be similar when trailer was fully loaded

ζ = 0.5

C_c = 2m w_n  = 2 × 1000 × 18.7 = 37400 N sec/m

∴             C = 0.5 × C_c = 0.5 × 37400 = 18700 N sec/m

The critical damping for empty trailer is following

C_c  = 2m ω_n = 2 × 250 × 37.4 =18700

∴             ζ = 18700/18700 = 1

This is a case of critical damping.

For critical damping amplitude ration = 1.