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Mechanical Vibrations
Prof. Paulo J. Paupitz Gonçalves
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Shock Absorber for a MotorcycleAn underdamped shock absorber is to be designed for a motorcycle of mass 200 kg. When the shock absorber is subjected to an initial vertical velocity due to a road bump, the resulting displacement-time curve is shown in the figure.
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Shock Absorber for a MotorcycleFind the necessary stiffness and damping constants of the shock absorber if the damped period of vibration is to be 2s and the amplitude x1 is to be reduced to one-fourth in one half cycle (i.e., x
1.5 = x
1/4). Also find the
minimum initial velocity that leads to a maximum displacement of 250 mm.
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Shock Absorber for a Motorcycle
Solution:
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Shock Absorber for a Motorcycle
Solution:
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Shock Absorber for a Motorcycle
Critical Damping:
Damping Constant:
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Shock Absorber for a Motorcycle
The displacement is a maximum at
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Shock Absorber for a Motorcycle
The envelope passing through the maximum points
Since
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Shock Absorber for a Motorcycle
The velocity of the mass can be obtained
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Analysis of Cannon
When the gun is fired, high pressure gases accelerate the projectile inside the barrel to a very high velocity. The reaction force pushes the gun barrel in the direction opposite that of the projectile.
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Analysis of Cannon
When the gun is fired, high pressure gases accelerate the projectile inside the barrel to a very high velocity. The reaction force pushes the gun barrel in the direction opposite that of the projectile.
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Analysis of Cannon
Since it is desirable to bring the gun barrel to rest in the shortest time without oscillation, it is made to translate backward against a critically damped spring-damper system called the recoil mechanism.
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Analysis of Cannon
In a particular case, the gun barrel and the recoil mechanism have a mass of 500 kg with a recoil spring of stiffness 10,000 N/m. The gun recoils 0.4 m upon firing.
Find
(1) the critical damping coefficient of the damper(2) the initial recoil velocity of the gun(3) the time taken by the gun to return to a position 0.1 m from its initial position.
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Analysis of CannonThe undamped natural frequency
The critical damping coefficient
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Analysis of Cannon
The response of critically damped system
To find the time when the displacement is a maximum
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Analysis of Cannon
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Analysis of Cannon
Initial Velocity
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Analysis of Cannon
the time taken by the gun to return to a position 0.1 m from its initial position.
Use a numerical root findMethod to obtain the solution
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Stability of Mechanical Systems
Considering the spring-mass-damper system
whose characteristic equation can be expressed as
or
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Stability of Mechanical Systems
or
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Stability of Mechanical Systems
The solution can be written as
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Stability of Mechanical Systems