EG260 Damping - A.K. Slone EG-260 Dynamics (1) ©a.k.slone...

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Unformatted text preview: A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2008 1 of 14 EG-260 DYNAMICS I – Damping EG-260 DYNAMICS I – Damping..........................1 1. Introduction ..............................................2 1.1. The idealised dashpot ...............................3 2. Underdamped motion ..............................7 3. Overdamped motion .................................9 4. Critically damped ...................................11 5. Summary .................................................12 A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2008 2 of 14 1. Introduction Observation of real life oscillatory systems shows that for most the oscillation eventually dies so that eventually there is zero motion. The equation: ( ) ( ) ( ) ( ) t x f t kx t x m = + & & (1) has a solution of the form: ( ) ( ) φ ω + = t A t x n sin (2) which shows no decay in the oscillations and so equation (1) needs a further term to account for the decaying motion. From theory of differential equations adding a term that is proportional to the velocity of the system i.e. viscous damping will decay the motion, thus dissipating energy. Thus the equation of motion is: ( ) ( ) ( ) = + + t kx t x c t x m & & & (3) The constant c is the damping coefficient and has units Ns/m or kg/s and the damping force is: ( ) ( ) t x c t f c & = (4) The following notes give an outline explanation of the method of viscous damping , but you will still need to attend the lectures to fully understand damping as applied in mechanics. A.K. Slone EG-260 Dynamics (1) ©a.k.slone 2008 3 of 14 In most real life situations the damping coefficient of a system can not be measure in the same way as the mass or stiffness, but is estimated using mathematical techniques such as Finite Element analysis. For an analytical approach, the physical representation for dissipating energy is the dashpot or damper . 1.1. The idealised dashpot The dashpot is modelled as a piston inside an oil filled cylinder as seen in Figure 1. Consider a block of mass, m, resting on a frictionless horizontal surface, which is attached to a vertical wall by a mass-less spring as in SHM notes, but which now is subject to damping, see Figure 2 The dashpot is indicated by Mounting point Mounting Seal Casing Piston Oil Orifice x(t) Figure 1 Schematic of dashpot A.K. Slone A....
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This note was uploaded on 05/18/2010 for the course ENGINEERIN EG 260 taught by Professor Stone during the Spring '10 term at Swansea UK.

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EG260 Damping - A.K. Slone EG-260 Dynamics (1) ©a.k.slone...

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