Phasor%20analysis%20of%20mechanical%20systems

Phasor%20analysis%20of%20mechanical%20systems - Phasor...

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Phasor Analysis of Linear Mechanical Systems and Linear Differential Equations ME104, Prof. B. Paden In this set of notes, we aim to imitate for linear mechanical systems and linear differential equations, the phasor analysis we learned for electric circuits. Recall how we derived the complex impedance for an inductor. Starting with the differential equation for the V-I characteristic for an inductor I dt d L V = ( 1 ) we substitute complex sinusoids t j e V V ω ˆ ( 2 ) t j e I I ˆ ( 3 ) So that equation (1) becomes ( t j t j e I dt d L e V ˆ ˆ = ) ( 4 ) Differentiating and solving yields I L j V ˆ ˆ = ( 5 ) and the impedance of the inductor is defined by ˆ ˆ V Z jL I = ± ( 6 ) where “ ” denotes “defined equal to”. Having done this calculation once, we see that we can jump directly from (1) to (5) by making the substitution ± j dt d ( 7 ) Phasor Analysis of Linear Mechanical Systems Consider a mechanical damper (a.k.a. shock absorber) which produces a velocity- dependent force according to the linear differential equation x dt d b f = ( 8 ) Making the substitution j dt d , we get
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x b j f ˆ ˆ ω = ( 9 ) And defining the mechanical impedance to be the ratio of force to displacement, we have ˆ ˆ f Z jb x = ± (Newtons/meter) (10) Note that the damper has a low stiffness at low frequencies, and a high stiffness at high frequencies. The units of impedance are Newtons/meter in mechanical systems and volts/amp = Ohms in electrical systems. For a mass, m , we have 2 2 dd d f mx m dt dt dt ⎛⎞ == ⎜⎟ ⎝⎠ x ( 1 1 ) Substituting j dt d yields () 2 2 ˆ ˆ f j m x ωω ( 1 2 ) The impedance of a mass increases very rapidly with frequency. This explains why anvils
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Phasor%20analysis%20of%20mechanical%20systems - Phasor...

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