High power piezoelectric axial shockwave generation

9.2 Stress impulse generation by a ramming rodUsually a stress impulse can be generated by theimpact of two cylindrical rods. An example are theimpacts of flying piston, ram and shaft of the HiltiEP-tool (Electropneumatic Percussion). Here nowthe fundamental difference between the realhammer and the piezo-hammer is based: a hammeris ramming - the piezo-actuator is impacting.If the surfaces of contact are spherical the contacttheory of Hertz (Johnson) can be applied. Thecontact of ideal plane surfaces has to be treatedacoustically or as the numerical limit of very largecontact radii.Definitions:The indices 1 and 2 are related to the rod 1 androd 2, respectively.E is the Young's modulus, ν is Poisson's ratio. Forthe effective modulus E eff :E eff = (1 - ν 12 )/E 1 + (1 - ν 22 )/E 2holds. For the effective contact radius r eff theequation hold:1/r eff = 1/r 1 + 1/r 2With E eff und r eff the stiffness constant of theHertzian contact k Hertz is writtenk Hertz = 4/3 E eff r eff1/2The force of contact is given byf contact (w) = k Hertz w 3/2where w is the interpenetration of the two surfaces.Dynamically the Hertzian contact between rod 1und rod 2 with the effective mass m effand the dynamic force1/m eff = 1/m 1 + 1/m 2 )f contact (w(t)) = k Hertz w(t) 3/2lasts during the contact time T contactT contact = 2.9432 (15/16) 2/5 (1/E eff ) 2/5 (m eff ) 2/5 (1/ reff ) 1/5 (1/v) 1/5with v beeing the relative velocity of the two rodsbefore impact.Before impact, the rod (e.g. the flying piston) isflying freely and can be represented in terms of twostress impulses one to the right σ right and a secondto the left σ left both with the same amount σ 0σ 0 = Iv 0 /2but with opposite sign. For the particle velocity therelation holds1/I(σ rechts - σ links ) = v 0During the contact energy and impulse momentumare exchanged.For a dynamic Hertzian contact the force of contactis in good approximation given by an Gaussiansignal form where impulse maximum and impulsewidth depend on the given parameters. In theexample of figure 13 the maximum is assumed to50 kN and the width to 50 µs.20