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On longitudinal impact II

Elastic impact of bars with cylindrical sections of different diameters and of bars with rounded ends

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Summary

The graphodynamical theory developed in a previous paper is used to treat elastic impact of a cylindrical hammer on a bar with sudden or stepwise increase or reduction of area, with neck or swell or with capblock. The influence of area ratios in these cases and for impact on a cylindrical bar of hammers consisting of two cylindrical parts of different diameters is given in formulae and diagrams. For elastic impact of bars with rounded ends a graphical equivalent to the numerical iteration method of Sears is given and compared with a simpler graphodynamical method. Some examples indicate the influence of impact velocity and radii of curvature on the form of the pulse front. In a number of cases the theoretical results are compared with oscillograms, and fairly good agreement is found.

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Abbreviations

A :

Cross sectional area

a, b, c...:

Different states (combinations of force and velocity values)

D :

Diameter of bar

E :

Young's modulus

L :

Length

P :

Force

q :

Reflection constant \(q = \frac{{1 - r}}{{1 + r}},{\text{ }}q1 = \frac{{1 - r_1 }}{{1 + r_1 }},{\text{ }}q2 = \frac{{1 - r_2 }}{{1 + r_2 }}\)

r :

Total area ratio r=A 3/A 1 etc. In the general case where different materials are considered r is the ratio of “dynamic stiffnesses” \(r = \frac{{A_3 E_3 /u_3 }}{{A_1 E_1 /u_1 }}etc.\)

r i :

Area ratio between consecutive steps in case of several steps. \(r_1 = \frac{{A_3 }}{{A_2 }} = \frac{{cot \alpha }}{{cot \beta }},{\text{ }}r_2 = \frac{{A_4 }}{{A_3 }} = \frac{{cot \beta }}{{cot \gamma }}etc.\)Total area ratio r=r 1·r 2. ...

s :

Displacement

t :

Time

T :

Duration of elementary pulse T=2L 1/u 1

u :

Velocity of disturbance propagation (of sound)

v :

Particle velocity

W :

Energy

x :

Coordinate along bar

α, β, γ :

Directrix angles with ν-axis in diagram of state

ρ :

Density

σ :

Stress

τ :

Time unit for method of Sears, τ=L 1/u

References

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Author information

Authors and Affiliations

  1. Research Laboratory of Atlas Copco AB, Stockholm, Sweden

    H. C. Fischer

Authors
  1. H. C. Fischer
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Fischer, H.C. On longitudinal impact II. Appl. sci. Res. 8, 278–308 (1959). https://doi.org/10.1007/BF00411755

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  • DOI: https://doi.org/10.1007/BF00411755

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