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Vibration problems of the spring mill discussed in my former paper for the first time may now be shown by graphical solution. A stability problem will be soluble so easily by this method that it becomes more advantageous method from this viewpoint. The stability problem treated here is of the phase stability, only and others are not mentioned. Consider a system having crank radius α, circular frequency of spring p and driven mass m. Let x be a displacement from neutral position of mass, and ω be an angular velocity of crank, then the equation of motion may be written as follows, x^^..+p^2x=ap^2sin (wt+ε) (1) Change the variable x to Z, we put x=Z-a/u^2-1sin (wt+ε) (2) x^^.=Z^^.-aw/u^2-1cos (wt+e) (3) Z^^..+p^2Z=0 (4) From (4) z=A sin (pt+λ) (5) Y≡pZ=Ap sin (pt+λ) (6) Z^^.=Ap con (pt+λ) (7) ∴Y^2+Z^2=A^2P^2 (8) From (8), [YZ^^.] is shown as a circle of radius Ap. Since the conditions t=0 : x=b (Y=Y_1), x=v_1 (Z^^.=Z^^._1) t=t : x=b (Y=Y_2), x^^.=v_2 (Z^^.=Z^^._2) hold, we get Y_1=pb+ap/u^2-1sin ε (9) Z^^._1+apu/u^2-1cos ε (10) Y_2=pb+ap/u^2-1sin (wt+ε) (11) Z^^.=v_2+apu/u^2-1cos (wt+ε) (12) Graphical solution may be attained when we use the formulas (6)〜(12), in which (9)〜(12) serve as rebounding conditions of mass. Thus gained [Y_2Z^^._2] offeres the use of value v_2,and with the aid of v_2=-cv_1 (13) the initial condition of next cycle may be determined. I am grateful to Mr. Watanae for his kind advice.
一般社団法人日本機械学会の論文
1951-12-15

ばね衝撃機

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