Brard 



III. POSSIBLE FURTHER DEVELOPMENTS - 

 GENERAL CONCLUSIONS 



The purpose of the present section is to examine to what extent the ideas 

 outlined above, may influence some problems of importance from a practical 

 point of view. 



17. Stability of the Steady Motions 



Two cases should be studied. In the first case, one assumes that the con- 

 trol devices are "OFF." In the second case, the control devices are "ON" and 

 act so as to keep constant the characteristics of the steady motion; the pilot 

 system is included in the chain and the loop is closed. 



Let \. be the parameters which define the motion, a. the parameters 

 which define the action of the control devices. Let k.°a?, the values of \. , a. 

 in the steady motion. The method generally used to study the stability consists 

 in the research of solutions of the form 



SA.: = k. - k. 



St 



a. e 



By substituting in the equations of the motion, one obtains an "equation in s"; 

 the steady motion is stable when all the roots of this equation are negative or 

 have a negative real part. 



When the equation of the motion contains terms of the form 



t' 



J k^(r') <^(t'-T')dT' , 



the substitution leads to terms 



no 



which are the Laplace Transforms of the derivatives c?i,0, 0',cp introduced in 

 pars. 5 and 12. 



The "equation in s" so obtained is no more algebraic and its study is more 

 difficult. But of course, it is not impossible. 



The stability of the steady motions gives rise to various comments: 



1*) The steady motions in the vertical plane are necessarily rectilinear. 

 The problem may be completely solved by measurements of the forces in steady 

 and harmonic forced motions in the (z,x) -plane. But it is possible also to use 

 other ways (at the French Navy Tank, we generally carry out tests on a semi- 

 model with its (z,x)-plane on the free surface of the tank). 



898 



