TM NO. 377 



APPENDIX A 



; SUPPLEMENTARY DISCUSSIONS 



Dynamics of a Ducted Meter 



Response to a Step Input — In disciissing the dynaiaic characteristics of 

 the ducted meter in chapter 11^ there is frequent mention of the terms "time 

 constant''^ "response time"^ and "frequency response"6 Since these terms are 

 often used (and also misused) in discussions of fluid flow sensors^ it is 

 desirable to clarify these meanings as they apply to the wave meters o The 

 first step is to derive the relation for instrument response to a step in- 

 crease in the driving force function. 



The equation of motion representing the balance of fttrces on a ducted 

 meter system can be written as t 



A B C 



where I is the moment of inertia of the impeller (gm cm^)^ tA) is the instan-= 

 taneoxis angular velocity (sec° )^ u is the velocity of fluid through the meter^ 

 and K- and K2 are proportionality constants o 



Equation (A-l) represents the inertial force (term a) as equal to the sum 

 of the drag force (B) produced by the iH5)eller as it spins in the fluid and 

 the dynamic thrust (C) caused by the moving fluid on the impeller » In other 

 words., the drag force (B) is merely the reaction to the impressed force (C), 



To investigate the solution Uj{t)g one must rewrite equation (A^I) in 

 the form: 



The initial conditions are given by: 



CO (t) = u(t) 3 t<0 I and 



(A=3) 

 u(t ) = Up = constant t ^ O « 



These conditions imply that the driving velocity Urp is applied instantaneously 

 to the impeller at time t = 0. The solution to equation (A-2) is obtained by 

 substitution of: 



K 



A"1 



