MIT OpenCourseWare http ocw mit edu 2 004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use visit http ocw mit edu terms Massachusetts Institute of Technology Department of Mechanical Engineering 2 004 Dynamics and Control II Spring Term 2008 Lecture 151 Reading 1 Class Handout Modeling Part 1 Energy and Power Flow in Linear Systems Sec 3 Class Handout Modeling Part 2 Summary of One Port Primitive Elements Nise Secs 2 4 and 2 6 Rotational Systems continued Example 1 The diagram shows a mechanical tachometer that uses frictional drag plates to create a torque proportional to angular velocity di erence The angular velocity is indicated by the displacement of a torsional spring K q B W J J fr ic tio n a l d r a g p la te s W in Find the transfer function relating the displacement of the indicator to the input angular velocity in s H s in s and show that for a constant input angular velocity the steady state indicated speed ss i Solution There are two distinct angular velocities and the system graph is 1 c D Rowell 2008 copyright 15 1 B J a c r o s s v a r ia b le s o u rc e W K in 0 W Using impedances redraw the graph combining the inertia and the spring into a single impedance Z2 Z 1 W J T W Z in W Z Z 2 1 1 2 B 1 1 J s s K 1 J K s K 1 J s 0 Then Z2 in s Z1 Z 2 s Js2 K in s 1 B s Js2 K Bs in s Js2 Bs K J But the angular displacement s J s s so that H s Js2 B Bs K If the input velocity is a step function at t 0 the in s s and the steady state indicated value will be ss lim t lim s s lim sH s t s 0 s 0 B s K show that the indicated speed is proportional to the input angular velocity This example is repeated using mesh equations below 15 2 2 Transfer Function Generation Using Mesh Loop Equations This method is useful when the system contains an across variable source We will develop the method using the following mechanical system as an example K K V t K 1 B m 2 1 m B 1 v K o 3 B 1 B V t 2 B 2 m 1 1 v 3 B 2 m o 2 2 The input is a velocity source V t and the output is the velocity of the mass m2 Step 1 This step is optional Do some simple impedance combinations to reduce the complexity being careful not to eliminate any nodes or branches that specify the output variable Z Z Z 5 Z 1 Z V t v 3 Z 2 o Z 4 K 5 B 1 m K 2 s 1 3 B s 1 m 1 1 4 Z 1 1 B 3 Z v 0 2 Z s 1 1 2 s 2 Step 2 De ne a set of closed loops meshes making sure that each branch is covered by at least one loop Z Z 5 3 1 V t Step 3 Z 1 Z 2 2 v 3 Z 4 v 0 Write a loop equation for each loop vZ1 vZ2 V 0 vZ3 vZ4 vZ2 0 vZ5 vZ3 vZ1 0 15 3 o Step 4 At this point we assume that each loop has a continuous through variable with it that is loop n has Fn associated with it When a branch is contacted by more that one loop the branch through variable is the sum of all of the contacting loop through variables We use impedance relationships and substitute loop through variable expressions for the across variables Z1 F1 F3 Z2 F1 F2 V 0 Z3 F2 F3 Z4 F2 Z2 F1 F2 0 Z5 F3 Z3 F2 F3 Z1 F1 F3 0 Step 5 Rewrite these equations collecting terms in the loop through variables to create a set of linear equations Z1 Z2 F1 Z2 F2 Z1 F3 V Z2 F1 Z2 Z3 Z4 F2 Z3 F3 0 Z1 F1 Z3 F2 Z1 Z3 Z5 F3 0 It may be useful to express these equations in matrix form Z1 Z2 Z2 Z1 F1 V F2 0 Z2 Z2 Z3 Z4 Z3 Z1 Z3 Z1 Z3 Z5 F3 0 Step 6 Identify how the output variable is related to the loop through variables In this case vout F2 Z4 Step 7 Solve the set of linear equations for the variable s identi ed in Step 6 in this case F2 using any method and substitute in the output equation Step 6 we will not do this step here Example 2 Repeat the mechanical tachometer problem of Example 1 this time using mesh equations B K q 1 B W J 2 J W fr ic tio n a l d r a g p la te s W J in 15 4 K in W 0 With the system graph and two loops de ned as above the loop equations are B J in 0 K J 0 Substitute for the across variables ZB T1 ZK T1 T2 in ZJ T2 ZK T1 T2 0 Rearrange ZB ZK T1 ZK T2 in ZK T1 ZJ ZK T2 0 and in matrix form ZB Z K ZK ZK ZJ Z K T1 T2 in 0 The output variable is J ZJ T2 therefore use Cramer s rule to solve for T2 ZB ZK in ZK 0 ZK in T2 ZB ZJ Z B ZK Z K ZJ ZK ZB Z K ZK ZJ Z K and since J ZJ T2 ZJ 1 Js ZK s K and ZB 1 B J ZJ ZK in Bs 2 in ZB ZJ Z B ZK Z K ZJ Js Bs K and as before s J s giving H s 3 s B 2 in s Js Bs K Transfer Function Generation Using Node Equations This method is useful when the system contains a through variable source We will demonstrate the method using the following graph 15 5 Z 2 a b Z F 1 Z 4 s Z 3 c Step 1 Write node equations for each node except the reference node let f be a generalized through variable Fs fZ1 fZ2 0 f Z2 f Z4 f Z3 0 node a node b Step 2 Express the branch through variables in terms of admittances Y 1 Z and across variables va vb Y2 va vc Y1 Fs va vb Y2 vb vc Y4 vb vc Y3 0 Step 3 Collect terms in the across variables and form a set of linear equations note that node c is the reference node therefore vc 0 Y1 Y2 va Y2 vb Fs Y2 va Y2 Y3 Y4 vb 0 Step 4 Identify the output variable in terms of nodal across variables and solve the …
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