“Translational and rotational mechanical systems”;
1.5 Translational and rotational mechanical systems
Let us write the differential equations governing the mechanical rotational system for the figure below.
And also we will xdiscuss about torque-voltage and torque-current electrical analogous circuits. Finally we shall verify it through mesh and node equations.
Solution:
Differential equations:
The opposing torques are,
Free body diagram of J1
Tk1 = k1(θ1 – θ2).
By newton’s second law,
Tj1 + TB1 + Tk1 = T
The opposing torque of J2 are Τj2, Tk1, TB1, TB2.
By Newton’s second law,
Τj2 + Tk1 + TB1 + TB2 = 0
The opposing torque of J3 are TB2 and ΤK3.
By Newton’s second law,
TB2+ TK3 = 0
On replacing θ = ∫ω dt in (1), (2) and (3), equations will be rewritten as
Torque-Voltage Analogous Circuit
The electrical analogous elements for elements of mechanical rotational system:
ω3 → i3 J3 → L3.
By Kirchoff s voltage law equation (A) can be written as:
Torque-Current Analogous Circuit:
The electrical analogous elements for the elements of mechanical rotational system are:
T→ i(t) ; ω1 → V1 ;J1 → C1
Β1 → 1/R1 ; K1 → 1/L1
ω2 → V2; J2→ C2; B2 → I/R2; K3 → I/L3
ω3 → V3 ; J3 → C3.
By using Kirchoff's current law,
Let us derive analogous electrical system for the below mechanical system using Force-voltage analogy
Solution:
By using Force – Voltage Analogy,
The Parameters for F-V Analogy are f(t) = e(t); B=R; M=L and K=1/C ; displacement x(t)= electric charge q(t). Substituting in the above equation,
From the above equation, Force-Voltage analogy can be drawn as,
Let us see the steps to convert the given Mechanical system into Force Voltage and Force current analogy
.
Solution:
a) Force Voltage analogy of the given mechanical system is
b) Force Current analogy of the given mechanical system is
Let us draw the voltage and current analogy for the following mechanical system.
Solution:
Force Voltage Analogy
Force Current Analogy:
1.5.1 Rotational Mechanical System based problems:
Here we shall discuss the differential equations governing the mechanical rotational system for the below figure. Also let us draw the torque-current electrical analogous circuit and verify by writing node equations.
Solution:
The differential equation governing the mechanical rotational system is given by,
(We know that the angular frequency w = dθ/dt)
Mechanical Rotational System
Torque-Current Analogous
Torque-Voltage Analogous
Torque (T)
Current i(t)
Voltage v(t)
Moment of Inertia (J)
Capacitance (C)
Inductance (L)
Displacement (θ)
flux (φ)
Charge (q)
Viscous friction coefficient (B)
Conductance (1/R)
Resistance (R)
Angular velocity (w)
voltage v(t)
Current i(t)
Stiffness coefficient (K)
Reciprocal of Inductance (1/L)
Reciprocal of Capacitance (1/C)
For the given Torque-Current Analogous circuit, (we know that v = dφ/dt)
The node equation is given by,
Let us draw the torque-voltage and torque-current electrical analogous circuit for the below mechanical system.
.
Solution:
Torque-Voltage Analogous circuit
Torque-Current Analogous circuit
Here we shall obtain the torque-voltage electrical analogous circuit for the following mechanical system shown.
Solution:
Torque-Voltage Analogous circuit
1.6 Transfer function
Transfer function is one of the types of modeling a system. Using first principle, differential equation is obtained. Laplace Transform is applied to the equation assuming zero initial conditions. Ratio of LT (output) to LT (input) is expressed as a ratio of polynomial in s in the transfer function.
Problems:
Let us derive the transfer function for the below diagram.
Solution:
Step 1: Reduce the feedback loops.
Step 2: Cascade the blocks.
Step 3: Reduce the feedback loop.
Lets see the transfer function for the system whose block diagram is shown below.
Solution :
A unity feedback system has the forward transfer function
Let us determine the transfer function for the below network.
Solution:
V0(t) = Voltage drop account C2
KVL for loop (1)
The output C(s) due to R(s) and disturbance D(s) for the below figure is as follows.
Solution:
Lets see the overall transfer function of the following block diagram using signal flow graph method.
Solution:
Let us obtain the transfer function of the following electrical network
Solution:
The above circuit can be drawn in laplace domain, then by voltage division rule,
The Transfer function is,
Hint : Convert the diagram into laplace transform, then use voltage divider formula to find e1(s).
Converting circuit into laplace is nothing but use same circuit, replace e(t) to e(s) and C to 1/Cs, L to Ls.
The transfer function for the below network can be derived as follows.
Solution:
In Laplace domain, the ckt can be drawn as,
There are two loops in the above circuit by writing Mesh equations,
Let us calculate the transfer function Y2(s) / F(s) for the below system.
Solution:
1.7 Synchros
It consists of two electro-mechanical devices. Synchros are simply variable transformers. Each synchro contains:
i. A rotor, which is similar in appearance to the armature in a motor.
ii. A stator, which corresponds to the field in a motor.
The stator consists of a balanced three phase winding and is star connected. The rotor is of dumb-bell type construction and is wound with a coil to produce a magnetic field. When no voltage is applied to the winding of the rotor, a magnetic field is produced. The coils in the stator link with this sinusoidal distributed magnetic flux.
Voltages are induced in the three coils due to transformer action. Three voltages are in time phase with each other and the rotor voltage. The magnitudes of the voltages are proportional to the cosine of the angle between the rotor position and its respective coil axis.
1.8 Block Diagram Reduction Techniques
A block diagram of a system is a pictorial representation of the functions performed by each component of the system and shows the flow of signals.
1.8.1 The basic elements of block diagram are Block, Branch point and Summing point.
Block
The block is the symbolic representation of transfer function of that element. A complete control system can be represented with a required number of interconnection of such blocks.
Block
Summing point
Different input signals are applied to same block. Here, resultant input signal is the summation of all input signals applied. Summation of the input signals can be represented by a point called summing point which is shown in the figure below by crossed circle.
Summing point
1.8.2 Block diagram rules
Let us see the rules in block diagram reduction techniques.
Cascaded blocks
Moving a summer beyond the block
Moving a summer ahead of block
Moving a pick-off ahead of block
Moving a pick-off behind a block
Eliminating a feedback loop
Cascaded Subsystems
Parallel Subsystems
Problems:
Let us calculate the closed-loop transfer function C/R using block diagram reduction technique whose block diagram is shown below.
Solution:
1. Shift the summing point S1 after block G1
2. Interchanging S1 and S2
3. Eliminate S1 point and S2
Using block diagram reduction method, lets see the output of the below system.
Solution:
Transfer function of the below block diagram is derived as follows.
Solution:
Step 1: Shift the take off point before the G4 block to a point after the block.
Step 2: Combine cascade blocks of G3, G4 and eliminate the Closed loops formed by G3 G4 and H1.
Step 3: Combine the Blocks in cascade with G2 and then, Eliminate the Closed loop formed by them with H2/G4.
Step 4: Combine the Blocks in cased with G1 and
Step 5: Eliminate the feedback loop formed by H3. The Transfer function of the given block diagram is,
Let us calculate C(s)/R(s) for the below figure by reducing the block.
Solution:
Step (a) Shift the take off point after the block G3 to before G3.
Step (b) Combine the cascade blocks G3, G4 and eliminate the parallel blocks.
Step (c) Eliminate the feedback loop formed by G2 and H1G3.
Step (d) Combine the cascade blocks.
Step (e) Eliminate the feedback loop.
By Reducing the block diagram for below figure, let us obtain its closed loop transfer function C(S)/R(S).
Solution:
Step (a): Eliminate the closed loop formed by G2 and H2, then the circuit diagram becomes,
Step (b): Shift the take off point H1 before the blockto a point after the block,
Step (c): Combine the cascade blocks G1 and , then
Step(d): Eliminate the Closed loop formed by , then
Step (e): Eliminate the closed loop formed by
1.9 Signal flow graphs
A signal flow graph is a diagram that represents a set of simultaneous linear algebraic equations. By taking Laplace transform, the time domain differential equations can be transferred to a set of algebraic equation in s-domain. A signal-flow graph consists of a network in which nodes are connected by directed branches. It depicts the flow of signals from one point of a system to another and gives the relationships among the signals.
Properties of Signal Flow Graph:
1) Signal flow applies only to linear systems.
2) The equations based on which a signal flow graph is drawn must be algebraic equations in the form of effects as a function of causes.
3) Signals travel along the direction described by the arrows of the branches.
Procedure for Determining Transfer Function Using Signal Flow Graph:
1. Determine the number of forward paths and determine the forward path gain correspondingly.
2. From the given graph find the possible individual loop and corresponding gains for them. Similarly find the number of two, three etc., non-touching loops and its gain.
3. Calculate the value of Δ and ΔK. Substitute all the gains in the Mason’s gain formula to determine the transfer function.
Mason’s gain formula
where, T = T(s) = Transfer function of the system.
ΡK = Forward path gain of Kth forward path.
K = Number of forward paths in the signal flow graph.
Δ = 1 – [Sum of individual loop gains]+[Sum of gain product of all possible combination of two non-touching loops]-[Sum of gain product of all possible combination of three non-touching loops]+…
Comparison between block diagram and signal flow graph methods:
Block Diagram
Signal Flow Graph
(1)
Each element is represented by a block.
Each variable is represented by a separate node.
(2)
Transfer function of the element is given inside the block.
Transfer function is shown along the branches connecting the nodes.
(3)
Summing points and take off points are separate.
Any node can have number of incoming and outgoing branches.
(4)
Feedback path is present.
Feedback loops are considered for the analysis.
(5)
Self loop concept is absent.
Self loop can exist.
(6)
Applicable to known invariant systems.
Applicable to linear time invariant systems.
Problems:
Let us calculate the transfer function of the transistor’s hybrid model below using signal flow graph.
.
Solution:
We know that the hybrid parameter equation for a transistor is,
V1 = h11I1 + h12V2
I2 = h21I1 + h22V2
Output voltage V2 = I2RL
The above model can be represented in signal flow graph as
From the signal flow graph,
There is only one forward path
There are two loops, Loop formed by node 1,
Loop formed by node 3, L2 = h22RL
All the loops touching the forward path, path factor Δ1 = 1
By using Mason’s gain formula, the transfer function of transistor hybrid model is given by,
Calculation of C(s)/R(s) for the signal flow graph is done as follows
Solution:
Step (a) The number of forward path k = 1, Forward path gain P1 = G1G2G3G4G5.
Step (b) Number of individual loops:
L1 = – G1H1
L2 = – G1G2G3H2
L3 = – G3H3
L4 = – G4H4
L5 = – G5H5
Step (c) Number of two non-touching loops at a time :
L1L3 = G1 H1 G3H3
L1L4 = G1H1 G4H4
L1L5 = G1H1 G5H5
L2L4 = G1G2G3H2 G4H4
L2L5 = G1G2G3H2 G5H5
L3L5 = G3H3 G5H5
Step (d) There is only one, three non-touching loops at a time.
L1 L3L5 = -G1H1 G3H3 G5H5
Step (e) By using mason’s gain formula, the transfer function becomes,
Let us obtain the transfer function of the below figure using Mason’s gain formula.
Solution:
Let us construct the equivalent signal flow graph and C/R using Mason’s gain formula for the below block diagram.
Solution: