Basics of Control System-2 Study notes For EE/EC

Restrictions on Systems Studied

Systems considered in this article generally under the category of linear systems.A lengthier description is "Linear, Time-Invariant, Continuous-Time, Dynamic Systems." We will discuss these definitions one by one.

• Linear System: Essentially a Linear system is one which follows the principle of Superposition & Homogeneity in their response of the system.

Consider a system with input f(t) and output x(t)

Now if the input is changed to g(t) the output is y(t)

If the system is linear, then an input of h(t)=g(t)+f(t) yields an output z(t)=x(t)+y(t)

• Example of a nonlinear system: Now consider the same situation when the system is nonlinear for example a squaring function.

Now we can no longer find the output to the complex function, h(t), by adding the responses of the system to the simpler function.

• Time-Invariant: In time invariant system, the physical parameters of the system do not change with time. The classic example of a non-time invariant system is a rocket whose mass changes with time (a time-invariant rocket would have constant mass).
• Continuous-Time: The Continuous time systems are time is a continuous, or real-valued, variable. On the other hand, discrete-time systems have time that moves in discrete steps.

Examples of discrete time systems include weekly closing stock prices (updated weekly), the sound on a standard audio CD (updated 44,100 times per second).

Analogous System

An analogous electrical and mechanical system will have differential equations of the same form. There are two analogues that are used to go between electrical and mechanical system.

To understand the analogy more clearly.The entries for the mechanical analogues are formed by substituting the analogous quantities into the equations for the electrical elements.For example, the electrical version of Ohm's law is e=iR.The Mechanical I analog stipulates that e is replaced by v, I by f and R by 1/B, which yields v=f/B.

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Force-Current 'F-I' Analogy (Electrical to Mechanical)

The important relationship when converting from a circuit to the Mechanical 1 analog is that between Kirchoff's Current Law and D'Alemberts Law (with inertial forces included).

Procedure for Conversion from Electrical to Mechanical.

• Start with an electrical circuit.Label all node voltages.
• Write a node equations for each node voltage.
• Rewrite the equations using analogs making substitutions from the table, with each electrical node being replaced by a position.
• Draw the mechanical system that corresponds with the equations.

Example: Draw the mechanical equivalent circuit of the given system.

Solution: By following the steps in given Procedure

Alternative method: Another way from electrical to mechanical simply redrawing the electrical circuit using mechanical components.

• Draw over the circuit, replacing electrical elements with their analogs; current sources replaced by force generators, voltage sources by input velocities, resistors with friction elements, inductors with springs, and capacitors (which must be grounded) by masses. Each node becomes a position (or velocity)
• Label currents, positions, and mechanical elements as they were in the original electrical circuits.

Force-Current 'F-I'to Electrical to Mechanical

The procedure to go from Mechanical to Electrical is simply the reverse of Electrical to Mechanical. Either a mathematical method can be used as in the previous example, Electrical to Mechanical conversion can be understand by reading the table from bottom to top, or a simple visual method can be used where force generators are replaced by current sources, friction elements by resistors, springs by inductors, and masses by capacitors (which are grounded). Each position becomes a node in the circuit.

Procedure for 'F-I' analogy for Electrical to Mechanical Conversion

• Start with the mechanical system.Label all positions.
• Draw the circuit by replacing mechanical elements with their analogs; force generators by current sources, input velocities by voltage sources, friction elements by resistors, springs by inductors, and masses by capacitors (which are grounded).Each position becomes a node.
• Label nodes and electrical elements as they were in the original mechanical system.

Force-Voltage 'F-V' Analogy (From Electrical to Mechanical)

The important relationship when converting from a circuit to the Mechanical analog is that between Kirchoff's Voltage Law and D'Alemberts Law (with inertial forces included).

Procedure for Conversion from Electrical to Mechanical

• Start with an electrical circuit. Label all currents. Choose currents so that only one current flows through inductors.
• Write a loop equations for each loop.
• Rewrite the equations using analogs, making substitutions from the table, with each electrical loop being replaced by a position.
• Draw the mechanical system that corresponds with the equations.

Example: Draw the Mechanical equivalent system of the Electrical Circuit.

Solution: By following the procedure given for force voltage analogy.

Rotating Mechanical Systems

Gear System

Gears perform many functions, here we look at gears that increase or reduce angular velocity (while simultaneously decreasing or increasing torque, such that energy is conserved).

• If we consider two gears in equilibrium and in contact with each other, we can two very useful relationships.

• First, we note the geometric relationship that results from the path that the arc lengths along their circumference must be equal as the gears turn.

Since the arc lengths (shown with a heavy blue line) must be equal ⇒ arc length⇒ r₁θ₁ = r₂θ₂ arc length

• Now we can derive the second relationship from a torque balance.Here we must define a force between the gears termed a "contact force."This force must be equal and opposite across the interface between the two gears, but its direction is arbitrary.

Since the contact force is tangent to both gears and so produces a torque that is equal to the radius times the force.

We can do a torque balance on each of the two gears

For Gear 1: torque τ₁ = f_cr_{1 or} f_c = τ₁/r₁ & For Gear 2: torque τ₂ = f_cr_{2 or} f_c = τ₂/r₂

From the above two equation we concluded that fc = τ₁/r₁ = τ₂/r₂

• In the system below, a torque, τ_a, is applied to gear 1 (with a moment of inertia J₁). It, in turn, is connected to gear 2 (with a moment of inertia J₂) and a rotational friction B_r. The angle θ₁ is defined positive clockwise, θ₂ is defined positive clockwise. The torque acts in the direction of θ₁.

• We start by drawing free body diagrams, including a contact force that we will arbitrarily choose to be down on J₁ and up on J₂. The directions of the reaction forces due to inertia and friction are chosen, as always, opposite to the defined positive direction.

This yields the two equations of motion

τ_a + f_cr₁ - J₁θ₁ =0

f_cr₂-J₂θ₂ +B_rθ₂ = 0

• We can easily solve for fc and eliminate it from the equations, but we also need to eliminate θ₂. To do this we use the relationship between θ₁ and θ₂ (from equal arc lengths).

r₁θ₁ = -r₂θ₂

• Note that we have a negative sign here because of the way θ₁ and θ₂ were defined (if θ₁ moves in the positive direction, then θ₂ is negative). When you use the arc length expression you must be careful of signs.

f_cr₂-J₂θ₂ - B_rθ₂ = 0

and θ₂ = - r₁/r₂(θ₁)

we can we write it as f_cr₂-J₂{r₁/r₂(θ₁)} =0

fc = -J₂{r₁/(r₂)²}θ₁ - Br {r₁/(r₂)²}θ₁

We can put this into the equation for J1 and solve (in standard form with the output (θ₁) on the left, and the input (τ_a) on the right.

τ_a + f_cr₁ - J₁θ₁ =0

or τ_a +[-J₂{r₁/(r₂)²}θ₁ - Br {r₁/(r₂)²}θ₁r₁] - J₁θ₁ =0

or {J₁+ J₂(r₁/r₂)²}θ₁ + _{Br(r1/r2)²}θ1} = τ_a

So we get θ₁ = r₂/r₁(θ₂) & ω₁ = r₂/r₁(ω₂)

& Since from the relation τ₁= r₁/r₂(τ₂)

we concluded that

τ₁ω₁ = {r₁/r₂(τ₂)}{r₂/r₁(ω₂)} = τ₂ω₂

All the Best.

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