FC 5 Mechanics of Translation

8/13/2019 FC 5 Mechanics of Translation

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Mechanics of Translation

Friction exists in physical systems whenever mechanical surfacesare operated in sliding contact.

Friction encountered in physical systems may be of many types.

1. Coulomb friction force: It is the force of sliding friction betweendry surfaces. This force is substantially constant.

2. Viscous friction force: It is the force of friction between movingsurfaces separated by viscous fluid, or the force between a solidbody and a fluid medium. This force is approximately linearlyproportional to velocity over a certain limited velocity range.

3. Stiction: It is force required to initiate motion between two

contacting surfaces ( which is obviously more than the forcerequired to maintain them in relative motion).


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Analogous Circuits

Analogous circuits represent systems for which

the differential equations have the same form.

The corresponding variables and parameters in

two circuits represented by equations of the

same form are called analogs.

An electric circuit can be drawn that looks like

the mechanical circuit and is represented bynode equations that have the same mathematical

form as the mechanical equations.


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In this table the force f and the current i are analogs and areclassified as through variables. There is a physical similaritybetween the two, because a measuring instrument must be placedin series in both cases; i.e. , an ammeter and a force indicator mustbe placed in series with the system.

Also, the velocity across a mechanical element is analogous tovoltage across an electrical element. Again, there is physicalsimilarity, because a measuring instrument must be placed acrossthe system in both cases.

A voltmeter must be placed across a circuit to measure voltage; itmust have a point of reference.

A velocity indicator must also have a point of reference.

Nodes in the mechanical network are analogous to nodes in theelectric network.


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G(s) = transfer function of the system

Thus the transfer function of a time-invariant system is

the ratio of the Laplace transforms of its output and

input variables, assuming zero initial conditions.


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Note that whenever a transfer function is used to describe asystem, the system is always implicitly assumed to be linear, time-invariant, and relaxed at to=0.

There are two different ways in which transfer function models areusually obtained

1. For lumped linear time-invariant systems, mathematical modelbuilding based on physical laws normally results in a set of (first-order, and second-order) differential equations. Applying Laplacetransform to the differential equations results in a transfer functionmodel of the system.

2. The transfer function model of a system can be identified fromexperimentally obtained input-output data. Generally theidentification methods can be considered to be a type of curvefitting, where the transfer function is fitted to the available data insome optimal manner.


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The methods may be based on frequency

response (sinusoidal input), step response

(step input), impulse response (pulse input;

pulse of small width approximating an

impulse), or the response to more general

inputs.


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(b) Free-body diagram : The displacement x(t) and velocity v(t)

are state variables.

A systematic way of setting up dynamical equations for mass-

springdamper systems is to draw a free-body diagram.

In a free-body diagram, each mass is isolated from the rest of

the system: forces acting on each free-body are due to the rest

of the system, including external forces. The free-bode mass

moves under the action of the resultant force.


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State Variable Models

Consider the resistanceinductance(RLC) network

The input is a voltage source.

The desired output is usually the voltages andcurrents associated with various elements of thenetwork.

The information at time t can be obtained if the

voltage across the capacitor and the currentthrough the inductor of the network at that timeare known, in addition to the values of the input.

State Variable Models


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The voltage e(t) across the capacitor and the current i(t) throughthe inductor thus constitute a set of characterizing variables of thenetwork.

The selection of characterizing variables is linked with the energyconcept.

At time t, energy stored in the capacitor is Ce2(t) and energystored in the inductor is Li2(t).

Dynamical changes in characterizing variables are caused by theredistribution of energy within the network.

The number of independent energy storage possibilities thus

equals the number of characterizing variables of the network. The values of the characterizing variables at time t describe the

state of the stateof the network at that time; these variables aretherefore called state variablesof the network.


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Network analysis usually requires setting up of

dynamical equations (using KVL and KCL) in

terms of rates of change of capacitor voltages

and inductor currents.

The solution of these equations describes the

state of the network at time t.

Desired output information is then obtained

from the state using an algebraic relation.


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FC 5 Mechanics of Translation