Showing posts with label Mechanics. Show all posts
Showing posts with label Mechanics. Show all posts

Properties of Drilling Fluids

Density of mud:

Density is defined as weight per unit volume. It is expressed either in pounds per gallon (lb/gal) or pounds per cubic foot (lb/ft"), or in kilograms per cubic meter (kg/m³), or compared to the weight of an equal volume of water, as specific gravity (SG). The pressure exerted by a static mud column depends on both the density and the depth; therefore, it is convenient to express density in terms of pounds per square inch per foot (psi/ft), or kilograms per square centimeter per meter (kg/cm/m).
In order to prevent the inflow of formation fluids and to lay down a thin, low-permeability filter cake on the walls of the hole, the pressure of the mud column must exceed the pore pressure-the pressure exerted by the fluids in the pores of the formation-by at least 200 psi (14 kg/ern'). The pore pressure depends on the depth of the porous formation, the density of the formation fluids, and the geological conditions.

Normally pressured formations, which have a self-supporting structure of solid particles (so the pore pressure depends only on the weight of the overlying pore fluids), and abnormally pressured or geopressured formations, which are not fully compacted into a self-supporting structure (so the pore fluids must bear the weight of some of the overlying sediments as well as the weight of the overlying fluids). The hydrostatic pressure gradient of formation fluids varies from 0.43 psi/ft to over 0.52 psi/ft (0.1 to 0.12 kg/cm/m), depending on the salinity of the water.

The bulk density of partially compacted sediments increases with depth, but an average (SG) of 2.3 is usually accepted, so that the overburden (or geostatic or litholostatic) pressure gradient is about 1 psi/ft (0.23 kg/cm2/m), and the pore pressure of geopressured formations is somewhere between the normal and the overburden pressure gradients, depending on the degree of compaction.
Besides controlling pore fluids, the pressure of the mud column on the walls of the hole helps maintain borehole stability. In the case of plastic formations, such as rock salt and unconsolidated clays, the pressure of the mud is crucial.
The buoyant effect of the mud on the drill cuttings increases with its density, helping transport them in the annulus, but retarding settling at the surface. Very rarely is an increase in mud density justified as a means of improving cutting-carrying capacity.

In the interest of well safety, there is a natural tendency to carry a mud density well above that actually needed to control the formation fluids, but this policy has several major disadvantages. In the first place, excessive mud density may increase the pressure on the borehole walls so much that the hole fails in tension. This failure is known as induced fracturing.

In induced fracturing, mud is lost into the fracture so formed, and the level in the annulus falls until equilibrium conditions are reached. The problem of maintaining mud density high enough to control formation fluids, but not so high as to induce a fracture becomes acute when normally pressured and geopressured formations are exposed at the same time. Under these circumstances, it i generally necessary to set a string of casing to separate the two zones. Several methods have been developed for predicting the occurrence of geopressures.' Knowledge of the expected pore pressure and fracture gradients.

Another disadvantage of excessive mud densities is their influence on drilling rate (rate of penetration R.O.P). Laboratory experiments and field experience have shown that the rate of penetration is reduced by mud overbalance pressure (the differential between the mud pressure and the pore pressure when drilling in permeable rocks) and by the absolute pressure of the mud column when drilling rocks of very low permeability. A high overbalance pressure also increases the risk of sticking the drill pipe.

Lastly, excessive mud densities are a disadvantage because they unnecessarily increase mud costs. Mud costs are not a very important consideration when drilling in normally pressured formations, because adequate densities are automatically obtained from the formation solids that are dispersed into the mud by the action of the bit. Mud densities greater than about I lb/gal (1.32 SG) cannot be obtained with formation solids because the increase in viscosity is too great. Higher densities are obtained with barite which has a specific gravity of about 4.25, as compared to about (2.6) for formation solids, so that much less of solids by volume is required to obtain a given density. Mud costs are increased not only by the initial cost of the barite, but also, and to a greater extent, by the increased cost of maintaining suitable properties, particularly flow properties. Because of the incorporation of drilled solids, the viscosity continuously increases as drilling proceeds, and must be reduced from time to time by the addition of water and more barite to restore the density.

Flow Properties:
The flow properties of the drilling fluid playa vital role in the success of the drilling operation. These properties are primarily responsible for removal of the drill cuttings, but influence drilling progress in many other ways. Unsatisfactory performance can lead to such serious problems as bridging the hole, filling the bottom of the hole with drill cuttings, reduced penetration rate, hole enlargement, stuck pipe, loss of circulation, and even a blowout.

The flow behavior of fluids is governed by flow regimes, the relationships between pressure and velocity. There are two such flow regimes, namely laminar flow, which prevails at low flow velocities and is a function of the viscous properties of the fluid, and turbulent flow, which is governed by the inertial properties of the fluid and is only indirectly influenced by the viscosity. Pressure increases with velocity increase much more rapidly when flow is turbulent than when it is laminar.

Laminar Flow:

Laminar flow in a round pipe may be visualized as infinitely thin cylinders sliding over each other. The velocity of the cylinders increases from zero at the pipe wall to a maximum at the axis of the pipe. The difference in velocity between any two such cylinders, divided by the distance between them. Defines the shear rate. The axial force divided by the surface area. Of a cylinder defines the shear stress. The ratio of shear stress to shear rate is called the viscosity. And is a measure of the resistance to flow of the fluid. The unit of viscosity is the (poise) the shear stress in dynes/m- divided by the shear rate in reciprocal seconds gives the viscosity in poises. The unit employed in mud viscometry is the centipoises (cp), which is one hundredth of a poise.

Turbulent Flow:

Flow in a pipe changes from laminar to turbulent when the flow velocity exceeds a certain critical value. Instead of layers of water sliding smoothly over each other, flow changes locally in velocity and direction, while maintaining an overall direction parallel to the axis of the pipe. Laminar flow may be compared to a river flowing smoothly over a plain, and turbulent flow to flow over rapids where interaction with irregularities on the bottom causes vortices and eddies.
Friction factor, which is a function of the Reynolds number and the roughness of the pipe wall.

PH:

The relative acidity or alkalinity of a liquid is conveniently expressed as pH.
Defined as the negative logarithm (to the base 10) of the hydrogen-ion concentration, pH units decrease with increasing acidity by a factor of 10. For example, the hydrogen ion concentration of a solution having a pH of 3 is ten times that of a solution of pH 4. At pH of 7, the hydrogen-ion concentration is equal to the hydroxyl-ion concentration and the liquid is neutral, as with pure water. Above pH 7, the hydroxyl-ion concentration increases by a factor of 10 with each pH unit; thus, the hydroxyl-ion concentration at pH 11 is ten times that at pH 10 (hydrogen ion concentration is one tenth).

The optimum control of some mud systems is based on pH, as is the detection and treatment of certain contaminants. A mud made with bentonite and fresh water, for example, will have a pH of 8 to 9. Contamination by cement will raise the pH to 10 to 11, and treatment with an acidic poly phosphate will bring the pH back to 8 or 9, other reasons for pH control include maintenance of lime-treated mud's, mitigation of corrosion, and effective use of thinners.

Measurement of pH is routinely made by comparing the color developed on immersing a paper strip impregnated with certain dyes (indicators) with the color of reference standards. If the liquid has a high concentration of dissolved salts, or is deeply colored (such as by tannins and lignite), the colorimetric method is not satisfactory, but an electrometric method employing the glass electrode can be used to give reliable results in most mud's. If the sodium-ion concentration is very high, a special glass electrode may be needed.

Alkalinity:
Alkalinity measurements are made to determine the amount of lime in lime" treated mud's. The mud is titrated to determine the total amount of lime, soluble and insoluble, in the system (Pm) The filtrate is titrated to determine the amount of lime in solution (Pt). The amount of undissolved lime is calculated from Pm Pt. Measurements of the alkalinity of water samples, and of filtrates of very lightly chemically treated mud's, can be used to calculate the concentration of hydroxyl (OH), carbonate (C03), and bicarbonate (HC03) ions in solution.

Cation Exchange Capacity:

Methylene Blue Test.

The methylene blue test serves to indicate the amount of active clay in a mud system or a sample of shale. The test measures the total cation exchange capacity of the clays present and is useful in conjunction with the determination of solids content as an indication of the colloidal characteristics of the clay minerals. Similarly, shale cuttings can be characterized and some estimations can be made regarding mud-making properties and possible effects on hole stability. Organic materials, if present in the sample, are destroyed by oxidation with hydrogen peroxide. The sample is titrated with standard methylene blue solution until the adsorptive capacity is satisfied, as shown by the appearance of a blue color in the water in which the sample is suspended. If other adsorptive materials are not present in significant amounts, the bentonite content can be estimated, based on an exchange capacity of 75 mill equivalents per 100 grams of dry bentonite.

Viscosity:

Although calculated from measurements at relatively low shear rates, the plastic viscosity is an indicator of high shear rate viscosities. Consequently, it tells us something about the expected behavior of the mud at the bit. One of our design criteria was to minimize the high shear rate viscosity. To accomplish this, we should minimize the plastic viscosity. A decrease in plastic viscosity should signal a corresponding decrease in the viscosity at the bit, resulting in higher penetration rate.

Increasing the plastic viscosity is not a desirable means of increasing the hole-cleaning ability of a mud. In fact, the increase in pressure drop down the drill string, caused by an increase in PV, would reduce the available flow rate and tend to offset any increase in lifting ability. In general, high plastic viscosity is never desirable and should be maintained as low as practical.
The plastic viscosity is primarily a function of the viscosity of the liquid phase and the volume of solids contained in a mud. The viscosity of the liquid phase is increased by addition of any soluble material. Many of the water-soluble polymers used for fluid-loss control are quite effective in increasing the plastic viscosity. Saturated salt water has twice the viscosity of fresh water. Diesel oil, which is commonly used as the liquid phase of oil-base mud's, has three times the viscosity of fresh water. Both salt water mud's and oil mud's tend to have high plastic viscosities.

The volume of solids in a mud, is the dry volume of solids plus the increase in volume due to hydration. The water of hydration actually becomes a part of the solid so far as its effect on viscosity is concerned. In other words, the plastic viscosity is increased by addition of any type of solid; but solids such as clays, which hydrate, will further increase the plastic viscosity as their volume is increased by hydration. This makes the hydration and dispersion of shale particles particularly detrimental.

As long as these particles are large and relatively unhydrated, their effect on viscosity is small. However, time, temperature, and agitation tend to disperse and allow hydration of the individual clay platelets, which results in increased viscosities. In order to combat the tendency of shale particles to disperse and hydrate, the "inhibitive" mud's were designed. Materials such as lime, gypsum, lignosulfonate, and polymers are added to inhibit the rate of dispersion and hydration. These materials do cause inhibition, but if the inhibited particles are not removed from the system, the solids content will continue to build. In time, the plastic viscosity will be as high or higher than before and other mud properties such as filter cake thickness will suffer.

Minimum plastic viscosities can be achieved only to the degree that the mud is kept free of drilled solids. (Figure) shows guidelines for plastic viscosity of water-base mud's at various mud weights. The lower curve represents mud's that contain only barite and sufficient bentonite to suspend the barite. This curve should represent minimum plastic viscosities for good mud performance. The upper curve is an average for many field mud's that have been checked.
Plastic viscosity decreases with increasing temperature, due to thinning of water. If the mud is checked at 130°F, the PV will be about 10 percent lower than at 120°F; if it is checked at 110°F, it will be about 10 percent higher. For this reason, all mud tests should be made at the same temperature, 120°F.

The desired viscosity of a mud is influenced by several factors, including:
I. Mud density;
II. Hole size;
III. Pump rate;
IV. Drilling rate;
V. pressure;
VI. Hole condition.

The viscosity of a mud is a function of three components:
I. Viscosity of the base liquid or continuous phase;
II. The size shapes and number of solids particles in the mud (plastic viscosity);
III. Inter-particle forces (yield point).

Plastic viscosity:

Is that part of the resistance to flow in mud caused by the friction between suspended particles and the viscosity of the base liquid. The plastic viscosity is a measure of the internal resistance to flow due to the amount, type and size of solids present in the mud. It is due to mechanical friction of the solids in the mud as they come in contact with Each other and with the liquid phase of the mud. The plastic viscosity depends on the concentration and size of solids present. The solids present in the mud can be considered either active or inactive. An example of an inactive solid would be drilled solids incorporated in the mud while drilling. Increasing the percentage by volume of solids in the mud can increase the plastic viscosity. If the volume percent solids remain constant, then reducing the size of the solid would also increase the plastic viscosity due to the increased surface area exposed. This increased surface area allows for more frictional contact. To reduce the plastic viscosity, either the solid concentration can be reduced or a flocculant can be added to increase the size of the particles thereby reducing the available surface area. In the field the reduction is usually made by dilution with water or separation with mechanical solids removal.

Funnel viscosity:
Routine field measurements of drilling mud viscosity are made with a Marsh Funnel, which measures a timed rate of flow. The values obtained are called “apparent viscosity”.

Forced Vibration

Forced Vibration

Forced vibration is when an alternating force or motion is applied to a mechanical system. Examples of this type of vibration include a shaking washing machining due to an imbalance, transportation vibration (caused by truck engine, springs, road, etc), or the vibration of a building during an earthquake. In forced vibration the frequency of the vibration is the frequency of the force or motion applied, with order of magnitude being dependent on the actual mechanical system.

Musical instruments and other objects are set into vibration at their natural frequency when a person hits, strikes, strums, plucks or somehow disturbs the object. For instance, a guitar string is strummed or plucked; a piano string is hit with a hammer when a pedal is played; and the tines of a tuning fork are hit with a rubber mallet. Whatever the case, a person or thing puts energy into the instrument by direct contact with it. This input of energy disturbs the particles and forces the object into vibrational motion - at its natural frequency.

If you were to take a guitar string and stretch it to a given length and a given tightness and have a friend pluck it, you would hear a noise; but the noise would not even be close in comparison to the loudness produced by an acoustic guitar. On the other hand, if the string is attached to the sound box of the guitar, the vibrating string is capable of forcing the sound box into vibrating at that same natural frequency. The sound box in turn forces air particles inside the box into vibrational motion at the same natural frequency as the string. The entire system (string, guitar, and enclosed air) begins vibrating and forces surrounding air particles into vibrational motion. The tendency of one object to force another adjoining or interconnected object into vibrational motion is referred to as a forced vibration. In the case of the guitar string mounted to the sound box, the fact that the surface area of the sound box is greater than the surface area of the string, means that more surrounding air particles will be forced into vibration. This causes an increase in the amplitude and thus loudness of the sound.


This same principle of a forced vibration is often demonstrated in a Physics classroom using a tuning fork. If the tuning fork is held in your hand and hit with a rubber mallet, a sound is produced as the tines of the tuning fork set surrounding air particles into vibrational motion. The sound produced by the tuning fork is barely audible to students in the back rows of the room. However, if the tuning fork is set upon the whiteboard panel or the glass panel of the overhead projector, the panel begins vibrating at the same natural frequency of the tuning fork. The tuning fork forces surrounding glass (or vinyl) particles into vibrational motion. The vibrating whiteboard or overhead projector panel in turn forces surrounding air particles into vibrational motion and the result is an increase in the amplitude and thus loudness of the sound. This principle of forced vibration explains why demonstration tuning forks are mounted on a sound box, why a commercial music box mechanism is mounted on a sounding board, why a guitar utilizes a sound box, and why a piano string is attached to a sounding board. A louder sound is always produced when an accompanying object of greater surface area is forced into vibration at the same natural frequency.

Now consider a related situation which resembles another common Physics demonstration. Suppose that a tuning fork is mounted on a sound box and set upon the table; and suppose a second tuning fork/sound box system having the same natural frequency (say 256 Hz) is placed on the table near the first system. Neither of the tuning forks is vibrating. Suppose the first tuning fork is struck with a rubber mallet and the tines begin vibrating at its natural frequency - 256 Hz. These vibrations set its sound box and the air inside the sound box vibrating at the same natural frequency of 256 Hz. Surrounding air particles are set into vibrational motion at the same natural frequency of 256 Hz and every student in the classroom hears the sound. Then the tines of the tuning fork are grabbed to prevent their vibration and remarkably the sound of 256 Hz is still being heard. Only now the sound is being produced by the second tuning fork - the one which wasn't hit with the mallet. Amazing!! The demonstration is often repeated to assure that the same surprising results are observed. They are! What is happening?

In this demonstration, one tuning fork forces another tuning fork into vibrational motion at the same natural frequency. The two forks are connected by the surrounding air particles. As the air particles surrounding the first fork (and its connected sound box) begin vibrating, the pressure waves which it creates begin to impinge at a periodic and regular rate of 256 Hz upon the second tuning fork (and its connected sound box). The energy carried by this sound wave through the air is tuned to the frequency of the second tuning fork. Since the incoming sound waves share the same natural frequency as the second tuning fork, the tuning fork easily begins vibrating at its natural frequency. This is an example of resonance - when one object vibrating at the same natural frequency of a second object forces that second object into vibrational motion.

The result of resonance is always a large vibration. Regardless of the vibrating system, if resonance occurs, a large vibration results. This is often demonstrated in a Physics class with an odd-looking mechanical system resembling an inverted pendulum. The apparatus consists of three sets of two identical plastic bobs mounted on a very elastic metal pole, which arere in turn mounted to a metal bar. Each metal pole and attached bob has a different length, thus giving it a different natural frequency of vibration. The bobs are often color coded to distinguish between them; they are colored red, blue and green (a set of three colors which will be significant later in The Physics Classroom Tutorial). The red bobs are mounted on the longer poles and they have the lowest natural frequency of vibration. The blue bobs are mounted on the shorter poles and have the highest natural frequency of vibration. (Note the length-wavelength-frequency relationship that was discussed earlier.) When the red bob is disturbed, it begins vibrating at its natural frequency. This in turn forces the attached bar to vibrate at the same frequency; and this forces the other attached red bob into vibrating at the same natural frequency. This is resonance - one bob vibrating at a given frequency forcing a second object with the same natural frequency into vibrational motion. While the green and the blue bobs were disturbed by the vibrations transmitted through the metal bar, only the red bob would resonate. This is because the frequency of the first red bob is tuned to the frequency of the second red bob; they share the same natural frequency. The result is that the second red bob begins vibrating with a huge amplitude.

Another common classroom demonstration of resonance involves a plastic tube containing an air column. The length of the air column was adjusted by raising and lowering a reservoir of water (dyed red). The raising and lowering of the reservoir adjusts the height of water in the open-air tube, and thus adjusts the length of the air column inside the tube. As the length of the air column is decreased, the natural frequency of the air column is increased. (Again note the length-wavelength-frequency relationship that was discussed earlier.) While adjusting the height of the liquid in the tube, a vibrating tuning fork is held above the air column of the tube. When the natural frequency of the air column is tuned to the frequency of the vibrating tuning fork, resonance occurs and a loud sound results. Quite amazingly, the vibrating tuning fork forces air particles within the air column into vibrational motion. Once more in this resonance situation, the tuning fork and the air column share the same vibrational frequency.

In conclusion, resonance occurs when two interconnected objects share the same vibrational frequency. When one of the objects is vibrating, it forces the second object into vibrational motion. The result is a large vibration. And if a sound wave within the audible range of human hearing is produced, a loud sound is heard.

Truss

Truss

In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight slender members whose ends are connected at joints referred to as nodes. External forces and reactions to those forces are considered to act only at the nodes and result in forces in the members which are either tensile or compressive forces. Moments (torsional forces) are explicitly excluded because, and only because, all the joints in a truss are treated as revolutes.

A planar truss is one where all the members and nodes lie within a two dimensional plane, while a space truss has members and nodes extending into three dimensions.

Characteristics of trusses

A truss is composed of triangles because of the structural stability of that shape and design. A triangle is the simplest geometric figure that will not change shape when the lengths of the sides are fixed.In comparison, both the angles and the lengths of a square must be fixed for it to retain its shape.

The simplest form of a truss is one single triangle. This type of truss is seen in a framed roof consisting of rafters and a ceiling joist. Because of the stability of this shape and the methods of analysis used to calculate the forces within it, a truss composed entirely of triangles is known as a simple truss.

A planar truss lies in a single plane. Planar trusses are typically used in parallel to form roofs and bridges. A space truss is a three-dimensional framework of members pinned at their ends. A tetrahedron shape is the simplest space truss, consisting of six members which meet at four joints.

The depth of a truss, or the height between the upper and lower chords, is what makes it an efficient structural form. A solid girder or beam of equal strength would have substantial weight and material cost as compared to a truss. For a given span length, a deeper truss will require less material in the chords and greater material in the verticals and diagonals. An optimum depth of the truss will maximize the efficiency.



Truss types

There are two basic types of truss:

* The pitched truss, or common truss, is characterized by its triangular shape. It is most often used for roof construction. Some common trusses are named according to their web configuration. The chord size and web configuration are determined by span, load and spacing.
* The parallel chord truss, or flat truss, gets its name from its parallel top and bottom chords. It is often used for floor construction.

A combination of the two is a truncated truss, used in hip roof construction. A metal plate-connected wood truss is a roof or floor truss whose wood members are connected with metal connector plates.

Pratt truss

Vierendeel Truss The Pratt truss was patented in 1844 by two Boston railway engineers; Caleb Pratt and his son Thomas Willis Pratt. The design uses vertical beams for compression and horizontal beams to respond to tension. What is remarkable about this style is that it remained popular even as wood gave way to iron, and even still as iron gave way to steel.

The Southern Pacific Railroad bridge in Tempe, Arizona is a 393 meter (1291 foot) long truss bridge built in 1912. The structure is composed of nine Pratt truss spans of varying lengths. The bridge is still in use today

Bow string roof truss

Named for its vicissitudal shape, thousands of bow strings were used during World War II for aircraft hangars and other military buildings.

King post truss

One of the simplest truss styles to implement, the king post consists of two angled supports leaning into a common vertical support.
Queen Post Truss

The queen post truss, sometimes queenpost or queenspost, is similar to a king post truss in that the outer supports are angled towards the center of the structure. The primary difference is the horizontal extension at the centre which relies on beam action to provide mechanical stability. This truss style is only suitable for relatively short spans.

Lenticular Truss

American Lenticular Truss Bridges have the top and bottom chords of the truss arched forming a lens shape. Patented in 1878 by William Douglas.

Town's lattice truss

American architect Ithiel Town designed Town's Lattice Truss as an alternative to heavy-timber bridges. His design, patented in 1835, uses easy-to-handle planks arranged diagonally with short spaces in between them.

Vierendeel truss

The Vierendeel truss is a truss where the members are not triangulated but form rectangular openings, and is a frame with fixed joints that are capable of transferring and resisting bending moments. Regular trusses comprise members that are commonly assumed to have pinned joints with the implication that no moments exist at the jointed ends. This style of truss was named after the Belgian engineer Arthur Vierendeel, who developed the design in 1896. Its use for bridges is rare due to higher costs compared to a triangulated truss.

The utility of this type of truss in buildings is that there is no diagonal bracing, the creation of rectangular openings for windows and doors is simplified and in cases the need for compensating shear walls is reduced or eliminated.

After being damaged by the impact of a plane hitting the building, parts of the framed curtain walls of the Twin Towers of the World Trade Center resisted collapse by Vierendeel action displayed by the remaining portions of the frame.

Statics of trusses

A truss that is assumed to comprise members that are connected by means of pin joints, and which is supported at both ends by means of hinged joints or rollers, is described as being statically determinate. Newton's Laws apply to the structure as a whole, as well as to each node or joint. In order for any node that may be subject to an external load or force to remain static in space, the following conditions must hold: the sums of all horizontal forces, all vertical forces, as well as all moments acting about the node equal zero. Analysis of these conditions at each node yields the magnitude of the forces in each member of the truss. These may be compression or tension forces.

Forces in members

On the right is a simple, statically determinate flat truss with 9 joints and (2 x 9) − 3 = 15 members. External loads are concentrated in the outer joints. Since this is a symmetrical truss with symmetrical vertical loads, it is clear to see that the reactions at A and B are equal, vertical and half the total load.

The internal forces in the members of the truss can be calculated in a variety of ways including the graphical methods:

* Cremona diagram
* Culmann diagram
* the analytical Ritter method (method of sections).

Free body diagram

Free body diagram

A free body diagram is a pictorial representation often used by physicists and engineers to analyze the forces acting on a free body. A free body diagram shows all contact and non-contact forces acting on the body. Drawing such a diagram can aid in solving for the unknown forces or the equations of motion of the body. Creating a free body diagram can make it easier to understand the forces, and moments, in relation to one another and suggest the proper concepts to apply in order to find the solution to a problem. The diagrams are also used as a conceptual device to help identify the internal forces—for example, shear forces and bending moments in beams—which are developed within structures.

Construction

A free body diagram consists primarily of a sketch of the body in question and arrows representing the forces applied to it. The selection of the body to sketch may be the first important decision in the problem solving process. For example, to find the forces on the pivot joint of a simple pair of pliers, it is helpful to draw a free body diagram of just one of the two pieces, not the entire system, replacing the second half with the forces it would apply to the first half.

What is included

The sketch of the free body need include only as much detail as necessary. Often a simple outline is sufficient. Depending on the analysis to be performed and the model being employed, just a single point may be the most appropriate.

All external contacts, constraints, and body forces are indicated by vector arrows labeled with appropriate descriptions. The arrows show the direction and magnitude of the various forces. To the extent possible or practical, the arrows should indicate the point of application of the force they represent.

Only the forces acting on the object are included. These may include forces such as friction, gravity, normal force, drag, or simply contact force due to pushing. When in a non-inertial reference frame, fictitious forces, such as centrifugal force may be appropriate.

A coordinate system is usually included, according to convenience. This may make defining the vectors simpler when writing the equations of motion. The x direction might be chosen to point down the ramp in an inclined plane problem, for example. In that case the friction force only has an x component, and the normal force only has a y component. The force of gravity will still have components in both the x and y direction: mgsin(theta) in the x and mgcos(theta) in the y, where theta is the angle between the ramp and the horizontal.

What is excluded

All external contacts and constraints are left out and replaced with force arrows as described above.

Forces which the free body applies to other objects are not included. For example, if a ball rests on a table, the ball applies a force to the table, and the table applies an equal and opposite force to the ball. The FBD of the ball only includes the force that the table causes on the ball.

Internal forces, forces between varies parts that make up the system that is being treated as a single body, are omitted. For example, if an entire truss is being analyzed to find the reaction forces at the supports, the forces between the individual truss members are not included.

Any velocity or acceleration is left out. These may be indicated instead on a companion diagram, called "Kinetic diagrams", "Inertial response diagrams", or the equivalent, depending on the author.

Assumptions

The free body diagram reflects the assumption and simplifications made in order to analyze the system. If the body in question is a satellite in orbit for example, and all that is required is to find its velocity, then a single point may be the best representation. On the other hand, the brake dive of a motorcycle cannot be found from a single point, and a sketch with finite dimensions is required.

Force vectors must be carefully located and labeled to avoid assumptions that presuppose a result. For example, in the accompanying diagram of a block on a ramp, the exact location of the resulting normal force of the ramp on the block can only be found after analyzing the motion or by assuming equilibrium.

Other simplifying assumptions that may be considered include two-force members and three-force members.

Engineers (and now you) often make simple, though still perfectly good, sketches called Free Body Diagrams to show the position of all the forces acting on an object. They get their name from the fact that they have been cut free from their surroundings, allowing a close examination of the forces acting on them. In our example at right, we've broken down the forces on a hanging obelisk (don't ask for an explanation, for the idea or the picture) to three simple ones: gravity pulling it down and the two ropes keeping it up. However, all forces are represented the same in the free body diagram. By simplifying the forces like this, it becomes possible to solve a system using math, which we'll explore later in Calculating Equilibrium.

A free-body diagram is a sketch of an object of interest with all the surrounding objects stripped away and all of the forces acting on the body shown. The drawing of a free-body diagram is an important step in the solving of mechanics problems since it helps to visualize all the forces acting on a single object. The net external force acting on the object must be obtained in order to apply Newton's Second Law to the motion of the object.
A free-body diagram or isolated-body diagram is useful in problems involving equilibrium of forces.

Free-body diagrams are useful for setting up standard mechanics problems.