Mechanical Design Handbook

Standards of limits and fits for mating parts

noreply@blogger.com (Suparerg Suksai) — Mon, 12 Oct 2009 16:56:00 +0000

METRIC STANDARDS FOR LIMITS & FITS

Definitions
1. Basic size is the size to which limits or deviations are assigned and is the same for both members of a fit. It is measured in millimeters.
2. Deviation is the algebraic difference between a size and the corresponding basic size.
3. Upper deviation is the algebraic difference between the maximum limit and the corresponding basic size.
4. Lower deviation is the algebraic difference between the minimum limit and the corresponding basic size.
5. Fundamental deviation is either the upper or the lower deviation, depending on which is closest to the basic size.
6. Tolerance is the difference between the maximum and minimum size limits of a part.
7. International tolerance grade (IT) is a group of tolerances which have the same relative level of accuracy but which vary depending on the basic size.
8. Hole basis represents a system of fits corresponding to a basic hole size.
9. Shaft basis represents a system of fits corresponding to a basic shaft size.

International Tolerance Grades
The variation in part size, also called the magnitude of the tolerance zone, is expressed in grade or IT numbers. Seven grade numbers are used for high-precision parts; these are

IT01, IT0, IT1, IT2, IT3, IT4, IT5

The most commonly used grade numbers are IT6 through IT16. For these, the basic equation is

where D is the geometric mean of the size range under consideration and is obtained from the formula

Basic size ranges (sizes are for over the lower limit and including the upper limits in millimeters.
0-3; for this range use D_min = 1 mm
3-6
6-10
10-18
18-30
30-50
50-80
80-120
120-180
180-250
250-315
315-400
400-500
500-630
630-800
800-1000

Formulas for finding tolerance grades.
Grade - Formula
IT5 - 7i
IT6 - 10i
IT7 - 16i
IT8 - 25i
IT9 - 40i
IT10 - 64i
IT11 - 100i
IT12 - 160i
IT13 - 250i
IT14 - 400i
IT15 - 640i
IT16 - 1000i

Deviations
Fundamental deviations are expressed by tolerance position letters using capital letters for internal dimensions (holes) e.g. 20G7, 40F8, etc. and lowercase letters for external dimensions (shafts) e.g. 20h6, 16g7, etc.

The formula for the fundamental deviation for shafts is
Fundamental deviation = a + (bD^g)/1000
where those 3 coefficients can be obtained from the separate table (not shown here).

Shaft Deviations.
For shafts designated a through h, the upper deviation is equal to the fundamental deviation. Subtract the IT grade from the fundamental deviation to get the lower deviation. Remember, the deviations are defined as algebraic, so be careful with signs.

Shafts designated j through zc have the lower deviation equal to the fundamental deviation. For these, the upper deviation is the sum of the IT grade and the fundamental deviation.

Hole Deviations.
Holes designated A through H have a lower deviation equal to the negative of the upper deviation for shafts. Holes designated as J through ZC have an upper deviation equal to the negative of the lower deviation for shafts.

An exception to the rule occurs for a hole designated as N having an IT grade from 9 to 16 inclusive and a size over 3 mm. For these, the fundamental deviation is zero.

A second exception occurs for holes J, K, M, and N up to grade IT8 inclusive and holes P through ZC up to grade 7 inclusive for sizes over 3 mm. For these, the upper deviation of the hole is equal to the negative of the lower deviation of the shaft plus the change in tolerance of that grade and the next finer grade.

source: google books

We will see more examples with excel file later in the next post.

Mechanical Power Transmission using Belt Drives and Chain Drives

noreply@blogger.com (Suparerg Suksai) — Wed, 19 Aug 2009 13:40:00 +0000

Major types of flexible mechanical power transmission are belts and chains. Belts operate on pulleys or sheaves, whereas chains operate on toothed wheels called sprockets.

When to use chain drives or belt drives?

Electric motors typically operate at too high speed e.g. 1500 rpm and deliver too low torque e.g. 1.8 N.m to be appropriate for the final drive application. These figures are taken from 0.25 kW motor specs of some manufacturers just to get an idea. For a given power transmission, the torque is increased in proportion to the amount that rotational speed is reduced. So the method of speed reduction is usually required for normal mechanical power transmission system.

Usually, we use belt drives for first stage reduction because of high speed of the motor. A smaller drive pulley is attached to the motor shaft which runs at high speed, while a larger diameter pulley is attached to the parallel shaft that operates at a correspondingly lower speed.

" Usually, we use belt drives for first stage reduction because of high speed of the motor..."

However, we can imagine that for the very large ratios of speed reduction, gear reducers are desirable because they can typically accomplish large reductions in a rather small package. The output shaft of the gear-type speed reducer is generally at low speed and high torque. If both speed and torque are satisfactory for the application, it could be directly coupled to the driven machine.

However, the output of gear reducers must often be reduced more before meeting the requirements of the machine because the gear reducers are available only at discrete reduction ratios. At the low-speed, high-torque condition, chain drives become desirable. The high torque causes high tensile force in the chain. The chains normally made of metal and they can withstand the high forces.

" At the low-speed, high-torque condition, chain drives become desirable..."

In general, belt drives are used where the rotational speeds are relatively high, which results in relatively low tensile forces in the belt. But at lower speed, the tensile force in the belt becomes too large for typical belt cross sections and may lead to slipping between sides of the belt and pulleys (or sheaves)

Philosophy of a safe design

noreply@blogger.com (Suparerg Suksai) — Tue, 18 Aug 2009 14:51:00 +0000

Every design approach, we must ensure that the stress level is below the yield in ductile materials, automatically ensuring that the part will not break under a static load.

For brittle materials, we must ensure that the stress levels are well below the ultimate tensile strength.

Two other failure modes that apply to machine members are fatigue and wear. Fatigue is the response of a part subjected to repeated loads. Wear often happens where two parts are in contact with each other such as gears, bearings, and chains, for which it is a major concern.

source: Machine Elements in Mechanical Design, Robert L. Mott

Chain Sprockets

noreply@blogger.com (Suparerg Suksai) — Mon, 17 Aug 2009 14:53:00 +0000

Chain Sprockets are fabricated from a variety of materials; this would depend upon the application of the drive. Large fabricated steel chain sprockets are manufactured with holes to reduce the weight of the chain sprocket on the equipment. Because roller chain drives sometimes have restricted spaces for their installation or mounting, the hubs are made in several different styles.

Type A chain sprockets are flat and have no hub at all. They are usually mounted on flanges or hubs of the device that they are driving. This is accomplished through a series of holes that are either plain or tapered.

Type B chain sprockets has a hub on one side and extend slightly on the other side. The hub is extended to one side to allow the sprocket to be fitted close to the machinery that it is being mounted on. This eliminates a large overhung load on the bearings of the equipment.

Type C chain sprockets are extended on both sides of the plate surface. They are usually used on the driven sprocket where the pitch diameter is larger and where there is more weight to support on the shaft. Remember this the larger the load is, the larger the hub should be.

Type D chain sprockets use an "A" chain sprocket mounted on a solid or split hub. The type A chain sprocket is split and bolted to the hub. This is done for ease of removal and not practicality. It allows the speed ratio to be changed easily by simply unbolting the sprocket and changing it without having the remove bearings or other equipment.

Chain Drives - Conveyor Roller Chain

noreply@blogger.com (Suparerg Suksai) — Mon, 17 Aug 2009 13:18:00 +0000

Chain drives are an important part of a conveyor system. Chain drives are normally used to transmit power between a drive unit and a driven unit of the conveyor system. Chain drives can consist of one or multiple strand chains, depending on the load that the unit must transmit. The chains need to be the matched with the sprocket type, and they must be tight enough to prevent slippage.

" Chain is sized by the pitch or the center-to-center distance between the pins. This is done in 1/8" increments. "

Conveyor Roller Chain
Roller chains are made up of roller chain link that are joined with pin links. The roller chain links are made up of two side bars, two rollers, and two bushings. The roller reduces the friction between the chain and the sprocket, thereby increasing the life of the unit.

Roller chains can operate at faster speeds than plain chains, and properly maintained, they will offer years of reliable service. Some roller chains come with a double pitch, meaning that the pitch is double that of a standard chain, but the width and roller size remains the same. Double-pitch chain can be used on standard sprockets, but double-pitch sprockets are also available.

The main advantage to the double-pitch chain is that it is cheaper than the standard pitch chain. So, they are often used for applications that require slow speeds, as in for lifting pieces of equipment in a hot press application.

Roller chain is ordinarily hooked up using a master link (also known as a connecting link), which typically has one pin held by a C clip rather than friction fit, allowing it to be inserted or removed with simple tools. Half links (also known as offsets) are available and are used to increase the length of the chain by a single roller.

Roller chain is made in several sizes, the most common American National Standards Institute (ANSI0 standards being 40, 50, 60, and 80. The first digit(s) indicate the pitch of the chain in eighths of an inch, with the last digit being 0 for standard chain, 1 for lightweight chain, and 5 for bushed chain with no rollers.

Roller chain is used in low- to mid-speed drives at around 600 to 800 feet per minute; however, at higher speeds, around 2,000 to 3,000 feet per minute, V-belts are normally used due to wear and noise issues.

It is advisable either to monitor the exact length of a drive chain (the generally accepted rule of thumb is to replace a roller chain which has elongated 3% on an adjustable drive or 1.5% on a fixed-center drive), or just replace it at established intervals of use to minimize wear on the sprockets. Thus, any savings in maintenance costs from skimping on lubrication result in increased costs for monitoring wear and for replacement. This need for frequent maintenance, comprising lubrication, assessing wear, and replacement of the chain and/or the sprockets, represents the major drawback of the utilization of roller chain.

The lengthening of a chain is calculated by the following formula:

% = ((M − (S * P)) / (S * P)) * 100)

M = the length of a number of links measured
S = the number of links measured
P = Pitch

Some contents from wikipedia.org

Dowel Pins and Locating Pins

noreply@blogger.com (Suparerg Suksai) — Sun, 16 Aug 2009 16:42:00 +0000

Dowel pins are the fasteners used to secure two parts together. They are available in both Metric and English sizes, and carry specifications such as diameter, length, and materials. Most dowel pins are made of stainless steel, plastic, , hardened steel, or ground steel. Plastic dowel pins are made of thermoplastic or thermosetting polymers with high molecular weight. Stainless dowel pins are chemical and corrosion resistant, and have relatively high pressure ratings.

Dowel pins are often used as precise locating devices in machinery. Stainless dowel pins are machined to tight tolerances, as are the corresponding holes, which are typically reamed. A dowel pin may have a larger diameter so that it must be pressed into its hole or a smaller diameter than its hole so that it freely slips in.

When mechanical design engineers design the mechanical components, typically they use dowel holes as reference points to control positioning variations and attain repeatable assembly quality. If no dowel pins are used for alignment e.g., components are mated by bolts only, there can be significant variation, or "play," in component alignment. Typical drilling and milling operations, as well as manufacturing practices for bolt threads, introduce at least 0.2 mm play for bolts up to 10 mm. If dowel pins are used in addition to bolts, the play is reduced to approximately 0.01 mm.

In automatic machinery, dowel pins are used when precise mating alignment is required, such as in differential gear casings, engines, transmissions and indexing mill. Not only high precision will be achieved, but also it can reduce time to exchange the machine parts. Imagine if you have 50 units of product holders that are required to mount into the indexing mill, without dowel pins, you have to take most of the time to adjust the position of the product holders online. With the dowel pins, you can easily set the units off-line and just mount them into the indexing mill. They are widely used for SMED concept (Single Minute Exchange of Die).

Single Minute Exchange of Die (SMED) is one of the many lean production methods for reducing waste in a manufacturing process. It provides a rapid and efficient way of converting a manufacturing process from running the current product to running the next product. This rapid changeover is key to reducing production lot sizes and thereby improving flow

To locate the parts precisely, normally diamond locating pins are used in conjunction with a round locating pin. The round locating pin holds the part in position, and the diamond pin hold the part to keep it from rotating around the round locating pin.

It is poor design practice to use two round pin four-way locators, as the tolerance stack-up from the center of one pin to the other pin will make mounting the part impossible. Two round locating pins should only be used when one is place in a hole and one is place in a slot. If two holes are to be used as locating features, use one round and one diamond pin.

The long axis of the diamond locating pin should be positioned perpendicular to a line drawn between the center of the round locating pin and the center of the diamond locating pin. Any other orientation will allow the part to swing from side to side and produce inaccurate results.

Stepper Motors and Linear Induction Motors

noreply@blogger.com (Suparerg Suksai) — Wed, 29 Jul 2009 16:12:00 +0000

A linear induction motor is made up of an inductor which is made of individual cores with a concentrated poly phase. Linear induction motor can be directly substituted for ball screw drives, hydraulic drives, pneumatic drives, or cam drives.

A linear induction motor is basically what is referred to by experts as a “rotating squirrel cage” induction motor. The difference is that the motor is opened out flat. Instead of producing rotary torque from a cylindrical machine it produces linear force from a flat machine. The shape and the way it produces motion is changed, however it is still the same as its cylindrical counterpart. There are no moving parts, however and most experts don’t like that. It does have a silent operation and reduced maintenance as well as a compact size, which appeals many engineers. There is also a universal agreement that it has an ease of control and installation. These are all important considerations when thinking about what type of device you want to create. The linear induction motor thrusts ratio varies depending mainly on the size and rating. Speeds of the linear induction motor vary from zero to many meters per second. Speed can be controlled. Stopping, starting and reversing are all easy. Linear induction motors are improving constantly and with improved control, lower life cycle cost, reduced maintenance and higher performance they are becoming the choice of the experts. Linear induction motors are simple to control and easy to use. They have a fast response and high acceleration. Their speed is not dependant on contact friction so it is easier to pick up speed quickly.

Stepper motors are a special kind of motor that moves in discrete steps. When one set of windings is energized the motor moves a step in one direction and when another set of windings is energized the motor moves a step in the other direction. The advantage of stepper motors that the position of the motor is "known". Zero position can be determined, if the original position is known.

Stepper motors come in a wide range of angular resolution and the coarsest motors typically turn 90 degrees per step. High resolution permanent magnet motors are only able to handle about 18 degrees less than that. With the right controller stepper motors can be run in half-steps, which is amazing.

The main complaint about the stepper motors is that it usually draws more power than a standard DC motor and maneuvering is also difficult.

The followings are from wikipedia.org ...

Stepper motors operate differently from normal DC motors, which rotate when voltage is applied to their terminals. Stepper motors, on the other hand, effectively have multiple "toothed" electromagnets arranged around a central gear-shaped piece of iron. The electromagnets are energized by an external control circuit, such as a micro controller. To make the motor shaft turn, first one electromagnet is given power, which makes the gear's teeth magnetically attracted to the electromagnet's teeth. When the gear's teeth are thus aligned to the first electromagnet, they are slightly offset from the next electromagnet.

So when the next electromagnet is turned on and the first is turned off, the gear rotates slightly to align with the next one, and from there the process is repeated. Each of those slight rotations is called a "step," with an integer number of steps making a full rotation. In that way, the motor can be turned by a precise angle.

Linear Actuators and Linear Motion

noreply@blogger.com (Suparerg Suksai) — Wed, 29 Jul 2009 14:32:00 +0000

Mechanical energy is an area of science that is making strides every day. The study of how actuators produce mechanical motion by converting various forms of energy into mechanical energy is a source of great exploration. Science finds new ways to make use of actuators every day including for medical purposes. Many scientists believe that the more they study these seemingly simple machines, the more they will discover ways of helping mankind.

The way in which a linear actuator works is that there is a motor that rotates a drive screw using a synchronous timing belt drive. Some linear actuators can also use a worm gear drive or direct drive. Whichever the choice, the turning of the screw pushes a drive nut along the screw, which in turn pushes the rod out and the rotating the screw in the opposite direction will retract the rod. According to the Association of Sciences, the drive screw is either an ACME or ball thread or is belt-driven which is what gives the machine its motion. A cover tube protects the screw nut from environmental elements and contamination thus allowing for the machines use continually without the chance of it getting gummed up. Radial thrust bearings permit the screw to rotate freely under loaded conditions and gives the linear actuator its strength.

Linear actuators usually serve as part of motion control systems. These days most are run by computers. Control systems, a device that you find linear actuators in, move or control objects. This is made possible by the actuators.

There are various forms of energy that run actuators. These forms of energy include, hydraulic, pneumatic, mechanical and electrical. Linear actuators are used a lot in robotics and factory automation.

Linear motion is when an object moves in a straight line. This is the basic concept that drives the linear actuator. One has to stop and consider when choosing a linear actuator which type they need to fit the purpose of their project. Some things to keep in mind are the speed, stroke length and load rating of the actuator. Programmability of the actuator is also a factor especially when the application will be one that requires specialized detail. A linear actuator can be used in just about any forum. Ask yourself some questions when attempting to choose the right one for your project such as are there particular safety mechanisms required, environmental concerns to be addressed or space issues?

Automotive Braking Methods

noreply@blogger.com (Suparerg Suksai) — Tue, 21 Jul 2009 14:57:00 +0000

Modern automotive brake was invented in the late 19th century, around the same time as the tyre. Up until then, vehicles had wooden wheels that were stopped by large wooden blocks, lowered into position by the driver using a simple lever system. When tyres were invented, the wooden block system was not good enough to stop them at the higher speeds they could achieve, which meant that a new braking system had to be invented.

To see the basic principles of modern automotive braking, it is easiest to look at a bicycle. Basically, when you put pressure on the brakes, the pressure is transferred through cables to pull small brake pads onto the side of the tyres, and the force of the friction against the tyres causes them to stop.

In fact, cars originally used this very same cable system, but it was found not to work so well at high speeds. Instead, the cables were replaced with hydraulic fluid, which works to transfer the pressure the driver puts on the pedal to the brakes. This works because the fluid cannot get much smaller when pressure is put on it, meaning that pressure at one end is transferred to the other – much like water flowing through a pipe. However, if this brake fluid leaks even a little, then the brakes may not work properly any more, which is why it’s very important to check your brake fluid regularly.

Of course, in modern cars, there are other mechanisms apart from pure pressure to help you brake. Most cars now have a vacuum system to create more friction in the brakes, and a servo system that uses the car’s own speed to help your pressure have more of an impact.

One word of warning, though: some cars now have fully computerized brakes, where pushing on the pedal sends an electrical signal to turn on electrically-powered brakes. While this makes it much easier to brake, it is also more prone to failure, meaning that if your car’s computer breaks you might find it impossible to stop. Until this technology has been around a little longer, it’s probably best to stick to traditional mechanical braking methods.

Advantage Of Disc Brake Pads

noreply@blogger.com (Suparerg Suksai) — Tue, 21 Jul 2009 13:37:00 +0000

Whether you drive a Chevrolet, a cycle, a van or a pickup truck, you probably have disc brake pads on your vehicle. And even though you probably never think about their function, they are the single most important function on your vehicle. Though there are several types of motor brakes, disc brakes, drum brake, caliper brakes, etc. but the disk brakes are more commonly used. Disc brakes are far better than drum brakes because of their powerful stopping ability. Disc brake pads handle substantially better in wet weather conditions. Why choose anything but the best?

What are Disc Brakes?

Put simply, disc brakes consist of two disc brake pads that grasp a rotating disk. The disk, or rotor, connects to the wheels by an axle. You control the grasping power. When you pull on the brake, the clamps come together on the disk, forcing it to stop spinning and causing your vehicle to slow down and eventually stop.

How Do You Control Disk Brakes?

In a car, controlling your disk brakes is as simple as pressing the brake pedal or pulling up on the emergency brake. For motorcycles, however, there are two ways to slow it down. You can use the right hand lever or the rear left foot lever. They actually work better when you use them together to better the efficiency and lengthen the life of the disc brake pads.

How To Maintain Disc Brake Pads?

Regardless of the type of vehicle you drive, you will probably need to consider disc brake maintenance or replacement at some point. It is important to check the thickness of your disc brake pads. If these disc brake pads are bare they can cause pricey damage to your disc brakes.

You should also keep an eye on your vehicle’s brake fluid. Your vehicle will run more efficiently with the occasional dose of fresh brake fluid.

Replacing the disc brake pads and the disc brakes fairly easily on your own. Don’t hesitate to get help if you are unsure though. A simple mistake like a poorly fitted disc brake pads can cause scarring to your disk brake.

What Type of Damage is Possible To Your Disc Brakes?

There are several ways your disc brake pads can show damage. They can warp, scar or crack. It’s best if you can catch these signs of damage early on and repair them as quickly as possible to limit further damage to your disc brake pads. Unfortunately, once they crack, the disc brake pads are not repairable. It also helps to get the help of a certified professional when it comes to making repairs to your disc brakes.

How Are Disc Brakes Designed?

These days, the designs of the disc brakes vary greatly. Some are made in classic solid steel, but others have special hallowed out sections that allow the extra built up heat to escape. These slotted steel wheels may help prolong the life of the discs because they reduce built-up heat and cut back on the possibility of warping. The creative designs are endless and each design has a different effect on the performance of your braking system.

Ball Bearings

noreply@blogger.com (Suparerg Suksai) — Tue, 07 Jul 2009 15:12:00 +0000

Many bearings look very similar, whether they are ball bearings, roller bearings or other bearings. What?! Other bearings?

What is a ball bearing, anyway?

Ball bearings are formed with an outer ring, an inner ring, a cage or a retainer inside, and a rolling element inside, typically a ball (which is why they are called ball bearings). Roller bearings are formed using a roller instead of a ball, which is why they are called roller bearings (Yes, finally something that makes sense!). Other bearings look just like metal tubes, called plain bearings or bush bearings. They look like sawed off pipe or tube.

The principle of bearings is the same principle behind the wheel: things move better by rolling than by sliding. They are called "bearings" because they bear the weight of the object, such as an inline skate or the head of dentist's drill, allowing the object to glide over them with incredible ease and speed. Unlike wheels, they don't turn on an axel; they turn on themselves.

You can see this in action with some great cut-away pictures of bearings.

The balls or rollers spin on themselves inside the bearing, reducing friction for the machine parts attached to them. It's much neater than using a bucket of oil, especially in dental equipment, and significantly more reliable than hamsters on a wheel.

Once upon a time, all bearings were metal – like a metal tube or pipe with metal balls stuck inside. These days, more and more are made of ceramic or even plastic (like everything else in this world!).

How to Build a Robot

noreply@blogger.com (Suparerg Suksai) — Mon, 06 Jul 2009 16:57:00 +0000

Robots as we all know are considered as friendly creature created by human beings as we are created by God. They are created for human being to simplify life even more basically for our daily chores with the specified sequence and even by military for the purpose of doing things which has the danger to life of human beings and thus they are developed over years to substitute human beings in all the fields.

Many of us are not that qualified to make a robot by ourselves and that why we all are anxious to know how to make a robot and even depends upon the task we want to create it for. We all have the tendencies of exploring whatever new comes in the field of science and hence a basic prototype robot can be created knowing few basic high end programming stuffs.

Robots are almost 30% programming and hence if we target one specific purpose and program it well enough then it serves our purpose and the program mostly used for this is Unix and for beginner's Lego Mindstorms series is the best and how complicated your robot might turn up to be depends upon your technical acumen.

Sure, Lego Mindstorms NXT is a toy, but it is an important toy, like a piano or a chemistry set. It's one of those items that engages an imagination and possibly opens doors to new interests. Since our future is surely to be shared with robots--it's already started happening, just look at Roomba--those robots will need, at least initially, humans to program and maintain them. Those people, years from now, will likely remember their experiences with Lego Mindstorms.

While learning how to make a robot we should always keep in mind that fewer the moving parts be of the robot better it is for the beginner's as for start up we might just want it to move from here and there or hold something and sort of stuff. We should link if-then statement well and it should be taken care of that battery is never less then 50% and if so happens it should be re charged.

Thus we now understand that knowing how to make robot can never be known as there is no limit to what can be achieved with the knowledge of science and development of robots can never end.

Sir Isaac Newton and the 3 Laws of Motion

noreply@blogger.com (Suparerg Suksai) — Mon, 06 Jul 2009 12:52:00 +0000

Sir Isaac Newton first presented his three laws of motion in the "Principia Mathematica Philosophiae Naturalis" in 1686.

Let's start with the First law of Newton, which states: In the absence of external influences, a material body remains in a condition of rest or continues in uniform and rectilinear movement through inertia. This law is also known as "the law of inertia". And what is inertia? As a matter of fact, it describes the ability of a body to preserve the initial parameters of its own motion.

The formula of the Newton's second law is: F = m • a, where F = the size of the external force, m = size of inert mass, a = size of the acceleration of a body. If we rewrite this as: a = F / m it becomes obvious, that the larger the mass of a body, the greater external effort is required to apply the same acceleration to it. Actually, inertial mass here acts as a measure of its own internal resistance to the influence of the external force.

The third law of Newton states that any external influence on a body causes an equal and opposite action from the body. In other words, to every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary part.

Finite Element Analysis (FEA): Post-processing

noreply@blogger.com (Suparerg Suksai) — Sun, 05 Jul 2009 15:22:00 +0000

The following four-article series was published in a newsletter of the American Society of Mechanical Engineers (ASME). It serves as an introduction to the recent analysis discipline known as the finite element (FEM). The author is an engineering consultant and expert witness specializing in finite element analysis.

FINITE ELEMENT ANALYSIS: Post-processing
by Steve Roensch, President, Roensch & Associates

Last in a four-part series

After a finite element model has been prepared and checked, boundary conditions have been applied, and the model has been solved, it is time to investigate the results of the analysis. This activity is known as the post-processing phase of the finite element method.

Post-processing begins with a thorough check for problems that may have occurred during solution. Most solvers provide a log file, which should be searched for warnings or errors, and which will also provide a quantitative measure of how well-behaved the numerical procedures were during solution. Next, reaction loads at restrained nodes should be summed and examined as a "sanity check". Reaction loads that do not closely balance the applied load resultant for a linear static analysis should cast doubt on the validity of other results. Error norms such as strain energy density and stress deviation among adjacent elements might be looked at next, but for h-code analyses these quantities are best used to target subsequent adaptive remeshing.

Once the solution is verified to be free of numerical problems, the quantities of interest may be examined. Many display options are available, the choice of which depends on the mathematical form of the quantity as well as its physical meaning. For example, the displacement of a solid linear brick element's node is a 3-component spatial vector, and the model's overall displacement is often displayed by superposing the deformed shape over the undeformed shape. Dynamic viewing and animation capabilities aid greatly in obtaining an understanding of the deformation pattern. Stresses, being tensor quantities, currently lack a good single visualization technique, and thus derived stress quantities are extracted and displayed. Principal stress vectors may be displayed as color-coded arrows, indicating both direction and magnitude. The magnitude of principal stresses or of a scalar failure stress such as the Von Mises stress may be displayed on the model as colored bands. When this type of display is treated as a 3D object subjected to light sources, the resulting image is known as a shaded image stress plot. Displacement magnitude may also be displayed by colored bands, but this can lead to misinterpretation as a stress plot.

An area of post-processing that is rapidly gaining popularity is that of adaptive remeshing. Error norms such as strain energy density are used to remesh the model, placing a denser mesh in regions needing improvement and a coarser mesh in areas of overkill. Adaptivity requires an associative link between the model and the underlying CAD geometry, and works best if boundary conditions may be applied directly to the geometry, as well. Adaptive remeshing is a recent demonstration of the iterative nature of h-code analysis.

Optimization is another area enjoying recent advancement. Based on the values of various results, the model is modified automatically in an attempt to satisfy certain performance criteria and is solved again. The process iterates until some convergence criterion is met. In its scalar form, optimization modifies beam cross-sectional properties, thin shell thicknesses and/or material properties in an attempt to meet maximum stress constraints, maximum deflection constraints, and/or vibrational frequency constraints. Shape optimization is more complex, with the actual 3D model boundaries being modified. This is best accomplished by using the driving dimensions as optimization parameters, but mesh quality at each iteration can be a concern.

Another direction clearly visible in the finite element field is the integration of FEA packages with so-called "mechanism" packages, which analyze motion and forces of large-displacement multi-body systems. A long-term goal would be real-time computation and display of displacements and stresses in a multi-body system undergoing large displacement motion, with frictional effects and fluid flow taken into account when necessary. It is difficult to estimate the increase in computing power necessary to accomplish this feat, but 2 or 3 orders of magnitude is probably close. Algorithms to integrate these fields of analysis may be expected to follow the computing power increases.

In summary, the finite element method is a relatively recent discipline that has quickly become a mature method, especially for structural and thermal analysis. The costs of applying this technology to everyday design tasks have been dropping, while the capabilities delivered by the method expand constantly. With education in the technique and in the commercial software packages becoming more and more available, the question has moved from "Why apply FEA?" to "Why not?". The method is fully capable of delivering higher quality products in a shorter design cycle with a reduced chance of field failure, provided it is applied by a capable analyst. It is also a valid indication of thorough design practices, should an unexpected litigation crop up. The time is now for industry to make greater use of this and other analysis techniques.

Finite Element Analysis (FEA): Solution

noreply@blogger.com (Suparerg Suksai) — Sun, 05 Jul 2009 15:06:00 +0000

The following four-article series was published in a newsletter of the American Society of Mechanical Engineers (ASME). It serves as an introduction to the recent analysis discipline known as the finite element method (FEM). The author is an engineering consultant and expert witness specializing in finite element analysis.

FINITE ELEMENT ANALYSIS: Solution
by Steve Roensch, President, Roensch & Associates

Third in a four-part series

While the pre-processing and post-processing phases of the finite element method are interactive and time-consuming for the analyst, the solution is often a batch process, and is demanding of computer resource. The governing equations are assembled into matrix form and are solved numerically. The assembly process depends not only on the type of analysis (e.g. static or dynamic), but also on the model's element types and properties, material properties and boundary conditions.

In the case of a linear static structural analysis, the assembled equation is of the form Kd = r, where K is the system stiffness matrix, d is the nodal degree of freedom (dof) displacement vector, and r is the applied nodal load vector. To appreciate this equation, one must begin with the underlying elasticity theory. The strain-displacement relation may be introduced into the stress-strain relation to express stress in terms of displacement. Under the assumption of compatibility, the differential equations of equilibrium in concert with the boundary conditions then determine a unique displacement field solution, which in turn determines the strain and stress fields. The chances of directly solving these equations are slim to none for anything but the most trivial geometries, hence the need for approximate numerical techniques presents itself.

A finite element mesh is actually a displacement-nodal displacement relation, which, through the element interpolation scheme, determines the displacement anywhere in an element given the values of its nodal dof. Introducing this relation into the strain-displacement relation, we may express strain in terms of the nodal displacement, element interpolation scheme and differential operator matrix. Recalling that the expression for the potential energy of an elastic body includes an integral for strain energy stored (dependent upon the strain field) and integrals for work done by external forces (dependent upon the displacement field), we can therefore express system potential energy in terms of nodal displacement.

Applying the principle of minimum potential energy, we may set the partial derivative of potential energy with respect to the nodal dof vector to zero, resulting in: a summation of element stiffness integrals, multiplied by the nodal displacement vector, equals a summation of load integrals. Each stiffness integral results in an element stiffness matrix, which sum to produce the system stiffness matrix, and the summation of load integrals yields the applied load vector, resulting in Kd = r. In practice, integration rules are applied to elements, loads appear in the r vector, and nodal dof boundary conditions may appear in the d vector or may be partitioned out of the equation.

Solution methods for finite element matrix equations are plentiful. In the case of the linear static Kd = r, inverting K is computationally expensive and numerically unstable. A better technique is Cholesky factorization, a form of Gauss elimination, and a minor variation on the "LDU" factorization theme. The K matrix may be efficiently factored into LDU, where L is lower triangular, D is diagonal, and U is upper triangular, resulting in LDUd = r. Since L and D are easily inverted, and U is upper triangular, d may be determined by back-substitution. Another popular approach is the wavefront method, which assembles and reduces the equations at the same time. Some of the best modern solution methods employ sparse matrix techniques. Because node-to-node stiffnesses are non-zero only for nearby node pairs, the stiffness matrix has a large number of zero entries. This can be exploited to reduce solution time and storage by a factor of 10 or more. Improved solution methods are continually being developed. The key point is that the analyst must understand the solution technique being applied.

Dynamic analysis for too many analysts means normal modes. Knowledge of the natural frequencies and mode shapes of a design may be enough in the case of a single-frequency vibration of an existing product or prototype, with FEA being used to investigate the effects of mass, stiffness and damping modifications. When investigating a future product, or an existing design with multiple modes excited, forced response modeling should be used to apply the expected transient or frequency environment to estimate the displacement and even dynamic stress at each time step.

This discussion has assumed h-code elements, for which the order of the interpolation polynomials is fixed. Another technique, p-code, increases the order iteratively until convergence, with error estimates available after one analysis. Finally, the boundary element method places elements only along the geometrical boundary. These techniques have limitations, but expect to see more of them in the near future.

Finite Element Analysis (FEA): Pre-processing

noreply@blogger.com (Suparerg Suksai) — Sun, 05 Jul 2009 14:12:00 +0000

The following four-article series was published in a newsletter of the American Society of Mechanical Engineers (ASME). It serves as an introduction to the recent analysis discipline known as the finite element method (FEM). The author is an engineering consultant and expert witness specializing in finite element analysis.

FINITE ELEMENT ANALYSIS: Pre-processing
by Steve Roensch, President, Roensch & Associates

Second in a four-part series

As discussed in Finite Element Analysis (FEA): Introduction, finite element analysis is comprised of pre-processing, solution and post-processing phases. The goals of pre-processing are to develop an appropriate finite element mesh, assign suitable material properties, and apply boundary conditions in the form of restraints and loads.

The finite element mesh subdivides the geometry into elements, upon which are found nodes. The nodes, which are really just point locations in space, are generally located at the element corners and perhaps near each midside. For a two-dimensional (2D) analysis, or a three-dimensional (3D) thin shell analysis, the elements are essentially 2D, but may be "warped" slightly to conform to a 3D surface. An example is the thin shell linear quadrilateral; thin shell implies essentially classical shell theory, linear defines the interpolation of mathematical quantities across the element, and quadrilateral describes the geometry. For a 3D solid analysis, the elements have physical thickness in all three dimensions. Common examples include solid linear brick and solid parabolic tetrahedral elements. In addition, there are many special elements, such as axisymmetric elements for situations in which the geometry, material and boundary conditions are all symmetric about an axis.

The model's degrees of freedom (dof) are assigned at the nodes. Solid elements generally have three translational dof per node. Rotations are accomplished through translations of groups of nodes relative to other nodes. Thin shell elements, on the other hand, have six dof per node: three translations and three rotations. The addition of rotational dof allows for evaluation of quantities through the shell, such as bending stresses due to rotation of one node relative to another. Thus, for structures in which classical thin shell theory is a valid approximation, carrying extra dof at each node bypasses the necessity of modeling the physical thickness. The assignment of nodal dof also depends on the class of analysis. For a thermal analysis, for example, only one temperature dof exists at each node.

Developing the mesh is usually the most time-consuming task in FEA. In the past, node locations were keyed in manually to approximate the geometry. The more modern approach is to develop the mesh directly on the CAD geometry, which will be (1) wireframe, with points and curves representing edges, (2) surfaced, with surfaces defining boundaries, or (3) solid, defining where the material is. Solid geometry is preferred, but often a surfacing package can create a complex blend that a solids package will not handle. As far as geometric detail, an underlying rule of FEA is to "model what is there", and yet simplifying assumptions simply must be applied to avoid huge models. Analyst experience is of the essence.

The geometry is meshed with a mapping algorithm or an automatic free-meshing algorithm. The first maps a rectangular grid onto a geometric region, which must therefore have the correct number of sides. Mapped meshes can use the accurate and cheap solid linear brick 3D element, but can be very time-consuming, if not impossible, to apply to complex geometries. Free-meshing automatically subdivides meshing regions into elements, with the advantages of fast meshing, easy mesh-size transitioning (for a denser mesh in regions of large gradient), and adaptive capabilities. Disadvantages include generation of huge models, generation of distorted elements, and, in 3D, the use of the rather expensive solid parabolic tetrahedral element. It is always important to check elemental distortion prior to solution. A badly distorted element will cause a matrix singularity, killing the solution. A less distorted element may solve, but can deliver very poor answers. Acceptable levels of distortion are dependent upon the solver being used.

Material properties required vary with the type of solution. A linear statics analysis, for example, will require an elastic modulus, Poisson's ratio and perhaps a density for each material. Thermal properties are required for a thermal analysis. Examples of restraints are declaring a nodal translation or temperature. Loads include forces, pressures and heat flux. It is preferable to apply boundary conditions to the CAD geometry, with the FEA package transferring them to the underlying model, to allow for simpler application of adaptive and optimization algorithms. It is worth noting that the largest error in the entire process is often in the boundary conditions. Running multiple cases as a sensitivity analysis may be required.

Finite Element Analysis (FEA): Introduction

noreply@blogger.com (Suparerg Suksai) — Sun, 05 Jul 2009 13:41:00 +0000

The following four-article series was published in a newsletter of the American Society of Mechanical Engineers (ASME). It serves as an introduction to the recent analysis discipline known as the finite element method (FEM). The author is an engineering consultant and expert witness specializing in finite element analysis.

FINITE ELEMENT ANALYSIS: Introduction
by Steve Roensch, President, Roensch & Associates

First in a four-part series

Finite element analysis (FEA) is a fairly recent discipline crossing the boundaries of mathematics, physics, engineering and computer science. The method has wide application and enjoys extensive utilization in the structural, thermal and fluid analysis areas. The finite element method is comprised of three major phases:

(1) pre-processing, in which the analyst develops a finite element mesh to divide the subject geometry into subdomains for mathematical analysis, and applies material properties and boundary conditions.

(2) solution, during which the program derives the governing matrix equations from the model and solves for the primary quantities.

(3) post-processing, in which the analyst checks the validity of the solution, examines the values of primary quantities (such as displacements and stresses), and derives and examines additional quantities (such as specialized stresses and error indicators).

The advantages of FEA are numerous and important. A new design concept may be modeled to determine its real world behavior under various load environments, and may therefore be refined prior to the creation of drawings, when few dollars have been committed and changes are inexpensive. Once a detailed CAD model has been developed, FEA can analyze the design in detail, saving time and money by reducing the number of prototypes required. An existing product which is experiencing a field problem, or is simply being improved, can be analyzed to speed an engineering change and reduce its cost. In addition, FEA can be performed on increasingly affordable computer workstations and personal computers, and professional assistance is available.

It is also important to recognize the limitations of FEA. Commercial software packages and the required hardware, which have seen substantial price reductions, still require a significant investment. The method can reduce product testing, but cannot totally replace it. Probably most important, an inexperienced user can deliver incorrect answers, upon which expensive decisions will be based. FEA is a demanding tool, in that the analyst must be proficient not only in elasticity or fluids, but also in mathematics, computer science, and especially the finite element method itself.

Which FEA package to use is a subject that cannot possibly be covered in this short discussion, and the choice involves personal preferences as well as package functionality. Where to run the package depends on the type of analyses being performed. A typical finite element solution requires a fast, modern disk subsystem for acceptable performance. Memory requirements are of course dependent on the code, but in the interest of performance, the more the better, with 512 Mbytes to 8 Gbytes per user a representative range. Processing power is the final link in the performance chain, with clock speed, cache, pipelining and multi-processing all contributing to the bottom line. These analyses can run for hours on the fastest systems, so computing power is of the essence.

One aspect often overlooked when entering the finite element area is education. Without adequate training on the finite element method and the specific FEA package, a new user will not be productive in a reasonable amount of time, and may in fact fail miserably. Expect to dedicate one to two weeks up front, and another one to two weeks over the first year, to either classroom or self-help education. It is also important that the user have a basic understanding of the computer's operating system.

CE Marking to EU Directives

noreply@blogger.com (Suparerg Suksai) — Fri, 26 Jun 2009 17:45:00 +0000

When designing the machine for the customers in Europe, I have to make sure that the machine has full CE compliance.

By the way, what is CE?

I search for more details about CE and put them here for more understanding.

The CE marking is an acronym for the French "Conformité Européenne". By affixing the CE marking, the manufacturer, or in certain cases another legal person responsible for the product, asserts that the item meets all the essential "Health and Safety" requirements of the relevant European Directive(s) that provide for the CE marking. Examples of European Directives requiring CE marking include toy safety, machinery, low-voltage equipment, medical devices and electromagnetic compatibility.

CE Marking Procedure

The "New Approach" to conformity enables manufacturers to use what is called as "SELF DECLARATION" where the manufacturer himself declares conformity by signing the "Declaration of Conformity (DOC)" and then affixes the CE Mark on his product.

The following simple steps are involved
STEP 1: Identify Applicable "DIRECTIVES"
STEP2 : Identify Applicable "Conformity Assessment Module"
Although CE Marking follows the Self Declaration principle, depending upon product complexity and risk to human life, various conformity assesment module are prescribed:-
Module 'A' (Internal Production control).
Applicable for products falling under EMC and Low Voltage Directives. Manufacturer tests the product from third party. After compliance with the tests, his production process ensures continued conformance. He maintains "Technical Documentation" as a proof of compliance. There is No mandatory involvement of European Lab (i.e. Notified Body).
Module 'B' to 'H'
Mandatory involvement of European Lab is required which issues "Type Examination Certificate", certifies documentation (called "Techncal Construction File"(TCF) and carries out inspections.
STEP3 : Identify Applicable "Standard"
STEP4 : Test one sample of the productEither yourself or from test lab.
STEP5 : Compile "Technical Documentation"
STEP 6 : Sign the EC "Declaration of Conformity"
STEP 7 : Affix "CE Mark" on the product.

Source: http://en.wikipedia.org/wiki/CE_mark

There are 21 European "New Approach" Directives against which a wide range of products are required to be CE marked against before they are "placed on the market" in Europe. When placing the CE mark on the product the manufacturer or "his authorized representative in the European union" is declaring that the product complies with all applicable directives. The list of 21 applicable directives can be found on the European Commission website. The EU Directives are not themselves law as they are all taken into the the law of each member state. Whilst the directive and standards may be harmonized across Europe manufacturers still need to be aware of country specific requirements in safety standards, permitted frequency bands and other areas that might affect their product.

Some directives requirements are more onerous than others, but a general guide is that a manufacturer must

identify all directives applicable to their product
assess their product against the requirements of a directive - usually through a combination of technical argument and testing. Testing can be done in house by the manufacturer or by 3rd party specialist test laboratories
create a Technical File containing full details of the product and all assessments carried out to demonstrate compliance. This technical file becomes a living document and needs to be kept up-to-date as changes occur to the product or to the standards against which the product was assessed
depending on the directive there is an optional or mandatory requirement to have the Technical File assessed by a Notified Body
Draw up and sign a Declaration of Conformity and affix CE mark to the product and/or packaging

Charlie is a CE marking consultant at DheaniSulis Ltd who specialise in helping companies navigate their way through these product approval requirements in a timely and cost efficient manner.

Article Source: http://EzineArticles.com/?expert=Charlie_Blackham

Sankyo AD Alpha Series - Dial Index

noreply@blogger.com (Suparerg Suksai) — Sun, 14 Jun 2009 03:35:00 +0000

If you are looking for a strong indexing box for your dial application, you can have a look at Sankyo AD Alpha Series.

Apart from its product specifications, you'll find useful technical information regarding indexing drives.

Sankyo's AD/Alpha Series features a low profile design cast iron housing with a ground pilot flange for mounting dial plates, hub less gears, or weldment fixtures. Above the indexing dial, a second dial mounting surface is stationary and hollow to allow air lines, wiring or bearings for a shaft to be routed through the tolerance bore. A globoidal cam rotates Sankyo made needle bearing type cam followers. Optimizing the cam follower diameter inherently maximizes torque transfer and extends the life hours due to fewer rotations with increased surface contact.

Multiple index periods offer continuous or on demand duty cycles with the indexing flange locking in position within 30 arc seconds during the dwell period. The output flange turret is mounted with top and bottom tapered bearings which supports large bending moments, maintain rigidity and reliability. These versatile units index in both directions, oscillate or perform non-patterned motions with a continuous lead cam using servo driven motors. Most units are driven with geared motors that are within the height of the index unit housing. There’s no need for spacers or burning a hole in the mounting plate to accommodate for drive height interference. An optional torque limiter safety clutch mounts between the index and indexing dial plate. Torque thresholds are easily adjusted without any disassembly and a proximity sensor detects overloads.

Timing cam packages are available with inexpensive photo beam sensors or proximity switches for wash down (IP67/NEMA-6/6P) applications. The AR/Alpha ServoDEX Series shares the same housing as the AD/Alpha series but uses a constant lead cam for programmable indexing stops with a servo motor drive. Non-patterned motions, oscillating, variable or continuous speeds can be programmed to suit your current application needs or reprogrammed for the future. Repetitive accuracy is within 30 arc seconds or 0.0014” per inch of radius.

Features

2 to 32 incremental output stops, 1-dwell cam equals; 1-output stop to 1-input shaft revolution, 2-dwells for 16 to 32 stops
Dial diameters range from 280mm to 3600mm (1 to 12 feet) with second stationary dial mounting
120° to 330° index periods with Sankyo modified sine or constant velocity (~50% CV) cam curves
Maximum 200 cycles per minute, bi-directional indexing, oscillating motions (non-pattern for servo driven AR/ServoDEX)
±30 arcsec indexing accuracy (average ±9 arcsec), 30 arcsec repetitive, ±60 arcsec for 2-dwell cams for 16 to 32 stops
Painted cast iron housings are available in 8 standard sizes ranging from 70 to 450mm input to output shaft centers
Standard stationary hollow flange output with tolerance bore, available with optional torque limiting clutch
Index mounts in any position with right angle gearmotor mounting in 90° increments on either side of the index
Standard brushless dual voltage (230/460) gearmotors are UL & CE certified, brake motor or wash down optional
Variable frequency drive controller utilizes electronic braking and variable speed for gearmotors in 115-1ph, 230-1/3ph, 460-3ph volts, 50/60 Hz, UL & CE certified
Stop/start frequency up to 60~90 per minute (output load/reducer dependant), clutch brake or servo drives are available
Standard IP50 photo eye timing cam sensor package supplies dial “in position” signal to stop the motor, IP67 proximity type optional
Output torque limiting clutch option features quick-adjust torque setting, automatic reset every 360° & overload detect prox.

Stationary Output Flange Riser Option

This option is used if part handlers or measurement equipment are required to be mounted above the rotary indexing dial. The riser is flanged on both ends and offered in multiple lengths with a slightly smaller hollow bore than the index bore. Each riser incorporates a tolerance male centering pilot on the index output flange and for the dial plate or fixture mounting flange. The
riser is mounted to the index output flange with (4) bolts and also includes the grease fitting to be extended for easy access. Standard units are painted steel to match the beige index color but custom materials and coatings are available.

Index is shown with a riser for mounting a second fixed dial above the indexing dial. Each riser is hollow to mate to the index hollow output flange. A center pilot flange makes dial mounting concentric and precise. Dowel holes are an option.

Find more details about Sankyo AD Alpha Series - Dial Index at http://www.sankyoamerica.com

SANDEX Indexing Box

noreply@blogger.com (Suparerg Suksai) — Thu, 04 Jun 2009 17:14:00 +0000

In my design, I've been using several indexing boxes from the Japanese company "Sankyo Seisakusho". The model of the indexing box is "SANDEX". It is a generic name for our cam units that combine a simple and ideal indexing mechanism, the "Roller gear cam mechanism". Sankyo's original cam technology, consisting of a roller gear cam and a follower spider (turret) is used to complete the units. SANDEX indexing units have always enjoyed high sales and are responding to industrial needs where the new technologies are evolving.

The indexing units must accommodate three types of basic motion -- that is the indexing drives with an intermittent rotating motion, the oscillating drives with their oscillating rotating motion, and the roller drives with a continuous rotating motion. SANDEX units have a higher number of indexes, index angle, and degree of design freedom than other indexing mechanisms. They are more rigid and have higher output values relative to the size of the unit, and they can eliminate backslash completely, performing at high-speed. A broad lineup of models and sizes will meet the needs of automated rotary machines of all sizes and specifications from the world smallest miniature indexes to the largest class of super-size indexes.

In addition, one model with parallel input/output shafts is also available, using a parallel mechanism as an "ideal flat cam". This is optimum for Conveyer drives with a large feed pitch or indexing drives with relatively long stopping interval.

This is the example of indexing movement (intermittent rotation).

More details about how to select the indexing drive and calculation example will be shared later.

Source: http://www.sankyo-seisakusho.co.jp/english/2index.htm

How to calculate spring force online

noreply@blogger.com (Suparerg Suksai) — Sun, 31 May 2009 13:48:00 +0000

I found a very useful web site about computation. Try it and you'll love it.

Wolfram|Alpha's long-term goal is to make all systematic knowledge immediately computable and accessible to everyone. We aim to collect and curate all objective data; implement every known model, method, and algorithm; and make it possible to compute whatever can be computed about anything. Our goal is to build on the achievements of science and other systematization of knowledge to provide a single source that can be relied on by everyone for definitive answers to factual queries.

Wolfram|Alpha aims to bring expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. Our goal is to accept completely free-form input, and to serve as a knowledge engine that generates powerful results and presents them with maximum clarity.

Wolfram|Alpha is an ambitious, long-term intellectual endeavor that we intend will deliver increasing capabilities over the years and decades to come. With a world-class team and participation from top outside experts in countless fields, our goal is to create something that will stand as a major milestone of 21st century intellectual achievement.

Today's Wolfram|Alpha is the first step in an ambitious, long-term project to make all systematic knowledge immediately computable by anyone. You enter your question or calculation, and Wolfram|Alpha uses its built-in algorithms and growing collection of data to compute the answer. Based on a new kind of knowledge-based computing.

Numerical Methods - The Newton-Raphson Method to Solve Mechanical Design Problems Part II

noreply@blogger.com (Suparerg Suksai) — Wed, 27 May 2009 02:45:00 +0000

In the previous post, we talked about several root finding techniques. In this post, we're going to see how we can use Microsoft Excel VBA to find the roots using Newton-Raphson Method.

As we know that Newton-Raphson Method is the most widely used of all root-locating formulas. The Newton-Raphson method uses the slope (first derivative) of the function to find the root. That means, in the VBA code, we have to find the first derivative of the function in order to find the root. We already have the VBA code to find the first derivative and we're going to use it again in Newton-Raphson Method.

The Newton-Raphson Method uses 2 terms of taylor series to approximate delta x as shown below.

f(x) = f(x₀) + (x_N - x₀)f'(x₀) = 0
(x_N - x₀)f'(x₀) = -f(x₀)
x_N - x₀ = -f(x₀)/f'(x₀)

delta x = -f(x₀)/f'(x₀)

The followings are the procedure to find the root using Newton-Raphson Method.

Guess the initial value of the root --> select x₀
Calculate delta x using the formula as shown above
Calculate x_N using x_N = x₀ + delta x
Check whether the error of x_N is within the allowable tolerance or not --> abs(x_N - x₀) < tolerance

Then the VBA code will be like this.

' ================================================
' Created by Ake Blogger
' Mechanical Design Handbook
' http://mechanical-design-handbook.blogspot.com
'
' The Newton-Raphson Method - Numerical Methods
' ================================================

Function f(x As Double) As Double
f=Exp(-x)-x ' This is the function we want to find the root
End Function

Function fDeriv(x As Double) As Double
...
...
End Function

Function RootNewtonRaphson(x0 As Double)
...
...
...
End Function

To calculate the root of e^-x - x = 0 we then type the following formula in the Microsoft Excel spreadsheet,

In any cell, type "=RootNewtonRaphson(1)", where 1 is the guessing value of the root.
The program will then show the result of 0.567143290409784.

We can recheck the calculation result by entering the following formula into excel,

=exp(-0.567143290409784)-0.567143290409784

Then the result is 0. That means one of the roots of of e^-x - x = 0 is 0.567143290409784

FREE DOWNLOAD AN EXCEL FILE WITH VBA CODE OF NEWTON-RAPHSON ROOT FINDGING METHOD

Numerical Methods - First derivative using Excel VBA code

noreply@blogger.com (Suparerg Suksai) — Sat, 02 May 2009 14:38:00 +0000

In the previous post, Numerical Methods - First derivative using Excel formula, we know how to calculate the first derivative of functions using formula in Microsoft Excel. In this post, let's see how we can simplify it by using Excel VBA code.

The VBA is just like this.

' ================================================
' Created by Ake Blogger
' Mechanical Design Handbook
' http://mechanical-design-handbook.blogspot.com
'
' First Derivative - Numerical Methods
' ================================================

Function f(x As Double) As Double

f = 5 * (x - 3) ^ 3 - 4 * x ^ 2 - Sin(2 * x)
End Function Function fDeriv(x As Double) As Double

Const h = 0.00001 ' Step size
fDeriv = (-f(x + 2 * h) + 8 * f(x + h) - 8 * f(x - h) + f(x - 2 * h)) / 12 / h
End Function

The highlighted (yellow) text is the function that we want to calculate for the first derivative. We can simply replace it with new function. The remaining code still remains the same.

We have to put this code into the Microsoft Visual Basic Editor in Microsoft Excel (Alt+F11). Now we're going to test this Derivative code. Because the first derivative of the function in this example can be determined by direct method. Then we can use it to recheck the calculation result.

The first derivative of 5(x-3)³-4x²-sin(2x) is 15(x-3)²-8x-2cos(2x). Therefore the first derivative of f(x) where x = 7 must be 15(7-3)²-8(7)-2cos(2(7)) = 183.7265256

Here how to use

In excel spreadsheet, enter =fDeriv(7) and see the result.

We'll use this First Derivative Function in the Newton-Raphson Root Finding Method later.

FREE DOWNLOAD EXAMPLE EXCEL FILE

Numerical Methods - The Newton-Raphson Method to Solve Mechanical Design Problems Part I

noreply@blogger.com (Suparerg Suksai) — Sat, 25 Apr 2009 14:16:00 +0000

Machine designers have to deal with several number of equations in their design projects. There are a number of ways to solve for roots of algebraic and transcendental equations. Sometimes, the roots could be obtained by direct methods. However, there are many more that could not. The classical equation such as f(x) = e^-x - x cannot be solved analytically. For this case, the only alternative is an approximate solution technique.

There are several methods available to solve the root finding problem such as "bracketing methods" and "Open methods".

The bracketing methods require 2 initial guesses for the root. These guesses must "bracket" the root. The numerical methods using bracketing methods consist of the following techniques:

The Bisection Method: The idea of this technique is incremental search that related to the sign change. Sometimes, this technique is called binary chopping, or Bolzano's method.
The False-Position Method: It's the improved technique from the bisection methods. It replaces the curve f(x) by a straight line and gives a false-position of the root. The false-position method is also called the linear interpolation method.

However, my favorite technique in root finding method is not the bracketing methods but the open methods.

The bracketing methods are said to be convergent. In contrast, the open methods are based on formulas that require a single starting value of x. They sometimes diverge. However, when the open methods converge they usually do so much more quickly than the bracketing methods.

The followings are the root finding techniques using open methods.

Simple One-Point Iteration
The Newton-Raphson Method: The most widely used of all root-locating formulas. The Newton-Raphson method uses the slope (first derivative) of the function to find the root. It's my favorite one.
The Secant Method: Instead of using the first derivative of the function to find the slope as in The Newton-Raphson Method, the first derivative in secant method can be approximated by a finite divided difference.

In the next post, let's see how we can use Microsoft Excel VBA again to solve the root finding problems using the Newton-Raphson Method. I'll show you the simple VBA code that you can copy & paste into your excel file and use it as the module to calculate for the root of your equation.

Numerical Methods - First derivative using Excel formula

noreply@blogger.com (Suparerg Suksai) — Mon, 20 Apr 2009 15:07:00 +0000

Numerical methods can be very helpful for calculation of mechanical engineering design. What I would like to introduce in this post is about calculation of the first derivative using numerical methods in Microsoft Excel without using any math software.

From the textbook, there are a lot of formula and explanation about how we get this formula. So I would skip that part and directly start with the selected formula as an example.

We know that the first derivative of function f(x) can be calculated from the following equation

It is the formula to calculate first derivative using interior points that give the error of h⁴
That means we will use the values of function at other 4 points around the center of interest to calculate the first derivative and the smaller the step size (h), the higher accuracy of calculation result.

The following is the equation of favorite cam curve - Cycloid

The cam follower will slowly move at the beginning of time and start accelerating, then decelerating in the middle of time and slowly stops at the end of movement. This give smooth motion of cam mechanism.

We can calculate the velocity of the cam follower using the following formula.

What we have to do is to make a table in excel and enter the formula of f(x), for this case, it's the cycloid motion function.

The above table is calculated using 1.25 deg for each step. But the real value of h should be calculated from 1.25 x Pi/180 = 0.021816616 rad.

Then we can calculate the velocity at each point using the formula explained above.
Example: to calculate the first derivative of cycloid motion at 15 deg
If camshaft speed = 50 rpm, then omega = 2 x Pi x 50/60 = 5.235987756 rad/s

Then the velocity at 15 deg can be calculated from

f'(15 deg) = [-4.20358 + 8x3.48065 - 8x2.23829 + 1.72606]/(12x0.021816616) x 5.235987756 = 149.2272246 mm/s = 0.1492 m/s

If we do the same calculation for the remaining points, we can then get the velocity profile of cycloid motion. Try it yourself and get the result as follows,

No need to use any math software. Excel can easily help you calculate first derivative using knowledge of numerical methods.