Mechanical Design Handbook

Links for 2012-05-14 [del.icio.us]

Tue, 15 May 2012 00:00:00 PDT

Torque limiter: Overload Clutch

Ball Detent Torque Limiter: Overload Clutch

akeblogger@gmail.com (Suparerg Suksai) — Sun, 13 May 2012 07:11:00 +0000

A torque limiter is an automatic overload clutch that provides machine protection and reduces repair time during jamming load conditions. This is done to protect expensive machines and prevent physical injuries. A torque limiter may limit the torque by slipping (as in a friction plate slip-clutch), or uncouple the load entirely (as in a shear pin).

Ideally the torque limiter should be placed as close as possible to the source of the jam. This will allow the system inertia and torque to be quickly and effectively disconnected from the jammed section. The system can then be allowed to stop without causing further machine damage. A mechanical torque limiter will provide faster response times and better protection than typical electronic methods at high crash rates.
There are several disconnect types available, but we will focus at the Ball Detent type.

A ball detent type torque limiter transmits force through hardened balls which rest in detents on the shaft and are held in place with springs. An over-torque condition pushes the balls out of their detents, thereby decoupling the shaft. It can have single or multiple detent positions, or a snap acting spring which requires a manual reset. There may be a compression adjustment to adjust the torque limit. Unlike friction style or shear pin type torque limiters, ball detent torque limiter can provide an accurate method of resetting the torque with no operator intervention. A single position clutch will reengage in the exact rotational position each time. This is often necessary for system timing in bottling, packaging, and paper converting type applications.

This is how it works.

When the set limit torque is reached, the clutch disengages; the torque drops immediately
After the cause of overload has been removed, the clutch re-engages automatically after 360 angular degrees. Other cycle sequences, for example 180 degrees, are also available.
The clutch is ready for operation again

The following video clip is from mayr showing how it works to transmit and limit torque. Examples of torque limiter application are provided in the link below with calculation example as a guideline for selecting the right torque limiter model for your mechanical design project.

Source:

http://www.mayr.com/en/products/torque-limiters/eas-compact/

http://en.wikipedia.org/wiki/Torque_limiter

Keyless Bushings for power transmission

akeblogger@gmail.com (Suparerg Suksai) — Sun, 29 Apr 2012 13:48:00 +0000

There are several methods to connect shaft and hub together for power transmission. Let's find the advantage of using keyless bushings from Fenner compared with other traditional connection methods.

Traditional Connection Methods

Interference Fits (Shrink and Press)
A shrink fit is a procedure whereby heat is used to facilitate a mechanical interference fit between two pieces of metal, such as a steel shaft and hub. Extreme heat is applied to the hub, causing it to expand and increasing the size of its machined bore. The expanded hub is removed from the heat source and quickly positioned onto the shaft. As the hub cools, its bore contracts back to its original machined dimension, effectively “shrinking” the hub onto the shaft.

A press fit achieves the same end as a shrink fit — a mechanical interference fit between a steel shaft and hub — but does so through different means. Press fits rely on the application of simple brute force to “press” the hub onto the shaft. Interference fits offer several advantages, such as zero backlash and uniform fit pressures, but these advantages come at a price. High capacity interference fits require long fit lengths, close tolerances, expensive and sometimes hazardous heat sources or hydraulic presses, and field maintenance is extremely difficult. Finally, separated components can rarely be re-used.

Keys, Keyways and Splines
The centuries-old industry standard shaft-to-hub mounting technique is the key and keyway. While ubiquitous and intuitively easy to understand, the key and keyway is a remarkably ineffective technology. Machining a keyway into a shaft is not inexpensive, nor is the equipment required to do so, though these costs are often unknown or overlooked. Keyways introduce notch factors, which account for the reduced effective cross section and abridged fatigue life that occurs when a shaft is keyed and lead, in turn, to systematic over-sizing of shaft diameters. This translates to more shaft material and weight, larger bearings and other drive components, and increased cost.

The combined effect of these clearances is backlash. In applications with frequent starts/stops, direction changes, and/or shock overloads, this backlash can lead to pounded out keyways, fatigue failures, fretting corrosion or some combination of these failure modes. Nor do keys and keyways lend themselves to motion control applications, since backlash erodes the accuracy of motion profiles over time.

A splined connection is simply a series of keys and keyways that suffers the same limitations and drawbacks associated with a single keyed connection. Manufacturing costs are high, especially on hollow shafts, and special surface treatment is often required to increase strength.

Why Go Keyless

Today’s global marketplace demands precise, efficient machines that optimize productivity while minimizing material and fabrication costs. When compared to traditional connection methods, Fenner Drives Keyless Bushings offer the following advantages:

A mechanical interference fit with a uniform pressure distribution similar to that achieved through a shrink or press fit.
A true zero backlash shaft-to-hub connection with none of the operational drawbacks of keyways or splines.
The ability to mount on plain shafting, which need not be over-sized to compensate for notch factors. This allows the use  of smaller shafts and bearings for more cost effective designs.
The flexibility to mount over existing keyways if desired.
Straight bore machining of the mounted component, generous machining tolerances and as-turned surface finishes.
Complete axial and radial adjustability.
Simple installation, adjustment and removal, even in the field.

Principles of Operation
Though offered in many shapes and sizes, Fenner Drives Keyless Bushings and Specialty Locking Devices all operate using the simple wedge principle. An axial force is applied — by either a hex nut or a series of annular screws — to engage circular steel rings with mating tapers. In the case of keyless bushings, the resulting wedge action creates a radial force on the tapered rings, one of which contracts to squeeze the shaft while the other expands and presses into the component bore.

In the case of specialty locking devices, similar tapered geometry generates a radial force that is concentrated (in the case of Shrink Discs) around a solid steel hub, squeezing so tightly that the hub “shrinks” onto the underlying shaft, or (in the case of WK Series Couplings) simultaneously onto two solid shaft ends to form a high- capacity rigid coupling.

In all cases, the product of the radial force applied to the shaft, the radius of that shaft and the coefficient of friction between the surfaces being joined equals the rated torque capacity of the connection.

Source: http://www.fennerdrives.com/

Column Design (Part 6)

akeblogger@gmail.com (Suparerg Suksai) — Sat, 24 Mar 2012 12:27:00 +0000

From Column Design Part 1 to 5, we talked about the formulas to calculate the critical buckling load. This time we're going to use the excel spreadsheet program to help calculate. Let's use the following design problem as an example.

The machine designer would like to calculate the allowable load for his steel column having rectangular cross section. The column has section 80 mm x 30 mm, and 380 mm long. It's proposed to use AISI 1040 hot-rolled steel. The upper end is pinned and the lower end of the column is inserted into a close-fitting socket and is welded securely as can in the picture.

To calculate the critical load for the column we need to do as following.

Solid rectangular section, 80 mm x 30 mm.
The area moment of inertia, I = 1/12 x 80 x 30³ = 180000 mm⁴ -- The least I (for this case, around X-X axis) is used.
The cross sectional area, A = 80 x 30 = 2400 mm²
Then the radius of gyration, r = (180000/2400)^1/2 = 8.66 mm
Since the column is pinned at the upper end and welded at the lower end, then the end fixity is fixed-pinned.
The practical value of constant K = 0.8 is used for this kind of end fixity.
The effective of the column, Le = KL = 0.8 x 380 = 304 mm
The slenderness ratio = Le/rmin = 304/8.66 = 35.1

The material AISI 1040 hot-rolled steel has yield strength (Sy) and modulus of elasticity (E) as follows.
Sy = 290 MPa
E = 207 GPa

Then we can calculate the column constant, Cc = (2p² x 207x10⁹ / 290x10⁶)^1/2 = 118.7

Because the slenderness ratio (35.1) is less than the column constant (118.7), the column is short. The J.B. Johnson formula should be used.

The critical buckling load becomes,\
Pcr = (2400x10^-6) x 290x10⁶[1-(290x10⁶x35.1²)/(4p² x 207x10⁹)] = 665571 N

With the safety factor N = 3, the allowable load for this column is
P_allow = Pcr/N = 665571/3 = 221.8 kN

As usual, mechanical design handbook provides free excel program that can help you solve your column design quickly. Please download the file for FREE at the link below.

Please make sure that macro is enabled so that you can run the program. Once the program is launched, (1) select the end fixity (2) enter values of cross section, column material and safety factor (3) Click calculate to see the result.

Download FREE Excel File of Column Design.

Column Design (Part 5)

akeblogger@gmail.com (Suparerg Suksai) — Mon, 05 Dec 2011 09:01:00 +0000

From [Column Design (Part 4)], the Euler formula has been introduced. But when the slenderness ratio KL/r is less than the transition value Cc, then column is short, and the J.B. Johnson formula should be used. If we use Euler formula for the short column, it would predict too high critical load than it really is.

The J.B. Johnson formula is as follows:

From the J.B. Johnson formula we can see that the critical load for the short column is affected by the strength of the material (Sy) in addition to its stiffness (E). But for the long column as Euler formula is used, the strength of material (Sy) is not a factor for the critical load.

Let's see how we can make the excel file to help calculate critical load for both short and long columns in the next post.

Column Design (Part 4)

akeblogger@gmail.com (Suparerg Suksai) — Sat, 22 Oct 2011 04:09:00 +0000

From Column Design (Part 3), we compute the value of column constant (Cc) and slenderness ratio (KL/r_min) to check whether the column is long or short.

If the column is long, Euler formula will be used for calculation. The Euler formula is defined as,

Another form of this equation can be calculated form substituting r² = I/A into the above equation. Then we get,

Notice that the buckling load is dependent only on the length (L), cross section (I) and the stiffness of material (E) of the column. The strength of the material is not involved at all. Therefore, in a long column application, there is no benefit to use a high-strength material. A low-strength material having the same modulus of elasticity (E) would perform well.

See the J.B. Johnson formula in the next post.

Reference:

Column Design (Part 3)

akeblogger@gmail.com (Suparerg Suksai) — Thu, 18 Aug 2011 16:07:00 +0000

Let's continue from the previous post ...
The criteria to select whether we should use the Euler formula or the J. B. Johnson formula to calculate for the critical load (Pcr) of the column is related to the value of the actual slenderness ratio or column constant, Cc. It is defined as

where:

E = Modulus of elasticity of the material of the column

Sy = Yield strength of the column material

The use of the above column constant (Cc) is as follows,

Determine the length and end fixity of the column
Define the value of the constant (K) according to the type of end fixity
Compute the effective length (Le) from Le = KL
From the cross section shape and dimensions, compute the radius of gyration (r) from r = sqrt(I/A)
Compute the slenderness ratio from Slenderness ratio = Le/r_min = KL/r_min
From the material of the column, compute the column constant (Cc) as per the above formula
Check whether KL/r > Cc?

If yes, the column is Long: Use the Euler formula
If no, the column is Short: Use the J. B. Johnson formula

In the next post, let's check the formulas of critical load (Pcr) of Euler and J. B. Johnson.

Column Design (Part 2)

akeblogger@gmail.com (Suparerg Suksai) — Wed, 17 Aug 2011 14:16:00 +0000

From Column Design (Part 1), we know that a column will tend to buckle about the axis for which the radius of gyration (r) and the moment of inertia (I) are minimum. Another important parameter for column design is the effective length (Le) of the column. The effective length is defined as

Le = KL

where:
L = Actual length of the column between its supports
K = Constant value dependent on the end fixity of the column as following.

A pinned-end column is guided so that the end of the column cannot sway from side to side, but it can rotate with no resistance at the end.
A fixed-end column is held against rotation at the support.

The higher constant value of K as shown as the "practical values" in the above table is recommended because in reality it is particularly difficult to achieve a true fixed-end column because of lacking of rigidity of the support.

The slenderness ratio is the ratio of the effective length of the column to the least radius of gyration.

Slenderness ratio = Le/r_min = KL/r_min

The slenderness ratio will be used to select the method of performing the analysis of straight, centrally loaded columns. Two methods will be presented in the next post.

The Euler formula for long, slender columns
The J. B. Johnson formula for short columns

Source:

Machine Elements in Mechanical Design (4th Edition)

Column Design (Part 1)

akeblogger@gmail.com (Suparerg Suksai) — Sun, 24 Apr 2011 15:43:00 +0000

A column in the definition of mechanical engineering does not have to be in vertical. The column is a structural member that carries an axial compressive load, and that tends to fail by elastic instability or buckling rather than by crushing the material. Buckling or elastic instability is the the failure condition in which the shape of the column is not sufficient enough to hold it straight under axial compressive load. At the point of buckling, a radical deflection of the axis of the column occurs suddenly. Then if the load is not reduced, the column will collapse. It's obviously that this kind of failure must be avoided in our machine elements design.

Columns that tends to buckle are ideally straight and relatively long and slender. If a compression member is so short, the normal failure analysis must be used rather than the method that we're going to discuss in this post.

How will we know when a member is long and slender?
The tendency for a column to buckle is dependent on the shape and the dimensions of its cross section and how it is supported.
If we take a look at the cross section of the column, the followings are important properties for buckling.

The cross sectional area, A.
The moment of inertia of the cross section, I, with respect to the axis about which the value of I is minimum.
The least value of the radius of gyration of the cross section, r.

The radius of gyration is computed from

r = sqrt(I/A)

A column tends to buckle about the axis for which the radius of gyration and the moment of inertia are minimum.

From the above picture (thin plate h x t) , we can calculate the value of radius of gyration about x-x axis and y-y axis as shown above. From calculation, we can see that r_y-y is less than r_x-x because t < h. Therefore, the expected axis of buckling is y-y. We can imagine that we press a common ruler with an axial load of sufficient magnitude to cause buckling, and we can easily imagine how it will bend. The formula of radius of gyration is the tool to predict this phenomenon.

Let's explore more in the next post and make excel sheet to calculate.

Chain Drives Design (Part 3)

akeblogger@gmail.com (Suparerg Suksai) — Tue, 29 Mar 2011 14:33:00 +0000

Let's take a look at the formulas related to chain design.
The pitch diameter of a sprocket with N teeth for a chain with a pitch of p is determined by

Note: the angle of sine function must be degree (not radian)

The center distance, C, is the distance between the center of the driver and the driven sprockets. It's the distance between the two shafts coupled by the chain drive. In typical applications, the center distance should be in the following range:

The chain length, L, is the total length of the chain. Because the chain is comprised of interconnected links, the chain length must be an integral multiple of the pitch.

"It's preferable to have and odd number of teeth on the driving sprocket and an even number of pitches (links) in the chain to avoid a special link"

The chain length is expressed in number of links, or pitches (not in mm or inches!), can be computed as

The center distance for a given chain length can be computed as

Please note that the computed center distance assumes no sag in either the tight or the slack side of the chain, and thus it is the maximum center distance. Negative adjustment and adjustment for wear must be provided.

The angle of contact, θ, is a measure of the angular engagement of the chain on each sprocket. The arc of contact θ₁ is for the chain on the smaller sprocket and it should be greater than 120^o. It can be computed as

And the angle of contact of the chain on the larger sprocket, θ₂, can be computed using

Source:

Machine Elements in Mechanical Design, Fourth edition, Robert L. Mott

Chain Drives Design (Part 2)

akeblogger@gmail.com (Suparerg Suksai) — Tue, 15 Mar 2011 15:40:00 +0000

GENERAL CHAIN DESIGN CONSIDERATIONS

Nominal Tensile Load
The main consideration for all types of chain is the nominal tensile load that is required to perform the basic function. The nominal tensile load generally fluctuates in a regular cycle. For example, the chain tension from the nominal load in a chain drive increases as the chain moves around the driven sprocket. The tension remains basically constant at a high level as the chain runs through the tight strand. Tension then decreases as the chain moves around the driver sprocket. It then remains basically constant at a low level as it runs through the slack strand. This cycle then repeats again and again.

Shock Load
Shock loads are caused by the characteristics of the power source and the driven machinery. They occur repeatedly in a regular cycle, usually one or more times in each shaft revolution. They usually must be added to the nominal tensile load. Service factors are used to account for commonly known shock loads in most chain drives and conveyors.

Inertia Load
As the term is used here, inertia loads are different from shock loads. Inertia loads are the occasional loads imposed on the chain by unusual, and often unexpected, events. They may come from starting a heavily loaded conveyor or a drive with a large flywheel. Or they may be caused by a sudden momentary jam in the driven machine or conveyor. The drive or conveyor designer should calculate expected starting loads and be sure that they are never more than the yield strength of the chain.

Centrifugal Tension
In high-speed drives, centrifugal force is generated as the chain travels around the sprockets. Centrifugal force also may be generated by the chain’s travel over a curved path between sprockets. The tensile load from centrifugal force may have to be added to the nominal tension when appropriate.

Catenary Tension
The weight of that portion of the chain that hangs in a catenary generates additional tensile loads in the chain. The tensile load from the catenary tension must also be added to the nominal tension when appropriate. Catenary tension is usually a minor consideration in drives, but it may be a major consideration in conveyors.

Chordal Action
As the chain wraps a sprocket, it effectively forms a regular polygon. That causes the chain strand to rise and fall each time a joint engages a sprocket tooth. This motion is called chordal action. Chordal action also causes the chain speed to increase and decrease each time a joint engages a sprocket tooth.

Vibration
Chain vibration can cause very large increases in chain tensile loading if the vibration occurs at or
near the natural frequency of the chain. The added tension from vibration can sometimes be as
large as the nominal tensile load.

Source: Standard Handbook of Chains: Chains for Power Transmission and Material Handling, Second Edition (Dekker Mechanical Engineering)

Chain Drives Design (Part 1)

akeblogger@gmail.com (Suparerg Suksai) — Wed, 23 Feb 2011 15:38:00 +0000

Chain drives are used to transmit rotational motion and torque from one shaft to another, smoothly, quietly and inexpensively. Chain drives provide the flexibility of a belt drive with the positive engagement like a gear drive. Therefore, the chain drives are suitable for applications with large distances between shafts, slow speed and high torque.

Usually, chain is an economical part of power transmission machines for low speeds and large loads. However, it is also possible to use chain in high-speed conditions like automobile engine camshaft drives. This is accomplished by devising a method of operation and lubrication.

Compare to other forms of power transmission, chain drives have the following advantages:

Chain drives have flexible shaft center distance, whereas gear drives are restricted. The greater the shaft center distance, the more practical the use of chain and belt, rather than gears. Chain can accommodate long shaft-center distances (less than 4 m), and is more versatile.
Chain drives require little adjustment, whereas belts require frequent adjustment.
Chain drives are less expensive than gear drives.
Chain drives have no slippage, as with belts, and provide a more efficient power transmission.
Chain drives are more effective at lower speeds than belts.
Chain drives have lower loads on the shaft bearings because initial tension is not required as with belts.
Chain drives have longer life service life and do not deteriorate with factors such as heat, oil, or age, as do belts. Sprockets are subject to less wear than gears because sprockets distribute the loading over their many teeth.
The sprocket diameter for a chain system may be smaller than a belt pulley, while transmitting the same torque.

Points of Notice:

Chain has a speed variation, called chordal action, which is caused by the polygonal effect of the sprockets.
Chain needs lubrication.
Chain wears and elongates.
Chain is weak when subjected to loads from the side. It needs proper alignment.

Standardization of chains under the American National Standards Institute (ANSI), the International Standardization Organization (ISO), and the Japanese Industrial Standards (JIS) allow ease of selection.

The followings are some example of chains from British & ISO standard.

BRITISH STANDARD ROLLER CHAINS
BS 228, ISO R606, DIN 8187

ISO Chain no.
The first two digits are related to the chain pitch. See below table.

Source:

http://www.fptgroup.com/Chain/
Machines & Mechanisms Applied Kinematic Analysis, 3rd edition, David H. Myszka
http://chain-guide.com *** Recommended web site for chain drives.
http://tsubakimoto.com/product/drive-chains/lube-free/lube-free/class3/10/2/2/

Solving System of Equations using Gauss Elimination Method (Part 6)

akeblogger@gmail.com (Suparerg Suksai) — Mon, 21 Feb 2011 16:54:00 +0000

In this post, you'll find the video clip to show how to use the excel program to solve system of equations using Gauss Elimination method and download link.

You can find the links related to this series of post below:

Free download excel file of equations solver with Gauss elimination method

Links for 2011-02-20 [del.icio.us]

Mon, 21 Feb 2011 00:00:00 PST

Solving System of Equations using Gauss Elimination Method with VBA Excel Program
Free download excel program to solve system of linear equations using Gauss Elimination Method.
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Solving System of Equations using Gauss Elimination Method (Part 5)

akeblogger@gmail.com (Suparerg Suksai) — Mon, 21 Feb 2011 04:39:00 +0000

Let's continue from [Solving System of Equations using Gauss Elimination Method (Part 4)]. Now you know how to enter data and solve a set of linear equations using our program. Now it's time to see how to setup another set of equations. Let's try to solve system of equations with 10 unknowns.

From the following screen, click "Main Menu".

Program will move to main screen with pop-up windows. Enter number of equations to be solved, for this example, enter 10 and click OK.

Warning screen will appear as follows. Please note that the program allows to keep only 1 set of equations at a time. Existing equations will be deleted we you set new equations. If you wish to solve new set of equations, click "Yes".

Program will move back to the calculation screen. You'll find matrix [A] with 10x10 dimensions, vector {B} with 10x1 dimensions and vector {X} with 10x1 dimensions. The program erased all existing data and create new table automatically.

Try enter the following values.

And click "Solve" to see the results.

You can recheck the result by multiplying matrix [A] with vector {X} using "=MMULT($B3:$K3,$L$3:$L$12)" in cell M3 of the following screen. I just copy it from excel program to another excel workbook and put those formula to confirm the computation results. You will find that the results are the same as vector {B}.

Let's watch the video clip and get a download link for this Gauss Elimination Equations Solver in the next post.

Solving System of Equations using Gauss Elimination Method (Part 4)

akeblogger@gmail.com (Suparerg Suksai) — Sun, 20 Feb 2011 15:21:00 +0000

After you download an excel program Gauss Elimination Method from our web site and open it, you'll find the following screen.

You have to click "Options..." and "Enable this content" to enable the VBA otherwise the VBA code will be blocked and can't run the program.

The program will show a form asking for your agreement confirmation.

After you've agreed, you'll find the main screen, then click Start.

You'll find the menu whether you need to review or recalculate the previous equations or you want to add new equations. Let's try review the earlier equations by clicking at "Existing equations" button.

The program will move to another screen where you can change values in matrix [A] or vector {B} and recalculation to solve the equations with the same number of equations. For this example, the dimension of matrix [A] is 3x3. So if you want to solve another set of linear equations with 3 unknowns then you can modify this existing table or you can either create new table which we will explain later.

In this screen, you will be able to change values only in matrix [A] and vector {B}. Other cells are locked. The program will show the values of solution vector {X} as soon as the calculation is complete.

Now, let's test the case of "division by zero" that we explained in [Solving System of Equations using Gauss Elimination Method (Part 3)] by swapping equation (1) and (3) as follows then click "Solve" button. The screen will show the same result in solution vector {X}.

Let's try to change equations and click "Solve". If there's nothing wrong with equations, you'll find solutions.

In case something wrong with equations, the program will shows the error message. The following equations have no solution and program terminated automatically.

Let's see how to setup the program to solve more equations such as 10 unknowns in the next post.

Solving System of Equations using Gauss Elimination Method (Part 3)

akeblogger@gmail.com (Suparerg Suksai) — Sun, 20 Feb 2011 11:53:00 +0000

From post [Solving System of Equations using Gauss Elimination Method (Part 2)], we know the procedures of Gauss Elimination method to solve system of linear equations. But if we write excel VBA program based on those procedures, we may encounter with problems and cannot get the calculation result. You can find more details regarding problems and how to correct them from [Numerical methods in engineering]. We will explain one problem that may occur. It's division by zero.

For example, we want to solve a set of simultaneous equations as follows using Gauss Elimination method.

The first step of Gauss Elimination method is to divide equation (1) with a coefficient of x₁ which is 5. This is fine to do that. But imagine if the equation (3) and (1) are swapped as follows, we will encounter with "division by zero" problem.

Actually we will have problem if the value in diagonal term of matrix [A] is zero or has very small value. So we can avoid this problem by swapping equations to get the highest absolute value of coefficient of x in diagonal term.

In the next post, let's have a look the excel program to solve system of equations using Gauss Elimination method.

Solving System of Equations using Gauss Elimination Method (Part 2)

akeblogger@gmail.com (Suparerg Suksai) — Sat, 19 Feb 2011 15:17:00 +0000

The the previous post [Solving System of Equations using Gauss Elimination Method (Part 1)], the basic information regarding Gauss Elimination Method has been shared. In this post, we will have at more details about Gauss Elimination method.

Why called "elimination"?

The general form of system of equations is like this.

The Gauss Elimination method starts from forward elimination by dividing equation (1) with coefficient of x₁. Equation (1) now becomes:

Multiply equation (1) with coefficient of x₁ from equation (2) then we get:

Then we subtract equation (2) with equation (1). Equation (2) becomes:

Or we can write it as

Repeat the same procedures for the remaining equations and we get the system of equations as follows:

We can see that the first term in equation (2) to (n) are eliminated for this round of calculation. For the next round, we will repeat the same procedure, only we change the coefficient to equation (2) i.e. dividing equation (2) with a'₂₂ and multiplying it with a'₃₂ ... As soon as we do the forward elimination until round (n-1) we will get the system of linear equations that is ready for backward substitution as follows.

Where: the superscript ', ", ... ^(n-1) means number of round of forward elimination
We can find from the above system of equations that, first, we can compute the value of x_n from equation (n) from:

Then we can calculate for x_n-1, x_n-2, ..., x₂, x₁ by backward substitution using the following equation.

The excel program we developed to solve system of equations using Gauss Elimination method is based on the above equations with some improvements. We will discuss later in the next post about the limitation of Gauss Elimination method with improvement method.

Further reading:

Solving System of Equations using Gauss Elimination Method (Part 1)

akeblogger@gmail.com (Suparerg Suksai) — Fri, 18 Feb 2011 19:21:00 +0000

Sometimes engineers need to solve a system of equations to obtain solutions for their engineering design works. This kind of problems consists of several number of unknowns and equations that cannot be solved easily without good knowledge of system of equations solving method and understanding of computer programming. If we need to solve a set of linear equations with 2 - 3 unknowns, it will be easily to solve by hand calculation. There are several methods available for solving them. But when we need to solve much more unknowns e.g. 20 unknowns or 1,000 unknowns, computer software is unavoidable.

System of equations with "n" linear equations can be expressed as follows:

We can write it the matrix form as [A]{X} = {B}

The dimension of matrix [A] is (nxn), vector {X} is (nx1) and vector {B} is (nx1).
There are several methods to solve a set of simultaneous equations e.g. Cramer's rule, Gauss elimination or Gaussian Elimination method, Gauss-Jordon method, matrix inversion method, LU decomposition method, Cholesky decomposition method, etc.

Cramer's rule can easily solve a set of equations with small number of equation. The Cramer's rule uses determinant to solve for the unknowns. The unknowns can be solved from:

However, the number of calculations using Cramer's rule is approx. (n-1)(n+1)!, where n is number of equations to be solved. Therefore, if we want to solve a set of 10 linear equations, the number of calculations will be approx 360,000,000 times. But if we use Gauss Elimination method for 100 linear equations, the number of calculations is only about 700,000. Thus, practically, Gauss Elimination Method is the most favorite method for general use.

Gauss Elimination Method or Gaussian Elimination Method is widely used for solving most of engineering problems. It can be used to program in the computer easily. Gauss Elimination method can be categorized as follows:

Forward elimination
Back substitution

If you're interested in more details, please check the link of article sources below.

Mechanical Design Handbook has developed a VBA excel program to solve a set of simultaneous equations using original FORTRAN code from "Numerical Methods in Engineering by Dr. Pramote Dechaumphai". The program is written in VBA excel with simple user interface. It can solve up to 10 numbers of equations. Mechanical Design Handbook is going to share more details about how to use the program and download link in later posts [Solving System of Equations using Gauss Elimination Method (Part 2)].

Here are some pictures of the program that we will discuss in more details later.

Main screen

User interface

Input and Solution Screen

Source:

Peaucellier–Lipkin and Sarrus Straight-line Mechanism

akeblogger@gmail.com (Suparerg Suksai) — Mon, 07 Feb 2011 17:50:00 +0000

The Peaucellier–Lipkin linkage (or Peaucellier–Lipkin cell), invented in 1864, was the first planar linkage capable of transforming rotary motion into perfect straight-line motion, and vice versa. It is named after Charles-Nicolas Peaucellier, a French army officer, and Yom Tov Lipman Lipkin, a Lithuanian Jew and son of the famed Rabbi Israel Salanter.

Until this invention, no planar method existed of producing straight motion without reference guideways, making the linkage especially important as a machine component and for manufacturing. In particular, a piston head needs to keep a good seal with the shaft in order to retain the driving (or driven) medium. The Peaucellier linkage was important in the development of the steam engine.

The mathematics of the Peaucellier–Lipkin linkage is directly related to the inversion of a circle.

There is an earlier straight-line mechanism, whose history is not well known, called "Sarrus linkage". This linkage predates the Peaucellier–Lipkin linkage by 11 years and consists of a series of hinged rectangular plates, two of which remain parallel but can be moved normally to each other. Sarrus' linkage is of a three-dimensional class sometimes known as a space crank, unlike the Peaucellier–Lipkin linkage which is a planar mechanism.

The Sarrus linkage, invented in 1853 by Pierre Frédéric Sarrus, is a mechanical linkage to convert a limited circular motion to a linear motion without reference guideways. The linkage uses two perpendicular hinged rectangular plates positioned parallel over each other. The Sarrus linkage is of a three-dimensional class sometimes known as a space crank, unlike the Peaucellier–Lipkin linkage which is a planar mechanism.

Source:

More information about Peaucellier–Lipkin linkage

Hoekens Straight-line Mechanism

akeblogger@gmail.com (Suparerg Suksai) — Sun, 06 Feb 2011 16:52:00 +0000

The Hoekens linkage is a four-bar mechanism that converts rotational motion to approximate straight-line motion. The Hoekens linkage is a cognate linkage of the Chebyshev linkage.

"DESIGN OF MACHINERY" by Robert L. Norton shows the link ratios that give the smallest possible structural error in either position or velocity over values of Δβ from 20° to 180°.

The followings are some interesting examples of Hoekens straight-line mechanism from youtube.

Walking robot

Marble machine

Source:

Chebyschev Straight-line Mechanism

akeblogger@gmail.com (Suparerg Suksai) — Sun, 19 Dec 2010 11:51:00 +0000

The Chebyschev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion.

It was invented by the 19th century mathematician Pafnuty Chebyschev who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight line motion. This was also studied by James Watt in his improvements to the steam engine. (Read more info about Watt Straight-line Mechanism)

The straight-line linkage of Chebyschev confines the point P — the midpoint on the link AB — on a straight line at the two extremes and at the center of travel. Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are

O₂O₄ : O₂A : AB = 200 : 250 : 100 = 4 : 5 : 2

Point P is in the middle of AB. This relationship assures that the link AB lies vertically when it is at the extremes of its travel.

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

Watt Straight-Line Mechanism

akeblogger@gmail.com (Suparerg Suksai) — Sat, 04 Dec 2010 15:33:00 +0000

Watt's linkage (also known as the parallel linkage) is a type of mechanical linkage invented by James Watt to constrain the movement of a steam engine piston in a straight line. The idea of its genesis using links is contained in a letter he wrote to Matthew Boulton in June 1784.

"I have got a glimpse of a method of causing a piston rod to move up and down perpendicularly by only fixing it to a piece of iron upon the beam, without chains or perpendicular guides [...] and one of the most ingenious simple pieces of mechanics I have invented."

This linkage does not generate a true straight line motion, and indeed Watt did not claim it did so.

Watt's straight-line mechanism is used in the rear axle of some car suspensions. It intends to prevent relative sideways motion between the axle and body of the car. Watt’s linkage approximates a vertical straight line motion more closely, and does so while locating the center of the axle rather than toward one side of the vehicle.

It consists of two horizontal rods of equal length mounted at each side of the chassis. In between these two rods, a short vertical bar is connected. The center of this short vertical rod – the point which is constrained in a straight line motion - is mounted to the center of the axle. All pivoting points are free to rotate in a vertical plane.

Here is the video of Watt straight-line mechanism in Solid Edge ST 2D model.

Source:

http://en.wikipedia.org/wiki/Watt's_linkage

Roberts straight-line mechanism

akeblogger@gmail.com (Suparerg Suksai) — Sun, 28 Nov 2010 15:27:00 +0000

Many engineering applications require things move in a linear fashion or "straight-line motion". We can use a linear motion guide that can guide a device accurately along a straight line. Manufacturing know-how of most linear guide manufacturers has let us keep expanding the range of linear guidance. The picture shown here is an example of commercially available linear guides from THK. This Linear Ball Slide is a lightweight, compact, limited stroke linear guide unit that operates with very low sliding resistance. It excels in high-speed responsive performance due to its very small frictional factor and low inertia.

In the late seventeenth century, before the development of the milling machine, it was extremely difficult to machine straight, flat surfaces. For this reason, good prismatic pairs without backlash were not easy to make. During that era, much thought was given to the problem of attaining a straight-line motion as a part of the coupler curve of a linkage having only revolute connection. Probably the best-known result of this search is the straight line mechanism development by Watt for guiding the piston of early steam engines. Although it does not generate an exact straight line, a good approximation is achieved over a considerable distance of travel.

In this post, we show approximated straight-line mechanism discovered by Richard Roberts (1789-1864). He discovered the Roberts' Straight-line mechanism.

Lengths:
O₂A = 100
O₄B = 100
AB = 100
AC = 100
BC = 100
O₂O₄ = 200

We can find from the following video clip that point C moves as an approximated straight line. Though it is not an exact straight-line motion, but it's good as a starting point. In later post, we will explore more straight-line mechanisms that can give better straight-line approximation.

Source:

3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 3

akeblogger@gmail.com (Suparerg Suksai) — Sun, 14 Nov 2010 07:09:00 +0000

In [3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 2], we've shown an example to do three-position synthesis of a four-bar linkage using inversion method in Unigraphics NX4 sketch.

We can make a quick motion simulation using "animate dimension" command in Unigraphics (UG) NX4 sketch. Just draw lines as per a sketch and add one driving dimension as shown below. Then use animate dimension command to set the lower and upper limits, for this case they're minimum and maximum angles.

3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 2

akeblogger@gmail.com (Suparerg Suksai) — Sat, 13 Nov 2010 05:38:00 +0000

In previous posts, the fixed pivot points were determined from the moving pivot points. We can get result that can't be fitted in our design due to space limit. The principle of inversion can be applied to solve this problem. The first step is to find the three positions of the ground plan that correspond to the three desired coupler positions.

We start with our desired positions of fixed pivot points.

1) Draw desired fixed pivots (O₂ and O₄) and moving pivot points. Red lines are three desired positions of links (moving pivots).

2) Draw lines to make fixed relations between the ground plane (O₂O₄) and the second coupler position.

3) Transfer the ground position to the first coupler position using same relations developed in previous step as shown in dashed lines. Name new ground positions as O'₂ and O'₄ respectively.

4) Draw lines to make fixed relations between the ground plane (O₂O₄) and the third coupler position (A₃B₃).

5) Transfer the ground position to the first coupler position using same relations developed in previous step as shown in dashed lines. Name new ground positions as O''₂ and O''₄ respectively.

6) Draw lines O₂O'₂ and O'₂O''₂, bisect both lines and extend the perpendicular bisectors until they intersect. Label the intersection G.

6) Draw lines O₄O'₄ and O'₄O''₄, bisect both lines and extend the perpendicular bisectors until they intersect. Label the intersection H.

7) Draw O₂G, GH and O₄H. Now we get G and H as inverted fixed pivot points of moving link O₂O₄.

8) Re-invert the linkage to return to the original arrangement.

Let's see how it moves in [3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 3]

Further reading:

3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Introduction

akeblogger@gmail.com (Suparerg Suksai) — Sun, 31 Oct 2010 16:04:00 +0000

Example [3-Position Motion Generation Synthesis with Alternate Moving Pivots using Unigraphics NX4 Sketch] shows how to synthesize four-bar linkages according to required moving pivots. By doing this, we first define desired locations of moving pivots then get positions of fixed pivots O₂ and O₄. However, sometimes it may be more useful to define the location of fixed pivots O₂ and O₄first, then find other linkages that can move according to 3 desired positions of moving pivots.

An Inversion method of four-bar linkage will be introduced in [3-Position Synthesis with Inversion Method using Unigraphics NX4 Sketch - Part 2]

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Wed, 28 Jul 2010 00:00:00 PDT

Introduction of Polynomial Cam Function
Normally the motion from cam cannot be defined by only a single mathematical expression, it consists of different equation on each different segment. Any discontinuity in the acceleration function will lead to infinite spikes (derivative of acceleration). The derivative of acceleration is called "jerk". And infinite jerk is unacceptable because it will lead to high vibration of the mechanism.

Links for 2009-10-21 [del.icio.us]

Thu, 22 Oct 2009 00:00:00 PDT

Standards of limits and fits using Excel Formula
Read how to use excel formulas to calculate lower deviation and upper deviation of the shaft and holes e.g. 25h7, 40E9, etc. Excel download is available for FREE at http://mechanical-design-handbook.blogspot.com/

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Mon, 02 Mar 2009 00:00:00 PST

BlackBerry Storm smartphone
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