Saturday, April 25, 2009

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

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.

Monday, April 20, 2009

Numerical Methods - First derivative using Excel formula

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.

Sunday, April 19, 2009

Design trick about selection of ball joints for fine adjustment of pull rod length

Normally we can find a lot of ball joints with pull rod at every machine. It is used to transmit power from a lever to another link in cam or even pneumatic driven mechanism. It's the part of the system that we can adjust its length due to imperfection in manufacturing of machine parts. For example, we may require high accuracy of mechanism at one position but we cannot achieve it with the fixed length of rod, therefore the use of adjusting rod like this is required.

This adjusting rod consists of right-handed ball joint at one side and left-handed ball joint at another side. This is because we need to adjust the distance between center of both ball joints by turning the pull rod then the screw threads will increase or decrease the distance depending on the turning direction of the pull rod. Of course, if we use right-handed ball joints at both sides, there will be no length changes from turning of the pull rod, but sliding of the pull rod to one side!

Normally, the coarse threaded ball joints will be used in most application because it's normal standard thread. But we then have the limitation of accuracy by this pitch of the thread.

Let's see the example. From the above picture, we use two M8 ball joint (LH & RH threads). The standard pitch of M8 screw thread is 1.25 mm. That means, by turning 1 revolution of the pull rod, the center distance between both ball joints (or the length of the adjusting rod) changes by 2 x 1.25 = 2.5 mm

How can we get more accuracy in each turn? Shall we change to smaller screw thread to reduce the pitch e.g. change to use M6 which has pitch of 1 mm?? Or shall we use fine thread??

We can do anything, but the trick in this post is to put additional M10 coarse thread in between those 2 ball joints and separate the pull rod into 2 pieces.

The pitch of M10 coarse thread is 1.5 mm. Now we can see that we turn "pull rod 1" by 1 revolution, it will pull ball joint M8 (RH) inside by 1.25 mm (= pitch of M8 coarse thread). At the same time, it will push "pull rod 2" to the same direction by 1.5 mm (= pitch of M10 coarse thread). Therefore, totally, the change in length of the adjusting rod is 1.5 - 1.25 = 0.25 mm!!
Of course, there should be a nut to lock the final position (not present in the picture)

With this simple technique, we can see that we can now get higher accuracy of length adjustment. Ten turns of this new rod change the length of adjusting rod the same as 1 turn of normal adjusting rod.

I hope this can be useful for you :)

Sunday, April 5, 2009

Tool Steels

As the designation implies, tool steels serve primarily for making tools used in manufacturing and in the trades for the working and forming of metals, wood, plastics, and other industrial materials. Tools must withstand high specific loads, often concentrated at exposed areas, may have to operate at elevated or rapidly changing temperatures and in continual contact with abrasive types of work materials, and are often subjected to shocks, or may have to perform under other varieties of adverse conditions. Nevertheless, when employed under circumstances that are regarded as normal operating conditions, the tool should not suffer major damage, untimely wear resulting in the dulling of the edges, or be susceptible to detrimental metallurgical changes.

Tools for less demanding uses, such as ordinary handtools, including hammers, chisels, files, mining bits, etc., are often made of standard AISI steels that are not considered as belonging to any of the tool steel categories.

The steel for most types of tools must be used in a heat-treated state, generally hardened and tempered, to provide the properties needed for the particular application. The adaptability to heat treatment with a minimurn of harmful effects, which dependably results in the intended beneficial changes in material properties, is still another requirement that tool steels must satisfy.

To meet such varied requirements, steel types of different chemical composition, often produced by special metallurgical processes, have been developed. Due to the large number of tool steel types produced by the steel mills, which generally are made available with proprietary designations, it is rather difficult for the user to select those types that are most suitable for any specific application, unless the recommendations of a particular steel producer or producers are obtained.

The Properties of Tool Steels.—Tool steels must possess certain properties to a higher than ordinary degree to make them adaptable for uses that require the ability to sustain heavy loads and perform dependably even under adverse conditions.

Tool and die design tips to reduce breakage in heat treatment

The extent and the types of loads, the characteristics of the operating conditions, and the expected performance with regard to both the duration and the level of consistency are the
principal considerations, in combination with the aspects of cost, that govern the selection of tool steels for specific applications.

Although it is not possible to define and apply exact parameters for measuring significant tool steel characteristics, certain properties can be determined that may greatly assist in appraising the suitability of various types of tool steels for specific uses.

Saturday, April 4, 2009

Cam Follower Selection

Needle Roller Cam Followers

Needle Roller Cam Followers have a heavy outer ring cross section and a full complement of needle rollers. They offer high dynamic and static load carrying capability, and anti-friction performance, in a compact design. They are used as track rollers, cam followers, and in a wide range of linear motion systems.

Standard Stud cam followers offer the mounting convenience of a threaded stud and are designed to accommodate moderate loads. They are available with and without seals. Standard stud cam followers are also available with crowned outer rings for applications where misalignment is a problem.

Heavy Stud cam followers are designed to provide additional stud strength for applications with
high loading or shock loads. Heavy stud cam followers are available with and without seals, and with crowned outer rings.

Yoke Type cam followers are intended primarily for applications where loading conditions
exceed the capabilities of stud type cam followers, or where clevis mounting is desired. Clevis mounting provides support on both sides of the cam follower and permits use of a high strength pin. Yoke type cam followers are available with and without seals, and with crowned
outer rings.

CamCentric® adjustable cam followers (page 18) are used where accurate positioning is required. They are particularly useful for reducing clearance or backlash in opposed arrangements, and for assuring load sharing in multiple cam follower installations. Seals and hex socket are standard features of CamCentric® adjustable cam followers. Crowned outer rings are also available.

Crowned Outer Rings are used to minimize outer ring thrusting in applications where the axis of the cam follower is not parallel to the surface of the track or is skewed relative to the direction of travel. Crowned outer rings are a good selection for use with curved or circular tracks. In well aligned applications, crowned outer rings can cause accelerated track wear.

Cam Followers, Yoke, and Track Rollers

1. Difference from Standard Bearings

The outer rings of regular ball and roller bearings are typically mounted in rigid housings providing support around the entire circumference. Individual roller forces are transmitted through the outer ring directly into the housing with no major deformations.

By contrast, cam followers and yoke rollers are supported at a single point on their circumference. Individual roller forces produce bending moments on the outer ring around the point of contact. The effects are outer ring deformation with reversed bending stresses in dynamic applications, a reduced load zone, and a higher maximum roller load (see Fig. 1.).

To keep deformation to a minimum, the outer ring of a cam follower must have a considerably heavier cross section than a standard bearing. This requirement conflicts with the desire for maximum dynamic bearing capacity which needs as large a roller diameter as possible. RBC cam followers and yoke rollers provide an optimum compromise between outer ring strength and theoretical bearing capacity.

2. Capacity and Load Limits

Evaluation of the expected service life and limit loads of cam followers is more complex than with housed bearings. In addition to the static and dynamic capacity of the rolling elements, outer ring deformation, track capacity, and cam follower stud bending stress must be considered. In yoke rollers, the pin shear stress must be considered.

RBC lists the static bearing capacity for reference purposes only. Typically, the maximum allowable load is a function of the maximum permissible bending stress of the stud or the outer ring. For best results, the operating loads should not exceed the lower of track capacity or 50% of the dynamic capacity.

2.1 Capacity of Rolling Element Bearing

Equations for static and dynamic capacities of roller bearings are given in ANSI/ABMA Standard 11. The more recent revisions leave it up to the manufacturer to introduce factors which account for internal design features and operating conditions. For cam followers and yoke rollers RBC has chosen to apply a conservative rating system, so a direct comparison with capacity figures of competitive products may not be possible.

2.2 Track Capacity

Track capacity is that load which a track subject to a uniform contact stress can withstand without excessive plastic deformation. It is directly related to track hardness. The published track capacity is based on a hardness of HRc 40. For other track hardness values the track capacity must be modified with factors from Table 1.

Alternatively, contact stress can be easily calculated and compared directly to the strength of material. The equation for the Hertz contact stress between a cylindrical cam follower outer ring and a flat steel track is given by “Roark, Formulas for Stress and Strain” as:

It can be shown that for infinite life the ultimate tensile strength of track and roller must be at least equal to the maximum contact stress σC max

2.3. Bending and Shear Stresses

2.3.1 Cam Follower Stud Bending Stress

If the load over the width of the outer ring is evenly distributed, it may be replaced by a single
concentrated force F [lbf] acting at the center of the cam follower (see Fig.2). Assuming that the cam follower stud has been tightly mounted in a housing bore flush with the end plate, this concentrated force generates a bending moment Mb.

The bending moment generates a bending stress in the cam follower stud of approximate magnitude

Standard cam follower studs are heat treated to a hardness of HRc 58 min in the raceway area only.

The hardness in other areas of the stud is typically in a range of HRc 20 -22 with an ultimate strength of material of 110,000 -120,000 psi. RBC bases the maximum allowable load of stud type cam followers on a theoretical stud bending stress of 100,000 psi. Standard stud cam followers and heavy stud cam followers differ in stud diameter, which permits higher operating loads and more resistance to impact loading for the heavy stud version. High stud strength cam followers are available by special order.

In most applications the stud will deflect away from the load, which causes the point of attack to
shift toward the support, shortening the moment arm and reducing the effective bending moment (see Fig. 3). Tests show that this deflection yields a safety factor of at least 2 over RBC’s maximum allowable load. However, this effect may not be sufficient to avoid damage in severely misaligned applications where the load is applied at the very extreme outboard edge of the cam follower outer ring.

Where misalignment is a problem, RBC recommends crowned cam followers.

2.3.2 Yoke Roller Pin Shear Stress

Yoke rollers are mounted with a pin in a yoke. Under load the pin is subject to shear and bending
stresses. RBC recommends that the yoke arms are located as close to the yoke roller as possible, so that bending stress can be ignored. In case of widely spaced pin supports, the resulting pin deflection may cause yoke roller damage. The pin shear stress can be calculated with

The permissible stress depends on the pin material selection.


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