Grashof's Criterion: Crank-Rocker Four-Bar Mechanism

Grashof's criterion for Double Crank Four-Bar Mechanism is almost the same as Crank-Rocker Mechanism. They're both defined as S+L < P+Q. The difference is on the location of the shortest link (S). For double crank mechanism, S must be on the ground link (frame). But for the crank-rocker, S will be on the side link and it must be the input link.

We then move the shortest link from the frame to the side link on LH. With the help of SAM 7.0 Professional software (by Artas Engineering), it can display the paths of desired nodes as shown in the picture.

Grashof's Criterion: Crank-Rocker Mechanism using SAM 7.0 Professional Software

The pink curves represent paths of desired nodes. The input link which is now the shortest link is able to make a full revolution. It's called a crank. And the output (can be either L, P or Q) will only oscillate around its pivoting point. It's called a rocker. But if we change the driver to node 4 and let it drives from the link which is not the shortest link, it can't be reversed. The shortest link can't make a full revolution. What we have to do is to swap as shown in the following picture and it becomes a crank-rocker mechanism again which is now driving from the RH pivoting point.

Grashof's Criterion: Crank-Rocker Mechanism Driver on RH side

How can we see the velocity? From the path display, we can see how they move. But to see the velocity, we can choose to display the hodograph.

Path is the line that a moving point describes in the fixed reference system.

Velocity Hodograph is the locus of the arrowhead of the velocity vectors (rotated 90 degree) of a moving point. 

 The hodograph in SAM 7.0 is shown as follows.

Velocity Hodograph in SAM 7.0 The Ultimate Mechanism Designer

The lines on the outside of the path represent CCW direction of the node. And the lines at the inside of the path represent CW direction of the node. From the above picture, we can see that the input link of the crank-rocker mechanism rotates at a constant speed in CCW direction since the hodograph displays uniform lines at outside of the circular path. However, the output link oscillates back and forth with changes in velocity since there are lines of the hodograph both on the outside and inside of the its curve path. And we can quickly point out the position where the mechanism has highest velocity from longest line on the hodograph. SAM 7.0 is also able to plot various design parameters of the mechanism e.g. velocity, displacement, acceleration, length, etc. In the above picture, LH graph shows the velocity profile of the output link which can be traced manually. The following shows the hodograph of the double crank mechanism for a comparison.

Hodograph of Double Crank Mechanism

The following is the video showing how to use SAM 7.0 Software to simulate the double crank and crank-rocker mechanisms.

Grashof's Criterion: Four-bar Mechanism Double Crank

One of the most commonly used linkages is the four-bar linkage. It consists of 3 moving links and 1 ground link (also called a frame). There are 4 pin joints connected between those links. And from Gruebler's Equation, it's the mechanism with 1 degree of freedom (1DOF) which requires only 1 actuator to drive and control position of all linkages. The four-bar mechanism consists of the following components.

Four-bar mechanism components

The link that connects to the driver or power source is called the input link. The other link connected to the fixed pivot is called the output link. The remaining moving link connected between input and output links is called the coupler. It couples the motion of the input link to the output link.

There are different configurations of lengths of the four-bar mechanism and it results in different movements of the mechanism. Grashof's Criterion helps classifies into the following categories:

1. Double Crank --- also called a drag link mechanism
2. Crank-Rocker
3. Double Rocker
4. Change Point
5. Triple Rocker

In this post, we're going to explore the case of Double Crank which both input and output links are able to rotate through a full revolution. We use Autodesk ForceEffect app on Google chrome to illustrate and demonstrate how the mechanism moves. The links can be setup easily by just dragging. The exact length dimension can also be specified. Once we setup all required items i.e. links, pivots and drive, we can play and see the animation of the linkages with the path of desired tracing points.

Autodesk ForceEffect app on Google chrome

Grashof's criterion states that a four-bar mechanism has at least one revolving link if:

S + L ≤ P + Q

where:
S = length of the shortest link
L = length of the longest link
P = length of one of the intermediate length links
Q = length of the other intermediate length link

For the Double Crank category, the following criteria must be satisfied:

Double Crank:

• S + L < P + Q
• S is the length of the frame (ground link)

So we setup the links in ForceEffect app as follows.

Grashof's Criteria for Double Crank 4-bar Mechanism

The shortest link (S) is the frame which is 400 mm long. The length of the longest link (L) is 1300 mm. The remaining 2 intermediate links (P and Q) have length 700 mm and 1200 mm.

This satisfies Grashof's criteria since (S)400 + (L)1300 < (P)700 + (Q)1200. And this is how this mechanism moves.

Paths of double crank four-bar mechanism

Both input and output links can make a full revolution as desired. Let's find more details of other categories in later post.

The following video shows how to use ForceEffect app to simulate the motion of double crank four-bar linkage.

Reference:

• Machines & Mechanisms 3rd edition, David H. Myszka
• Autodesk ForceEffect

Gruebler's Equation for calculating Degrees of Freedom of the Mechanism

Mechanism with 1 degree of freedom

To design the mechanism, the first thing we should check is the number of degrees of freedom (DOF) of the mechanism. The degree of freedom is the number of inputs required to control the position of all links of the mechanism. It's usually the number of actuators needed to operate the mechanism. 

We can use Gruebler's equation to calculate the number of degrees of freedom of the mechanism as follows.

Gruebler's Equation

where:

F = number of degrees of freedom

n = total number of links in the mechanism

L = total number of lower pairs (1 DOF such as pins and sliding joints)

H = total number of higher pairs (2 DOF such as cam and gear joints)

4-bar linkage

In the machine, we often require one degree of freedom which we can position all linkages with only 1 actuator. The four-bar linkage as shown in the picture is the example of the mechanism with 1 DOF. It has 4 links (3 bars with 1 ground link) and 4 revolute joints which the degree of freedom (F) can be calculated as follows.

• n = 4 --- 4 links
• L = 4 --- 4 revoulte joints
• H = 0 --- no higher pairs
• F = 3(4-1) - 2(4) - 0 = 1

Another example of 1 DOF mechanism is the slider-crank mechanism where it has the following number of links and joints.

Slider-Crank Mechanism

• n = 4 --- 2 links + 1 ground link + 1 slider
• L = 4 --- 3 pins + 1 slider
• H = 0 --- no higher pairs
• F = 3(4-1) - 2(4) - 0 = 1

The gears mechanism also has 1 DOF since it has the following values.

Gears Mechanism

• n = 3 --- 2 gears + 1 ground link
• L = 2 --- 2 revolute joints
• H = 1 --- 1 gear joint
• F = 3(3-1) - 2(2) - 1 = 1

These 3 examples have 1 degree of freedom which requires only 1 actuator to move the mechanism. It could be a motor, an air cylinder, etc.

If we change the slider joint of the slider-crank mechanism to the fixed pin joint, the mechanism will be locked since it has 0 DOF which is considered as a structure. The calculation using Gruebler's equation is as follows.

Structure: Degree of freedom = 0

• n = 3 --- 2 links + 1 ground link
• L = 3 --- 3 pin joints
• H = 0 --- no cam or gear joints
• F = 3(3-1) - 2(3) - 0 = 0  --- F=0 then it can't move.

If one of the pin joint of the 4-bar linkage changes to the slider joint, it will increase both the number of links and number of lower pairs. This makes the mechanism unconstrained because it has 2 DOF and required 2 actuators to control the position of the mechanism.

• n = 5 --- 3 links + 1 slider + 1 ground link
• L = 5 --- 4 pin joints + 1 slider joint
• H = 0 --- no higher pairs
• F = 3(5-1) - 2(5) - 0 = 2 --- F > 1, the mechanism is unconstrained.

References:

• Machines & Mechanisms 3rd Edition, David H. Myszka
• 4 Basic Kinematics of Constrained Rigid Bodies
• Linkages sketch and animation using SAM 7.0 Mechanism Design Software

Stiffness of a lever with eccentric loading - Part 2

Let's continue from the previous post. We clearly see that the small post must be shortened because it the weakest point and creates torsional load to the lever arm. We're going to try redesigning with a bent lever arm so that the loading point stays in the middle of the center line of the arm.

Lever bending to eliminate torsion

The eccentric distance "r₂" is shorter than previous design (r₁). It's now the eccentric dimension with respect to the bent portion of the lever. Torsion still exists at end portion which has shorter length. This gives much less torsion compared with the earlier design. It also eliminate the torsion from the longer part of the lever since the loading point stays at the center line of the lever.

Two lever designs at the same loading point

From the overlay, we can see that the pull rod connecting point is still at the same location. The lever arm still connect to the same hub. But we can reduce the length of the small post which is the pin for the pull rod connection. The longer portion of the lever is now under bending only. Most torsion has been eliminated. It exists at the small portion as explained.

Finite element analysis - displacement of the lever with bent end

Shortening the small post and keeping the loading point on the center line of the lever can reduce displacement from 2.4 mm to 1.6 mm (33% reduction). Small twisting is present at the end but there is no twisting over the long portion of the lever.

von Mises stress of the bent lever

The von Mises stress reduces about 77%. This improvement can be used in most cases even when the lever shaft is short and the pull rod location is beyond the lever hub location and it's stiffer than the big lever with small post. The following picture is the example of bent levers on the machine.

Example of a bent lever on the machine

Stiffness of a lever with eccentric loading - Part 1

A lever is a mechanical part that has arms and a fixed pivot used to transmit torque. The cross section of the arm is usually a rectangular shape which makes it stiff against bending. However, its stiffness may become lower if the loading point is not on the right place. We're going to see the effect of torsion on the lever with the improvement idea.

The following picture is the components of a general cam and lever system. The lever has the hub in the middle. It's where bearings are placed inside and it's the fixed pivoting point on a lever shaft (not shown). There are 2 arms to the left and the right. They're both welded to the hub and rotate together. The right arm is connected to a cam follower that rolls over cam surface. When the cam rotates the lever will swing up and down since there will be spring forces pushing the cam follower to keep contact with the cam surface all the time. The left arm has another point to connect to other machine parts, which usually is a pull rod. It will rotate at the same angle (degrees) with the right arm, but the distance (mm) may be different depending on a lever ratio.

Components of cam & lever system

The pull rod connecting point on the left in this example is not good since it has a long distance from the arm and the arm will not only be subjected to bending but also torsion. The following is the lever in the machine.

Lever with eccentric pull rod connecting point

The eccentric distance "r" usually comes from space limitation on the lever shaft. There may be some other machine parts blocking the lever and it can't move any further. The hub stays at the same location and the lever is just straight from the hub. Because of this, there will be a gap between the lever and the pull rod. That's why the designer add that small connection post at the end.

Load from the pull rod

The small post will be subjected to bending and the lever arm will also subjected to bending as well as torsion. If the distance "r₁" is much less then it could improve the stiffness. Here is the finite element analysis result of this lever.

Finite element analysis - displacement of the lever under eccentric load

We can see from the finite element analysis result that the lever bends down due to the load and it also twists to the right because of the eccentric load. However, the small post that connects to the pull rod reduces the overall stiffness since it also bends down.

von Mises stress of the lever under torsional load

The highest von Mises stress is at the weakest point on the small post as shown in the picture.

To reduce the overall displacement, the overall stiffness will improve. We can change some designs to eliminate the torsion away from the lever arm and reduce the length of the post. Let's see how it's done in the next post.

Moment of inertia

The Moment of Inertia or Mass Moment of Inertia is the measure of a body's resistance to change in it's rotational speed. The moment of inertia must be specified with respect to a chosen rotational axis.
The moment of inertia depends on the body's mass distribution and the rotational axis chosen. The larger moment of inertia requiring more torque to change the body's rotational speed.

A point mass
The moment of inertia is the mass times the radius from the rotational axis squared.

Moment of Inertia of Point Mass

A collection of point mass
The moment of inertia is just the sum of the point mass moment of inertia.

Moment of Inertia of Collection of Point Mass

A continuous mass distributions 
The moment of inertia require an infinite sum of all the point mass moment of inertia which make up the whole part. This can be calculated by an integration over the whole mass.

Moment of Inertia of Continuous Mass Distributions

A common shape
For a common shape, we usually calculate the moment of inertia about the certain point and use it to apply for another rotation axis by use the parallel axis theorem.

Moment of Inertia of Common Shape Body

Parallel Axis Theorem
The moment of inertia about any rotation axis which parallel to a certain axis at center of mass (I_parallel)  is the moment of inertia about the center of mass (I_cm) plus the product of mass times the distance between the center of mass and the rotation axis squared.

Parallel Axis Theorem

Parallel Axis Theorem can also apply to any rotation axis which parallel to  any certain axis at a point that already know the standard moment of inertia (I_xx , I_yy or I_zz).

Parallel Axis Theorem & Radius of Gyration

Radius of gyration [K]
It is the distance from a rotation axis to a certain point which the mass of a body may be assumed to be concentrated and at which the moment of inertia will be equal to the moment of inertia of the actual mass about the rotation axis.

Differential screw for fine adjustments of precision equipment

Differential screw components

A differential screw is a mechanism that provides very fine motions of machine parts. There are several forms of its configuration.

The picture shows one common form of the differential screws. There are 3 main components as follows:

• Spindle (differential screw) -- The spindle has two different threads on the same axis.
• Base -- It's the base of the whole mechanism which has one threaded hole.
• Nut -- It has one threaded hole with sliding joint. This is the end mover where we will get fine motion. This part may be connected to other machine components to provide precise motion.

How it works

Different threads on the same spindle / distance between marks

The spindle has two different thread sizes. In this example, the larger one (thread A) has M12 coarse thread which has a pitch of 1.75 mm. Another thread (B) is M10 coarse thread which has a pitch of 1.5 mm.

Pitches (leads) of the differential screw

The pitch (or lead) of a screw is distance the screw advances when it turns one revolution. Therefore, when the handle turns one revolution, thread A rotates one revolution and moves in a distance equal to the pitch of thread A (1.75 mm). Since thread B is on the same spindle, it also moves together with thread A (1.75 mm) and also rotates one revolution. However, thread B connects to the nut which is unable to rotate. So, the nut retracts a distance equal to the pitch of thread B which is 1.5 mm. Hence, the motion of the nut is the advance distance of thread A minus the retracted distance. It is the difference between the pitch of threads. This is why it is called the differential screw.

Differential screw displacement formula

where:

• ΔS_nut = travelling distance of the nut (mm)
• L_A = pitch of thread A (mm)
• L_B = pitch of thread B (mm)
• Δθ_screw = number of turns of the screw (rev)

From this example, we have ΔS_nut = 1.75 - 1.5 = 0.25 mm. That means the nut travels 0.25 mm per each turn of the spindle. As shown in the above picture, the distance between 2 marks is 2.5 mm. Then we need to turn the spindle 10 revolutions so that the nut will travel 2.5 mm.

As we can see from the formula, if we need the nut to move 0.1 mm per one turn of the spindle, we need to select the different screw threads. Since we know that the standard metric coarse threads have the following values:

• M5, pitch = 0.8 mm
• M4, pitch = 0.7 mm

The difference between pitches is 0.8 - 0.7 = 0.1 mm which is as per the requirement. So, thread A will be M5 and thread B is M4 and we will get 0.1 mm per turn.

Watch the following video to see how it moves. We use Unigraphics NX4 motion simulation to show all motions.

Reference:

• Machines & Mechanisms Third Edition by David H. Myszka

A pull rod for position adjustment of a cam-driven mechanism

Kinematic diagram of a cam-driven mechanism

A cam-driven mechanism is commonly used in most production machines since all motions and timings can be controlled. Not only the displacement is controlled, but also the velocity and acceleration as well as jerk can be controlled. Cam-driven mechanism allows overlapping motion between machine parts since the positions of all relevant parts can be determined from the timing diagram which is desirable for high speed application.

A simple cam-driven mechanism consist of the following parts as shown in the kinematic diagram:

• Cam: for motion generation (displacement, velocity, acceleration and timing).
• Cam follower: rolling part mounted on a lever.
• Spring (not shown): to keep contact between cam surface and cam follower.
• Lever: to transfer continuous cam rotation to swinging motion.
• Pull rod: to transfer the motion from the lever to the slider.
• Slider: end equipment (processing equipment)

In this post, we're going to focus on the pull rod (also known as push rod or tie rod) which is one of the common parts for most machines. The pull rod allows position adjustment of its connected parts since its length can be adjusted. Normally, the pull rod consists of the following parts:

• Pull rod
• Rod end bearing RH thread
• Rod end bearing LH thread
• Nut RH thread
• Nut LH thread

Pull rod with both female rod end bearings

The pull rod usually made of a hexagonal post. The mechanic can use a wrench on the hexagonal part to tighten or loosen the pull rod from the rod end bearings. Rod end bearings must have RH thread on one side and LH thread on the other side otherwise the distance between the rod ends will remain the same.

Example of pull rods on the machine

Adjustment of the pull rod length usually happens when both sides of the rod end bearings are already connected to other parts in the machine (in this example, it is connected to the lever and the slider already). To adjust the length, no need to disconnect the rod end bearing, first we have to loosen both RH and LH nuts so that the pull rod can be turned. Then turn the pull rod in either direction and its length will change. By doing this, we can then adjust the position of the connected parts which, in the case, is the slider. After the slider is at the desired position, tighten both nuts.

The male rod end bearings version is also available. We can use the same nuts, but the hexagonal post will have threaded holes instead (see the following picture).

Pull rod with both male rod end bearings

The increment of the pull rod length (distance between both rod end bearings) is determined by the pitch of the thread on the rod end bearings. For this example, the M10 thread has a pitch of 1.5 mm. One turn of the pull rod will change the distance of each rod end bearing by 1.5 mm. Therefore, the increment is 2 times the pitch (2 x 1.5 = 3 mm/turn).

The pull rod length is increased or decreased according to the following directions.

Pull rod length extension and retraction according to the turning direction

The following is the animated picture showing how the slider position can be adjusted by turning the pull rod.

Animated picture of pull rod length adjustment

Watch the following video for how the cam driven-mechanism works and where the pull rod is used in the system. The simulation uses NX4 motion simulation module.