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Design of four bar mechanisms

The synthesis, or design, of four bar mechanisms is important when aiming to produce a desired output motion for a specific input motion. In order to maximize cost and efficiency, a designer will choose the simplest mechanism possible to accomplish the desired motion. When selecting a mechanism type to be designed, link lengths must be determined by a process called dimensional synthesis. Dimensional synthesis involves an iterate-and-analyze methodology which in certain circumstances can be an inefficient process; however, in unique scenarios, exact and detailed procedures to design an accurate mechanism may not exist.

Time Ratio

The time ratio (Q) of a four bar mechanism is a measure of its quick return and is defined as follows:

Q = ${\tfrac {Timeofslowerstroke}{Timeofquickerstroke}}$ ≥ 1

With four bar mechanisms there are two strokes, the forward and return, which when added together create a cycle. Each stroke may be identical or have different average speeds. The time ratio numerically defines how fast the forward stroke is compared to the quicker return stroke. The total cycle time (Δt_cycle) for a mechanism is:

Δt_cycle = Time of slower stroke + Time of quicker stroke

Most four bar mechanisms are driven by a rotational actuator, or crank, that requires a specific constant speed. This required speed (ω_crank)is related to the cycle time as follows:

ω_crank = (Δt_cycle)^-1

Some mechanisms that produce reciprocating, or repeating, motion are designed to produce symmetrical motion. That is, the forward stroke of the machine moves at the same pace as the return stroke. These mechanisms, which are often referred to as in-line design, usually do work in both directions, as they exert the same force in both directions.

Examples of symmetrical motion mechanisms include:

• Windshield wipers
• Engine mechanisms or pistons
• Automobile window crank

Other applications require that the mechanism-to-be-designed has a faster average speed in one direction than the other. This category of mechanism is most desired for design when work is only required to operate in one direction. The speed at which this one stroke operates is also very important in certain machine applications. In general, the return and work-non-intensive stroke should be accomplished as fast as possible. This is so the majority of time in each cycle is allotted for the work-intensive stroke. These quick-return mechanisms are often referred to as offset.

Examples of offset mechanisms include:

• Cutting machines
• Package-moving devices

With offset mechanisms, it is very important to understand how and to what degree the offset affects the time ratio. To relate the geometry of a specific linkage to the timing of the stroke, an imbalance angle (β) is used. This angle is related to the time ratio, Q, as follows:

Q = (180° + β) ÷ (180° - β)

Through simple algebraic rearrangement, this equation can be rewritten to solve for β:

β = 180° $\times$ ${\tfrac {Q-1}{Q+1}}$

Timing Charts

Timing charts are often used to synchronize the motion between two or more mechanisms. They graphically display information showing where and when each mechanism is stationary or performing its forward and return strokes. Timing charts allow designers to qualitatively describe the required kinematic behavior of a mechanism.

These charts are also used to estimate the velocities and accelerations of certain four bar links. The velocity of a link is the time rate at which its position is changing, while the link's acceleration is the time rate at which its velocity is changing. Both velocity and acceleration are vector quantities, in that they have both magnitude and direction; however, only their magnitudes are used in timing charts. When used with two mechanisms, timing charts assume constant acceleration. This assumption produces polynomial equations for velocity as a function of time. Constant acceleration allows for the velocity vs. time graph to appear as straight lines, thus designating a relationship between displacement (ΔR), maximum velocity (v_peak), acceleration (a), and time(Δt). The following equations show this.

ΔR = ${\tfrac {1}{2}}$v_peakΔt

ΔR = ${\tfrac {1}{4}}$a(Δt)^2

Given the displacement and time, both the maximum velocity and acceleration of each mechanism in a given pair can be calculated.

Design of Slider-Crank Mechanism

Slider-crank mechanisms involve both rotational and linear motion. For most of these mechanisms, a crank rotates at constant speed in order to repeatedly move an object in a linear motion to perform some task. This device is a simple way to convert rotational motion to linear motion.

With engines, for example, a crank continuously rotates which forces many pistons to move linearly back and forth through cylindrical chambers.

There are two types of slider-cranks: in-line and offset. There are also two methods to design each type: graphical and analytical.

In-line Design

In-line Slider Crank animation

An in-line crank slider is oriented in a way in which the pivot point of the crank is coincident with the axis of the linear movement. The follower arm, which is the link that connects the crank arm to the slider, connects to a pin in the center of sliding object. This pin is considered to be on the linear movement axis. Therefore, to be considered an in-line crank slider, the pivot point of the crank arm must be in-line with this pin point. The stroke((ΔR₄)_max) of an in-line crank slider is defined as the maximum linear distance the slider may travel between the two extreme points of its motion. With an in-line crank slider, the motion of the crank and follower links is symmetric about the sliding axis. This means that the crank angle required to execute a forward stroke is equivalent to the angle required to perform a reverse stroke. For this reason, the in-line slider-crank mechanism produces balanced motion. This balanced motion implies other ideas as well. Assuming the crank arm is driven by a constant force and therefore constant velocity, the time it takes to perform a forward stroke is equal to the time it takes to perform a reverse stroke.

Graphical Approach

The graphical method of designing an in-line slider-crank mechanism involves the usage of hand-drawn or computerized diagrams. These diagrams are drawn to scale in order for easy evaluation and successful design. Basic trigonometry, the practice of analyzing the relationship between triangle features in order to determine any unknown values, can be used with a graphical compass and protractor along side these diagrams to determine the required stroke or link lengths.

When the stroke of a mechanism needs to be calculated, first identify the ground level for the specified slider-crank mechanism. This ground level is the axis on which both the crank arm pivot-point and the slider pin are positioned. Draw the crank arm pivot point anywhere on this ground level. Once the pin positions are correctly placed, set a graphical compass to the given link length of the crank arm. Positioning the compass point on the pivot point of the crank arm, rotate the compass to produce a circle with radius equal to the length of the crank arm. This newly drawn circle represents the potential motion of the crank arm. Next, draw two models of the mechanism. These models will be oriented in a way that displays both the extreme positions of the slider. Once both diagrams are drawn, the linear distance between the retracted slider and the extended slider can be easily measured to determine the slider-crank stroke.

The retracted position of the slider is determined by further graphical evaluation. Now that the crank path is found, draw the crank slider arm in the position that places it as far away as possible from the slider. Once drawn, the crank arm should be coincident with the ground level axis that was initially drawn. Next, from the free point on the crank arm, draw the follower link using its measured or given length. Draw this length coincident with the ground level axis but in the direction toward the slider. The unhinged end of the follower will now be at the fully retracted position of the slider. Next, the extended position of the slider needs to be determined. From the pivot point of the crank arm, draw a new crank arm coincident with the ground level axis but in a position closest to the slider. This position should put the new crank arm at an angle of 180 degrees away from the retracted crank arm. Then draw the follower link with its given length in the same manner as previously mentioned. The unhinged point of the new follower will now be at the fully extended position of the slider.

Both the retracted and extended positions of the slider should now be known. Using a measuring ruler, measure the distance between these two points. This distance will be the mechanism stroke, ((ΔR₄)_max).

Analytical Approach

To analytically design an in-line crank slider and achieve the desired stroke, the appropriate lengths of the two links, the crank and follower, need to be determined . For this case, the crank arm will be referred to as L₂, and the follower link will be referred to as L₃. With all in-line slider-crank mechanisms, the stroke is twice the length of the crank arm. Therefore, given the stroke, the length of the crank arm can be determined. This relationship is represented as:

L₂ = (ΔR₄)_max ÷ 2

Once L₂ is found, the follower length (L₃) can be determined. However, because the stroke of the mechanism only depends on the crank arm length, the follower length is somewhat insignificant. As a general rule, the length of the follower link should be at least 3 times the length of the crank arm. This is to account for an often undesired increased acceleration yield, or output, of the connecting arm.

Offset Design

With an offset slider-crank mechanism, an offset distance is introduced. This offset distance is referred to as L₁ and is the fixed distance between the crank arm pivot point and the slider axis. This offset distance means that the slider-crank motion is no longer symmetrical about the sliding axis. In addition, the required crank angles of the forward and reverse strokes are no longer equivalent. An offset slider-crank provides a quick return when a slower working stroke is desired.

With offset slider-cranks, the stroke is always twice the crank length, and as the offset distance increases, the stroke also becomes larger. The potential range for the offset distance can be written in relation to the other mechanism lengths, L₂and L₃, as the equation:

L₁ < L₃ - L₂

The design of an in-line crank slider mechanism involves finding the two link lengths, L₂and L₃, and an appropriate offset distance,L₁, in order to achieve the wanted stroke,(ΔR₄)_max, and imbalance angle, β.

Analytical Approach

The analytical method for designing an offset crank slider mechanism is the process by which triangular geometry is evaluated in order to determine generalized relationships among certain lengths, distances, and angles. These generalized relationships are displayed in the form of 3 equations and can be used to determine unknown values for almost any offset slider-crank. These equations express the link lengths, L₁, L₂, and L₃, as a function of the stroke,(ΔR₄)_max, the imbalance angle, β, and the angle of an arbitrary line M, θ_M. Arbitrary line M is a designer-unique line that runs through the crank pivot point and the extreme retracted slider position. The 3 equations are as follows:

L₁ = (ΔR₄)_max × [(sin(θ_M)sin(θ_M - β)) / sin(β)]

L₂ = (ΔR₄)_max × [(sin(θ_M) - sin(θ_M - β)) / 2sin(β)]

L₃ = (ΔR₄)_max × [(sin(θ_M) + sin(θ_M - β)) / 2sin(β)]

With these relationships, the 3 link lengths can be calculated and any related unknown values can be determined.