Mechanism design  visual and programmable approaches  visual and programmable approaches

Introduction to Kinematics

Kinematics

Kinematics focuses on the analysis of motion without factoring in the forces involved It involves calculating positions, displacements, velocities, and accelerations of mechanical system components independently of the loads affecting them This approach contrasts with other engineering disciplines, such as statics, which adheres to Newton’s first law (∑ F=∑ M=0) by considering both motion and governing loads, and dynamics, which also examines motion in relation to these forces.

Newton's second law, represented by the equations ∑ F = m a and ∑ M I = α, is foundational in engineering design, particularly in kinematics Effective mechanical system design necessitates an understanding of not just the motion of components, but also the forces acting on them This includes analyzing material stress and strain responses (stress analysis) and determining the necessary dimensions for components to withstand working stresses (machine design) Consequently, kinematic analyses are essential precursors to these critical evaluations.

The progression of mechanical design involves a systematic approach, starting with kinematics, followed by the analysis of static and dynamic loads on system components Once the feasibility of these loads is established, the next step is to evaluate the resulting stresses and strains within the components Finally, machine design principles are applied to ensure that the materials and dimensions of the mechanical system meet the necessary standards for the identified working stresses.

Kinematics serves as the foundational discipline in engineering design, as depicted in Figure 1.1 When kinematic principles are not properly applied, issues often manifest in other engineering areas For instance, a flawed displacement profile from kinematic analysis can lead to excessive acceleration during dynamic analysis, resulting in high dynamic forces These forces may cause significant stress, necessitating impractical material choices or component dimensions that compromise the overall design's feasibility for its intended application.

In contemporary engineering design, it is essential to consider not only traditional factors but also modern elements such as producibility, cost, environmental impact, disposal, aesthetics, ergonomics, and human factors, as these aspects are equally significant.

Kinematic analysis involves calculating the positions, displacements, velocities, and accelerations of known mechanical system components, aiming to understand their kinematic behavior Conversely, kinematic synthesis focuses on determining the necessary dimensions of mechanical systems to achieve specified positions, displacements, velocities, and accelerations The goal of kinematic synthesis is to compute the dimensions required to meet or approximate these known quantities This topic is first introduced in Chapter 4 and explored in detail throughout Chapters 4 to 8.

Kinematic Chains and Mechanisms

This textbook delves into the kinematic design of mechanical systems, commonly known as mechanisms A kinematic chain, which encompasses mechanisms, consists of interconnected links that influence each other's motion through joints For instance, complex mechanical systems like automobile engines may consist of multiple kinematic chains, while simpler tools can operate with just one An example illustrated in Figure 1.2 is a pair of shears, where moving one handle (link L2) affects the motion of the other handle (link L1) and the cutting link (L3), resulting in a cutting action The ability of one link to control the motion of another is crucial, as the primary goal in mechanical system design is to achieve a controlled output motion in response to an input motion.

* Because a mechanism is an assembly of links, it is also called a linkage.

† Links are generally assumed to be nondeforming or rigid in kinematics.

(Component dimensions for working stresses)

FIGURE 1.1 Kinematics in relation to other associated engineering design disciplines.

Mechanisms are uniquely identified by the presence of at least one grounded link, which is fixed to a specific frame of reference Unlike other kinematic chains, mechanisms can have links that remain permanently grounded through various means such as friction, gravity, or coupling members like bolts and welds In the case of shears, the grounded link can be determined based on individual preferences.

Mobility

The mobility of a mechanism refers to the number of independent parameters needed to uniquely determine its position in space, which is crucial for developing mathematical models for kinematic analysis or synthesis These models must incorporate sufficient parameters to accurately represent the motion of each component In two-dimensional space, defining a body's position at a specific moment requires three independent parameters For example, when considering an automobile on a road, its position relative to the X–Y coordinate frame can be established using these parameters.

In order to define the position of an automobile in the X–Y plane, three independent parameters are necessary: the linear coordinates (px, py) of the point p on the automobile and its angular position θ about the Z-axis Consequently, a mechanism link designed for planar motion can achieve a maximum mobility of three.

To accurately define an object's position in three-dimensional space at any given moment, six independent parameters are necessary For an aircraft in flight, its position relative to the X–Y–Z coordinate frame is determined by the linear coordinates (X, Y, Z) of a specific point on the aircraft, along with its angular orientation around the X, Y, and Z axes (θx, θy, θz) Consequently, any mechanism designed for spatial motion can possess a maximum mobility of six degrees of freedom.

FIGURE 1.2 Cutting tool in (a) open and (b) closed positions.

The concept mechanism depicted in Figure E.1.1 demonstrates two positions utilized for stamping parts By rotating link a0-a1 to position a0-aj, the mechanism transitions from its initial to final stamping position During the stamping process, a reaction force, F_stamp, acts on the mechanism at point pj, while the driving torque ensures the mechanism remains in a state of static equilibrium.

The evaluation of structural integrity for member b0–bj during stamping involves the application of kinematics to analyze motion and forces, while statics helps in understanding the equilibrium of forces acting on the member Stress analysis is crucial for assessing the material's response to applied loads, ensuring it can withstand the stamping process without failure Additionally, principles of machine design are employed to optimize the member's geometry and material selection, enhancing its durability and performance under operational conditions Together, these disciplines provide a comprehensive framework for ensuring the reliability and safety of the stamping process.

Known Information: Figure E.1.1, background knowledge of kinematics, statics, stress analysis, and machine design principles.

Solution Approach: The mechanism in Figure E.1.1 can be modeled as a planar four-bar mechanism.*

Kinematic analysis involves calculating the angular rotation of the driving link needed to reach the final stamping positions (a j and b j) using the coordinates of points a 0, a 1, b 0, and b 1 This calculation is derived from the displacement equations applicable to a planar four-bar mechanism.

* Kinematic displacement, velocity, and acceleration equations for the planar four-bar mechanism are introduced in Chapter 3.

FIGURE 1.3 Automobile and aircraft with maximum (a) planar and (b) spatial degrees of freedom.

Static analysis: A static equilibrium equation can be formulated to calculate the columnar force (which we will call F b b

[2] For example, summing the moments about a j produces the equation

Stress analysis involves calculating the normal stress (σ) on component b0-bj using the formula σ b b0 j = F b b0 j A b b0 j, where F represents the force and A denotes the cross-sectional area Furthermore, it is possible to determine the buckling load for component b0-bj, which behaves like a column with pinned ends.

In machine design, it may be necessary to adjust the cross-sectional dimensions and material type of component b0-bj based on the calculated values for normal stress and buckling load, ensuring an acceptable level of structural integrity for its intended use.

Summary

Kinematics, the study of motion independent of forces, is a crucial discipline in mechanical system design It is essential to assess kinematic feasibility prior to exploring other engineering areas, including statics, dynamics, stress analysis, and machine design.

Mechanical systems consist of kinematic chains, which are assemblies of interconnected links that enable controlled motion transfer The primary goal in designing these systems is to achieve a precise output motion that responds effectively to an input motion.

The stamping mechanism's initial and final positions are crucial for understanding its operation Mechanisms, often referred to as linkages, fall under the broader category of chains A key distinguishing feature of mechanisms is the presence of a fixed ground link, which sets them apart from kinematic chains.

The mobility of a mechanism refers to the number of independent parameters needed to uniquely define its position in space Understanding a mechanism's mobility is crucial for developing mathematical models for its kinematics or synthesis, as these models must incorporate sufficient variables to accurately describe the motion of each component For instance, a link constrained to planar motion can possess up to 3 degrees of freedom, while a link limited to spatial motion can have as many as 6 degrees of freedom.

1 Norton, R.L 2008 Design of Machinery, 4th edn., pp 30–40 New York: McGraw-Hill.

2 Wilson, C.E and J Peter Sadler 2003 Kinematics and Dynamics of Machinery, 3rd edn., Chapter 9 Upper Saddle River, NJ: Prentice Hall.

3 Ugural, A.C and S.K Fenster 2009 Advanced Strength and Applied Elasticity, 4th edn., Chapter 11 Englewood Cliffs, NJ: Prentice Hall.

Myszka, D.H 2005 Machines and Mechanisms: Applied Kinematic Analysis, 3rd edn., Chapter 1 Upper Saddle River, NJ: Prentice Hall.

Waldron, K.J and G.L Kinzel 2004 Kinematics, Dynamics and Design of Machinery, 2nd edn., Chapter 1 Upper Saddle River, NJ: Prentice Hall.

Wilson, C.E and J Peter Sadler 2003 Kinematics and Dynamics of Machinery, 3rd edn., Chapter 1 Upper Saddle River, NJ: Prentice Hall.

Mobility of Mechanisms

Planar Mechanism Types

The planar four-bar mechanism, depicted in Figure 2.1a, consists of four interconnected links: the crank (driving link), coupler, follower, and ground When an input rotation is applied to the crank, it drives the motion of the coupler and follower links, both of which are connected to the ground and experience pure rotation This mechanism is widely used in everyday devices, as shown in Figures 2.1b through d, including locking pliers, folding chairs, and doorways.

The slider-crank mechanism, although briefly discussed in this textbook, can be theoretically represented as a planar four-bar mechanism with a follower link of infinite length This mechanism is one of the most commonly used kinematic chains in everyday applications, notably in the crankshaft-connecting rod-piston linkage, which is a crucial subsystem of the internal combustion engine.

Attaching a grounded link pair, or dyad, to the coupler of a planar four-bar mechanism creates a multiloop planar six-bar mechanism, specifically known as the Stephenson type III mechanism While it is not as frequently used in everyday applications as the planar four-bar mechanism, the Stephenson III features two intermediate links that facilitate complex motion This design allows for dual and simultaneous motion, as well as path generation capabilities Further details on multiloop planar six-bar motion and path generation can be found in Chapter 5.

The Stephenson type III mechanism in Figure 2.4 features loops a-b-c-d-e and g-f-c-d-e that can be considered as independent kinematic chains, effectively forming planar five-bar mechanisms Notably, the additional link in these planar five-bar mechanisms allows for the production of link paths with a greater maximum order, resulting in more complex curvature compared to the link paths generated by planar four-bar mechanisms.

* Because mechanisms are comprised of links, they are also called linkages.

† The coupler undergoes complex motion—a combination of simultaneous rotation and translation.

The planar five-bar mechanism, characterized by its two degrees of freedom, often incorporates a gear pair to eliminate one degree, resulting in what is known as a geared five-bar mechanism According to Wunderlich (1963), the maximum order of a coupler curve (m) for a mechanism with n links connected solely by revolute joints is calculated using the formula m = 3/2(n - 1).

FIGURE 2.1 (a) Planar four-bar mechanisms as (b) lock pliers, (c) folding chair, and (d) doorway linkages.

FIGURE 2.3 Slider-crank mechanism as crankshaft-connecting rod-piston linkage.

FIGURE 2.2 (a) Slider-crank mechanism and (b) four-bar mechanism as slider-crank mechanism.

Links, Joints, and Mechanism Mobility

A mechanism consists of an assembly of links and joints, where the attachment points to adjacent links are referred to as nodes Links can be classified based on the number of nodes they possess; a link with two nodes is termed a binary link, while a link with three nodes is known as a ternary link The Stephenson III six-bar mechanism features a ternary link as the ground and a movable intermediate ternary link, with all other links in the mechanism being binary.

FIGURE 2.4 Multiloop planar six-bar mechanism (Stephenson type III mechanism).

FIGURE 2.5 Link types in the Stephenson III mechanism.

Adjacent mechanism links are interconnected at their nodes through various joint types, which differ in the number and type of degrees of freedom (DOF) Among planar mechanisms, the revolute joint (R) is the most prevalent, offering one rotational DOF Other common joints include the prismatic joint (P) with one translational DOF, the cylindrical joint (C) with two DOFs (one rotational and one translational), and the spherical joint (S) with three rotational DOFs The cylindrical and spherical joints are primarily used in spatial mechanisms discussed in Chapters 7 and 8 Joints like R, P, C, and S, referred to as lower pairs or full joints, involve surface contact, such as a ball in a socket for the S joint or a pin in a hole for the R joint.

Gruebler's equation is essential for calculating the mobility, or degrees of freedom (DOF), of a mechanism Specifically, Equation 2.1 is utilized for determining the mobility of planar mechanisms, where each individual link in a planar mechanism is limited to a maximum number of degrees of freedom.

3 degrees of freedom, the maximum mobility of a planar mechanism with L links is 3L Because the ground link is fully constrained, its mobility is subtracted from

In mechanical systems, the mobility of a spatial mechanism can be calculated using Equation 2.2, which indicates that the maximum mobility is 6L for L links However, since the ground link is fully constrained, the effective mobility is adjusted to 6(L-1) Each one-degree-of-freedom (1 DOF) joint reduces the mobility by 5, leading to a total decrease of 5J1, where J1 represents the number of 1 DOF joints Additionally, higher DOF joints, such as 2 DOF and 3 DOF joints, further decrease mobility by 4 and 3 degrees of freedom, respectively Understanding these relationships is crucial for analyzing the functionality of various joint types, including revolute, prismatic, cylindrical, and spherical joints.

J 4 = J 5 = 0 since 4 and 5 DOF joints are not utilized.

The primary goal of mechanical system design is to achieve precise controlled output motions based on given input motions This control is most effective in systems with a single degree of freedom Understanding the mobility of a mechanism allows designers to assess whether additional constraints are necessary to achieve the desired mobility and to identify the number of constraints required.

Problem Statement: Determine the mobility of the planar and spatial mechanisms illustrated in Figure E.2.1.

Known Information: Figure E.2.1, Equations 2.1 and 2.2.

The planar mechanism depicted in Figure E.2.1 consists of eight links and ten joints, including eight revolute and two prismatic joints, resulting in a mobility of 1 according to Gruebler’s equation for planar mechanisms In contrast, the spatial linkage shown in the same figure comprises six links and ten joints, featuring three revolute, three spherical, and one cylindrical joint, which yields a mobility of 2 when calculated using Gruebler’s equation for spatial mechanisms.

The joint type J 2, as outlined in Equation 2.1, encompasses cam joints, gear joints, and roller joints, which are also referred to as half joints These joints can exhibit a mobility of 1 or 2, contingent upon the occurrence of rolling and/or sliding motion Consequently, they can eliminate either 2 or 1 degree(s) of freedom (DOF) from the overall mobility of the mechanism.

Number Synthesis

Mechanism mobility can be calculated using Gruebler’s equation, which considers the number and order of mechanism links and joints An inverse application of this equation helps identify various link and joint combinations that yield alternate mechanism solutions for a specified mobility Number synthesis is the process of determining these alternative mechanisms, revealing that for any given mobility, there are countless combinations of links and joints By expressing Gruebler’s equations for planar and spatial mechanisms, it becomes evident that different configurations can achieve the same mobility This systematic approach not only aids in the creative design of mechanisms but also allows for the generation of tables listing conceptual mechanism solutions by adjusting the link and joint variables in Gruebler’s equation.

Problem Statement: Compile a table of single DOF planar mechanisms having

Known Information: Equation 2.1, DOF PLANAR = 1, L = 2, 3, and 4.

FIGURE E.2.1 (a) Planar and (b) spatial mechanisms.

In the solution approach, for every value of variable L, variable J1 is incrementally increased, and for each combination of L and J1, the corresponding value of the remaining unknown J2 in Equation 2.1 is determined This systematic method yields the mechanism solutions presented in Table E.2.1.

Figure E.2.2 showcases various mechanism configurations that correspond to mechanism solutions 1, 2, 5, 9, and 10 listed in Table E.2.1 It is important to note that each mechanism solution may encompass multiple configurations; for instance, mechanism solution 9 in Table E.2.1 includes two distinct configurations as depicted in Figure E.2.2.

FIGURE E.2.2 Example 2, 3, and 4-link single DOF planar mechanism configurations.

TABLE E.2.1 Two, Three, and Four-Link Single DOF Planar Mechanisms

Grashof’s Criteria and Transmission Angle

The Grashof criteria are essential for assessing the rotational behavior of four-bar mechanisms, focusing on the lengths of the crank, coupler, follower, and ground links These criteria categorize mechanisms into Grashof and non-Grashof classifications, as outlined in Table 2.1 In this classification, S represents the shortest link length, L denotes the longest link length, while P and Q are the lengths of the remaining two links.

Figure 2.7 illustrates the link rotations for the crank-rocker, double-crank (also called drag-link), double-rocker, and triple-rocker mechanisms As illustrated, only

TABLE 2.1 Grashof and Non-Grashof Mechanisms

Crank-rocker S + L < P + Q Crank Double-crank (drag-link) S + L < P + Q Ground Double-rocker S + L < P + Q Coupler

Grashof mechanisms exhibit distinct link rotations, including the crank-rocker, double-crank, double-rocker, and non-Grashof triple-rocker configurations In a crank-rocker mechanism, the crank link is capable of a full rotation, while both the crank and follower links can rotate completely in a double-crank mechanism The double-rocker mechanism features the coupler link as the driving link, which completes a full rotation, whereas no link achieves full rotation in the triple-rocker configuration Additionally, in a change point mechanism, all links align simultaneously twice per revolution, resulting in unpredictable output behavior during this theoretical state.

The double-rocker mechanism differs from crank-rocker and drag-link mechanisms in that its driving link is not fixed to the ground, which can limit its practical applications in design Designers must address the technical challenges of attaching a drive system to a movable joint while ensuring its effective operation.

In four-bar mechanism design, understanding the transmission angle is essential for effective force and torque transmission between the crank and follower links The transmission angle, represented as τ, is the angle formed between the coupler and follower links When an input torque (T in) is applied to the crank link, it transmits force to the coupler, which then transfers that force to the follower (F follower) This follower force has two components: one that is perpendicular to the follower link, contributing to the output torque (T out), and another that acts along the length of the follower as a columnar load (F follower cos τ) Both components are influenced by the transmission angle, and minimizing the columnar component of the follower force is a common objective in four-bar mechanism design.

* Figure 2.7c illustrates the path achieved by the shortest link in the double-rocker when a driver is applied to the leftmost joint of the shortest link.

† Affixing a drive system (such as a motor or manual crank) to a grounded link joint (and operating such a system) can be more easily accomplished than to a nongrounded joint.

The planar four-bar mechanism, illustrated in Figure 2.8, is crucial in design applications involving significant forces and torques An optimal transmission angle of 90° is ideal, as it ensures that the follower force consists solely of a normal component, effectively reducing complications in force transmission Designers typically prefer a transmission angle range of 90° ± 50° Additionally, the equations presented in Figure 2.8b indicate that a decrease in the transmission angle leads to an increase in the columnar load component while the normal load component diminishes.

The analysis of the three planar four-bar mechanisms, as depicted in Figure E.2.3, involves determining the Grashof type based on the provided identical link lengths from mechanism #1 Each mechanism features driving links clearly marked with rotation arrows, which are essential for understanding their motion characteristics By evaluating the link lengths and the configuration of the mechanisms, we can classify each mechanism according to Grashof's criterion, which categorizes them as either Grashof or non-Grashof types.

Known Information: Figure E.2.3 and Table 2.1.

The analysis of the mechanisms reveals that mechanism #1 qualifies as a Grashof double-crank due to its shortest link being the ground link and satisfying the Grashof condition S + L < P + Q In contrast, mechanism #2 features the crank link as its shortest link.

The Grashof condition, defined as S + L < P + Q, indicates that the mechanism is a Grashof crank-rocker In mechanism #3, the coupler link is the shortest, and since the Grashof condition holds true, this mechanism is classified as a Grashof double-rocker.

Follower columnar loads, unlike typical follower normal loads that links with revolute joints are built to handle, can lead to follower buckling and excessive bearing forces within the follower revolute joints.

FIGURE E.2.3 Planar four-bar mechanism configurations (with dimensionless link lengths given for mechanism #1).

Summary

The planar four-bar mechanism is a fundamental kinematic chain commonly found in various everyday devices It consists of four interconnected links: the crank, coupler, follower, and ground links In this mechanism, the crank and follower links experience pure rotational motion, while the coupler link exhibits complex movement By attaching a dyad to the coupler, this mechanism can be transformed into the Stephenson configuration, enhancing its functionality.

The III mechanism is a specific type of multiloop planar six-bar mechanism, which, while less common than the planar four-bar mechanism in everyday applications, features at least two links that experience intricate motion This complexity allows multiloop planar six-bar mechanisms to offer enhanced kinematic synthesis capabilities compared to their four-bar counterparts.

Mechanism links are connected through various types of joints, including revolute joints, which provide a single rotational degree of freedom, as seen in the planar four-bar mechanism and the Stephenson III mechanism In contrast, prismatic joints offer a single translational degree of freedom, ideal for applications requiring sliding contact Additionally, this textbook explores cylindrical and spherical joints, which are essential for the functioning of spatial mechanisms.

Gruebler’s equation is essential for calculating the mobility of mechanisms, and number synthesis plays a crucial role in determining the necessary links and joints to achieve desired mobility By systematically applying Gruebler’s equation and incrementally specifying components, number synthesis aids in the creative design process of mechanisms, providing alternative solutions for engineers and designers.

The Grashof criteria are essential for analyzing the link rotation behavior of four-bar mechanisms, which is crucial in design, especially when connecting a drive system to the crank link Understanding these criteria helps engineers optimize mechanisms for efficient performance.

Understanding the transmission angle—the angle between the coupler and follower links—is crucial in the design of four-bar mechanisms, especially regarding force and torque transmission A decrease in the transmission angle leads to an increase in the columnar load on the follower link, while the load normal to the follower decreases, and vice versa Designers typically prefer a transmission angle range of 90° ± 50° for optimal performance.

2.1 Planar four-bar linkages have many everyday applications (some are illustrated

Figure 2.1) Identify and describe four additional everyday applications for the planar four-bar linkage.

2.2 (a) Why is it important to know if a mechanism has a single degree of freedom?

(b) Why is a crank-rocker mechanism more useful than a double-rocker mechanism?

(c) Should the transmission angle for the planar four-bar linkage be close to 0°? Explain.

2.3 For the two linkages illustrated in Figure P.2.1, which (if any) of the links can undergo a complete rotation relative to the other links? How do you know?

2.4 Determine the number of links and the mobility of each of the three planar mechanisms in Figure P.2.2.

2.5 For the planar four-bar linkage illustrated in Figure P.2.3, L L 2 1=1 5 and

L L 3 1=1 2 , find the range of L L 0 1 required for a drag link mechanism.

2.6 Compile a table of 1 DOF spatial mechanisms having 2, 3, and 4 links (let

J 4 = J 5 = 0 in Equation 2.2) Illustrate some of these mechanism solutions.

2.7 Figure P.2.4 illustrates nine spatial mechanisms that include revolute (R), pris- matic (P), cylindrical (C), and spherical (S) joints Calculate the mobility of these mechanisms.

FIGURE P.2.1 Planar four-bar linkages with dimensionless link lengths.

2.8 Euler’s buckling load ( )F for a columnar member with pinned ends is

To calculate the transmission angle of a columnar member, we utilize Euler’s equation represented as πL²/2, where E denotes the modulus of elasticity, I signifies the moment of inertia, and L represents the length of the column By analyzing the parameters outlined in Figure P.2.5, we can derive the necessary equation for determining the transmission angle.

FIGURE P.2.3 Planar four-bar linkage.

RRSC RSSR-SC Spatial slider-crank

To calculate the transmission angle for a follower link with a length of 12 inches (0.3048 m) and a square cross-section measuring ⅜ inch (0.635 cm), we consider the material properties of aluminum, which has a modulus of elasticity of 10,000,000 psi (68.05 GPa) The follower load applied is 250 lbf (1112.05 N) The analysis involves determining the spatial mechanisms formed by R, P, C, and S joints in relation to the follower link's buckling load.

2.9 Maverick mechanisms are mechanisms that defy Gruebler’s equation (Gruebler’s equation will produce misleading results for maverick mechanisms) Passive degrees of freedom are localized DOFs that have no effect on the overall mecha- nism kinematics Determine which mechanisms in Figure P.2.4 are maverick mech- anisms and locate the passive DOFs from among the mechanisms in Figure P.2.4.

2.10 Compile a table of 2 DOF planar mechanisms having 4 and 5 links (consider- ing cam or gear joints as J 2) Illustrate one four-bar mechanism solution and one five-bar mechanism solution.

1 Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and

Synthesis, Vol 2, p 4 Englewood Cliffs, NJ: Prentice Hall.

3 Norton, R.L 2008 Design of Machinery, 4th edn., p 123 New York: McGraw-Hill.

5 Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and

Synthesis, Vol 2, p 64 Englewood Cliffs, NJ: Prentice Hall.

7 Norton, R.L 2008 Design of Machinery, 4th edn., pp 42–46 New York: McGraw-Hill.

9 Wilson, C.E and J Peter Sadler 2003 Kinematics and Dynamics of Machinery, 3rd edn., p 33 Upper Saddle River, NJ: Prentice Hall.

FIGURE P.2.5 Planar four-bar mechanism with transmission angle and crank and follower loads.

Kinematics of Planar Mechanisms

Kinematic Analysis of Planar Mechanisms

Kinematic analysis involves determining the positions, displacements, velocities, and accelerations of mechanism links, either qualitatively or quantitatively This article focuses on quantitative kinematic analysis, utilizing equations that comprehensively describe the motion and higher-order quantities of each mechanism link Specifically, kinematic equations for the planar four-bar mechanism, slider-crank mechanism, and Stephenson III mechanism are developed Displacement equations are derived by summing the closed vector loops of each mechanism By taking the first and second derivatives of the vector loop sum displacement equation, velocity and acceleration equations for the mechanism links are ultimately produced.

Four-Bar Mechanism Analysis

The planar four-bar mechanism consists of four links interconnected by revolute joints As calculated from Gruebler’s equation for planar mechanisms (with L = 4 and

The planar four-bar mechanism, characterized by a single degree of freedom (DOF), is analyzed through the derivation of its displacement equation This equation is obtained by summing the vector loop of the mechanism, as illustrated in Figure 3.1, resulting in a clockwise vector loop sum.

After expanding Equation 3.1 and grouping its real and imaginary terms as separate equations, the resulting planar four-bar mechanism displacement equations become f W V U G f

* Qualitative methods include constructing and measuring mechanism schematics and polygons to determine the positions, velocities, and accelerations of mechanism link locations of interest.

† This procedure for formulating kinematic equations for mechanisms is called the method of vector loop closure.

In the planar four-bar displacement equations, all variables are user-defined except for the coupler and follower displacement angles, denoted as α j and γ j The equations f 1 (α γ j , j ) and f 2 (α γ j , j ) in Equation 3.2 incorporate these unknown angles, resulting in a system of nonlinear equations that allows for the calculation of α j and γ j.

A single planar four-bar velocity equation is derived by differentiating the planar four-bar displacement equation Differentiating Equation 3.1 with respect to time produces β j W 1 e i θ β α j V 1 e i ρ α γU 1 e σ γ 0 j i j j j

After expanding Equation 3.3 and grouping its real and imaginary terms as separate equations, the planar four-bar mechanism velocity equation in matrix form becomes

By incorporating the unknowns derived from the displacement equations alongside the defined mechanism variables in Equation 3.4, it becomes possible to directly compute the angular velocities of the coupler and follower, denoted as α˙ j and γ˙ j, respectively.

* While analytical solutions are possible for sets of linear equations (e.g., Cramer’s rule), a numerical root-finding method (e.g., Newton’s method) must be used to solve sets of nonlinear equations.

In the textbook "Design of Machinery" (Fourth Edition) by R L Norton, it is demonstrated that analytical equations for planar four-bar displacement can be derived through significant algebraic manipulation.

FIGURE 3.1 Planar four-bar mechanism displacement equation variables.

A planar four-bar acceleration equation is derived by differentiating the planar four- bar velocity equation Time differentiation of Equation 3.3 produces β j i θ β β i θ β α ρ α α ρ α j i j i e j e j e j e j

After expanding Equation 3.5 and grouping its real and imaginary terms as separate equations, the planar four-bar mechanism acceleration equation in matrix form becomes

Incorporating the unknowns derived from the displacement and velocity equations, along with the designated mechanism variables into Equation 3.4, allows for the direct calculation of the angular accelerations of the coupler and follower, denoted as α¨ j and γ¨ j, respectively.

Problem Statement: Determine if the two door linkage configurations in Table E.3.1

(Figure E.3.1a and b) will operate properly as the door closes.

Known Information: Equation 3.2, Table E.3.1, and displacement angle range for crank link.

To assess the operability of door linkage configurations, it is essential to calculate the displacement angles of the coupler link (or follower link) during the complete rotation of the crank link By analyzing these displacement angle values, one can identify any discontinuities that may affect the functionality of the door linkage system This process utilizes Equation 3.2 for accurate results.

* This discontinuity is often displayed numerically in commercial software as a complex number.

2 8, −90° 8, 1.4850° 9, −120.0198° 12.5, 0 calculate the coupler and follower link displacement angles α j and γ j , respectively

The crank rotation range corresponding to Figure E.3.1b is 0° (door fully open) to 90° (door fully closed).

Figure E.3.2 demonstrates the coupler link displacement angles in relation to the door rotation angle (β) during the closing process In configuration 1, a discontinuity occurs after the door reaches an 84° rotation, indicating a mechanism "lock-up" that necessitates disassembly to resolve Consequently, with configuration 1, the door fails to close fully Conversely, configuration 2 maintains a continuous coupler displacement angle throughout the entire 90° rotation, allowing the door to close completely.

* The lock-up state of the mechanism is associated with the circuit defect—a condition requiring mecha- nism disassembly and reassembly to pass through.

FIGURE E.3.1 (a) Door linkage and (b) kinematic model of four-bar door linkage.

Slider-Crank Mechanism Analysis

The planar slider-crank mechanism features four links connected by revolute joints, with the slider link attached to the ground via a prismatic joint According to Gruebler’s equation for planar mechanisms, this mechanism has a mobility of 1, indicating a single degree of freedom (DOF) Analyzing the clockwise sum of the vector loop in the slider-crank mechanism demonstrates its functional characteristics.

FIGURE E.3.2 Coupler displacement angles for configurations 1 and 2.

As illustrated in Figure 3.2, vector U 1 only has an imaginary component and vector

The slider-crank displacement equations reveal that G j consists solely of a real component, representing the slider's sliding distance in the x-direction By expanding Equation 3.7 and separating the real and imaginary terms, the equations can be clearly organized into distinct relationships, resulting in the formulation f G W V G f W.

In the slider-crank displacement equations, the coupler displacement angle (α j) and slider displacement magnitude (G jx) are the only variables not defined by the user Unlike Equation 3.2, which necessitates a numerical solution, Equation 3.8 allows for an analytical solution The imaginary term equation f 2 (α j) can be rearranged to express α θ β j in terms of y, ρ, and other variables.

The coupler angle solutions from Equation 3.9 are used in f 1 (α j ,G jx ) and the corresponding sliding distances G jx are calculated.

A slider-crank velocity equation is derived by differentiating the displacement vector loop sum equation Differentiating Equation 3.7 with respect to time produces β j W 1 e i θ β α j V 1 e i ρ α G 0 j j jx

After expanding Equation 3.10 and grouping the real and imaginary terms as sepa- rate equations, the slider-crank velocity equation in matrix form becomes

* Having a zero U 1 produces an in-line slider crank (a nonzero U 1 produces an offset slider crank).

FIGURE 3.2 Slider-crank mechanism displacement equation variables.

By incorporating the unknowns derived from the displacement equations along with the specified mechanism variables into Equation 3.11, it is possible to directly compute the coupler angular velocity (α˙ j) and the slider velocity (G˙ jx).

A slider-crank acceleration equation is derived by differentiating the velocity vector loop sum equation Time differentiation of Equation 3.10 produces β j i θ β β i θ β α ρ α α ρ α j i j i e j e j e j e j

After expanding Equation 3.12 and grouping the real and imaginary terms as sepa- rate equations, the slider-crank acceleration equation in matrix form becomes

By incorporating the unknowns derived from the displacement and velocity equations, along with the specified mechanism variables into Equation 3.13, one can directly compute the coupler angular acceleration (α¨ j) and the slider acceleration (G¨ jx).

Problem Statement: Plot the displacement, velocity, and acceleration profiles for the piston in the crankshaft-connecting rod-piston linkage (Figure E.3.3)

FIGURE E.3.3 Crankshaft-connecting rod-piston linkage.

The mechanism assembly configuration and driving link parameters are given in Table E.3.2.

Known Information: Equations 3.8, 3.9, 3.11, 3.13 and Table E.3.2.

The crankshaft-connecting rod-piston linkage functions as an in-line slider crank mechanism, indicated by U 1 = 0 This setup allows for the calculation and plotting of the piston’s displacement, velocity, and acceleration profiles, with variable G x representing the piston displacement.

G x , G˙ x , and G¨ x , respectively The outputs calculated from Equation 3.9 are used in Equation 3.11 to calculate G˙ x The outputs calculated from Equations 3.9 and 3.11 are used in Equation 3.13 to calculate G¨ x

Figure E.3.4 illustrates the piston displacement, velocity, and acceleration pro- files with respect to the crank rotation angle ( )β over a 720° crank rotation range.

FIGURE E.3.4 Piston (a) displacement, (b) velocity, and (c) acceleration profiles.

TABLE E.3.2 Slider-Crank Mechanism Assembly Configuration (Dimensionless Link Lengths)

Multiloop Mechanism Analysis

Figure 3.3 includes several planar multiloop six-bar mechanism classifications

The Watt and Stephenson mechanisms, as classified by their unique designs, demonstrate significant advantages over the planar four-bar mechanism by enabling at least two links to undergo complex motion This capability is particularly beneficial for tasks requiring rigid-body guidance in both motion-specific and path-specific applications Notably, the Watt II and Stephenson III mechanisms allow for three links to perform pure rotation, unlike the two links in the planar four-bar mechanism, while all these mechanisms maintain a single degree of freedom Consequently, the Watt and Stephenson six-bar mechanisms present effective solutions for tasks that necessitate the simultaneous guidance of multiple rigid bodies Detailed discussions on planar rigid-body guidance and six-bar rigid-body guidance can be found in the relevant chapters.

Stephenson I Stephenson II Stephenson III

FIGURE 3.3 Watt and Stephenson six-bar mechanisms.

Problem Statement: Derive displacement, velocity, and acceleration equations for the Stephenson III mechanism.

Known Information: Vector loop closure method and established notation for mechanism dyad variables.

To derive displacement equations for a mechanism using the vector loop closure method, it is essential to establish vectors for each mechanism loop and define all vector loop variables After defining these loops and variables, the next step involves calculating the vector loop sum to formulate a displacement equation for each loop, separating real and imaginary terms into distinct equations Differentiating the displacement equation yields the corresponding velocity equation, which can be further differentiated to obtain acceleration The Stephenson III mechanism, a planar four-bar mechanism with an additional dyad attached to the coupler link, can be modeled using the planar four-bar displacement equation along with the displacement equation for the additional dyad By taking the clockwise sum of the vector loop that includes variables U1, S1, V1*, U1*, and G1*, a comprehensive representation of the Stephenson III mechanism can be achieved.

Velocity and acceleration equations are derived by taking the first and second derivatives of the displacement equation, respectively Differentiating Equation 3.14 with respect to time produces f j U j j S j j α γ j j γ i σ γ α i ψ α α ρ α j e e V e i

FIGURE E.3.5 Stephenson III mechanism dyad displacement equation variables.

Differentiating Equation 3.15 with respect to time produces f α γ j j γ j U i σ γ γ j U i σ γ α S ψ α j i e j e j e j

By separating the real and imaginary terms in kinematic equations, we create sets of nonlinear simultaneous equations These equations enable the numerical calculation of unknown variables such as displacement, velocity, and acceleration.

Kinematics of Mechanism Locations of Interest

In kinematic analyses, calculating the positions, displacements, velocities, and accelerations of specific mechanism link locations is essential, even when these are not directly defined in the kinematic equations A prime example is the importance of understanding the trajectory of a point on the coupler link, which is crucial for the design and analysis of four-bar path generators.

To establish a displacement equation for a specific location within a mechanism, a vector loop equation that incorporates the desired location is developed Subsequently, the velocities and accelerations at this location can be determined using the velocity and acceleration equations, which are derived as the first and second derivatives of the vector loop displacement equation, respectively.

Problem Statement: Plot the path traced by point p on the level-luffing crane

(Figure E.3.6) The mechanism assembly configuration is given in Table E.3.3.

Known Information: Equation 3.2 and Table E.3.3.

The planar four-bar mechanism is classified as a non-Grashof type, which limits its ability to achieve full crank link rotation By applying Equation 3.2, we can determine the range of crank rotation and the corresponding range of coupler rotation.

The vector loop equation to define point p on the crane is p j = W 1 e i ( θ β + j ) + L 1 e i ( ρ α * + j ) (3.17)

After expanding Equation 3.17 and grouping the real and imaginary terms, the resulting displacement equation for the luffing crane point becomes p p xj j j yj j j

* A path generator (introduced in Chapter 5) is a mechanism designed to achieve or approximate a given series of coupler path points.

The prescribed β j values and calculated α j values from the four-bar displacement equations are utilized in Equation 3.18 to determine the path traced by point p In this mechanism configuration, the angles for V 1 and L 1 are identical due to the parallel nature of both vectors Figure E.3.7 demonstrates the path traced by the coupler point p for a crank displacement angle range of β j = 10 to -32 Additionally, Figure E.3.8 showcases the level-luffing crane configuration along with the calculated path, highlighting a lightly shaded section of the curve that is particularly beneficial for leveling applications as it maintains a near-constant level.

* The angles for V 1 and L 1 (angles ρ and ρ *) turn out to be identical in this example problem Identical values for these angles are not required.

Level-Luffing Crane Mechanism Assembly Configuration for

FIGURE E.3.6 Level-luffing crane mechanism.

Solution Method for Vector Loop Kinematic Equations

This chapter introduces displacement, velocity, and acceleration equations that consist of two nonlinear equations with two unknown variables To solve these kinematic equations numerically, a root-finding method is employed Among these methods, the Newton–Raphson technique is widely recognized for its effectiveness A flowchart illustrating the application of the Newton–Raphson method for a system of two equations and two unknowns is provided in Figure 3.4.

While a root-finding method could be utilized for the slider-crank displacement model discussed in Section 3.3, it is unnecessary due to the availability of an algebraic solution for determining the coupler angle.

FIGURE E.3.7 Path traced by point p on the level-luffing crane mechanism.

Section of curve for leveling applications p iY X

FIGURE E.3.8 Level-luffing crane mechanism with calculated coupler curve path.

In a system of two equations (f1 and f2) with two unknowns (V1 and V2), initial values for the unknowns are established, as illustrated in Figure 3.4 The residuals δV1 and δV2 are computed by multiplying the inverted Jacobian with the negative column matrix of f1 and f2 Updated values for the unknown variables are obtained by adding these residuals to the original variables, followed by recalculating f1 and f2 using the new values This iterative process continues until the results from the equations fall below a defined error threshold ε.

The kinematic models discussed in this chapter can be solved using codified Figure 3.4, but many modern mathematical analysis software programs, such as Mathcad®, Mathematica®, and MATLAB®, utilize the Newton-Raphson method and other root-finding algorithms for efficient computation.

Planar Kinematic Modeling in MATLAB ®

SimMechanics ® is a multibody mechanical modeling and simulation package that complements MATLAB A library of MATLAB and SimMechanics files is available

The article discusses the use of the Newton–Raphson method for analyzing displacement, velocity, and acceleration in mechanisms such as the planar four-bar, slider-crank, and Stephenson III mechanisms Users can download necessary files from www.crcpress.com, where they can input link dimensions and driving parameters to compute the mechanisms' performance The simulations allow for the visualization of motion and provide detailed user instructions for MATLAB and SimMechanics files in the appendices.

The MATLAB and SimMechanics files facilitate efficient kinematic analyses of planar mechanisms, including four-bar, slider-crank, and Stephenson III systems These analyses utilize the principle of vector loop closure to enhance accuracy and effectiveness.

To analyze the planar four-bar mechanism depicted in Figure E.3.9, we aim to plot the trajectory of the coupler vector L1 during a full rotation of the crank link Additionally, we will simulate the mechanism's motion to observe the dynamic behavior of the system The resulting path will provide insights into the kinematic characteristics of the mechanism throughout the crank's rotation.

The values for vectors W1, V1, U1, G1, and L1, as specified in Figure E.3.9 and the Appendix G.1 MATLAB file, were utilized in the simulation of a four-bar mechanism The default crank link rotation range of 1° to 360° and an increment of 1° were applied Upon executing the MATLAB file, a graphical user interface is generated, allowing for the simulation of the mechanism's motion This process produces files compatible with Microsoft® Excel, which contain detailed data on coupler and follower link displacement, velocity, and acceleration at each crank rotation increment Additionally, Figure E.3.10 showcases the planar four-bar mechanism and the trajectory of the calculated coupler point path.

FIGURE E.3.9 Planar four-bar mechanism.

In this analysis, we examine the motion of a slider-crank mechanism by plotting the slider's position, velocity, and acceleration against the crank's angular displacement for a full crank rotation The crank operates with an angular velocity of 1 rad/s and an angular acceleration of 0.1 rad/s² Additionally, we will simulate the mechanism's motion to visualize these dynamics effectively.

Known Information: Figure E.3.11, Appendix G.2 MATLAB, and SimMechanics files.

The solution approach utilizes specified values for vectors W1, V1, U1, and G1 from the Appendix G.2 MATLAB file, employing default settings for crank rotation ranging from 1° to 360° with a 1° increment The simulation of the slider-crank mechanism is initiated through the MATLAB file, resulting in a graphical user interface that generates compatible files for Microsoft Excel These files contain crucial data on the position, velocity, and acceleration of the slider link at each crank rotation increment Additionally, Figure E.3.12 presents the plots of slider position, velocity, and acceleration in relation to the crank displacement angle for the analyzed mechanism.

FIGURE E.3.10 Planar four-bar mechanism and coupler point path.

Problem Statement: Attaching the dyad to the planar four-bar mechanism in

Example 3.5 (resulting in the Stephenson III mechanism in Figure E.3.13), deter- mine the positions of vector V 1 * over a complete crank rotation (at 45° rotation increments) Also simulate the motion of this mechanism.

Known Information: Example 3.5, Figure E.3.13, Appendix G.3 MATLAB, and

The solution approach utilizes vector values from Example 3.5, along with the specified values for vectors V 1 * and U 1 * found in Figure E.3.13, as detailed in the Appendix G.3 MATLAB file The crank rotation is set to range from 1° to 360° with increments of 1° Upon executing the MATLAB file, a graphical user interface is generated, simulating the motion of the Stephenson III mechanism and producing files compatible with Microsoft Excel These files contain angular position, velocity, and acceleration data for the U 1 * − V 1 * dyad at each crank rotation increment To define the position of vector V 1 *, one can calculate its starting point (W 1 + L 1) and its displacement angle, with additional data for vector V 1 * at 45° crank rotation increments provided in Table E.3.4.

FIGURE E.3.12 Slider (a) position, (b) velocity, and (a) acceleration plots.

Summary

This chapter utilizes the vector loop closure method to derive kinematic equations for various mechanisms, including the planar four-bar, slider-crank, and Stephenson III mechanisms By representing the closed loop of mechanism link vectors in complex form, the method allows for the expansion and grouping of these vectors into real and imaginary components This results in a set of two equations that facilitate the calculation of two unknown variables within the mechanism Consequently, this approach provides comprehensive equations that detail the position, displacement, velocity, and acceleration of the mechanisms.

TABLE E.3.4 Position Data for Stephenson III Mechanism Dyad Vector V 1 *

The Stephenson III mechanism involves formulating each link of the mechanism and deriving its displacement equations By taking the first and second derivatives of these equations, we can obtain the corresponding velocity and acceleration equations The unknown variables derived from these kinematic equations can then be utilized in further vector loop equations to determine the positions, displacements, velocities, and accelerations of other significant locations within the mechanism.

The Newton–Raphson method is a traditional technique used to solve vector loop kinematic equations Users can download MATLAB and SimMechanics files from www.crcpress.com, enabling them to perform displacement, velocity, and acceleration analyses on various mechanisms, including planar four-bar, slider-crank, and Stephenson III mechanisms, as well as simulate their motion.

3.1 Derive vector loop-based displacement, velocity, and acceleration equations for the Watt II mechanism in Figure P.3.1 Include displacement velocity and acceleration equations for rigid-body points A and B.

3.2 Figure P.3.2 illustrates a Watt II mechanism used in a concept adjustable chair

To determine the angular displacements of the head-rest (V1) and leg-rest (V1*) components, analyze the Watt II mechanism as two distinct planar four-bar linkages Given a 20° angular displacement of the base-rest component, which consists of vectors (U1) and (W1*), calculate the corresponding angular movements for the other components.

3.3 Produce piston velocity versus crankshaft rotation plots for the engine linkage illustrated in Figure P.3.3 at crankshaft speeds of 1000, 1750, and 3500 rpm (104.72, 183.26, and 366.52 rad/s) The ratio of the coupler length to the crank length is 3:1.

3.4 Derive vector loop-based displacement, velocity, and acceleration equations for the Watt I mechanism in Figure P.3.4 Include displacement velocity and acceleration equations for rigid-body points A, B, and C.

3.5 Calculate the rotation range of the designated coupler link in the planar four- bar folding chair linkage from the fully opened position to the fully closed position illustrated in Figure P.3.5.

FIGURE P.3.1 Watt II mechanism (with rigid-body points A and B).

3.6 Determine if the two folding chair linkage designs illustrated in Figure P.3.6 will produce properly folded chairs and if not, why?

3.7 Produce a velocity magnitude versus crank angular displacement plot and an acceleration magnitude versus crank angular displacement plot for point p of the leveling crane illustrated in Figure P.3.7 over a −35° crank rotation range (where crank angular velocity and acceleration are −1 rad/s and −0.1 rad/s 2 , respectively).

FIGURE P.3.3 Slider-crank mechanism used in a crankshaft-connecting rod-piston mechanism.

Upright chair position Reclined chair position

FIGURE P.3.2 Watt II mechanism used in an adjustable chair.

FIGURE P.3.4 Watt I mechanism (with rigid-body points A, B, and C).

FIGURE P.3.5 Folding chair linkage in (left) fully opened and (right) fully closed positions.

FIGURE P.3.6 Folding chair linkage designs.

3.8 Derive vector loop-based displacement, velocity, and acceleration equations for the Stephenson I mechanism in Figure P.3.8 Include displacement velocity and acceleration equations for rigid-body points A, B, and C.

3.9 Figure P.3.9 illustrates a planar four-bar mechanism used as a brake Calculate the driver displacement angles corresponding to the minimum and maximum transmission angle (and subsequently the minimum and maximum buckling load) From Figure 3.1, a transmission angle (τ) equation is U V U V⋅ == cos τ

FIGURE P.3.8 Stephenson I mechanism (with rigid-body points A, B, and C).

Normal breaking force Brake pad

FIGURE P.3.9 Planar four-bar linkage used as a braking mechanism.

3.10 Derive vector loop-based displacement, velocity, and acceleration equations for the Stephenson II mechanism in Figure P.3.10 Include displacement veloc- ity and acceleration equations for rigid-body points A, B, and C.

1 Kimbrell, J.T 1991 Kinematics Analysis and Synthesis, Chapter 2 New York: McGraw-Hill.

2 Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and

Synthesis, Vol 2, pp 10–16 Englewood Cliffs, NJ: Prentice Hall.

3 Chapra, S.C and R.P Canale 1998 Numerical Methods for Engineers, Chapters 5–8 New York: McGraw-Hill.

Norton, R.L 2008 Design of Machinery, 4th edn., Chapter 4 New York: McGraw-Hill. Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and Synthesis, Vol 2, p 64 Englewood Cliffs, NJ: Prentice Hall.

Waldron, K.J and G.L Kinzel 2004 Kinematics, Dynamics and Design of Machinery, 2nd edn., Chapter 5 Upper Saddle River, NJ: Prentice Hall.

Wilson, C.E and J Peter Sadler 2003 Kinematics and Dynamics of Machinery, 3rd edn., Chapter 2 Upper Saddle River, NJ: Prentice Hall.

FIGURE P.3.10 Stephenson II mechanism (with rigid-body points A, B, and C).

Kinematic Synthesis and Planar Four-Bar Motion Generation

Introduction to Kinematic Synthesis

Kinematic synthesis focuses on determining the necessary dimensions of a mechanism to achieve specific output parameters, such as link lengths, positions, and joint coordinates These output parameters include link positions, path points, and displacement angles This process is often referred to as dimensional synthesis, as it primarily involves calculating mechanism dimensions based on desired outputs Unlike kinematic analysis, where the dimensions are known and outputs are calculated, kinematic synthesis starts with known output parameters to derive the corresponding dimensions.

Kinematic synthesis is divided into three main subcategories: motion generation, path generation, and function generation Motion generation focuses on calculating mechanism dimensions to achieve specific rigid-body positions, while path generation involves determining dimensions based on prescribed output parameters as rigid-body path points In function generation, the goal is to calculate dimensions to achieve specific link displacement angles The subsequent chapters (4 to 6) are dedicated to exploring planar motion, path generation, and function generation in detail.

Research in kinematic synthesis showcases a diverse range of qualitative and quantitative methods for generating motion, paths, and functions Qualitative methods, such as graphical techniques, offer extensive insights with minimal computational effort In contrast, quantitative methods involve mathematical models that can be solved through analytical or numerical approaches using solution algorithms and root-finding techniques The kinematic synthesis methods discussed in this textbook are exclusively quantitative, as intended by the authors.

* Another basic description of kinematic synthesis is the design of a mechanism to produce a desired output motion for a given input motion.

† Mechanism links are also called “rigid-bodies” because they are generally assumed rigid (nondeform- ing) in kinematic synthesis.

‡ In four-bar motion and path generation, the coupler link is the rigid-body for which positions and path points are prescribed.

Me cha ni sm o u t p ut para m e te rs

M ech anism output param eter s

FIGURE 4.1 (a) Kinematic analysis versus (b) kinematic synthesis.

Branch and Order Defects

Although motion and path generation ensure that the synthesized mechanisms will achieve or approximate prescribed rigid-body output parameters, they do not guar- antee the following:

• The synthesized mechanism will achieve or approximate the prescribed rigid-body output parameters without a change in its original assembly con- figuration (thus requiring mechanism disassembly).

• The synthesized mechanism will achieve or approximate the prescribed rigid-body output parameters in the intended order.

Ri g id- b od y path poi nt s

Li n k a ngu l a r d isp l acem e n ts where γ j = f(β j ) where γ j = f (β j )

FIGURE 4.2 Subcategories of kinematic synthesis.

These two uncertainties are defects inherent in motion and path generation: mechanism synthesis requiring rigid-body positions and rigid-body path points, respectively.

A branch defect in a planar four-bar mechanism occurs when different assembly configurations, such as a-b-c-d and a-b-c*-d, are needed to achieve specific rigid-body positions or path points This defect can create design challenges, as it necessitates disassembly and reassembly of the mechanism during operation to access all required outputs For instance, as illustrated in Figure 4.3b, the rigid-body positions 1-2-3 are reachable through configuration a-b-c-d, while position 1* is accessible only via a-b-c*-d Consequently, if the design requires the mechanism to continuously achieve the positions in the order 1*-1-2-3, it cannot be accomplished with a single assembly configuration due to the inherent branch defect.

The second identified issue is known as an order defect, which arises when the prescribed positions or path points for a rigid-body mechanism are not achieved in the intended sequence For instance, if the desired path points are specified as 1-2-3-4-1 but the mechanism follows the sequence 1-2-4-3-1, it can lead to significant problems, particularly when the order is crucial for tracing a specific coupler curve The sequence in which these rigid-body path points are executed plays a vital role in determining the resulting path profile.

Branch and order defect elimination by way of constraint equations, graphi- cal methods, or particular prescribed values are often employed in motion and

* The assembly configurations a-b-c-d and a-b-c*-d of the planar four-bar mechanism are called the open and crossed configurations, respectively.

† The branch defect is distinct from the circuit defect (which was briefly noted in Example 3.1 in Chapter

3) although the branch and circuit defect both require mechanism disassembly With the circuit defect, the mechanism is reassembled in another position of the same assembly configuration (and not from one assembly configuration to another as with the branch defect). b c a d

In this article, we explore four-bar assembly configurations and address the issue of branch defects We present path generation models designed to create motion and path generator solutions that are free from both branch and order defects Additionally, we introduce various approaches and constraints for eliminating these defects, which are further elaborated in Chapters 5, 7, and throughout this chapter.

Motion Generation: Three, Four, and Five Precision

Figure 4.5a depicts a planar four-bar mechanism, showcasing various coupler positions defined by the coordinates of rigid-body point p j and the rigid-body displacement angle α j In the context of quantitative motion generation, the dimensions of the mechanism are calculated to achieve precise positions Additionally, Figure 4.5b demonstrates the operational sequence of an aircraft landing gear, highlighting the importance of guiding the landing gear from within the airframe to its designated position for effective operation.

* Prescribed rigid-body positions are also called precision positions.

FIGURE 4.4 Paths associated with point orders (a) 1–2–3–4–1 and (b) 1–2–4–3–1.

The four-bar mechanism is essential for ensuring proper wheel contact with the ground, particularly in applications like aircraft landing gear This section discusses the planar four-bar motion generation equations, which are crucial for determining the necessary dimensions to achieve specific precision positions, whether three, four, or five Figure 4.6 demonstrates the dyads of a planar four-bar mechanism, showcasing both the starting (position 1) and displaced positions (position j) The left and right dyads consist of the vector chains W-Z and U-S, respectively By calculating the vector sum between these positions for each dyad, we derive the vector loop closure equations The counterclockwise vector sum for each dyad, beginning with W1 for the left-side dyad and U1 for the right-side dyad, leads to the formulation of these equations for the four-bar mechanism dyads.

After factoring the terms for the starting dyad position, the resulting standard form loop closure equations become

FIGURE 4.6 Four-bar mechanism in starting and displaced (dashed) positions.

Expanding Equation 4.3 and grouping the real and imaginary terms as separate equations produces the equation set

W 1cos cosθ( β j −1) − W 1sin sinθ β j + Z 1cos cosφ( α j −1) − Z 1sin sinφ α j = P j 11

1 1 1 1 1 1 cos sin cos cos sin sin cos cos s δ θ β θ β φ α φ j j j j

After specifying W 1cosθ =W 1 x , W 1sinθ =W 1 y , Z 1cosφ =Z 1 x , and Z 1sinφ =Z 1 y , Equation 4.5 becomes

1 1 cos sin cos sin cos cos β − β α α δ

Likewise, after separating the real and imaginary terms in the right-side dyad Equation 4.4 and specifying U 1cos σ =U 1 x , U 1 sin σ =U 1 y , S 1 cos ψ =S 1 x , and

1 1 cos sin cos sin cos cos γ − γ α α δ

When expressed in matrix form for N precision positions (where j = 2, 3, … N ), Equations 4.6 and 4.7 become the following equations, respectively: cos sin cos sin sin cos sin cos cos β β α α β β α α β j j j j j j j j

 sin cos sin sin cos sin cos β α α β β α α

 (4.8) cos sin cos sin sin cos sin cos cos γ γ α α γ γ α α γ j j j j j j j j

 sin cos sin sin cos sin cos γ α α γ γ α α

Although as written, Table 4.1 includes the design variables for the left-side dyad only, this table also corresponds to the right-side dyad (replacing variables W 1x , W 1y ,

With three precision positions (N = 3), Equation 4.8 generates four linear simultaneous equations with four scalar unknowns, solvable via Cramer’s rule The angles β2 and β3 serve as “free choices” alongside the precision positions, while the corresponding angles for the right-side dyad are γ2 and γ3 This configuration allows the mechanism to achieve the precision positions accurately, while also adhering to the specified dyad displacement angles β and γ, a process known as motion generation with prescribed timing By integrating prescribed timing, the order problem is effectively addressed, ensuring that the synthesized mechanism attains the precision positions in the desired sequence.

When dealing with more than three precision positions in motion generation, the number of simultaneous equations surpasses the scalar dyad unknowns, rendering Equations 4.8 and 4.9 unsolvable by Cramer’s rule For four precision positions, an algorithm outlined in Appendix A.1 allows for the analytical calculation of β3 and β4 based on a given β2 value By utilizing any four of the resulting six simultaneous equations from Equation 4.8, one can solve for W1x, W1y, Z1x, and Z1y A range of β2 values (0 ≤ β2 ≤ 2π) leads to corresponding calculations for W1 and Z1, revealing that ungrounded mechanism joints relate to -Z1 while grounded mechanism joints correspond to - (ZW1 + 1) This results in a locus of moving and fixed pivots, known as Burmester circle and center points, respectively, confirming the existence of infinite planar four-bar mechanism solutions for four precision positions, as detailed in Table 4.1.

A planar four-bar mechanism can be constructed from any two pairs of circle points

Maximum Number of Solutions for Equation 4.8 for Three, Four, and Five Precision Positions

Positions ( ) N Number of Equations Total Number of Unknowns

The article provides instructions for using a MATLAB® file to calculate Burmester circles and center points, with detailed user guidance available in Appendix A.2 The MATLAB file can be downloaded from www.crcpress.com.

Problem Statement: Synthesize a planar four-bar mechanism to guide the landing gear through the three precision positions in Figure E.4.1.

Known Information: Equations 4.8 and 4.9 and Table E.4.1.

To determine the dyad vectors for the planar four-bar mechanism, the prescribed coupler points \( p_j \), vectors \( P_{j1} \), and vector angles \( \delta_j \) are utilized These variables, along with the dyad displacement angles listed in Table E.4.1, are incorporated into Equations 4.8 and 4.9 By applying Cramer’s rule, the calculations yield the dyad vectors \( W_1 \), \( Z_1 \), \( U_1 \), and \( S_1 \).

Table E.4.2 presents the calculated components of the planar four-bar mechanism dyad, while Figure E.4.2 showcases the resulting mechanism This analytically derived mechanism ensures precise positioning.

* One circle point–center point pair corresponds to the W 1 − Z 1 dyad and the U 1 − S 1 dyad corresponds to the other pair.

† P j1 = p j − p 1 and δ j is the angle vector P j1 makes with the positive x-axis. p 3 p 2 p 1 iY p 21 p 31 α 3 α 2 X δ 3 δ 2

FIGURE E.4.1 Three precision positions and position parameters for a landing gear.

Problem Statement: Synthesize a planar four-bar mechanism to guide the landing gear through the four precision positions in Figure E.4.3.

Known Information: Appendix A.2 MATLAB file and Table E.4.3.

In the MATLAB file, precision position variables P j1, δ j (derived from the p j values in Table E.4.3), and α j are utilized The circle and center points depicted in Figure E.4.4 represent the β 2 - β 3 - β 4 solution for the first mechanism branch, contrasting with the alternate β 2 - β̃ 3 - β̃ 4 solution found in the second mechanism branch.

* The Appendix A.2 MATLAB file calculates circle and center point solutions for both mechanism branches.

FIGURE E.4.2 Synthesized planar four-bar motion generator.

Landing Gear Precision Position Parameters and Dyad Displacement Angles

Calculated Four-Bar Mechanism Dyad Variables

Table E.4.4 presents the mechanism dyad components for selected circle and center point pairs derived from calculated curves, as shown in Figure E.4.5 To reach positions 2, 3, and 4, vector W1, the designated driver link, is rotated by angles of 248°, 242.6542°, and 216.7028° (β2, β3, and β4) The resulting planar four-bar mechanism is depicted in Figure E.4.6 Since the circle and center points are calculated analytically, the mechanism achieves the desired precision positions accurately.

TABLE E.4.3 Landing Gear Precision Position Parameters and Dyad Displacement Angles

FIGURE E.4.3 Four precision positions and position parameters for a landing gear.

Selected Four-Bar Mechanism Dyad Variables

FIGURE E.4.4 Calculated circle and center points.

FIGURE E.4.5 Four-bar mechanism selection from circle and center points.

FIGURE E.4.6 Synthesized planar four-bar motion generator.

Cramer’s rule is not applicable for Equation 4.8 with five precision positions, leading to a finite number of solutions, as detailed in Table 4.1 An algorithm for analytically calculating β2 to β5 is provided in Appendix B.1, which indicates that motion generation with five precision positions can yield four, two, or no real values for β2 to β5 This results in corresponding pairs of circle-center points or motion generator solutions Additionally, Appendix B.2 offers user instructions for a MATLAB file designed for calculating circle and center points for five position synthesis, which can be downloaded at www.crcpress.com.

Problem Statement: Synthesize a planar four-bar mechanism to guide the landing gear through the five precision positions in Figure E.4.7.

Known Information: Appendix B.2 MATLAB file and Table E.4.5.

Solution Approach: Precision position variables P j1 and δ j (calculated from the p j values in Table E.4.5) and α j are used in the MATLAB file.

The analysis reveals a unique solution for the planar four-bar mechanism, as detailed in the Appendix B.2 MATLAB file results Table E.4.6 presents the calculated dyad components of the mechanism To attain positions 2, 3, 4, and 5, the designated driver link, vector W1, is rotated by angles of −15.8812°, −27.5064°, and −43.6940°.

−52.5036° ( β 2 , β 3 , β 4 , and β 5 ), respectively Figure E.4.8 illustrates the synthesized p 5 p 4 p 3 p 2 p 1 X iY

The five precision positions and corresponding parameters for a landing gear are based on a planar four-bar mechanism By employing analytical calculations for the circle and center points, the mechanism ensures that the precision positions are achieved with accuracy.

A circuit defect is identified between the third and fourth positions of the synthesized landing gear mechanism Figure E.4.9 illustrates the relationship between follower displacement angle and crank displacement angle during the motion sequence The observed discontinuity in this plot indicates the presence of the defect, necessitating the disassembly and reassembly of the mechanism between these positions to ensure all five precision positions are achieved.

An alternative approach for synthesizing a planar four-bar mechanism with five precision positions involves breaking down the five-position challenge into two or more four-position problems This method allows for the calculation of the loci of circles and center points for each of these simplified problems.

TABLE E.4.5 Landing Gear Precision Position Parameters and Dyad Displacement Angles

Calculated Four-Bar Mechanism Dyad Variables

The synthesized planar four-bar motion generator allows for the identification of common intersection points among various curves, indicating the presence of a solution that satisfies all five precision positions By segmenting the five precision positions from Example 4.3 into three synthesis problems—(1-2-3-4, 1-2-3-5, and 1-2-4-5)—and plotting the center point loci, two fixed pivot solutions emerge as common intersection points Additionally, plotting the circle point loci also reveals two moving pivot solutions, further confirming the findings in Example 4.3.

Pos 5 Crank displacement angle (deg.)

FIGURE E.4.9 Follower versus crank displacement angle for synthesized planar four-bar motion generator.

FIGURE E.4.10 Center point solutions for five-position synthesis problem in Example 4.3.

FIGURE E.4.11 Circle point solutions for five-position synthesis problem in Example 4.3.

Branch and Order Defect Elimination: Three, Four, and Five Precision Positions

THREE, FOUR, AND FIVE PRECISION POSITIONS

Branch and order defects are common challenges in motion and path generation; however, various conventional practices and construction methods can effectively produce defect-free motion and path generators Key techniques include prescribed timing, Filemon's construction, and Waldron's construction, all of which help ensure optimal performance in generating motion paths.

Motion and path generation with prescribed timing effectively prevents order defects by ensuring that each coupler position and its corresponding dyad displacement angles are predetermined The planar four-bar motion generation equations outlined in Section 4.3 (Equations 4.8 and 4.9) incorporate displacement angle variables for both mechanism dyads, specifically angles β and α for dyad W-Z, and angles γ and α for dyad U-S By defining both the precision positions and their related dyad displacement angles, the sequence in which these precision positions are reached is consistently maintained, as demonstrated in Example 4.1.

In the early 1970s, Filemon developed a construction method that ensures branch defect-free planar four-bar motion generator solutions derived from circle and centerpoints Her research demonstrates that if the moving pivot of the mechanism's driving link is chosen outside the wedge-shaped region defined by her method, the planar four-bar motion generator will successfully navigate all precision positions without disassembly, resulting in a nonbranching solution This construction presupposes the selection of an output link, with the follower link's moving pivot and fixed pivot denoted as b1 and b0, respectively The relative inverse displacements of the follower link create a planar wedge-shaped region, and the possible positions of the follower's fixed pivot relative to the moving pivot are calculated using the formula b0j = Mj b0.

Equation 4.10 calculates the ground link displacements of the inverted four-bar mechanism using the precise position variables α j and P j1 When choosing the input link, it is essential to first construct the wedge-shaped region and identify the moving pivot for the input.

* Filemon’s construction ensures that there will be no change of branch if the motion generator solution is a Grashof crank-rocker or drag-link mechanism.

If the angle of a wedge-shaped region is 180° or greater, it occupies the entire 2D space, making Filemon’s construction inapplicable To utilize this construction effectively, a link must be selected from outside the region Filemon’s method can be employed for motion generation with three, four, or five precision positions, and is extendable to more complex scenarios.

Problem Statement: Calculate a nonbranching landing gear mechanism for

Known Information: Example 4.1 and Equation 4.10.

In Example 4.1, the calculated follower (U S 1 − 1) serves as the selected output link, with the coupler displacement angles α j and the x- and y-components of the coupler point vectors P j1 provided in Table E.4.1 Utilizing this information in Equation 4.10, the two positions of the fixed pivot b 0 (b 02 and b 03) are determined, establishing the borders of a planar wedge-shaped region formed by the lines connecting b 1 to b 02 and b 03 It is essential for the moving pivot of the driving link dyad (W Z 1 − 1) to remain outside this wedge-shaped area Figure E.4.13 demonstrates the (W 1 − Z 1) solution from Equation 4.8 with angles β 2 = 10° and β 3 = 40°, depicting a Grashof crank-rocker four-bar motion generator that is free from branch defects.

* Because P j1 = p j − p 1 , if p 1 is specified to p 1 = ( ) 0 0 , , then the x- and y-components of the precision point vectors P j1 are identical to the x- and y-components of the precision points p j b 1

FIGURE E.4.12 Wedge-shaped region borders formed by b 02 and b 03 (form U 1 ).

In the mid-1970s, Waldron developed a method for constructing feasible regions to select nonbranching planar four-bar motion generators for three precision positions His research indicates that follower link moving pivots chosen outside the three circles generated by his method, as well as those in overlapping regions of any two circles, lead to nonbranching motion generator solutions Once a suitable follower link dyad is determined using Waldron’s construction method, a corresponding crank link dyad can then be calculated based on Filemon’s construction.

In the context of three precision positions, there are three rotation centers, known as poles, that facilitate the rotation from position 1 to position 2 (pole p12), from position 1 to position 3 (pole p13), and from position 2 to position 3 (pole p23) These poles create a geometric structure referred to as a pole triangle (∆p p p12 13 23) Additionally, reflecting pole p23 across the side connecting poles p12 and p13 generates a new pole, termed the image pole (p′23).

In Waldron’s construction, each side of the triangle ∆p p p 12 13 23 is equal to a diameter of each circle.

The rotation pole equation corresponding to the two rigid-body positions in Figure 4.6 is defined as polep jk j i k i j i

1 δ δ δ α α (4.12) where P 11= =δ1 α1=0 [19] Equation 4.12 is used to calculate pole p 12 , pole p 13 , and pole p 23

FIGURE E.4.13 Wedge-shaped region with calculated dyads W 1 − Z 1 and U 1 − S 1

Problem Statement: Calculate a nonbranching landing gear mechanism for

Example 4.1 using Waldron’s construction and Filemon’s construction.

Known Information: Example 4.1 and Equation 4.12.

The solution approach outlined in Table E.4.7 details the calculation of poles p 12, p 13, and p 23 using Equation 4.12, based on the data from Example 4.1 By reflecting pole p 23 across the triangle edge that connects poles p 12 and p 13, we obtain the image pole p ′ 23, as illustrated in Figure E.4.14 and Table E.4.7 Additionally, Figure E.4.15 presents the three-circle diagram derived from Waldron’s construction, indicating that suitable follower link moving pivots are located either outside the three circles or within the overlapping regions of any two circles.

The analysis of Figure E.4.15 indicates that the follower link dyad (U1 - S1) solution from Example 4.1 is effective, as its moving pivot is positioned outside the three circles Additionally, the crank link dyad (W1 - Z1) solution from Example 4.4 is also deemed satisfactory since its moving pivot remains outside the wedge-shaped area Consequently, the resulting four-bar motion generator, identified as a Grashof crank-rocker, is free from branch defects.

FIGURE E.4.14 Calculated poles with image pole.

Summary

Kinematic analysis focuses on calculating the displacements, velocities, and accelerations of a mechanism when its dimensions are known In contrast, kinematic synthesis aims to determine the necessary dimensions of a mechanism to achieve specified displacements, velocities, and accelerations This process is divided into three subcategories: motion generation, path generation, and function generation Motion generation involves setting specific positions for mechanism links, particularly the coupler link positions in a planar four-bar mechanism Path generation in a planar four-bar system requires defining the coupler link path points, while function generation prescribes the crank and follower displacement angles The defined coupler positions are often referred to as precision positions.

Branch and order defects are fundamental issues in planar four-bar motion generation A mechanism with a branch defect fails to reach its precision positions in one assembly configuration, while one with an order defect does not achieve these positions in the correct sequence To address these defects, techniques such as prescribed timing, Filemon’s construction, and Waldron’s construction are commonly utilized.

The formulation of simultaneous equation sets for planar four-bar motion generation utilizes vector loop closure for both the initial and displaced positions of a mechanism dyad These equations are relevant for synthesizing mechanisms with three, four, or five precision positions Detailed solution algorithms for four and five position synthesis can be found in the MATLAB files located in Appendix A.2 and B.2, respectively.

A locus of fixed and moving pivots (called Burmester center and circle points) can be produced given four precision positions Burmester points confirm the existence

The three-circle diagram illustrates the calculated dyad U1 - S1, showcasing an infinite array of planar four-bar mechanism solutions tailored for four-position synthesis Additionally, the loci of the center and circle points can facilitate the development of four-bar mechanisms for five-position synthesis.

4.1 Synthesize two planar four-bar mechanisms for the three hatch positions in

Figure P.4.1 Through displacement analyses, determine if the two mecha- nisms are circuit and branch defect–free over the crank rotation ranges.

4.2 Synthesize two planar four-bar mechanisms for the three load–unload bucket positions in Figure P.4.2 Through displacement analyses, determine if the two mechanisms are circuit and branch defect–free over the crank rotation ranges.

4.3 Synthesize two planar four-bar mechanisms for the three wing positions in

Figure P.4.3 Through displacement analyses, determine if the two mecha- nisms are circuit and branch defect–free over the crank rotation ranges.

4.4 Synthesize two planar four-bar mechanisms for the three lower cutting blade positions in Figure P.4.4 Through displacement analyses, determine if the two mechanisms are circuit and branch defect–free over the crank rotation ranges.

4.5 Synthesize two planar four-bar mechanisms for the three brake pad positions in Figure P.4.5 Through displacement analyses, determine if the two mecha- nisms are circuit and branch defect–free over the crank rotation ranges.

4.6 Synthesize two planar four-bar mechanisms for the three stamping part posi- tions in Figure P.4.6 Through displacement analyses, determine if the two mechanisms are circuit and branch defect–free over the crank rotation ranges.

4.7 Produce and plot the circle and center points (for both mechanism branches) for the precision positions given in Table P.4.1.

4.8 Produce and plot the circle and center points (for the first mechanism branch) at the β2 values β2= °15 30 150, °… ° for the folding wing positions in Figure P.4.7 Determine if a mechanism solution exists among the circle and center points that is order defect–free (hint: consider results in the file beta_vs_theta.csv).

FIGURE P.4.2 Three load–unload bucket positions.

4.9 Synthesize a planar four-bar mechanism for the four loading–unloading bucket positions in Figure P.4.8 using Burmester curves and Filemon’s construction Verify that the mechanism is branch defect–free and determine (through a dis- placement analysis) if it is circuit defect–free over the crank rotation range.

4.10 Synthesize a planar four-bar mechanism for the four hatch positions in

In Figure P.4.9, Burmester curves and Filemon’s construction are utilized to analyze the mechanism It is essential to confirm that the mechanism is free from branch defects Additionally, a displacement analysis must be conducted to verify whether the mechanism is also free from circuit defects throughout the crank's rotation range.

FIGURE P.4.5 Three brake pad positions.

Lower blade (tool closed) Upper blade p 1 = (0, 0) p 2 = (0.6947, 0.3449) p 3 = (1.0111, 0.6790)

FIGURE P.4.4 Three lower cutting blade positions.

4.11 Repeat Problem 4.4 using Waldron’s construction to verify that the mecha- nisms are branch defect–free.

4.12 Repeat Problem 4.5 using Waldron’s construction to verify that the mecha- nisms are branch defect–free.

4.13 Synthesize a planar four-bar mechanism for the five digger bucket positions in

Figure P.4.10 Determine (through a displacement analysis and/or Filemon’s construction) if the mechanism is branch, order, and circuit defect–free over the crank rotation range.

4.14 Synthesize a planar four-bar mechanism for the five latch positions in

Figure P.4.11 Determine (through a displacement analysis and/or Filemon’s construction) if the mechanism is branch, order, and circuit defect–free over the crank rotation range.

FIGURE P.4.6 Three stamping part positions.

FIGURE P.4.8 Four load–unload bucket positions.

FIGURE P.4.10 Five digger bucket positions.

4.15 Synthesize a planar four-bar mechanism for the five gripper positions in

Figure P.4.12 Determine (through a displacement analysis and/or Filemon’s construction) if the mechanism is branch, order, and circuit defect–free over the crank rotation range.

4.16 Synthesize a planar four-bar mechanism for the five assembly component positions in Figure P.4.13 Determine (through a displacement analysis and/or Filemon’s construction) if the mechanism is branch, order, and circuit defect– free over the crank rotation range.

4.17 Synthesize a planar four-bar mechanism for the five stamping part positions in Figure P.4.14 Determine (through a displacement analysis and/or Filemon’s construction) if the mechanism is branch, order, and circuit defect–free over the crank rotation range.

4.18 Verify the solution in Problem 4.13 by overlapping Burmester curves for preci- sion positions 1–2–3–4, 1–2–3–5, and 1–2–4–5.

FIGURE P.4.14 Five stamping part positions.

FIGURE P.4.13 Five assembly component positions.

4.19 Verify the solution in Problem 4.15 by overlapping Burmester curves for preci- sion positions 1–2–3–4, 1–2–3–5, and 1–2–4–5.

4.20 Verify the solution in Problem 4.17 by overlapping Burmester curves for preci- sion positions 1–2–3–4, 1–2–3–5, and 1–2–4–5.

1 Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and

Synthesis, Vol 2, p 51 Englewood Cliffs, NJ: Prentice Hall.

2 Martin, G.H 1969 Kinematics and Dynamics of Machines, p 319 New York: McGraw-Hill.

3 Erdman, A.G 1993 Modern Kinematics: Developments in the Last Forty Years,

Chapters 4–5 New York: John Wiley and Sons.

4 Kimbrell, J.T 1991 Kinematics Analysis and Synthesis, Chapter 7 New York: McGraw-Hill.

6 Mallik, A.K., A Ghosh, and G Dittrich 1994 Kinematic Analysis and Synthesis of

Mechanisms, pp 306–307 Boca Raton, FL: CRC Press.

8 Norton, R.L 2008 Design of Machinery, 4th edn., Chapter 5 New York: McGraw-Hill.

9 Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and

Synthesis, Vol 2, pp 179–180 Englewood Cliffs, NJ: Prentice Hall.

11 Kimbrell, J.T 1991 Kinematics Analysis and Synthesis, pp 189–192 New York: McGraw-Hill.

12 Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and

Synthesis, Vol 2, pp 199–204, 261–263 Englewood Cliffs, NJ: Prentice Hall.

13 McCarthy, M.J 2000 Geometric Design of Linkages, pp 110–111 New York: Springer.

14 Filemon, E 1972 Useful ranges of centerpoint curves for design of crank-and-rocker linkages Mechanism and Machine Theory 7: 47–53.

15 McCarthy, M.J 2000 Geometric Design of Linkages, pp 111–112 New York: Springer.

16 Waldron, K.L 1976 Elimination of the branch problem in graphical Burmester mecha- nism synthesis for four finitely separated positions Transactions of the ASME, Journal of Engineering for Industry 98: 176–182.

17 Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and

Synthesis, Vol 2, p 76 Englewood Cliffs, NJ: Prentice Hall.

18 McCarthy, M.J 2000 Geometric Design of Linkages, pp 51–52 New York: Springer.

19 Sandor, G.N and A.G Erdman 1984 Advanced Mechanism Design: Analysis and

Synthesis, Vol 2, pp 198–199 Englewood Cliffs, NJ: Prentice Hall.

Planar Four-Bar and Multiloop Path and Motion Generation

Path Generation versus Motion Generation

Figure 5.1a depicts a planar four-bar mechanism along with the curve traced by the coupler point p1, which is essential for quantitative path generation This process involves calculating the necessary dimensions of the mechanism to achieve or approximate specific coupler path points To illustrate this concept further, Figure 5.1b shows a level-luffing crane, designed to keep its hook at a constant level during operation, requiring the crane's extremity to follow a horizontal path The planar four-bar path generation equations discussed in this chapter are instrumental in determining the dimensions needed for the planar four-bar mechanism to accurately approximate precision points.

There is a distinct difference between path generation and motion generation

In motion generation, precision positions are either achieved or approximated, while path generation focuses on achieving or approximating precision points Unlike motion generation, where coupler positions are critical, path generation primarily prescribes coupler path points, rendering coupler displacement angles less significant The planar four-bar motion generation equations from Chapter 4 can be effectively utilized for path generation involving three, four, or five precision points, as these mechanisms meet both prescribed coupler points and displacement angles Users have the flexibility to specify coupler displacement angles arbitrarily; however, path generation for more than five precision points necessitates numerical optimization methods due to the lack of analytical solutions for Equations 4.8 and 4.9 Additionally, the inclusion of coupler displacement angles as unknowns in these equations introduces nonlinearity, complicating the solution process This chapter presents path generation equations that incorporate coupler displacement angles as variables to be calculated.

* In path generation, the prescribed rigid-body path points are also called precision points.

Coupler Curves and Dwell Motion

The coupler links of planar four-bar and slider-crank mechanisms exhibit complex motion, combining simultaneous rotation and translation, unlike the pure rotation of crank and follower links or the pure translation of the slider link This intricate motion allows coupler links to trace higher-order curves, with the ability to produce coupler curves of up to order six.

The desired features of a coupler curve vary based on specific mechanical design applications; however, certain characteristics are commonly utilized across various designs Among the most fundamental and widely employed curve features are straight lines and circular arcs, which play a crucial role in enhancing design functionality.

Figure 5.2 includes the Robert’s straight-line linkage as well as a linkage that traces a circular arc [1] †

The shape of a coupler curve is influenced by the proportions of the mechanism link lengths and the specific location of the coupler point from which the curve is traced Consequently, a planar four-bar linkage can generate an infinite variety of unique coupler curves A coupler curve atlas showcases various coupler curves alongside their corresponding four-bar mechanism link proportions, offering designers valuable starting points for mechanism dimensioning in design applications Figure 5.3 presents several variations of planar four-bar coupler curves, highlighting different coupler point locations and link length proportions.

* As derived by Wunderlich (1963), the maximum order of a coupler curve m for a mechanism with n links connected with revolute joints only is m = ⋅ 3 2 ( ) n 2 1 −

† Other straight-line tracing linkages include the Watt, Chebyschev, Evans, Hart, Peaucellier, and Hoeken straight-line linkages. p N p j p 1 p

FIGURE 5.1 (a) Four-bar mechanism and coupler path points, and (b) level-luffing crane mechanism.

The dwell region is a notable feature of coupler curves, characterized by constant curvature, which can manifest as either a circular arc or a straight line segment In the context of a four-bar mechanism, both types of dwell regions can be traced, making them significant for designers The key advantage of dwell regions is their ability to produce zero radial displacement despite a nonzero rotational input from the crank link, a phenomenon known as dwell motion This motion is especially valuable in various design applications, with cam-follower systems being a common example However, linkage-based mechanisms also provide a viable alternative for achieving dwell motion.

* An arc has a constant finite radius while a straight line has constant radius of infinity.

FIGURE 5.2 (a) Robert’s approximate straight-line mechanism and (b) circular arc mechanism. q r c d a

FIGURE 5.3 Coupler curve variation with (a) coupler point location and (b) link–length ratio.

Approximate Four-Bar Path and Motion Generation

According to Table 4.1, users can define up to five precision positions utilizing the analytical motion/path generation equations from Chapter 4 However, there may be instances when five precision points are insufficient to effectively represent the desired coupler curve or position sequence.

For mechanisms with more than five precision positions, numerical optimization methods are essential for determining unknown dyad variables, as outlined in Equations 4.8 and 4.9 Analytical motion and path generation equations allow for algebraic calculation of mechanism solutions, ensuring precise achievement of specified positions In contrast, numerical optimization approaches yield approximations of these precision positions, leading to structural errors, which are the finite differences between the expected and actual mechanism outputs.

Numerical optimization is often preferred when analytical solution methods become increasingly complex or constrained For instance, the analytical solution algorithm for five position synthesis, detailed in Appendix B.1, requires extensive computations, and there is no available analytical solution algorithm for more than five precision positions.

Numerical optimization offers advantages by allowing the formulation of equation systems that incorporate constraints to address order and branch defects These defects, often seen in analytical motion and path generation, can be effectively mitigated through this approach, as discussed in Chapter 4.

The motion and path (and function) generation equation systems presented from this chapter through the rest of the textbook are either unconstrained or constrained

* As a result, the phrase approximate synthesis is sometimes used to describe synthesis via numerical optimization.

The planar four-bar coupler curve features distinct dwell regions, which are analyzed using nonlinear equation systems To address unconstrained nonlinear equations, the trust-region interior-reflective Newton method is utilized, while the trust-region interior-point method is applied for solving constrained nonlinear equation systems.

This textbook incorporates both methods within the commercial mathematical analysis software MATLAB® It features a comprehensive library of MATLAB files, available for download at www.crcpress.com, tailored for each type of numerical optimization problem discussed.

The objective function for the W-Z dyad is formulated by summing the squares of Equation 4.6 For N precision points, this function is represented as f(X) = (W1x(cos βj - 1) - W1y sin βj + Z1x(cos αj - 1) - Z1y sin αj - Pj1 cos δj).

W y cosβ j W x sinβ j Z y cosα j Z x sinα j P j sinδ j 22

The variable X is defined as (W 1x, W 1y, Z 1x, Z 1y, α2 … α4, β2 … β4) for four precision points and extends to (W 1x, W 1y, Z 1x, Z 1y, α2 … α5, β2 … β5) for five precision points For N precision points, X is represented as (W 1x, W 1y, Z 1x, Z 1y, α2 … αN, β2 … βN) Additionally, an objective function for the U-S dyad is formulated by summing the squares from Equation 4.7, which for N precision points is expressed as f(X) = (U 1x(cos γj - 1) - U 1y sin γj + S 1x(cos αj - 1) - S 1y sin αj - Pj1 cos δj).

U y cosγ j U x sinγ j S y cosα j S x sinα j P j sinδ j 22

(5.2) where X = (U 1x , U 1y , S 1x , S 1y , γ2 … γ4) when four precision points are prescribed and

When five precision points are defined, the variable X is represented as (U 1x, U 1y, S 1x, S 1y, γ2 … γ5) For N precision points, X is expressed as (U 1x, U 1y, S 1x, S 1y, γ2 … γN) The αj values obtained from Equation 5.1, which pertains to the left-side dyad, serve as the prescribed values in Equation 5.2, related to the right-side dyad.

Equations 5.1 and 5.2 facilitate path generation and motion generation by specifying coupler points and displacement angles For motion generation using Equation 5.1, the dyad displacement angles βj are determined based on the number of precision positions prescribed Specifically, when four precision positions are defined, the variables are represented as X = (W1x, W1y, Z1x, Z1y, β2 … β4) In contrast, for five precision positions, the representation expands to X = (W1x, W1y, Z1x, Z1y, β2 … β5) For N precision positions, the general form is X = (W1x, W1y, ).

* This method was also used to solve the sets of linear and nonlinear equations presented in Chapters 5 through 8.

† The unknown objective function variables to be calculated are denoted by X =( ) …

To eliminate order defects, Inequality (5.3) should accompany Equation 5.1 as an inequality constraint for the driver dyad: β β β π j j

As expressed, Inequality (5.3) constrains the crank to counterclockwise rotation To constrain the crank to clockwise rotation, the inequality β j