Chemical Engineering - Apllied Mechanics

Joints and linkage - Part 1

Divided by topic

Each=joint=is characterised by an index, often called=class, c i, that is the number of the free=independent= dofs of the joint, regardless of the fact that the free dofs are translations or rotations.=Besides, a specific name is associated with each type of=joint.= In technical literature there are many tables that summarize the=main=joints, their=class, name and type of= the free degrees of freedom, configuration, etc.. Here below some examples of such tables are shown.= However, these examples are not at all=exhaustive.= In order to make a joint, two surfaces that belong to the couple of interconnected=mechanical components must be in contact. These surfaces, which are called=mechanical elements=(elementi=cinematici),=may=have= an infinitesimal area of a finite extended area. If the contact area is=infinitesimal=the two mechanical= elements=are called=not-adhering. Conversely, if the contact area is=finite (extended), the two mechanical= elements=are called=adhering.= A cylinder, or a sphere, supported on a plane, forms a=not-adhering=joint: the contact surfaces are a line and a point, respectively. A=brick supported on a plane forms an adhering=joint.= It is important to remark that the same type of joint, in terms of class=and type of free dofs, can be obtained with different technical solutions having different configurations.= Here below, some examples of basic=(or primary) joints are shown.= = = = = Pin-joint= (or hinge)= = = == Prismatic joint= (or collar)= = = == Helical joint= = (rotation and= translation are correlated dofs )= = Figure=1.1 Examples of joints= = = = = = Figure=1.2 =Cylindrical joint=Figure=1.3 =Planar higher pair= = = = Figure=1.4 Spherical joint (or ball joint)=Figure=1.5 �Plane on plane joint�= = In the case of planar higher pairs (Figure=1.3), the link contours=(or surfaces) that are in contact in H are= conjugate, that is they have the same normal passing through the contact point H. Therefore, the two= conjugate contours also have the same tangent line at point H.==The contact between links n.1 and n.2 is= assumed to be kept during the relative motion of link n.2,=with respect to link n.1. Different relative= motions can occur, depending on the value of the projections of the relative velocity, V 2-1,=in the normal and=tangent directions (Figure=1.6).=The class of the joint shown in Figures 1.3 and=1.6, called=planar= higher pair, is=c i===2 as the two links can be subjected to a relative rotation and a sliding motion as well.= = Figure=1.6 Conjugate contours= = Relative m→tion 2 vs. 1= = () () 0 0 t n =  →  =  2-1 2-1 H H V V Rolling motion= = () () 0 0 t n ≠  →  =  2-1 2-1 H H V V Sliding motion (rolling possible as well)= = () () 0 0 t n =  →  ≠  2-1 2-1 H H V V Impact or Separation= = Ball joints=(Figure=1.7) are spherical bearings. A ball joint consists of a bearing stud and socket enclosed in a casing.=The bearing stud is tapered and threaded, and fits into a tapered hole in the steering knuckle. A protective encasing prevents=dirt from getting into the joint assembly. Usually, this is a rubber-like boot that allows movement and expansion of lubricant. Motion control ball joints tend to be retained with an internal spring, which helps to prevent vibration=problems in the linkage. A ball joint is used for allowing= free movements=in two planes at the same time, including rotating in those planes. Combining two such= joints with control arms enables motion in all three planes, allowing the front end of an automobile to be= steered and a spring and shock (damper) suspension to make the ride comfortable.= = = = Figure=1.7==Ball joint= = = = = Figure=1.8 Rear wheel drive vehicle, front suspension= with upper and lower ball joints and tie rod end shown=Figure =1.9 Ball joint configurations= = = = = Figure=1.10 Pin joint=Figure=1.11 Pin joint= = = = = Figure=1.12 Pin=joint=in a bridge pillar=Figure=1.13 Rigid joint= = = Figure 2.5 Open chain: industrial robot=Figure 2.6 Cartesian robot= = = = = Figure 2.7 Closed chain: �scissor� lift=Figure 2.8 Four -bar linkage=Figure 2.9 Mechanical linkage: locking pliers= = As said=above,=the movement of an ideal joint is generally associated with a subgroup of the group of Euclidean displacements. The number of parameters in the subgroup is called the degrees of freedom = (dof) of the joint. Mechanical linkages are usually designed to transform a given input force and movement into a desired output force and movement. The ratio of the output force to the input force is known as the= mechanical advantage=of the linkage, while the ratio of the input speed to the output speed= is known as the speed rati→. The speed ratio and mechanical advantage are defined so they yield the same number in an ideal linkage.= A kinematic chain, in which one link is fixed or stationary, is called a mechanism, and a linkage designed= to be stationary is called a structure.= The primary mathematical tool for the analysis of a linkage is known as the kinematics equations of the system. This is a sequence of rigid body transformation along a serial chain within the linkage that locates= a floating link relative=to the ground frame. Each serial chain within the linkage that connects this floating link to ground provides a set of equations that must be satisfied by the configuration parameters of the= system. The result is a set of non-linear equations that define=the configuration parameters of the system= for a set of values for the input parameters.= 2.1 =Degrees of freedom of a linkage= A linkage is characterized by the global number, N, of free degrees of freedom of the system. The parameter N depends on: i) the number,=m, of mechanical components the system is composed of; ii) the= number of joints having a specific=class=value,=c i=(with i===1,=2,=�, 5);=iii) the dimensional space (2-D or= 3-D) in which the linkage can move. It is important to remark that also the fixed frame of the system, if it= exists, must be taken into account when evaluating the number m=of the system mechanical components.= In the case of planar (2-D) mechanical linkages the number of free dofs of the system is given by:= = = Figure 2.18 Example=of=hydraulic actuators=Figure 2.19 Example=of=hydraulic actuators= = = = = Figure 2.20 Front loader=Figure 2.21 Front loader=sketch= = In the case of=3-D mechanical linkages the number of free dofs of the system is given by:= = ( ) 1 2 3 45 N 15 4 3 2 nm C C C C C = −−−−−− =with=n== 6=(2)= where C i=is the number of joints with class=c i===i (with i===1, 2, �, 5).= The industrial robot illustrated in Figures=2.22=and 2.23=is composed of=7 links, including the fixed frame, and 6 pin-joints that allow a relative rotation about one axis. Therefore, the number of dofs of this linkage= is 6. The end-effector=of the robot, that is the robot pliers,=can reach the points of the three dimensional space inside which the robot can=move=by controlling six independent=actuators.= = = = Figure 2.22 =Industrial robot=Figure 2.23 =Industrial robot= = = = = = = = = = == = = = = Figure 2.24 Examples of mechanical linkages= = = = Figure 2.25 Front loader= = = = = = = Figure 2.26 Escavator (or digger)= = = = = Figure 2.27 Slider-crank linkage= = = = Figure 2.28 Star (or radial) engine= = = = Figure 2.29 =Inverted slider-crank mechanism= = = = = Figure 2.30 Cutting machine= = = = = Figure 2.31 Cutting machine= = = Glossary= = Ordinary (in line) slider-crank mechanism=manovellismo ordinario centrat→= Offset slider-crank mechanism=manovellismo ordinario=deviat→= Crank and slotted link=glif→= Four-Bar mechanism (crank-rocker)=quadrilatero articolat→= = Piston rod=/ connecting rod=biella= Web=manovella (di albero a gomiti)= Driver link=(or input link)=manovella= Coupler-link=biella= Follower=bilanciere= Crankshaft=albero a gomiti= Wrist pin=spinotto= Flywheel==volano=