MT Chap3 Kinematic Analysis of Planar Mechanisms

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  • 1.
  • 2.3.1 Introduction 1. Purpose of Kinematic Analysis (1) The workspace of a mechanism is necessary by means of analysis of positions or tracing path Fig.3-1a shows a internal combustion engine, i n which the stroke of the piston can be used to design the length of the c ylinder, and path of the coupler can be used to design the internal dimen sions of the engine block. (2) Determine the velocities and accelerations of links to investigate the working characteristics of a machine Fig.3-1b shows a shaper mechanis m. The ram in the working stroke demands constant velocity approximat ely and the variation of accelerations is as little as possible. So the veloci ty analysis is very important to design a shaper mechanism. (3) Motion analysis is needed for the dynamic force calculation Once a position analysis is done, the next step is to determine the velocities of d riven links or tracing points of interest in the mechanism.
  • 3.Fig.3-1 Kinematic analysis of mechanisms(机构的运动分析)
  • 4.2. Methods of Motion Analysis 1) Instantaneous center method (1) Graphical method 2) Relative motion method (2) Analytical method We first establish a vector loop（or loops）around the mechanism, in which the links are represented as position vectors, then we can take the derivatives with respect to time to find the velocity and acce leration. (3) Experimental method We may install the displacement sensor, speed se nsor or acceleration sensor in the machine to measure the displacements, spe eds and accelerations, which are required. The experimental method is a con ventional method to analyze the performance of machines.
  • 5.3.2 Velocity Analysis with Instantaneous Centers 1. Concept of Instantaneous Center of Velocity (1) Instantaneous center of velocity An instantaneous center of velocity is a center of rotation of a moving link relat ive to another link. If a link is in motio n relative to a fixed link, the center is c alled as an absolute center; otherwise it is called as a relative center. Fig.3-2 Instantaneous center(速度瞬心)
  • 6.(2) The number of instantaneous centers (3) Locating instantaneous centers The following rules are use d when locating instantaneous centers. Fig.3-3 Primary instantaneous centers of two links linked by kinematic pair (两构件用运动副连接时的瞬心位置)
  • 7.1) Two links are connected by a kinematic pair. If two links ar e connected by a pivot joint, the center of the pivot is the insta ntaneous center; see the Fig.3-3a, b. If two links have sliding c ontact, the instantaneous center lies at infinity in a direction pe rpendicular to the path of the motion of the slider; see the Fig. 3-3c. If two links have pure rolling contact, the instantaneous center is the point of contact, this is because the two points of contact on the two bodies have the same linear velocity and th ere is no relative motion at the contact point. See the Fig.3-3d. If two links have rolling and sliding contact, the instantaneous center lies somewhere on the common normal of the contact p oint. See the Fig.3-3e.
  • 8.2) Two links having relative motion are not connected by kinematic pair. This instantaneous center can be determined by Kennedy theorem. Any three bodies in plane motion will have exactly three instantaneous centers, and they will be on the same straight line. This is known as Kennedy theorem. Fig.3-4 Kennedy theorem (三心定理)
  • 9.Example 3-1 Fig.3-5 shows a four bar linkage and a slider crank me chanism. Find all the instantaneous centers by graphical method. Fig.3-5 Instantaneous centers for four-bar mechanisms (四杆机构的瞬心)
  • 10.2. Velocity Analysis with Instantaneous Centers Example 3-2 Fig.3-6 shows a four bar linkage. The angular velocit y ω1of link 1 is known, as shown in the figure. Find the angular vel ocities ω 2 and ω3. Fig.3-6 Application of instantaneous centers for four-bar mechanism (瞬心法在铰链四杆机构速度分析中的应用)
  • 11.Example 3-3 Fig.3-7 shows a cam mechanism with a flat follower. The angular velocity of cam 1 is known and it is required to find the velocity of the follower 3. Fig.3-7 Application of instantaneous centers for cam mechanism (瞬心法在凸轮机构中的应用)
  • 12.3.3 Kinematic Analysis by Graphical Method 1. Principles of Relative Motions (1) Relative motion (velocity an d acceleration) of two points on the same link Let us consider a body which has plane motion; see the Fig.3-8. Fig.3-8 Relative velocity of two points on a link (同一构件上两点之间的速度关系)
  • 13.(2) Relative motion (velocity and acceleration)of two coincident points on different links In many mechanisms, such as in Fig.3-9, constraint of relative motion is achieved by guiding the slider 2 on the guider bar 1 along its path. The slider 2 is reciprocated along the guider bar 1, and they rotate about the pivot O together with an angular velocity ω1. Fig.3-9 Relative velocity of coincident point on separate links(两构件重合点处的运动关系)
  • 14.(3) Velocity image and acceleration image When we know the velocities or accelerations at two different points on a link, the vel ocity or acceleration of the third point can be determined by draw ing their images. While drawing the images, the following points should be kept in mind:1) The velocity image or acceleration image of a link is a scaled r eproduction of the link shape in the velocity diagram or acceleratio n diagram. 2) The order of the letters in the velocity image or acceleration im age is the same as in the link configuration.
  • 15.2. Graphical Method of Relative Motions The procedure of kinematic analysis of planar mechanism is as follows:1) Draw the scaled kinematic diagram. 2) Write the velocity vector equation and draw the velocity diagra m, then find out the unknown velocities or angular velocities. 3) Write the acceleration vector equation and draw the acceleration diagram, then find out the unknown accelerations or angular accele rations.
  • 16.Example 3-4 Fig.3-10a shows a four bar linkage; all the dimensio ns of the links and angular position of the driving link AB are know n. When the crank AB rotates counterclockwise with an angular vel ocity ω 1, determine the angular velocities ω2, ω3,the velocity of th e point E on the link 2 and angular accelerations α 2, α 3. Fig.3-10 Kinematic analysis of a four-bar linkage (铰链四杆机构的运动分析)
  • 17.Example 3-5 Fig.3-11a shows a guider bar mechanism; all the dim ensions of the links and angular position of the driving link AB are known. When the crank AB rotates counterclockwise with an angul ar velocity ω1,determine the angular velocities ω 2, ω3 and the ang ular acceleration α2 and α3. Fig.3-11 Kinematic analysis of a guide- bar mec hanism(导杆机构的运动分析)
  • 18.3. Some Key Points of Motion Analysis 1) The Coriolis acceleration of coin cident points on two different links must be discriminated correctly. 2) When we would establish the vel ocity equation or acceleration equat ion, the velocity and acceleration of the base point must be known. If we want to find out the angular velocit y of the link 3 in Fig.3-12, the point B must be considered to establish th e velocity equation, because the vel ocity of the point B on the link 1 is known. Fig.3-12 Expanded link(构件的扩大)
  • 19.3) When the mechanism is at its limited positions, the velocity polygon or acceleration polygon becomes simple, but sometimes it is difficult to analysis.Fig.3-13a shows a four bar linkage in which the crank and coupler are in collinear. In the guider bar mechanism shown in Fig.313b, the crank is perpendicular to the rocker. Their velocity polygon and acceleration polygon are very simple. Fig.3-13 Kinematic analysis in limited positions (特殊位置的运动分析)
  • 20.4)Hydraulic mechanism can be transformed into a relating guide r bar mechanism. Fig.3-14a shows a hydraulic mechanism. It ca n be transformed into a guider bar mechanism shown in Fig.3-1 4b ; they are equivalent mechanisms. Fig.3-14 Kinematic analysis of hydraulic mechanism (摆动液压缸机构运动分析)
  • 21.3.4 Kinematic Analysis by Algebraic Method 1. Fundamental Law of Analytical Method The procedure of motion analysis is asfollows:1) Establish a coordinate system in which the origin of the coordinat e system is coincident with the rotating center of the driving link an d the x axis is along the frame of the mechanism. 2) Establish position equation. 3) The vector loop equation which represents the position equation c an be written as two projective equations in the Cartesian coordinate. 4) Differentiate the position equation with respect to time. 5) The velocity equation is differentiated with respect to time again.
  • 22.2. Kinematic Analysis by Analytical Method Example 3-6 Fig.3-15 shows a four bar linkage. All the dimensions of the link s and angular position of the driving link AB are known. When the crank AB rot ates counterclockwise with an angular velocity ω1, determine the angular veloci ties ω2, ω3 and the velocity of the point E on the link 2. In the end, determine th e angular accelerations α2 and α3. Fig.3-15 Model of a four-bar linkage (铰链四杆机构的 数学模型)
  • 23.Example 3-7 In the following mechanism of Fig.3-16, the link 1 ro tates at a constant angular velocity ω1; its direction is counterclockw ise. When the link 1 is in the phase φ1, determine the displacement, velocity and acceleration of link 3. Fig.3-16 Model of a four-bar linkage with a sliding pair (含有移动副四杆机构的运动分析模型)
  • 24.3. Summary The key point of kinematical analysis with algebraic method is how to establish the vector loop. Fig.3-17 shows some closed vector loops in different mechanisms. Fig.3-17 Some closed vector loops in mechanisms(一些机构的封闭矢量环)