Understanding Degrees of Freedom in Mechanisms: A Kinematic Analysis, Study Guides, Projects, Research of Design

Download Understanding Degrees of Freedom in Mechanisms: A Kinematic Analysis and more Study Guides, Projects, Research Design in PDF only on Docsity! AME 352 KINEMATIC FUNDAMENTALS P.E. Nikravesh 1-1 1. KINEMATIC FUNDAMENTALS One of the objectives in the design of mechanisms is to ensure that the stresses in individual links do not exceed certain threshold. Stress analysis requires the knowledge of forces (applied and reaction) that act on each link. Determining the reaction forces requires complete understanding of the kinematics of the system. A kinematic analysis requires determining the position, velocity, and acceleration of all the links in a system. In this chapter some of the fundamental concepts and terminologies of kinematics and dynamics, particularly those for mechanisms, are reviewed. Reference Frame We will use a Cartesian x-y reference frame in our analyses whenever needed. Unless it is stated otherwise, we have the x-axis horizontally with the positive direction to the right. Therefore, the y-axis will be vertical, with the positive direction up. x y Degrees-of-Freedom One of the most important concepts in the analysis and design of a mechanical system is its mobility (M) or its degrees-of freedom (DoF). A mechanical system’s DoF is equal to the number of independent entities needed to uniquely define its position in space at any given time. A free particle (point) on a plane has 2 degrees-of-freedom (DoF): x y Move in the x direction. x y Move in the y direction. A free body (link) on a plane has 3 degrees-of-freedom. Any general motion (displacement) of a free planar body can be decomposed into three independent motions: x y Move in the x-direction. x y Move in the y-direction. x y z Rotate about the z-axis. Two free planar bodies have 2 × 3 = 6 DoF. x y n free planar bodies have n × 3 = 3n DoF. x y . . . Joints and Degrees-of-Freedom A kinematic joint connects two bodies (one of the bodies could be the ground). A joint, depending on its type, eliminates one or more degrees-of-freedom between the two bodies. A pin joint eliminates 2 DoF. A sliding joint eliminates 2 DoF. A pin-in-slot joint eliminates 1 DoF. AME 352 KINEMATIC FUNDAMENTALS P.E. Nikravesh 1-2 • Joints that eliminate 2 DoF are called full joints. • Joints that eliminate 1 DoF are called half joints. Examples A single pendulum is composed of 1 moving body pinned to the ground. The system has 1 DoF. A double pendulum consists of 2 moving bodies and 2 pin joints. The system has 2 DoF. A triple pendulum consists of 3 moving bodies and 3 pin joints—it has 3 DoF. A four-bar mechanism consists of 3 moving bodies and 4 pin joints (2 of the bodies are pinned to the ground). The mechanism has 1 DoF. Mobility Formula The mobility formula can help us determine the number of degrees of freedom for most planar systems: M = 3× Bmoving − 2 × J full − Jhalf where, M is the mobility or the number of DoF; Bmoving is the number of moving bodies; J full is the number of full joints; and Jhalf is the number of half joints. Examples Triple pendulum M = 3× 3− 2 × 3− 0 = 3 Six-bar mechanism M = 3× 5− 2 × (5+ 2)− 0 = 1 A structure This system contains 4 links and 6 pin joints. M = 3× 4 − 2 × 6 − 0 = 0 Zero DoF means a structure (none of the links can move). Slider-crank M = 3× 3− 2 × (3+1)− 0 = 1 AME 352 KINEMATIC FUNDAMENTALS P.E. Nikravesh 1-5 Inversion 1: ground link Inversion 2: ground link Inversion 3: ground link Inversion 4: ground link Slider-Crank Mechanisms A slider-crank is a four-bar where one of the pin joints is replaced by a sliding joint. The numbering system of the links is similar to a four-bar mechanism. The non-moving link may not be shown in a figure. If link (2) is the input link and (4), the slider, is the output link, the mechanism may be referred to as a “crank-slider”, but if the slider is the input link and link (2) is the output, such as in an internal combustion engine, the mechanism is referred to as a “slider-crank”. In our discussion, regardless of the input and output links, we will refer to this type of mechanism as a slider-cranks. (1) (2) (3) (4) (1) (2) (3) (4) AME 352 KINEMATIC FUNDAMENTALS P.E. Nikravesh 1-6 Inversions of A Slider-crank A slider-crank similar to a four-bar has four inversions. Inversion 1: Inversion 2: ground link ground link Inversion 3: Inversion 4: ground link this link rotates with the block ground link Toggle Position When two links of a mechanism become collinear (forming a straight line), the links are said to be in toggle. We consider a four-bar mechanism as an example. (1) (2) (3) (4) Links (2) and (3) are in toggle (2 cases): (2) (3) (2) (3) Links (2) and (1) are in toggle (2 cases): (1) (2) (2) (1) AME 352 KINEMATIC FUNDAMENTALS P.E. Nikravesh 1-7 Transmission Angle In a four-bar mechanism if link (2) is the input (driver) link and link (4) is the output link, then the angle between link (3) (the coupler) and the output link is called the transmission angle, μ . This angle is measured as the smaller angle between the two links; i.e., μ ≤ 90o . Therefore the optimal transmission angle is μ = 90o . μ (2) (1) (3) (4) μ (2) (1) (4) (3) In a slider-crank mechanism, if the slider is the input link, then the angle between the other two links is the transmission angle. In an internal combustion engine, in one complete cycle, the transmission angle becomes 0o twice and 90o twice. μ input link output link (4) (3) (2)