ME451 Kinematics and Dynamics of Machine Systems

ME451 ME451 的運動學和機械系統動力學的運動學和機械系統動力學授課老師 : 謝銘源組員 : 黃靖凱 莊沛語

Why/How do bodies move?Why?

The configuration of a mechanism changes in time based on forces and motions applied to its componentsForces

Internal (reaction forces)External, or applied forces (gravity, compliant forces, etc.)

MotionsSomebody prescribes the motion of a component of the mechanical system

Recall Finite Element Analysis, boundary conditions are of two types:Neumann, when the force is prescribedDirichlet, when the displacement is prescribed

How?They move in a way that obeys Newton’s second law

Caveat: there are additional conditions (constraints) that need to be satisfies by the time evolution of these bodies, and these constraints come from the joints that connect the bodies (to be covered in detail later…)

Putting it all together…

MECHANICAL SYSTEM = BODIES + JOINTS + FORCES

THE SYSTEM CHANGES ITS CONFIGURATION IN TIME

WE WANT TO BE ABLE TO PREDICT & CHANGE/CONTROL HOW SYSTEM EVOLVES

Examples of Mechanisms

快速返回插齒機構Quick-return shaper mechanism

雨刷機構Windshield wiper mechanism

當我說“機械系統”或“系統”這是什麼意思？What do I mean when I say “mechanical system”, or “system”?

雨刮器

搖桿耦合

左蹺板

右蹺板

曲軸耦合器

曲軸

切割行程

工作部件

齒輪

固定齒輪

轉動曲柄

路徑

連接

行程運動

鋸齒齒輪

雨刷連動方式 機構

More examples …

降低控制

支撐主軸裝配

平移關節

活塞杆

球形接頭

McPherson Strut Front Suspension麥弗遜式支柱前懸掛

Schematic of car suspension汽車懸架示意圖

麥弗遜式懸掛系統的優點是：結構簡單 ，懸掛重量輕和占用空間小，車輪跳動時前輪定位參數變化小，有良好的操縱穩定性。

懸挂係統的組成 一、彈簧 二、避震器 三、防傾桿 四、連桿

懸掛係統汽車，車架與車橋或車輪之間的一切傳力連接裝置的總稱，其功能是傳遞作用在車輪和車架之間的力和力矩，並且緩衝由不平路面傳給車架或車身的衝擊力，並衰減引起的震動，以保證汽車平順行駛。懸挂係統應有的功能是支持車身，改善乘坐的感覺，不同的懸挂設置會使駕駛者有不同的駕駛感受。外表看似簡單的懸挂係統綜合多種作用力，決定著轎車的穩定性、舒適性和安全性，十分關鍵之一。

More examples …Interest here is in controlling the time evolution of these

mechanical systems

機器人機械手Robotic Manipulator

發動機的截面Cross Section of Engine

Nomenclature( 命名法 )

Mechanical System, definition:A collection of interconnected rigid bodies that can move relative

to one another, consistent with joints that limit relative motions of pairs of bodies

Why type of analysis can one speak of in conjunction with a mechanical system?

Kinematics analysis ( 運動學分析 )Dynamics analysis ( 動力學分析 )Inverse Dynamics analysis ( 反向動力學分析 )Equilibrium analysis ( 均衡分析 )

Kinematics Analysis ( 運動學分析 )

Concerns the motion of the system independent of the forces that produce the motion

Typically, the time history of one body in the system is prescribed

We are interested in how the rest of the bodies in the system move

Requires the solution linear andnonlinear systems of equations

雨刷機械

Dynamics Analysis ( 動力學分析 )

Concerns the motion of the system that is due to the action of applied forces/torques

Typically, a set of forces acting on the system is provided. Motions can also be specified on some bodies

We are interested in how each body in the mechanism moves

Requires the solution of a combined system of differential and algebraic equations (DAEs)

Inverse Dynamics Analysis( 反向動力學分析 )It is a hybrid between Kinematics and DynamicsBasically, one wants to find the set of forces that lead to a certain desirable motion of the mechanismYour bread and butter in Controls…

這是一個運動學和動力學之間的混合Windshield wiper mechanism

機器人機械手Robotic Manipulator

End: Chapter 1 (Introduction)Begin: Review of Linear Algebra

結束：第 1 章（引言）開始審查線性代數

Why bother with vectors/matrices?

Kinematics (and later Dynamics), is all about being able to say at a given time where a point is in space, and how it is moving.

Vectors and matrices are extensively used to this end.

Vectors are used to locate points on a body.

Matrices are used to describe the orientation of a body.

Geometric Vectors

What is a Geometric Vector?A quantity that has two attributes:A directionA magnitude

VERY IMPORTANT:Geometric vectors are quantities that exist independently of any reference frame

ME451 deals almost entirely with planar kinematics and dynamicsWe assume that all the vectors are defined in the 2D plane

Geometric Vectors: Operations

What can you do with geometric vectors?Scale them

Add them (according to the parallelogram rule)Addition is commutative

–Multiply two of them• Inner product (leads to a number)• Outer product (leads to a vector, perpendicular on the plane)

–Measure the angle between two of them

單位坐標向量（短途旅行）Unit Coordinate Vectors(short excursion)單位坐標向量：一個用來表達所有其他向量的單位向量

在這個類中，以簡化我們的生活中，我們使用兩個正交的單位向量

一個向量分解為組件 和 沿 X 和 Y 軸

命名 : 和 被稱為笛卡爾向量的組件

符號約定：在本類中，向量 /矩陣粗體，標量不（通常是他們以斜體）

幾何向量 :操作Geometric Vectors: Operations

兩個向量的積

關於兩個向量之間的的角度，請注意

兩個向量的點積是可交換 由於坐標單位矢量之間的角度是 / 2 ：