Modeling and Energy Methods

Content

Two Methods of Modeling

Force balance
Energy methods

Energy Methods

Conservative system
Energy Conservation Method
Lagrange's Method

Reference
Log

Concepts

Modeling
Newton's 2nd Law
Euler's 2nd Law
Conservative system
Lagrange's Method

1. Two Methods of Modeling

如何得到系统的响应，我们需要建立系统的运动微分方程。在这个过程中，我们需要建模(Modeling)。

Modeling is the art or process of writing down an equation, or system of equations, to describe the motion of a physical device.[1]

01-modeling [1].JPG

这里，建模方法介绍两类：

Force balance: Newton's Second Law + Euler's 2nd Law
Energy methods

如上图中的弹簧质量系统，可通过Newton's Second Law (牛顿第二运动定律) 得到其运动微分方程。当然这个系统是 Conservative System。

Newton's Second Law

Newton’s second law states: the sum of forces acting on a body is equal to the body’s
mass times its acceleration.[1]

即 F = ma.

在一些旋转系统中，可以使用 Euler's Second Law。

Euler’s second law states: the rate of change of angular momentum is equal to the sum of external moments acting on the mass.[1]

即 ∑M = Jα. M为力矩， J 是转动惯量，α是角加速度。

在Free Vibration - 和 Harmonic Motion - 中基本就是使用 Force Balance 建模。

对于保守系统，我们还可以使用 Energy Methods.

2. Energy Methods

Energy Conservation Method
Lagrange's Method

Energy Conservation Method

Conservative System

保守系统: 系统动能和势能守恒，动能和势能相互转换，中间无能量消耗，当然这是最理想的系统，现实可没有这么理想，总是有能量损耗。不过先从简单，再到复杂，易于理解。

Energy Conservative System, 我们可以通过 Energy Conservation Method 建模。

** the principel of energy conservation**

the sum of the potential energy and kinetic energy of a particle remains constant at
each instant of time throughout the particle’s motion:

T + U = constant
U1 - U2 = T2 - T1
T_max = U_max

其他 T 是动能符号，U是势能符号。保守系统中，T 和 U 总能量守恒，系统没有能量损耗。 U1 和 T1 是 t1 时刻的势能和动能，U2 和 T2 是 t2 时刻的势能和动能.

The energy method can be used in two ways.

根据 T + U = constant，两边求时间 t 求导可以得到系统运动微分方程。
根据 T_max = U_max，再利用系统位移、速度和加速度的关系，就可以求解系统的Natural Frequency（固有频率）。这条路只限于求解保守系统固有频率。

Lagrange's Method

Lagrange’s method for conservative systems consists of defining the Lagrangian, L, of the system defined by L = T - U. Here T is the total kinetic energy of the system and U is the total potential energy in the system, both stated in terms of “generalized” coordinates.

哪里都能见到欧拉和拉格朗日啊！！！

拉格朗日方法其实也简单，就是使用一些方法来求系统的运动微分方程。不过该方法也只能用于保守系统中。

02-Lagrange method [1].JPG

最后列出一张表[1], 对比一下直线运动系统和旋转系统。

03-Comparisons [1].JPG

Reference

[1] Inman D J, Singh R C. Engineering vibration[M]. Upper Saddle River: Prentice Hall, 2014.

Log

@安然Anifacc
2017-01-07 11:30:42