2 2
x0 n tan v0
1
Undamped free response
Adding Damping
Damping
Damping is some form of friction! In solids, friction between molecules result in damping In fluids, viscosity is the form of damping that is most observed In this course, we will use the viscous damping model; i.e. damping proportional
Undamped Spring-Mass System
L
Unloaded Spring
k st m x
At equilibrium, kst = mg
Body in equilibrium (at rest)
m
Body in motion
Free Body Diagram
x
k( st + x )
Prerequisites: The most important prerequisite is ordinary differential equations. You should be prepared to review undergraduate differential equations if necessary.
………
考核办法—累加式
1. 大作业1 10%
2. 大作业2
3. 平时表现 4. 期终考试
10%
10% 70%
Preface
What is dynamics？
Dynamics focuses on understanding why objects move the way they do.
Dynamics = Kinematics + Kinetics
x(t ) ae
2
t
Into ODE you get the characteristic equation:

k t ae ae 0 m
t
Giving:
k m
2
k j m

The proposed solution becomes:
x(t ) a1e
机械系统动力学
机电工程学院 机械设计系
Chen Zhaobo (陈照波)
Tel: 86412057 E-mail: chenzb@ 机械楼一楼1020室
参考书
1. 闻邦椿等《机械振动理论及应用》 高等教育出版社
2. 胡海岩 《机械振动基础》 北京航空航天大学出版社
3. 师汉民 《机械振动系统》 华中科技大学出版社 4. W. T. Thomson 《振动理论及应用》 清华大学出版社
Critical Damping
No oscillation occurs.
2) 1, called over-damping two distinct real roots:
1,2 n n 2 1
x(t ) e
n t
(a1e
n t 2 1
Why to study dynamics of mechanical system? Higher speed Higher precision More flexible More complicated
Resonance
When a forcing frequency is equal to a natural frequency
x(t ) a1e jnt a2e- jnt
Note!
Natural frequency
In the previously obtained solution:

x(t ) A sin(nt ) The frequency of vibration is n
Three possibilities:
1)
1
Roots are repeated & equal.
Called critically damped
1 c ccr 2 mk 2m n
x(t ) a1e nt a2te nt Using the initial conditions: a1 x0 a2 v0 n x0
Chapter 1 Single degree of freedom systems
Objectives Recognize a SDOF system Be able to solve the free vibration equation of a SDOF system with and without damping Understand the effect of damping on the system vibration Apply numerical tools to obtain the time response of SDOF system
It depends only on the characteristics of the vibration system. That is why it is called the natural frequency of vibration.
k n m
Natural frequency
sin( ) sin( )cos( ) cos( )sin( )
Manipulating the solution
Solution we have: Rewriting:
x(t ) a1e
jnt
a2e
- jnt
Байду номын сангаас
x(t ) a1 (cos nt j sin nt ) +a2 (cos nt j sin nt ) (a1 a2 ) cos nt j (a1 - a2 )sin nt
=
mg
.. mx
Equations of motion
Sum forces:
F : mg - kst x m x
Rearrange to yield the familiar equation of motion:
Physical model
Spring Damping element
natural frequency from static deflection.
n

g
st
natural frequency from energy method.
Recall: Initial Conditions
Amplitude & phase from ICs
x0 A sin( n 0 ) A sin v0 A n cos( n 0 ) A n cos Solving yield v0 A x0 n
to velocity
Spring-mass-damper systems
From Newton’s Law:
mx(t ) cx(t ) kx(t ) 0 x(0) x0 x(0) v0
Solution (dates to 1743 by Euler)
Divide the equation of motion by m
COURSE GOALS:
1. To become proficient at modeling vibrating mechanical systems. 2. To perform dynamic analysis such as free and forced response of SDOF and MDOF systems. 3. To understand concepts of modal analysis. 4. To understand concepts in passive and active vibration control systems.
j
k t m
a2e
-j
k t m

For simplicity, let’s define:
k n m
jnt

Giving:
x(t ) a1e
a2e
- jnt
Let’s manipulate the solution
Recall
e
j
cos( ) j sin( )

Giving:
x(t ) A1 cos nt A2 sin nt
Further manipulation
Solution we have: Let:
x(t ) A1 cos nt A2 sin nt
A
A
2 A12 A2

A2
A1
cos A2 / A
sin A1 / A
2 x(t ) 2n x(t ) n x(t ) 0
Where the damping Ratio is given by: (dimensionless)
c 2 mk
Let x(t ) aet & substitute into equation of motion
a2 e
n t 2 1
)
where a1 a2 v0 ( 2 1) n x0 2 n 2 1 v0 ( 2 1) n x0 2 n 2 1