哈工大课件机械系统动力学Dynamics of Mechanical System-ch1

x(t ) A sin cos nt A cos sin nt
Giving:
x(t ) A sin(nt )
Different forms of the solution
x(t ) A sin(nt ) x(t ) A1 cos nt A2 sin nt
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

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
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
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
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
………
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 ) a1e jnt a2e- jnt
Note!
Natural frequency
In the previously obtained solution:

x(t ) A sin(nt ) The frequency of vibration is n
x(t ) ae
2
t
Into ODE you get the characteristic equation:

k t ae ae 0 m
t
Giving:
Байду номын сангаас
k m
2
k j m

The proposed solution becomes:
x(t ) a1e
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
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
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
Kinematics is the study of the motion of point masses or rigid bodies. Kinetics is the study of the forces which cause and affect motion.
What is mechanical system?
Structure
Mechanism
Machine
Development stages of dynamics of mechanical systems: Static analysis Kineto-static analysis Dynamic analysis Elasto-dynamic analysis
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 )
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
Single degree of freedom systems
• When one variable can describe the motion of a structure or a system of bodies, then we may call the system a 1-D system or a single degree of freedom (SDOF) system.
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.
sin( ) sin( )cos( ) cos( )sin( )
Manipulating the solution
Solution we have: Rewriting:
x(t ) a1e
jnt
a2e
- jnt
x(t ) a1 (cos nt j sin nt ) +a2 (cos nt j sin nt ) (a1 a2 ) cos nt j (a1 - a2 )sin nt
2 a 2et 2an et n aet 0
Which is now an algebraic equation in

1,2 n n 2 1
Here the discriminant , 2 1 , determines the nature of the roots.
Mathematical model
Restoring force
Damping force
Undamped Free Vibration
Differential equation
x(0) x0 , x(0) v0
Solving ODE
Proposed solution:
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@hit.edu.cn 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
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
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
=
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
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( )
Prerequisites: The most important prerequisite is ordinary differential equations. You should be prepared to review undergraduate differential equations if necessary.