Deeply understand the mode and vibration of mechanical system

One 、 Modal and vibration problems

Hello everyone , My training today is “ Mode and vibration of mechanical system ”, It's training , In fact, it's about talking about yourself for many years NVH Learning experience , Talking about NVH, Maybe many people think it's very “ Mysterious ”, Even in NVH People who have worked in the field for many years . I used to think it was very “ Mysterious ”, Encounter relevant NVH I'm always in a panic when I have a problem , Think NVH It's just “ Devil ”. After years of study and research , Especially after understanding the mode and vibration of mechanical system , Only then discovered NVH It turned out to be a “ The angel ”, quite a lot NVH Problems can be perfectly explained by science or physical mathematics , If you're right NVH have interest in , In the process of learning and research , You will feel the beauty of physics or Mathematics ！
Everything is difficult at the beginning , Let's start with the simplest phenomenon , This is a straight car .

Study the motion state of the car , We can build xy Coordinate system ,x Indicates the east-west direction ,y Indicates the north-south direction , Because the car is traveling in the east by north direction , Obviously , Establish a coordinate system along the direction of the car x’ It's a better choice .

Now increase the difficulty , If the car runs along a curve , Obviously , We just need to build along the vertical direction y’ The shaft can be .

Car in x’ Large displacement on , stay y’ Small displacement on , Usually only x’ Coordinate system , Accuracy will not
It has a big impact .

summary ：
Study the mechanical system , Although you can choose any coordinate system , But find the right
The coordinate system of can simplify the system model , It is convenient for us to see the problem 、 Understand the problem 、
solve the problem ;
When the object we study looks large on some coordinate axes of the coordinate system , Others sit
The scale axis looks very small , Such a coordinate system can be called the master coordinate system , Study for us
The problem studied affects the degree of freedom of the system corresponding to the smaller coordinate axis , We can delete , Conduct
Degrees of freedom are reduced .

What is the relationship between the principal coordinate system and the mode of the system , I'll talk about later .

Two 、 Single degree of freedom vibration

Now let's talk about vibration , Again , Let's start with the simplest phenomenon , Let's first look at the animation of a free vibrating ball .

We can see that the amplitude of the ball is getting smaller and smaller , besides , It's hard to find more with your eyes , What shall I do? ？ Test it , This is a “ measuring ” The displacement diagram of the ball obtained .

Except that the amplitude of the small ball gradually decreases , We can also find that the vibration frequency of the ball remains unchanged , yes 1Hz, in addition , Its amplitude decreases exponentially .

Now add a little more difficulty , Add some sinusoidal excitation to the ball , See what happens to the ball ？

We can see , The ball began to vibrate as if there were no rules , But in the end, it gradually stabilized , besides , It's hard to find more with your eyes , What shall I do? ？ Test it , This is a “ measuring ” The displacement diagram of the ball obtained .

We can find out , The final vibration amplitude of the ball remains unchanged , The frequency does not change , yes 2Hz

Why does the ball vibrate so much ？ To explore the mechanism of this phenomenon , We can use the mechanical knowledge we have learned .

First, analyze the force on the ball , Establish the force balance equation , Finally, it is transformed into a differential equation , The differential equation is the mathematical model of the ball vibration system , This is a one-dimensional differential equation , It's not difficult to solve .

I won't start the process of solving , Let's focus on the form of the result and some important parameters ,wn Is the undamped natural frequency of the system , This example is 1Hz,wd Is the damped natural frequency of the system ,w Is the frequency of the exciting force , This example is 2Hz,zeta Is the damping ratio . The result consists of three parts , The first part is the free vibration of the ball , It is determined by the initial displacement and initial velocity of the ball , Its amplitude decays exponentially , The vibration frequency is the damped natural frequency of the system ; The second part is still the free vibration of the ball , It is caused by motivation , It is usually called adjoint motion , Its amplitude decays exponentially , The vibration frequency is the damped natural frequency of the system ; The third part is steady-state forced vibration , The amplitude does not change , Frequency is the frequency of excitation , Its vibration amplitude is determined by excitation and system parameters .
Sometimes we pay more attention to free vibration , Such as vehicle start and stop 、 Acceleration and deceleration conditions, etc 、 Sometimes we pay more attention to the steady-state forced vibration , This requires transient simulation , For example, engine idling , The working condition of the vehicle running smoothly , This requires steady-state simulation .
One degree of freedom vibration system is all NVH The basis of the problem , The more you understand it, the better .

3、 ... and 、 Two degree of freedom vibration

Now let's talk about an example of two degree of freedom vibration ,

Study the car with two degrees of freedom , You can take the translational freedom and rotation of the center of gravity
freedom , Here I take the translational degrees of freedom at the front and rear wheels .
Let's first take a look at the free vibration of the car ,
Vibration state one ：

The front and rear wheels of the trolley vibrate in the same direction , The node is in front of the front wheel , The posture remains the same , The frequency does not change
This vibration is a first-order modal vibration , The vibration frequency is modal frequency , The amplitude composition of the front and rear wheels
Modal vector
Vibration state II ：

The front and rear wheels of the trolley vibrate in reverse , The node is between the front and rear wheels , The posture remains the same , The frequency does not change
This vibration is a second-order modal vibration , The vibration frequency is modal frequency , The amplitude composition of the front and rear wheels
Modal vector
Vibration state III ：

The vibration of the trolley transits between the first two vibration modes 、 Switch , There are no fixed nodes
The vibration frequency has 2 individual

Now let's analyze the vibration mechanism of the car , Mathematical modeling .

Similar to a one degree of freedom system , According to the theory of mechanics , We can establish force balance equations , For multidimensional problems , It is very convenient to describe with matrix ,M Represents the quality matrix ,K Represents the stiffness matrix ,q Represents the displacement vector ,q’’ Represents the acceleration vector .
For multi-dimensional system equations , We can use decoupling method to solve ,

The feature vector forms the coordinate axis of the feature coordinate system , The eigenvector is the modal vector ,
The characteristic coordinate system is the modal coordinate system

For our car in this case , After feature coordinate transformation , A system of differential equations with two degrees of freedom becomes 2 A differential equation with one degree of freedom , So we can use the algorithm mentioned above to solve , The resulting displacement is the modal displacement , Or modal participation factor 、 Or modal coordinates , Then it can be transformed into the physical coordinate system .

Let's explain what we've seen before by solving an example 3 Vibration phenomenon in animation .

Let's first look at the mode of the car ：

Look at the vibration state of the car again ：

As long as the initial displacement of the trolley is 1 On the first modal coordinate axis , The car is just 1 Do sinusoidal motion on the plane determined by the first-order modal coordinate axis and time axis .

Look at the vibration state of the car II ：

As long as the initial displacement of the trolley is 2 On the first modal coordinate axis , The car is just 2 Do sinusoidal motion on the plane determined by the first-order modal coordinate axis and time axis .

Then look at the vibration state of the trolley III ：

As long as the initial displacement of the trolley is not 1、2 On the first modal coordinate axis , The car is just 2 The vibration modes change .

Four 、 What is the mode ？

5、 ... and 、 Discussion on Problems

Be careful , there “ Main mode ” Refers to the principal coordinate system .