The phenomenon of * * vibration caused by forced vibration in mechanical manufacturing, construction engineering and other scientific and technological fields has aroused great concern of engineers and technicians. This phenomenon has both destructive effect and many practical values. Many electro-acoustic devices are designed and manufactured by using the vibration principle. In addition, "* * * vibration" is also an important research method in micro-scientific research, such as using nuclear magnetic resonance * * * vibration and paramagnetic gong to study material structure.

In this experiment, the amplitude-frequency characteristics and phase-frequency characteristics of mechanical forced vibration are quantitatively measured by Bohr vibrometer, and the dynamic physical quantity-phase difference is measured by stroboscopic method. The content of data processing and error analysis is also very rich.

First, the experimental purpose

1. The amplitude-frequency characteristics and phase-frequency characteristics of forced vibration of elastic balance wheel in Bohr vibrometer are studied.

2. Study the influence of different damping moments on forced vibration, and observe the phenomenon of * * * vibration.

3. Learn to use stroboscopic method to measure some quantities of moving objects, such as phase difference.

4. Correct learning system errors.

Second, the experimental principle

The vibration of an object under the continuous action of periodic external force is called forced vibration, and this periodic external force is called forced force. If the external force changes according to the law of simple harmonic vibration, then the steady forced vibration is also simple harmonic vibration. At this time, the amplitude remains unchanged, which is related to the frequency of forced force, the natural vibration frequency of the original vibration system without damping and the damping coefficient. In the state of forced vibration, the system is not only subjected to forced force, but also to restoring force and damping force. Therefore, in the steady state, the displacement, velocity change and force change of the object are not in phase, and there is a phase difference. When the frequency of the forcing force is the same as the natural frequency of the system, * * vibration occurs, and the amplitude is the largest at this time, and the phase difference is 90.

The experiment uses the balance wheel to swing freely under the action of elastic torque and do forced vibration under the action of electromagnetic damping torque, and studies the characteristics of forced vibration, which can directly display some physical phenomena in mechanical vibration.

When the balance wheel is subjected to periodic forced external torque and moves in the medium with air damping and electromagnetic damping (damping torque is 0), its motion equation is as follows

( 1)

Among them are the moment of inertia, elastic moment, amplitude of forced moment and circular frequency of forced force of the balance wheel.

Orders,

Then equation (1) becomes

(2)

When, Formula (2) is a damped vibration equation.

When there is no damping, equation (2) becomes a simple harmonic dynamic equation, which is the natural frequency of the system. The general solution of equation (2) is

(3)

As can be seen from Formula (3), forced vibration can be divided into two parts:

The first part indicates damping vibration, and the attenuation disappears after a certain time.

The second part explains that the forced torque really acts on the balance wheel, transferring energy to the vibrating body, and finally reaching a stable vibration state.

Amplitude (4)

The phase difference between it and the forced torque is

(5)

It can be seen from equations (4) and (5) that the values of amplitude and phase difference depend on four factors, namely, forced torque m, frequency, natural frequency of the system and damping coefficient, and have nothing to do with the initial state of vibration.

From the extreme conditions, it can be concluded that when the circular frequency of the force is forced, it will produce * * * vibration with a maximum value. If the circular frequency and amplitude of * * * vibration are expressed by and respectively, then

(6)

(7)

Equations (6) and (7) show that the smaller the damping coefficient, the closer the circular frequency is to the natural frequency of the system, and the greater the amplitude. Figures 1- 1 and 1-2 show the amplitude-frequency characteristics and phase-frequency characteristics of forced vibration at different times.

Third, experimental instruments.

ZKY-BG Bohr * * * vibrometer consists of vibrometer and electric control box. The vibrator part is shown in figure 1-3; by

β 1

β2

β3

β 1 & lt； β2 & lt； β3

ω/ωn

Figure 1- 1

ω/ωn

β 1

β2

β 1 & lt； β2

-π

-π/2

φ

Figure 1-2

The copper round balance wheel A is installed on the frame, one end of the spring B is connected with the shaft of the balance wheel A, and the other end can be fixed on the bracket of the frame. Under the elastic force of the spring, the balance wheel can swing back and forth freely around the shaft. On the periphery of the balance wheel, there is a slot-shaped gap, in which a long slot D is much longer. There is a photoelectric door H on the frame, which is aligned with the long slot and connected with the electric control box to measure the amplitude (angle value) and vibration period of the balance wheel. There is a pair of coils K with iron core under the frame, and the balance wheel A is just embedded in the gap of the iron core. According to the principle of electromagnetic induction, when DC current passes through the coil, the balance wheel is subjected to electromagnetic damping force. Changing the current value can make the damping change accordingly. So that the balance wheel a is forced to vibrate. An eccentric wheel is installed on the motor shaft, and the balance wheel A is driven by the connecting rod mechanism E. The motor shaft is equipped with a scribed plexiglass turntable F, which rotates with the motor. By adjusting the ten-turn motor speed adjustment knob on the control box, the voltage applied to the motor can be accurately changed, so that the motor speed can be continuously adjusted within the experimental range (30-45 rpm). Because the circuit adopts a special speed stabilizing device and the motor adopts a special motor with a tachometer generator, the inertia is small and the speed is extremely stable. The plexiglass turntable F of the motor is equipped with two hoods. There is also a photoelectric gate (stressed torque signal) 900 above the center of the angle reading dial G, which is connected with the control box to measure the period of stressed torque.

The phase difference between the balance wheel and the external torque during forced vibration is measured by a small flash lamp. The flashlight is controlled by the balance wheel signal photoelectric door. Whenever the long groove C on the balance wheel passes through the balance position, the photoelectric gate H will flash when it receives light. The flashlight is placed on the base, as shown in the figure (1-3). Don't hold it in your hand to directly illuminate the dial. In a stable situation, we can see that the plexiglass pointer F seems to be "stopped" on a certain scale under the irradiation of the flash. This phenomenon is called stroboscopic phenomenon, so it is convenient to read this value directly with an error of less than 20.

The amplitude of the balance wheel is to measure the number of notches on the ring at the reading a of the balance wheel by using the photoelectric gate H, and display this value directly on the liquid crystal display with an accuracy of 20.

The front panel and the rear panel of the electric control box of Niersball vibrometer are shown in Figure 1-4 and Figure 1-5 respectively.

The motor speed adjusting knob is a graduated ten-turn potentiometer. When this knob is adjusted, the motor speed can be accurately changed, that is, the period of forced torque can be changed. Calibration is only for experimental reference, so as to roughly determine the corresponding position of the stress torque cycle value on the multi-turn potentiometer.

Figure 1-3 Bohr vibrator

1. Photoelectric gate h; 2. long slot d; 3. short groove d; 4. copper balance wheel a; 5. rocker m; 6. coil spring b; 7. Supporting frame; 8. damping coil k; 9. connecting rod e; 10. rocker adjusting screw; 1 1. Photoelectric gate I; 12. Angle dial g; 13. Plexiglass turntable f; 14. Base; 15. spring clamping screw l; 16. Flash

Fig. 1-4 schematic diagram of the front panel of Nielspol * * * vibrometer.

1, LCD screen 2, direction control key 3, confirm key 4, reset key.

5. Power switch 6, flash switch 7, forced periodic adjustment potentiometer

Fig. 1-5 schematic diagram of the rear panel of Nielsbohr * * * vibrometer.

1, power socket (with safety) 2, flash interface 3, damping coil

4. Motor interface 5, amplitude input 6, period input 7, communication interface

The magnitude of DC current in the damping coil can be controlled by software to change the damping coefficient of the balance wheel system. The selector switch can be divided into four gears, the damping current of "damping 0" gear is zero, the damping current of "damping 1" gear is about 280mA, the damping current of "damping 2" gear is about 300mA, and the damping current of "damping 3" gear is the largest, about 320mA. The damping current is provided by a constant current source, which can be selected according to different situations in the experiment (you can choose "2" first, if *.

The flash switch is used to control whether the flash button flashes. When the flash button is pressed, the long gap of the balance wheel passes through the balance position, which will produce flash. Due to the stroboscopic phenomenon, you can see the reading that the scale line seems to be static from the phase difference reading disk (actually the scale line on plexiglass F has been rotating at a constant speed), thus reading out the phase difference value. In order to make the flash tube not easy to be damaged, a button switch is adopted, and the button is pressed only when measuring the phase difference.

Whether the motor rotates or not is controlled by software. When measuring the damping coefficient and the relationship between the natural frequency and amplitude of the balance wheel, the electric mechanism must be turned off.

The electric cabinet is connected with the flash lamp and Bohr vibrometer through various professional cables. And can not cause the defect of wiring error.

Fourth, the experimental content

1. Determination of damping coefficient β

Read out the amplitude values θ 1, θ2, θ 3...θ n and use the formula.

(8)

Find the β value, where n is the number of periods of damping vibration, θn is the amplitude of the nth vibration, and t is the average value of damping vibration periods. This value can measure 10 vibration period of the balance wheel, but take its average value.

In this experiment, the power supply of the motor must be cut off, the pointer f should be placed at 0, and θ0 is generally selected between 130- 150.

2. Measure the amplitude characteristic and phase-frequency characteristic curve of forced vibration.

Keep the damping gear unchanged, choose forced oscillation to carry out experiments, and change the motor speed, that is, change the frequency ω of forced external torque. After the forced vibration is stable, read the amplitude value of the balance wheel, and measure the phase difference between the forced vibration displacement and the forced force with a flash lamp (controlled at around 10).

The frequency of the forced torque can be calculated from the vibration period of the balance wheel, or it can be calculated by selecting 10 cycles with a period of "× 10" to directly measure the forced torque. When it reaches a steady state, these two values should be the same. The former has four significant figures and the latter has five significant figures.

Because the curve changes greatly near the * * * vibration point, the measured data are dense, and a small change in motor speed will cause a big change. The reading on the button of the motor speed selector (Example 2.50) is a reference value, so it is suggested to write down this value under different ω, so as to quickly find the reference for re-measurement in the experiment.

Verb (abbreviation of verb) How to use the control box of Bohr vibrometer

1, boot introduction

After pressing the power switch, a welcome interface appears on the screen, where 0000X is the number of the control box connected to the host. After a few seconds, the words "key description" will be displayed on the screen, as shown in figure 1. The symbol "t" means moving to the left; "u" means moving to the right; "p" means moving up; "q" moves down. The following symbols are not repeated.

2. Free vibration

Press Enter in the state of 1 to display the experiment type shown in Figure 2, with free oscillation selected by default and font highlighted as selected. (Note that free oscillation must be done before the experiment, and its purpose is to measure the relationship between the amplitude of the balance wheel and the natural vibration period. )

Key description

T u → select an item.

Pq → Change the working state

Confirm → Confirm the function item.

Tuyiyi

experimental procedure

Free oscillation damping oscillation forced oscillation

Figure ii

Damping 0 amplitude

After investigating 00, check and return.

Period x1= seconds (balance wheel)

Figure 3

Damping 0 amplitude 134

Measure 0 1 =↓ Press OK to return.

Period x1= 01.442 seconds (balance wheel)

Figure 4

Damping selection

Damping 1 damping 2 damping 3

Figure 5

10

Damping 1 amplitude

After investigating 00, check and return.

Period X = seconds (balance wheel)

Figure 6

Press Enter again, and the display will be as shown in Figure 3.

Turn the balance wheel by hand for about 160 degrees, press the "P" or "Q" key after releasing the hand, and the measuring state will change from "off" to "on". The control box will start to record the experimental data, and the effective numerical range of the amplitude is: 160-50 (if the amplitude is less than 160, the measurement will be automatically turned off. When the measurement display is closed, the data has been saved and sent to the host.

To query the experimental data, press the "T" or "U" key, select backtracking, and then press the confirmation key, as shown in Figure 4, which means that the amplitude recorded for the first time is 134, and the corresponding period is 1.442 seconds, and then press the "P" or "Q" key to view all the recorded data, that is, each measured amplitude.

3. Damping oscillation

In the state of Figure 2, according to the experimental requirements, press the "U" key to select damping oscillation, and press the OK key to display damping, as shown in Figure 5. There are three levels of damping, and 1 has the smallest damping. Select the damping gear according to your own experimental requirements, such as 1, and press Enter to display, as shown in Figure 6.

Turn the balance wheel about 160 degrees by hand, press the "P" or "Q" key after releasing the hand, change the measurement from "off" to "on" and record the data. After the instrument records ten sets of data, the measurement will be automatically turned off, and the amplitude is still changing, but the instrument has stopped counting.

The inspection of damping oscillation is similar to that of free oscillation. Please refer to the above operation. If the measured value of the damping gear changes, repeat the operation steps of the damping gear.

4. Forced vibration

When the instrument is in the state of Figure 2, select forced oscillation and press Enter to display it, as shown in Figure 7 (Note: the damping gear must be selected before forced oscillation, otherwise the experiment cannot be carried out. ) The motor is selected by default.

= seconds (balance wheel)

= seconds (motor)

Damping 1 amplitude

Measure off 00 cycles 1 return of electric organ.

Period x 1

Figure 7

10 = 14.252 seconds (balance wheel)

0 =14.252s (motor)

Damping 1 amplitude 122

Measure 10 motor return at 0 1 cycle.

X period

Figure VIII

Press the "P" or "Q" key to start the motor. However, the experiment cannot be carried out immediately, because the periods of the balance wheel and the motor are unstable at this time, and the measurement will be started after the periods are the same. Before measurement, select the period, and press the "P" or "Q" key to change the period from 1 (as shown in Figure 7) to 10 (as shown in Figure 8) (in order to reduce the error, the measurement cannot be started without changing the period). When the balance wheel and motor are stable, select the measurement, press the "P" or "Q" key to open the measurement and record the data, as shown in Figure 8. It is possible to measure different amplitudes for many times under the same damping, and keep the experimental data every time.

When measuring the phase, put the flash in front of the motor turntable, press the flash button, measure according to the stroboscopic phenomenon, and observe the phase carefully.

After the forced oscillation measurement is completed, press the "T" or "U" key to select Return, and press the OK key to return to the state shown in Figure 2.

Step 5 turn it off

In the state shown in Figure 2, press and hold the reset button, and after a few seconds, the instrument will automatically reset. At this time, all the experimental data will be cleared, and then press the power button to end the experiment.

Six, data recording and processing

1. Calculation of damping coefficient.

The measured data (table 1) is processed by differential method with formula (8), and the beta value is obtained.

Use equation (9) to find the beta value.

2. Measurement of amplitude-frequency characteristics and phase-frequency characteristics

Make the amplitude-frequency characteristic curve and find the β value from it. When the damping coefficient is very small (≤) and close to the * * * vibration position (), the conclusions can be drawn from equations (4) and (7):

If, that is, it can be obtained from the above formula.

This ω corresponds to two values ω 1, ω2 on the graph, from which it is concluded that:

(This content is generally not done)

Comparing and discussing the values obtained by this method with those obtained by successive difference method, the focus of this experiment should be the measurement of phase-frequency characteristic curve.

Table 1 damping gear

serial number

Amplitude (degree)

serial number

Amplitude (degree)

θ 1

θ6

θ2

θ7

θ3

θ8

θ4

θ9

θ5

θ 10

average value

10T= sec = sec

(9)

Table 2 Record Table of Measurement Data of Amplitude-frequency Characteristics and Phase-frequency Characteristics: Damping Switch Position

10 ton

T(s)

(0)

Theoretical value

θ(0)

Measured value

T/T0

Error analysis: Because this instrument uses the timing crystal as the timing element, the error of measuring period (circular frequency) can be ignored, and the error group comes from the determination of damping coefficient and the determination of the natural vibration frequency of the system when there is no damping vibration. The latter has a great influence on the experimental results.

In the previous principle part, we think that the elastic coefficient k of the spring is constant and has nothing to do with the torsion angle. In fact, due to the influence of manufacturing technology and material properties, the value of K changes slightly (about 3%) with the change of angle, resulting in the natural frequency of the system changing at different amplitudes. If the average value is taken, the theoretical value of phase difference will have great correlation with the experimental value near the vibration point. Therefore, the corresponding values of amplitude and natural frequency can be measured. In the formula, replacing T0 with a value corresponding to a certain amplitude can obviously reduce the system error.

The value of the amplitude corresponding to the * * * vibration frequency can be obtained by the following methods:

Cut off the power supply of the motor, set the pointer F of the angle dial to the "0" position, turn the pendulum wheel to a larger position (about 1400 ~ 1500) by hand, and then release it. The balance wheel attenuates vibration and reads out the corresponding swing period of each amplitude value. This method can be repeated several times.

Attachment: Adjustment method of ZKY-BG Boer vibroseis

All parts of the Bohr vibrometer have been calibrated, so please don't disassemble and replace them at will. The electric control box is connected with the host through a special cable, which will not be confused. Be sure to understand the functions of each switch and knob before use.

If it is found that the instrument is not working normally after transportation or experiment, it can be adjusted. The specific steps are as follows:

1, set the angle dial pointer f to "0".

2. Loosen the lock nut of the connecting rod, then turn the connecting rod E so that the rocker arm M is in a vertical position, and then fix the lock nut.

3. At this time, a long notch (marked with white paint line) on the balance wheel should be basically aligned with the pointer. If obvious deviation is found, you can slightly loosen the three fixing screws on the back of the balance wheel, hold the inner end of the spiral spring B with one hand, turn the balance wheel with the other hand to align the white paint line with the tip, and then tighten the three screws: generally, this adjustment is rarely made as long as the length of the spring B is not changed.

4. If the clamping screw L at the outer end of the joint between the spring B and the rocker M is loosened, the outer ring of the spring B can move at will (it can be shortened and lengthened), and the shortened distance should not be less than 6cm. When tightening the end clamping screw, the spring must be kept in the vertical plane, otherwise the experimental results will be obviously affected.

Align the center of the photoelectric door H with the white paint line (that is, the long slit) on the balance wheel, and keep the balance wheel swinging freely in the middle slit of the photoelectric door. At this time, you can select the damping switch "1" or "2" to turn on the motor. At this time, the balance wheel will be forced to vibrate. When it reaches a stable state, turn on the flash switch. At this time, you will see that the pointer F has a seemingly fixed reading on the phase difference dial, and the time difference between the two readings is 656. Within (no more than 2? When the experiment can be carried out), if a big difference is found, the position of the photoelectric gate can be adjusted. If the difference is more than 5? Above, you must repeat the above steps to readjust.

Due to the problems in the process of making the spring, there may be a phenomenon that the two ends of the pointer F overlap well, the middle is poor, or the middle is good and the two ends are poor when measuring the phase difference.

[Notes]

All parts of Bohr vibrometer are assembled accurately, so you can't touch them at will. The knobs and buttons on the control box and panel have many functions. After understanding its function, you must follow the rules.

Matters needing attention in Niels Bohr's vibration experiment:

1. When doing the free oscillation experiment, be sure to write down the balance period during the free oscillation experiment;

2. During the forced oscillation experiment, adjust the knob on the instrument panel to change different motor rotation periods, which must be done 3 ~ 1 1 times, including that the motor rotation period is the same as the free oscillation period in the free oscillation experiment.

3. In the forced oscillation experiment, the period of the motor and the balance wheel must be the same and the amplitude must be stable in order to record the experimental data.

4. After the students finish the experiment, the measurement data must be saved.