Velocity and acceleration experiment report

The main objective of the present report is to measure the velocity and acceleration by using different methods. The report consists of two parts and all the experiments are conducted in the laboratory workstation. The experiment is carried out under different circumstances and conditions to evaluate the rotational speed and beam vibration by using the piezoelectric accelerometer. Three different strategies are applied to measure velocity and the instruments used in the experiment include proximity sensor, incremental encoder, and toothed -gear (Ellis, 2012). To evaluate the beam vibration accelerometer and the accelerometer signal conditioner is used. The approximate time used in the whole experiment is 45 minutes.

The optical encoder is equipped with what is known as an “open collector output”. That is, the transducer does not, on its own, provide a voltage output. Instead, you must provide (in the lab you this was done for you) a natural frequencies that is connected between the high voltage source (that provides power for the encoder) and a photo transistors in the encoder. To learn more about this, search for information about open collector outputs. Show a schematic detailing how to connect an open collector output to your data acquisition system. Explain (in a paragraph or two)

The optical encoder is fully equipped with the open collector output in the experiment, therefore, the output voltage is considered as the transducer velocity of the beam. For the proper working of the encoder circuit, the load resistor is connected between the high voltage source and the photo-transistor in the encoder. The open-collector outputs are measured for the velocity of the material moving in the experimental process. 

The photo-transistor in the encoder is used with the open-collector outputs because the circuit is further connected to the signal detector oscilloscope. The schematic used in the experiment consists of point optical tachometer, incremental encoder, magnetic gear pickup, a motor speed controller, and connections for the encoder. In the setup, the motor speed controller is connected to the point optical tachometer to control the frequency. At the end of the point optical tachometer, the incremental encoder is used, and connections are done with the encoder. The schematic shown in figure 1 demonstrates the complete process of connecting the open-collector output with the data acquisition system.

Estimate the two different natural frequencies measured in the lab (in step 3 and 5)

On the contrary, beam vibration is measured by accelerometer and accelerometer signal conditioner. In this process, the vibration of mass ended aluminium beam is measured, this acceleration is then used to determine the natural frequency of the mass ended beam under two different sets of masses on the aluminium beam. In return, the natural frequency of the beam is measured in relation to the beam stiffnesses. In the field of measuring vibration, the accelerometer is the most widely used sensor and it is used to provide an accurate signal of acceleration (Valkenburg, 1672; Khorrami, Krishnamurthy, & Melkote, 2003). The mass ended aluminium shaft is connected with the different size of the mass. In case of end mass beam, one side of the beam is attached with a large mass. The system is then used as a second-order mass spring damper system to calculate the natural frequency of the system. The natural frequency of the system is measured by using the below equation,

ω_( n)  =  √(K/(M_( end)+  0.24 M_( beam) ))

where  K  =   (3 EI)/L^( 3) 

In the equation E shows elastic modulus of aluminium, "I" is the cross-sectional inertia of the beam, L is the length from base of the beam towards the centre of the mass that is connected at the end of the shaft. The displacement response is further measured by using second derivative 

x(t)  =  x_0   e^(-ξ ω t)  sin⁡〖( ω √(1 - ξ^2  t)〗

The accelerometer is further connected to the XDCR transducer input on the ICP that is integrated circuit piezoelectric signal conditioners. Three conditions of the gain settings using the signal conditioners are considered including x1, x10, x100. In the equation “g” define gravity and value of acceleration is 9.81 m/s2. 

Use the two different natural frequencies and the two different known masses to obtain the approximate stiffness of the beam. In your calculations, you should consider the mass of the beam; you can estimate that value if you know the cantilevered length of the beam its cross-sectional area and the density of aluminum. 

The first step is to measure the distance from the edge of the lab bench that is the point where the beam is clamped with the centre of mass clamped to the end of the aluminium beam. The length is measured as 11 inches.

L=11 inches

The connections are done for BNC-BNC cable to the signal conditioner output at channel 1 and the analogue discovery is measured using waveform software. The data is extracted by the oscilloscope by tracing the record and changing the frequency. The mass attached to the end of the beam is then changed with the different values of masses. The oscilloscope is used to measure the sufficient time and the voltage resolution for the peak value of the periods. Each peak value and the time are changed by changing the value of mass on the aluminium connector. The mass connected to the connector is changed from 200 g to 400 g for all the measurements.  

M_(end mass)    =  200 g 

E=  69 GPa  =10,000,000 psi

L =  11 

I =   12 mm =  0.012

K =   (3 EI)/L^3   

K =   (3 (69×10^(-9) )(0.012))/11^3 

K =  1.866×10^(-12)  

ω_( n)  =  √(K/(M_( end)+  0.24 M_( beam) ))

ω_( n)  =  √((  1.866×10^(-12)  )/(200+  0.24 (1)))   =   9.65×10^(-8)

ω_( n)  =  √((  1.866×10^(-12)  )/(400+  0.24 (1)))   =   6.83×10^(-8)

For each of the two tests, estimate the damping ratio of the beam using the log-decrement method outlined in the class notes. Use the damping ratio and the theoretical natural frequency as listed on the previous page to estimate the theoretical damped natural frequency. Compare the value to the measured natural frequency as estimated in (6) above 

The frequency of moving cantilever decreases with the increase of mass attached on the end of rod. The damping ratio and the theoretical natural frequency is measured by using the expression of natural frequency. The comparison shows high value for the low mass attached and low values for the high mass attached at the end of the rod. 

The velocity calculated by the encoder method is measured by conducting 10 experiments. The output velocity signal is given through oscilloscope. The average output is mentioned below in table 1.  

Test Frequency Speed = frequency x 30

Test 1 5.95 178.5

Test 2 6.00 180

Test 3 5.93 177.9

Test 4 5.81 174.3

Test 5 5.84 175.2

Test 6 5.74 172.2

Test 7 5.48 164.4

Test 8 5.28 158.4

Test 9 5.30 159

Test 10 5.45 163.5

Comparison of output:

The comparison for the output using three methods is mentioned below, 

Tachometer Direct Reading Encoder speed

1 98.6 100 178.5

2 218.12 220 180

3 391 390 177.9

4 593.3 592 174.3

5 802.6 803 175.2

6 979.6 979 172.2

7 1208 1209 164.4

8 1428 1429 158.4

9 1619 1617 159

10 1808 1809 163.5

In the present report, different methods are used to estimate the velocity and acceleration of the moving objects. The circuit used in the experiment is further connected to the oscilloscope and output voltage is provided on the oscilloscope through signals. The comparison of each method shows the reliability of methods, results and accuracy of the outcomes.

References of Velocity and Acceleration

Bartelt, T. L. (2012). Industrial Control Electronics. Cengage Learning.

Cruse, H., Dean, J., & Ritter, H. (2000). Prerational Intelligence: Adaptive Behavior and Intelligent Systems Without Symbols and Logic, Volume 1. Springer Science & Business.

Ellis, G. (2012). Control System Design Guide: Using Your Computer to Understand and Diagnose Feedback Controllers. Technology & Engineering. 

Khorrami, F., Krishnamurthy, P., & Melkote, H. (2003). Modelling and Adaptive Nonlinear Control of Electric Motors. Springer Science & Business.

Valkenburg, M. E. (1672). Reference Data for Engineers: Radio, Electronics, Computers and Communications. Technology & Engineering.