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Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring1 TitleForce vibration experiment – Resonance of spring – Dashpot system with spring2 List of Tables2 Table of Amplitude 2.1 Without damper ................................ separate sheet 2.1 Damped 150 mm (closed) ..........

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MEC 424-APPLIED MECHANICS LAB 1 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

1.0 Title Force vibration experiment – Resonance of spring – Dashpot system with spring

2.0 List of Tables 2.1 Table of Amplitude 2.1.1 Without damper ................................ separate sheet 2.1.2 Damped 150 mm (closed) ................ separate sheet 2.1.3 Damped 150 mm (open) .................. separate sheet 2.1.4 Damped 550 mm (closed) ............... separate sheet 2.2 Table of Tabulate data .........................................................page 8 - 20

3.0 List of Figures 3.1 Figure of universal system apparatus ................................. page 5 3.2 Figure of beam ...................................................................... page 6

MEC 424-APPLIED MECHANICS LAB 2 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

4.0 Introduction Forced vibration is vibration that undergoes the excitation of external forces. The system will vibrate at the excitation frequency when the excitation is oscillatory. Resonance will occur if the frequency of excitation coincides with one of the natural frequencies of the system and dangerously large oscillations may result. The failure of major structures such as buildings, bridges, or airplane wings is an awesome possibility under resonance.

4.1 Objective To determine the resonance of spring-dashpot system in different type of damping or damping condition.

MEC 424-APPLIED MECHANICS LAB 3 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

5.0 Theory As we know in the introduction section has been stated that resonance will occur if the frequency of excitation coincides with one of the natural frequencies of the system. This may result in the large oscillations occurs. The maximum amplitude of the oscillation for a certain body can be obtain by the large oscillations. So, this may result in failure of member of the system. This failure happen because the member may reach the maximum deformation and it can attain or soon will fracture. By conduct this experiment, we can obtain the relation of rotational frequency during the resonance. There is a formula to get the relation of rotational frequency during resonance which is, ω d=ωn

ω d=¿

The damped frequency

ω n=¿ The natural frequency

The unit of both frequency are in (rad/s).The resonance will occurs and may result in maximum amplitude for the oscillation because they are equal in unit. For the damper, the resulting force is proportional to the velocity because it resist motion via viscous friction but it acts in the opposite direction and may slowing the motion and absorbing energy. For information, it is usually been used in conjunction with a spring.

MEC 424-APPLIED MECHANICS LAB 4 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

6.0 Experimental Procedures Location of experiment : Dynamics laboratory , faculty of mechanical engineering , UiTM Shah alam .

6.1 Apparatus Universal system apparatus ( TM155 ) including : a. b. c. d. e. f. g.

Frame Beam Spring Damper Mechanical recorder Unbalanced exciter Control unit (TM150)

MEC 424-APPLIED MECHANICS LAB 5 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

Frame Spring (k=3.0N.mm) Beam

Mechanical recorder

Unbalance exciter

Control unit (TM150)

Figure 1 : Universal system vibration

MEC 424-APPLIED MECHANICS LAB 6 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

Figure 2 : Beam

The dimensions of the beam are as follows : Thickness ,h = 12mm

Width ,w = 25mm

Length ,L = 700mm

MEC 424-APPLIED MECHANICS LAB 7 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

6.2 Procedure

1. First , the apparatus was set up . The graph paper must be attached in roll on the mechanical recorder which is slotted into the holder . 2. Make sure the graph paper was in grip . A pen must be fixed and tight in the graph pointer slot so that it does not move when running the experiment . it is advisable to use blunt-tip pen or a gel pen . 3. The mechanical recorder was tested to ensure the graph was recorded clearly . 4. The control unit was turned on . 5. The unbalance exciter was controlled it’s frequency by using the control unit for no damper . 6. Before any reading is taken , the maximum amplitude at desired frequency must be predicted first as the beam is forced oscillation by the exciter . 7. Once we roughly know at what frequency the amplitude is at maximum , range of reading needed was tabulated . 8. The amplitude for desired frequency was recorded by mechanical recorder . 9. In this experiment, the amplitude for 5-14Hz was recorded . (5Hz ,6Hz ,7Hz ,8Hz, 8.1Hz, 8.2Hz, 8.3Hz, 8.4Hz, 8.5Hz, 8.6Hz, 8.8Hz, 8.9Hz, 9Hz, 10Hz, 11Hz, 12Hz, 13Hz and 14Hz) 10. Repeated steps 6-8 for damper 150 mm ( closed and open needle) and 550mm (closed needle) . Each frequency amplitude recorded needs to be labeled before move to next frequency to avoid mistake during record the data .

MEC 424-APPLIED MECHANICS LAB 8 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

7.0 Result Free body diagram of the beam P

k( 0.65  )

Support reaction 

150 mm

200 mm

200 mm

c ´y (experiment 2∧3)

100 mm

c ´y (experiment 4)

Experiment 1: Without damper Frequency, f (Hz) 5.0 6.0 7.0 8.0 8.5 9.0 10.0 11.0 12.0 13.0 14.0

2 x Amplitude, 2a (cm) 0.09 0.11 0.15 0.60 3.80 1.15 0.40 0.30 0.25 0.20 0.20

Amplitude, a (cm) 0.045 0.055 0.075 0.300 1.900 0.575 0.200 0.150 0.125 0.100 0.100

Experiment 2: Damper (needle open) at 150 mm from the left end of beam

50 mm

MEC 424-APPLIED MECHANICS LAB 9 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

Frequency, f (Hz) 5.0 6.0 7.0 8.0 8.5 9.0 10.0 11.0 12.0 13.0 14.0

2 x Amplitude, 2a (cm) 0.05 0.10 0.15 0.40 1.10 3.70 0.50 0.30 0.25 0.20 0.15

Amplitude, a (cm) 0.025 0.050 0.075 0.200 0.550 1.850 0.250 0.150 0.125 0.100 0.075

Experiment 3: Damper (needle close) at 150 mm from the left end of beam Frequency, f (Hz) 5.0 6.0 7.0 8.0 8.5 9.0 10.0 11.0 12.0 13.0 14.0

2 x Amplitude, 2a (cm) 0.05 0.10 0.20 0.45 1.30 3.60 0.50 0.30 0.20 0.20 0.20

Amplitude, a (cm) 0.025 0.050 0.100 0.225 0.650 1.800 0.250 0.150 0.100 0.100 0.100

Experiment 4: Damper (needle close) at 550 mm from the left end of beam Frequency, f (Hz) 5.0 6.0 7.0 8.0 8.5 9.0 10.0 11.0 12.0 13.0

2 x Amplitude, 2a (cm) 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

Amplitude, a (cm) 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050 0.050

MEC 424-APPLIED MECHANICS LAB 10 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

14.0

0.10

0.050

Below are the graphs plotting for the amplitude versus the frequency of the 4 experiments.

amplitude versus frequency graph 2 1.8 1.6 1.4 experiment 1 experiment 2 experiment 3 experiment 4

1.2 Amplitude,a (cm)

1 0.8 0.6 0.4 0.2 0 5

6

7

8

9

10

11

12

13

14

15

Frequency,f (Hz)

Sample calculation

First,we have to find , n and d = 2πf Example, = 2π(6.5)=40.84 rad/s To obtain n, Free body diagram (FBD) of the beam (experiment 1) k (0.65 )



150 mm

200 mm

200 mm

100 mm 50 mm

MEC 424-APPLIED MECHANICS LAB 11 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

Kinetic diagram k (0.65 ) Support reactions  150 mm

200 mm

200 mm mexciter (0.35 ❑ )

Equating the FBD and kinetic diagram,

mbeam(0.35 ❑ )

Clockwise direction of moment assumed to be positive,

-kȴ

α θ = IₒӪ

IₒӪ + kȴ α θ

= 0 (mathematical model)

Iₒ beam = IG + md²

IG =

=

1 12 1 12

m (a²+b²)

(1.68) (0.7² + 0.012²)

= 0.0686

Iₒ beam = 0.0686 + md² = 0.0686 + [(1.68)x(0.35²)] =0.2744

Iₒ exciter = IG + md² = 0 + 0.772(0.35²)

100 mm 50 mm

MEC 424-APPLIED MECHANICS LAB 12 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

=0.0946

Iₒ total = Iₒ beam + Iₒ exciter = 0.2744 + 0.0946 = 0.36899 Compared to the general equation; ωn =

=

√

√

2

´x +2❑n ´x +❑n x =

1 f m

k ȴ² Iₒ

3000 ( 0.652) 0.369

=58.6085 rad/s F=

ωn 2π

9.33 hertz #

For case 2, to find d,2, Free body diagram (FBD) of the beam (experiment 2) k (0.65 ) c (0.15 ❑ )



150 mm

200 mm

200 mm

100 mm 50 mm

Kinetic diagram k (0.65 ) Support reactions

150 mm

200 mm

200 mm

100 mm 50 mm

MEC 424-APPLIED MECHANICS LAB 13 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring 

mbeam(0.35 ❑ )

mexciter (0.35 ❑ )

c= 5 N.s/m Equating the FBD and kinetic diagram, Clockwise direction of moment assumed to be positive, Mo=I: ´ )(0.15)-k(0.65 )(0.65)=IO ❑ ´ -c(0.15 ❑ ´ + (5)(0.15 ❑ ´ )(0.15)+3000 (0.65 )(0.65)=0 0.36899❑

Compared to the general equation;

2❑n=

(5)(0.15)(0.15 ) 0.36899

¿

0.304885 2❑n

¿

0.304885 2(58.609) =2.601x10-3

❑d ,2 =❑n √1−❑

2

−3 2 ¿ 58.609 √ 1−(2.601 x 10 )

=58.6088 rad/s F =

ωn 2π

2

´x +2❑n ´x +❑n x =

1 m

MEC 424-APPLIED MECHANICS LAB 14 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

= 9.327 hertz

For experiment 3, to find d,3, Free body diagram (FBD) of the beam (experiment 3) k (0.65 ) c (0.15 ❑ )



150 mm

200 mm

200 mm

100 mm 50 mm

Kinetic diagram k (0.65 ) Support reactions  150 mm

200 mm

200 mm mexciter (0.35 ❑ ) mbeam(0.35 ❑ )

c= 15 N.s/m Comparing the FBD and kinetic diagram, Clockwise direction of moment assumed to be positive, Mo=I: ´ )(0.15)-k(0.65 )(0.65)=IO ❑ ´ -c(0.15 ❑ ´ + (15)(0.15 ❑ ´ )(0.15)+3000 (0.65 )(0.65)=0 0.36899❑

100 mm 50 mm

MEC 424-APPLIED MECHANICS LAB 15 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

Compared to the general equation;

2❑n=

2

´x +2❑n ´x +❑n x =

1 f m

(15 )( 0.15 )(0.15 ) 0.36899

¿

0.914655 2❑n

¿

0.914655 2(58.609)

=7.80303x10-3 ❑d ,3 =❑n √1−❑

2

−3 2 ¿ 58.609 √ 1−(7.80303 x 10 )

=58.6072 rad/s

For experiment 4, to find d,4, Free body diagram (FBD) of the beam (experiment 4) c (0.55 ❑ )

k (0.65 )



150 mm

200 mm

200 mm

100 mm 50 mm

Kinetic diagram k (0.65 ) Support reactions  150 mm

200 mm

200 mm

100 mm 50 mm

MEC 424-APPLIED MECHANICS LAB 16 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

mexciter (0.35 ❑ ) mbeam(0.35 ❑ )

c= 15 N.s/m Equating the FBD and kinetic diagram,

Clockwise direction of moment assumed to be positive, Mo=I: ´ )(0.55)-k(0.65 )(0.65)=IO ❑ ´ -c(0.55 ❑ ´ + (15)(0.55 ❑ ´ )(0.55)+3000 (0.65 )(0.65)=0 0.36899❑

Compared to the general equation; 2❑n=

(15 )( 0.55 )(0.55 ) 0.36899

¿

4.5375 2❑n

¿

4.5375 2(58.609)

=0.10491

❑d , 4=❑n √1−❑

2

¿ 58.609 √ 1−( 0.10491 )

2

=58.2856 rad/s F =

ωn 2π

=9.276 hertz Experiment 1: Without Damper

´x +2❑n ´x +❑n2 x =

1 f m

MEC 424-APPLIED MECHANICS LAB 17 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

Frequency, f (Hz) 5.0

Amplitude, a (cm) 0.045

Frequency ratio , ❑ ❑n

Rotational frequency,  (rad/s) 31.416

Natural frequency, n (rad/s) 58.6085

0.5360

6.0

0.055

37.699

58.6085

0.6432

7.0

0.075

43.982

58.6085

0.7504

8.0

0.300

50.265

58.6085

0.8576

8.5

1.900

53.407

58.6085

0.9112

9.0

0.575

56.548

58.6085

0.9648

10.0

0.200

62.832

58.6085

1.0720

11.0

0.150

69.115

58.6085

1.1792

12.0

0.125

75.398

58.6085

1.2865

13.0

0.100

81.681

58.6085

1.3937

14.0

0.100

87.964

58.6085

1.5009

Experiment 2: Damper (needle open) at 150 mm from the left end of beam Frequency, f (Hz) 5.0 6.0 7.0 8.0 8.5 9.0 10.0 11.0

Amplitude,

Rotational frequency,

Natural frequency,

a (cm) 0.025 0.050 0.075 0.200 0.550 1.850 0.250 0.150

 (rad/s) 31.416 37.699 43.982 50.265 53.407 56.548 62.832 69.115

d (rad/s) 58.6088 58.6088 58.6088 58.6088 58.6088 58.6088 58.6088 58.6088

Frequency ratio , ❑ ❑d 0.5360 0.6432 0.7504 0.8576 0.9112 0.9648 1.0720 1.1792

MEC 424-APPLIED MECHANICS LAB 18 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

12.0

0.125

75.398

58.6088

1.2865

13.0

0.100

81.681

58.6088

1.3937

14.0

0.075

87.964

58.6088

1.5009

Experiment 3: Damper (needle close) at 150 mm from the left end of beam Frequency,

Amplitude,

Rotational frequency,

Natural frequency,

f (Hz) 5.0 6.0 7.0 8.0 8.5 9.0 10.0 11.0 12.0 13.0

a (cm) 0.025 0.050 0.100 0.225 0.650 1.800 0.250 0.150 0.100 0.100

 (rad/s) 31.416 37.699 43.982 50.265 53.407 56.548 62.832 69.115 75.398 81.681

d (rad/s) 58.6072 58.6072 58.6072 58.6072 58.6072 58.6072 58.6072 58.6072 58.6072 58.6072

14.0

0.100

87.964

58.6072

Frequency ratio , ❑ ❑d 0.5360 0.6432 0.7504 0.8576 0.9112 0.9648 1.0720 1.1792 1.2865 1.3937 1.5009

MEC 424-APPLIED MECHANICS LAB 19 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

Experiment 4: Damper (needle close) at 550 mm from the left end of beam Frequency, f (Hz) 5.0 6.0 7.0 8.0 8.5 9.0 10.0 11.0 12.0 13.0 14.0

Amplitude,

Rotational frequency,

Natural frequency,

a (cm) 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

 (rad/s) 31.416 37.699 43.982 50.265 53.407 56.548 62.832 69.115 75.398 81.681

d (rad/s) 58.2856 58.2856 58.2856 58.2856 58.2856 58.2856 58.2856 58.2856 58.2856 58.2856

0.10

87.964

58.2856

Frequency ratio , ❑ ❑d 0.5390 0.6468 0.7546 0.8624 0.9163 0.9702 1.0780 1.1858 1.2936 1.4014 1.5092

MEC 424-APPLIED MECHANICS LAB 20 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

8.0 Discussions and Conclusions

8.1 Muhammad Arif bin Zakaria 2013301993 Discussion

Based on the experiment result, the external force did not affect the natural frequency for the same system. From the result that we got, the vibration will become slightly overwhelming when the frequency approaches the natural frequency of the system. Therefore the amplitude will be slightly increase. When the frequency is between 8.0 to 9.0 we can see the increasing of this amplitude. But, the graph will be drop after the frequency passed through the natural frequency of the system. This is happen because of the vibration become steadier as it passes the natural frequency. From our experiment, we also found that there are no vibration occurs throughout the experiment. This happen for experiment 4. We found there is no vibration occurs because the damper is to close with the source of the vibration. This will cause the damper absorbs the vibration and reduce the amplitude of the vibration. Besides that, when the damper in close needle valve condition, the value of amplitude is highly reduced. So, this will result only straight line will be recorded for the graph during experiment. However, there are several errors occur that affect the result of the experiment. For example we can say that the error occur when we try to set the frequency. This may happen because the control unit for the frequency is very sensitive and it may change the frequency automatically during the experiment. Beside that, parallax error also occur in this experiment. It happens when we measure the amplitude response. So, there are several precautions that can be take. Make sure to use the sharper marker so that it can react smoothly with the vibrations. Lastly, make sure there is no external vibration occur during the experiment that will cause the increasing the vibration towards the table. If this happens, it may affect the result.

MEC 424-APPLIED MECHANICS LAB 21 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

8.1 Muhammad Arif bin Zakaria 2013301993 Conclusion

From experiment we can conclude that the resonance of Spring-Dashpot System in different damping condition is obtained. Based on result from experiment we got the value of ω n and ω D for every that we got is 58.6085 rad / s experiment. The values for ω n and the value for ω D is58.6088 rad / s for experiment 2. . For experiment 3, ω D is58.6072 rad / s . For experiment 4 ω D is58.2856 rad / s . Therefore, based on the value that we got, we conclude that the objective of this experiment considered to be success.

MEC 424-APPLIED MECHANICS LAB 22 Dynamics – Forced vibration – Resonance of spring – Dashpot system with spring

8.2 Muhammad Syahmi Dzulkafly 2013554217 Discussion From the result above, we can say that the external force did not affect the natural frequency for the same system. As we can see, the vibration will become slightly overwhelming when the frequency approaches the natural frequency of the system. In the other word it will result sudden increase in amplitude. We can see that happen in the experiment during the frequency is be...