Introduction All system which possessing mass and elasticity are capable of undergoing free vibration. The main interest of such system is the natural frequency of the vibration.

Natural frequencies are a function of system stiffness and mass. Generally, all real systems have infinite number of natural frequencies. Resonance, large vibration, will occur in a system when the forcing frequency of the system is equal to one of the natural frequencies.

Generally, vibration is a form of wasted energy and normally is undesirable is most cases. It generates noise, instability and will consequently lead to system break down which is a disastrous effect.

This project studies a model vibration of a four-cylinder engine which is mounted on a cantilever. This system can be modeled as a mass spring system so that the vibration of the system can be studied. These vibrations are mainly due to the unbalance of primary or secondary forces and couples.

1. Background The model engine consists of four cylinders mounted of a crank shaft with different crank phase angles. A DC motor with a variable speed control drives the shaft and makes the four-cylinder to rotate. The shaft is rotating at a constant angular speed, however, producing an acceleration on the pistons. Hence there are some inertia forces acting on the crank shaft due to these accelerations.

The relative crank phase angles of the four cylinders can be adjusted by loosen three Allan screws to each point of the adjustment. Each crank shaft is to be dynamically balanced with the usual conventional allowance for the effect of its connecting rod. This helps the assembled crank shaft to remains substantially balanced for rotating effects.

The mass center G of the assembly is on the cantilever. There are two units of Linear Variable Displacement Transducer which are equipped at the free end if the cantilever beam and aligned perpendicularly to each other. The vertical LDVT measures bending displacement, horizontal LDVT measures the displacement caused by torsion. The outputs of the LDVT are shown in the oscilloscope. The tachometer measures the speed of the engine.

2. Objective The objective of this project is to: i) To study the vertical and torsional vibration of the model under the excitation of dynamic unbalance force and couple. ii) To study the approaches of dynamic balancing for reciprocating masses. . 3. Scope The scope of the project includes the study and analysis of vibrations of the four-cylinder engine. It examines the results and experiment carried out on each day of the experiment and study the problems suggested. It also discusses the open-end discussion found at the last page of the lab sheet and hence provides solutions to it.

1. Theory In this experiment, an electric motor assembly is mounted at the end of a single circular cantilever with the mass center G of the assembly on the axis of the cantilever. The assembly consists of four individual single crank-sliders with adjustable initial crank angle. Two units of Linear Variable Displacement Transducer (LVDT) are placed at the free end of the cantilever and aligned perpendicular to each other. The vertical LVDT measures the bending displacement while the horizontal one measures the displacement caused by torsion.
[pic]
Analyzing the individual single crank-slider mechanism,
[pic]

Derive the position vector of point B.
Position vector of point B = [pic] [pic] [pic][pic]
Derive the velocity and acceleration of point B if r is very small compared with l [pic] [pic] [pic] [pic]
So, we have
[pic]
[pic]

The force caused by[pic] of each cylinder on the shaft is:
[pic] (0 is higher order term)

So, if 4 cylinders are considered, the total shaking force will be:
[pic]=[pic]
Assuming all mass of each cylinder is the same, and [pic] is the crank phase angle
Substituting the trigonometry identity:
[pic]

[pic] [pic]
[pic]
The total shaking torque of the system is derived by taking moment about the mass centre G:

[pic]
[pic]
Where zi is the distance of each individual crank from the mass center G of the assembly as shown in the diagram above, which is -3d, -d, d and 3d.

2. Literature Review This literature review aimed to provide reviews on some of the works in free vibration and four-cylinder engine.

The William T.Thamson (1993) states that there should be some conventions of the multi cylinders for the measurements of these phase angles which should be take note.

Firstly, the first front cylinder should be reference as number 1 and phase angles will always be zero, so that it is a reference cylinder for all of the other cylinders. Secondly, the phase angles of all other cylinders will be measured with respect to that of the cylinder number 1. Thirdly, Phase angles are measured with internal to the crankshaft. This helps to provide further analysis of our experiment as the rotating coordinate system is embedded in the first crank throw. Lastly, it also states that all the cylinders will be numbered consecutively from front to back. In our experiment we have only four cylinders which make analysis easier after the conventions of measurements are taken.

It is also take note from William T.Thamson (1993) that for the maximum cancellation of inertia forces, which have a period of one revolution, the optimum phase delta angle will be : [pic] where n is the number of cylinders. All of the previous research done will aid in our understanding of the experiment as well as provide us with better guide in getting our datas while doing the experiment.
3. Experiment & Procedure 1. Equipment • Four Cylinder Model Engine • Two Transducers • Two Transducer Monitors • One Oscilloscope • One Motor Speed Controller • One Screw Driver • One Allan Key • One Tachometer [pic] [pic] Fig.1 Four Cylinder Model Engine Fig.2 Transducer

[pic] Fig.3 Experiment Equipments

2. Experiment The fundamental of this four cylinder model engine is the arrangement of crank throws on the crankshaft. Any arrangement of the crank phase angle can be chosen to study the system. To set the angle of the cylinder, the first cylinder will be number 1 and its phase angle will always be zero. It is the reference cylinder for all others. The phase angles of all other cylinders will be measured with respect to the crank throw for cylinder 1. Phase angles are measured internal to the crankshaft that is, with respect to a rotating coordinate system embedded in the first crank throw. Cylinder will be numbered consecutively from front to back of the engine. Four different experiments were carried out throughout this project. • To study the natural frequency characteristics of the vertical bending vibration of the system. • To study the natural frequency characteristics of the torsional vibration of the system • To study the resonance characteristics operation at crank angles [0,0,0,0], [0,0,180,180] and [0,180,0,180]. • To study the effect on the total shaking force and shaking torque with the variation of crank angles[0,0,0,0], [0,0,180,180], [0,180,0,180], [0,90,180,270] and [0,180,180,0].

3. Procedure 4 different experiments were carried out throughout this project.

i. To study the natural frequency characteristics of the vertical bending vibration of the system. During this experiment, motor is not turned on and the system is undergoing free vibration after a vertical impulse has been applied on the system. Step 1: Give an initial impulse to the system. Step 2: Observe the characteristics of the signal. Step 3: Measure the natural frequency and the amplitude of the vertical bending vibration.

ii. To study the natural frequency characteristics of the torsional vibration of the system. During this experiment, motor is not turned on and the system is undergoing free vibration after an impulse at an angle has been applied on the system. Step 1: Give an initial impulse to the block. Step 2: Observe the characteristics of the signal. Step 3: Measure the natural frequency and the amplitude of the torsional vibration. iii. To study the resonance characteristics operation at crank angles [0,0,0,0], [0,0,180,180] and [0,180,0,180]. During this experiment, the motor is turned on and the speed is slowly increased until resonance effect is observed when the motor is running at the natural frequency of the system. Step 1: Set the initial crank angles at [0,0,0,0]. Step 2: Let the frequency of the crank be increased slowly. Step 3: Measure the rotational frequency of the crank at the remarkable resonance for linear bending vibration. Step 4: Repeat the experiment for crank angles [0, 0,180,180] and [0, 180, 0,180].

iv. To study the effect on the total shaking force and shaking torque with the variation of crank angles[0,0,0,0], [0,0,180,180], [0,180,0,180], [0,90,180,270] and [0,180,180,0]. During this experiment, the motor is turned on and the speed is slowly increased until resonance effect is observed when the motor is running at the natural frequency of the system. This experiment allow a further understanding on the vibration of the system which is the vibration is caused by the total shaking force and the total shaking moment which is due to the motion of the four cylinders. Step 1: Set the initial crank angles at [0,0,0,0]. Step 2: Let the frequency of the crank be increased slowly. Step 3: Measure the rotational frequency of the crank at the remarkable resonance for linear bending vibration. Step 4: Repeat the experiment for crank angles [0,0,180,180], [0,180,0,180], [0,90,180,270] and [0,180,180,0]. Step 5: Observe any variation and trend in the results

4. Results 1. Results Given an initial vertical impulse to the block, using the signals of free bending vibration, the natural frequency of the vertical bending vibration of the system is measured.

Amplitude = 0.82 V Natural frequency, wn = 11.76 Hz

Given an initial angular impulse to the block, using the signals of free torsional vibration, the natural frequency of the torsional vibration of the system is measured.

Natural frequency, wn = 12.34 Hz

As the initial crank angles are set at 0,0,0,0, the frequency of the crank is increase slowly and the rotational frequency of crank at the remarkable resonance for linear bending vibration is measured.

Amplitude = 0.624 V Natural frequency, wn = 12.19 Hz Speed = 365 rpm

As follows, the initial crank angels are then set at 0,0,180,180, and 0,180,0,180, respectively. The frequency of crank is increase slowly, the rotational frequency of crank at the remarkable resonance for torsional vibration is measured.

[0,0,180,180] Amplitude = 18.51 V Natural frequency, wn = 18.51 Hz Speed = 1100 rpm

[0,180,0,180] Amplitude = 19.23 V Natural frequency, wn = 0.464 Hz Speed = 1168 rpm

2. Data Analysis The main concern of this project is the vibration which caused by the unbalance shaking force and shaking torque. Thus study and analysis the effect of the crank angle arrangement on the total shaking force and shaking torque is very important.

When the crank shaft is rotating with a contact angular velocity, ω, each cylinder will subjected to an inertia force due to the upward or downward acceleration which caused by the angular velocity. These inertia forces will caused a reaction force on the rotating shaft. These reaction forces are considered as unbalance forces acting on the shaft because these reaction forces acting at different direction at any instant due to the different crank angles. Thus causing the shaking force or shaking torque on the shaft and hence lead to vibration of the system. So, it is very important to determine the crank angles of the motor in order to minimize the vibration. The equation of calculating the shaking force and shaking moment are shown below:

Total shaking force: [pic] [pic] Total shaking moment: [pic] [pic] [pic]

In order to have no vibration, [pic] and [pic] have to be zero. From the equations above, we can conclude that the [pic] and [pic] will be equal to zero if [pic]，[pic]，[pic]，[pic]，[pic]，[pic]，[pic]，[pic] Thus, theoretically, finding a set of suitable crank angle and make the above terms to be zero can eliminate the shaking force and shaking moment, i.e. balanced the system so that no vibration occurs.

5 set of crank arrangement has been designed to study and analysis on the effect of total shaking force and shaking torque, which is: • [0, 0, 0, 0] • [0, 0, 180, 180] • [0, 180, 0, 180] • [0, 90, 180, 270] • [0, 180, 180, 0]

| |Set 1 |Set 2 |Set 3 |Set 4 |Set 5 | | | | [0, 0, 0, 0] | [0, 0, 180, 180] | [0, 180, 0, 180] | [0, 90,180, 270] | [0, 180, 180, 0] | |Pimary Shaking Force |ΣcosØi |4.00 |0.00 |0.00 |0.00 |0.00 | | |ΣsinØi |0.00 |0.00 |0.00 |0.00 |0.00 | |Secondary Shaking Force |Σcos2Øi |4.00 |4.00 |4.00 |0.00 |4.00 | | |Σcos2Øi |0.00 |0.00 |0.00 |0.00 |0.00 | |Pirimary Shaking Moment |Σzi cosØi |0.00 |8.00 |4.00 |4.00 |0.00 | | |Σzi sinØi |0.00 |0.00 |0.00 |4.00 |0.00 | |Secondary Shaking Moment |Σzi cos2Øi |0.00 |0.00 |0.00 |4.00 |0.00 | | |Σzi cos2Øi |0.00 |0.00 |0.00 |0.00 |0.00 | |Table 1. Effect on total shaking force and shaking torque

From the table above, we can conclude that: 1. For setting 1, which is having crank angles of (0, 0, 0, 0), primary and secondary excitation force dominant the vibration. 2. For setting 2, which is having crank angles of (0, 0, 180, 180), secondary force excitation and primary torque excitation dominant the vibration. 3. For setting 3, which is having crank angles of (0, 180, 0, 180), secondary force excitation and primary torque excitation dominant the vibration. 4. For setting 4, which is having crank angles of (0, 90, 180, 270), only primary and secondary torque excitation dominant the vibration. 5. For setting 5, which is having crank angles of (0, 180, 0, 180), secondary force excitation dominant the vibration.

From the result above, it is obviously that to obtain an optimized dynamic balance, i.e. to eliminating primary shaking force and shaking torque, the best combination of the crank phase angles will be set 5, which is 0, 180, 180, 0. Although the is some vibration due to secondary shaking force, however, the effect can be neglect as the contribution is less as compare to primary shaking force and primary shaking moment due to the [pic] terms. The higher the order the smaller the effect. Beside from the above table, we can also use the below method to prove that the best combination of crank phase angle will be [0, 180, 180, 0]. [pic]
Let [pic], [pic], [pic], [pic]

For dynamic balancing, total shaking force and total shaking moment should be equal to zero.
[pic]=0
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Thus, [pic]

[pic]
[pic]
[pic] [pic]
[pic][pic]
[pic]
As [pic]
So, [pic] [pic] [pic] [pic] [pic]
So,[pic], [pic], [pic], [pic]

5. Discussion 1) How to determine the natural frequency of the model in vertical oscillation? In the case of our model, we first set up the crank arrangements (example 0,0,0,0). The model is then powered up and the speed of the motor is gradually increased. As the speed increases, the model starts to vibrate. The vibration signals are picked up by probes located vertically and axially to the model crankshaft and read off through the use of an oscilloscope. The model is then noted to vibrate excessively in the range of 360 – 370rpm (reading taken off tachometer). On the oscilloscope, we noted the incoming signals were having giving a frequency reading of 12.05hz. Converting this frequency to oscillation will give approximately 725rpm which is about twice that of the tachometer readings. As we know that wave signals measured by oscilloscope gives the frequency of the vibration of the system, the phenomena that we just observed is known as the secondary resonance. We also noted this happened for all other crank arrangements with the exception of the (0,90,180,270) setup which will be further discussed in Q4. The vibrations then reduced when past the 370rpm mark and hits the peak again around the 725 – 745rpm which corresponds to the 12.05 – 12.5hz on the oscilloscope. This happened for all sets of experiment and therefore we can conclude that the frequency measured by the oscilloscope is the natural frequency.

2) How to determine the natural frequency of the model in torsional oscillation? The procedure is about the same as the above experiment except that we read off the signals on the oscilloscope provided by the axially located probe. Likewise the speed of the motor was increased till the vibrations hit the peak. We observed peak vibrations at the 1100 – 1145rpm range and that corresponds to the 18.5 – 18.9hz signals on the oscilloscope. This was noted for all the setups except the (0,0,0,0 & 0,180,180,0) arrangements. We can therefore conclude that this was the natural frequency of the model in torsional oscillation.

3) Justify the type of excitation that causes resonance at particular critical engine speed. The effect of a certain excitation at resonance will cause excessive vibrations throughout the system. The excitation forces can be categorized to 4 types : primary force; secondary force; primary torque and secondary torque. These 4 forces can be summed up by the following formulas : Total shaking Forces [pic] ------------( Primary Force [pic]-(Secondary Force

Total shaking Torque [pic] [pic]------(Primary Torque [pic]-(Secondary Torque

In the case of our experiment, the total primary force is a result of the movement of the individual cylinders (centrifugal and denoted by mrω²) and their phase shifts. It is to be noted that the largest shaking force will be generated at its natural frequency such that ω = ωn. The secondary resonance takes place when the crank reaches half the natural frequency and these phenomena may or may not occur (depending on phase arrangements) The shaking torque is a result of bending moments being created about the y-axis. This translates to a torque being generated on the cantilever beam which is set parallel to the y-axis. As with the shaking forces, max excitation takes place at its natural frequency. Do note that both natural frequencies are different.

4) How do you explain the fact that there are 4 most marked resonance frequencies? As mentioned in the above, the 4 most marked resonances are caused by the primary and secondary excitation forces and torque. In reality higher orders of harmonics such as the 4th and 6th terms do exist, however their coefficients are minute and therefore their contributions are often ignored. To explain this in detail, all we need to do is look at the shaking force and shaking torque formulas above. The secondary force and torque are lower in magnitude as compared to the primary as they have to multiplied by (r/l) which is generally around 1/3 depending on the machinery. Depending on the size of the engine involved, we can then decide on whether the secondary excitations will be of concern to the machinery. This fact is further enhanced from our experimental readings which shows the primary readings registering a increase of 0.2 – 1.2V in magnitude over the secondary readings.

6. Reference 1. William T. Thamson, “Theory of Vibration with Applications”, Prentice Hall, 1993.(2.1,2.2) 2. Robert L. Norton, “Design of Machinery”, McGraw Hill, Inc. 1992.(15.1-15.3, 15-5)

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Motor Speed Controller

Oscilloscope

Transducer Monitor