1s/m while for the closed damped at 150mm I 15 N. the natural frequency are the same as the no damped condition. damper (closed condition) positioned at 150mm of the beam and damper (closed condition) positioned at 550mm of the beam. which are no damper. 2 .0 Abstract The experiment is conducted to determine the natural frequency resonance of Spring-Dashpot System in different damping conditions. This experiment was conducted at 4 conditions/cases. For the 3 other conditions that are mentioned above. but nevertheless it is considered as a success experiment.817 Hz. the natural frequency is 10. but differs in the resonance frequency. At no damped condition.1. damper (open condition) positioned at 150mm of the beam. The experimental value is slightly different from the theoretical value.s/m. The experiment was done by using Control Unit that controls the Universal Vibration System Apparatus. The damping constant for open damped at 150mm is 5 N. 0 Conclusions 12.0 Abstract 2 2.0 Experimental Procedures 12 9.0 List of Figures 4 5.0 Table of Contents Title Page Title 1 1.0 Table of Contents 3 3.0 Results 10.0 Apparatus 11 8.0 Theory 6-10 7.0 Discussion 11.2.0 References 3 .0 Introduction 5 6.0 List of Tables 4 4. 0 List of Figures Figure 1 : Development of Motion Equation Figure 2 : Effect of Various Damping Factors Figure 3 : Amplitude vibration decreases in an actual case due to effect of air viscous force Figure 4 : Forces acting on a damped vibration case Figure 5 : Machine Diagram Figure 6 : Machine in Lab 4 .0 List of Tables Table 1 : No Damper Table 2 : Open Damper at 150mm Table 3 : Closed Damper at 150mm Table 4 : Closed Damper at 550mm 4.3. In general terms. 5 . the object that is being forced to vibrate at its natural frequency. we may need to define a suitable frequency interval to record the amplitude that will occurs. When the object is forced to vibrate at a particular frequency by a periodic input of force. If damping is absent. While objects which are free to vibrate will have one or more natural frequency at which they vibrate. the motion is called undamped.0 Introduction Vibration refers to the oscillation of a body or a mechanical system about its equilibrium position. As the reaction of the phenomenon’s happen in a short time. The system will vibrate at the excitation frequency when the excitation is oscillatory.5. Tabulate a table that consists of frequency i. resonance will occur and you will observe large amplitude vibrations. buildings. the motion is referred to as free vibration. we may generate the frequency of the system as we may need it for further progress of the experiment. and damped or undamped. Based on our learning of the resonance. The failure of major structures such as bridges. input frequency through the control unit.e. From the theoretical value of the natural frequency. the vibration is said to be forced. vibrations are categorized as forced or free. Damped vibrations refer to a system in which energy is being removed by friction or a viscous damper (resistance caused by the viscous drag or fluid). or airplane wings is an awesome possibility under resonance. 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 resonant frequency is fo. as variable value and amplitude as responding values. If no external force are driving the system. Forced vibration is a vibration that takes place under the excitation of external forces. this phenomenon only occurs if the frequency of the excitation coincides with the frequency of the system. unbalanced rotating motors will cause severe forces on the mounting brackets. For example. and since any periodic function can be expressed as a Fourier series. In this case.6. it is convenient to look at a forcing function of the type P(t) = Posinωt or P(t) = Pocosωt Considering the sin term. In general. forcing functions are periodic.0 Theory A spring-mass-damper system (damped) under free general equation of motion is vibration. it is common for a damped system to have an external harmonic force acting on the system. called a forcing function. Figure 1 : Development of Motion Equation However. the Similarly. A forced vibration is one in which the system is excited by an external. This system is called forced vibrations. x(t) = xc(t) + xp(t) 6 . for horizontal motion. the complementary or homogenous solution. the differential equation is The solution of this differential equations consists of two parts. time-varying force P. 7 . for an underdamped system.φ) Substituting the particular solution into the differential equation and take the appropriate derivatives gives D[(k-mω2)cosφ + cω sinφ] sinωt + D[cω cosφ . Recall.(k-mω2)sinφ] cosωt = Posinωt The sin and cos terms can be separated to get the two equations D[(k-mω2)cosφ + cω sinφ] = Posinωt D[cω cosφ . the particular solution will be of the form xp(t) = Dsin(ωt . the solution is Since the forcing function is sin. can be obtained as The D and φ terms can also be written using ωn and ζ terms.The complementary solution is the general solution of the homogeneous equation as presented in the damped free vibrations section. giving 8 .(k-mω2)sinφ] = 0 Now the constants. D and φ. ζ. systems with low damping have large displacement amplitudes. This ratio can be calculated as A plot for several damping ratios. Notice that when the external force frequency is close to the system natural frequency. 9 . This is commonly known as resonance. is shown below. . Damping causes continuous energy loss to the system. Here we have to consider effect of air viscous force on mass in order to simulate this problem more accurately.To better understand the solution. to the dynamic deflection. The total solution for the underdamped case is where the constants A and B must be determined from the initial conditions. D. Figure 2 : Effect of Various Damping Factors Amplitude of oscillation decreases with time and it finally dies out. it is convenient to consider the ration of the static deflection δp. as system loses its energy amplitude of oscillation decreases. Figure 4 : Forces acting on a damped vibration case 10 . Here spring force and viscous(damping) forces are acting on the mass. Effect of air viscous force can be represented as a single viscous dash port as shown.Figure 3 : Amplitude vibration decreases in an actual case due to effect of air viscous force Forces acting on mass in this case shown in figure below in blue arrows. damping force is proportional velocity of mass with opposite sign of velocity. rectangular.e. ω. In order to define the resonance of the system i. triangular etc. Next. the exciter will be used to give desired forced to the system. 11 . Spring-Dashpot System.we may know theoretically value of the natural frequency.Mathematical solution of this problem can be obtained by applying Newton's 2nd law of motion to the system.we need to find the natural frequency of the system in free vibration state. The effect of damping is to limit the maximum response amplitude and to reduce the sharpness of resonance.g. The differential equation so obtained will be It has got one more term compared to simple spring-mass system case. sine. which can be defined as occurring when the drive frequency Ω equals the natural frequency of the system. a term to incorporate viscous force on mass. By that. As we know exciter is capable to generate different type of forcing signal e. swept sine. Mechanical recorder 6. Damper 5. Control unit (TM150) Figure 5 : Machine Diagram Figure 6 : Machine in Lab 12 . Beam 3.0 Apparatus Universal system vibration apparatus (TM155) which includes: 1. Spring 4. Frame 2.7. Unbalanced exciter 1. followed by increasing 1Hz until 8 Hz. 7. Step 6 is repeated for 9 Hz until the end. 5.1Hz until 9 Hz. Once the reading is reached to 8 Hz. Precaution: Each chart paper plotted is need to be labeled every time before proceeding the frequency increment to avoid any mistakes during tabulation. The controller unit is switched on. Chart paper and plotter pen is fitted. 3. The frequency is first set on 5 Hz. 4. 9. 8. Distance b is taken from the following table. The excited oscillation of each set frequency is plotted on the chart paper by the mechnanical recorder. the increment is done by 0. 2. Step 6-8 is repeated for 3 other cases as mentioned above. 10.0 Experimental Procedure In this experiment. The mechanical recorder is then tested. 4 types of cases are to be examined: (i) No damper (ii) Damper (open condition) positioned at 150mm of the beam (iii) Damper (closed condition) positioned at 150mm of the beam (iv) Damper (closed condition) positioned at 550mm of the beam 1. The drum recorder and damper are assembled in line.8. 13 . to ensure if the graph is being recorded clearly and properly. 6. co.12. Theory of Damped Forced Vibration http://physicsnet.C Hibbeler Publisher : Pearson Prentice Hall 2. Kiusalaas Publisher : Thomson Learning 14 . Engineering Mechanics Dynamics. 2nd Edition In SI Edition by Pytel .0 References 1. Engineering Mechanics Dynamics. 11th Edition In SI Units by R.uk/a-level-physics-as-a2/further-mechanics/forced-vibrationsresonance/ 3.