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I would be grateful if you can answer the questions in page 5 and page 6.
Objective To find the stiffiness of a steel spring and determine the natural frequency of a mass spring vibrating system. Theory By Hooke's law, the deflection or extension of a spring is proportional to the force applied to it. From Figure 1, ×αF x=kF​ or F=kx Where F in N,x in m,k= stiffness of spring in N/m A rigid body of mass M under elastic restraint, supported by spring(s), forms the basis of all analysis of vibrations in mechanical systems. MEC4109 Dynamics - DY01 Page 1 of 6
The basic equation for undamped free vibration is of the form mx˙=−kx where k= stiffness of the spring m = mass of vibrating part (M) + effective mass of the spring The effective mass of spring is usually reglected if its value is small compared with M. The motion is known as Simple Harmonic of periodic time τ=2πkm​……(1)​τ2=4π2km​…⋯(2)​ where m in kg,k in N/m andt in second. The natural frequency f=1/τ Hz The effective mass of the spring can be found frem the intercept of the graph τ2 against M as shown in Figure 2 and the theoretical value should be 1/3 of the mass of the spring. Apparafus The set-up allows the helical spring to be suspended from the the top member of the portal frame. The lower end of the spring is bolted a rod and integral platform onto which bodies of mass 0.4 kg may be added. The deflection of the spring can measured with the graduating scale provided. A LVDT can be mounted on the top of the platform to record the vibration movement when the system is oscilating. The LVDT signal is recorded and stored in the Oscilloseope. The waveform and frequency of the vibration of the mass spring system can be obtained from the oscilloscope. Procedures PART I (Stiffness of spring) 1. Mount spring a on the frame and take down the initial reading of the pointer without any mass on the platform. 2. Add 0.8 kg of mass (M) on top of the platform. Record the reading of the pointer. 3. Without unloading the mass, add addition mass on the platform as indicated in table la. PART II (Undamped vibration) 1. Add 0.8 kg of mass (M) en top of the platform. Carcfully adjust the height of the LVDT so that the whole range of oscilation can be recorded. 2. Carefully set the system to vertical oscillation. Record the signal in the oscilloscope. 3. Determine the natural frequency from the oseilloscope. 4. Repeat steps 1 to 3 for different masses as indicated in table 2 .
Result PART 1 (Stiffness of spring) Table la for spring A. Stifiness of spring A=380 g Table 1 b for spring B stiftiness of spring B=160 g Plot the graph of M against x for both springs. Find the stiffiness of each spring frogs the slope of the graph. By Hooke's Law Fekx Mgi​kx M=(k/g)x Slope =k/R k= slope ∗ [BE CAREFUL. of units.] MECA109 Dynamics - DV0!
Part II (Undamped Firee Vibration) Table 2 a for spring a Intercept = kg Table 2b for spring b Intercept = k8 Erom equation (2), τ2=4π2km​ m=M+ mass of platform and guide rod assembly + effective mass of spring Plot the graph of τ2 against M for spring a and spring b on the same graph paper. The intercept of the best fit straight line on the M axis (Refer to figure 2) is the sum of mass of platform and guide rod assembly and the effective mass of spring. Given: Mass of spring A=0.16 kg Mass of spring B=0.38 kg Mass of the Rod and Integral Platform =1.53 kg  Discussion ​ 1. Discuss the factors that will affect the stiffness of a spring by comparing the stiffness of spring a and spring b. 2. Discuss the effect of changing the mass on the natural frequency of a mass spring vibrating system. 3. Discuss the effeet of the value of stiffness of spring on the natural frequency of a mass spring vibrating system. 4. Compare the effective mnss of the spring obtained with the generally accepted value, i.e. 1/3 mass of the spring and discuss the differences. MEC4109 Dynamics - DY01 Page 4 of 6
Discussion Discuss your result based on your data and calculations. You should support your argument with the cxperiment results. 1. Discuss the factors that will affect the stiffness of a spring by comparing the stiffiness of spring a and spring b. 2. Discuss the effect of changing the mass on the natural frequency of a mass spring vibrating system. 3. Discuss the effect of the value of stiffness of spring on the natural frequency of a mass spring vibrating system. 4. Compare the effective mass of the spring obtained with the generally accepted value, i.e. 1/3 mass of the spring and discuss the differences. MEC4109 Dynamics - DY01 Page 5 of 6
Summar Write bricfly what you have leamt in this experiment. VIEC:AlO9 Dynamies - DY II