ME 260W - Measurement TechniquesExperiment #4: Beam VibrationsA shaker is driven, with displacement y, to displace the fixed end of a cantilever beam and impart harmonic motion in the beam. Mounting two strain gages on the beam as shown in Figure 1 allows the measurement of maximum strain in the beam as the end of the beam oscillates in x direction. With virtual instrumentation, the computer operator can control the motion of the shaker and simultaneously record data from the strain gages. This data can be examined with two different approaches, through determination of the damping ratio and determination of the deflection of the end of the beam.Two metallic resistance type strain gages are used in conjunction with a half Wheatstone bridge as shown in Figure 2. Briefly, the gages are mounted on a specimen and under pre-load conditions; the balance potentiometer is adjusted such that eo is zero. Strain in the specimen elongates the strain gage, altering the electrical resistance in the gage. This change in the gage resistance unbalances the bridge, and results in a voltage at eo. This voltage at eo is proportional to the strain. Using two gages in this configuration results in a doubled bridge output, as compared to using a single gage, and also compensates for temperature effects, and torsional and axial components. As the beam vibrates with harmonic motion, the output from the strain gages is a sine wave with amplitude proportional to the strain and a period inversely proportional to the frequency of the vibrations.Exercise 2: Phase Angle measurementsTaking MeasurementsTaking MeasurementsShaker settingsDepartment of Mechanical EngineeringME 260W - Measurement TechniquesSpring 2005Experiment #4: Beam Vibrations1. OBJECTIVES- To determine the first three natural frequencies and mode shapes of a cantilever beam- To characterize a second-order system by it's damping frequency, and damping coefficient- To find the frequency response of a cantilever under sinusoidal excitation2. EQUIPMENT For this lab, we will use:- Power amplifier, CE 2000- Shaker, Model, VTS 100- Strain gage conditioner, P-3500- Digital storage oscilloscope, Tektronix TDS 210- Accelerometer transducer PCB 482A22- Signal conditioner, PCB 488A-01- Computer station with data acquisition A/D board 3. THEORETICAL ANALYSISA shaker is driven, with displacement y, to displace the fixed end of a cantilever beam andimpart harmonic motion in the beam. Mounting two strain gages on the beam as shown inFigure 1 allows the measurement of maximum strain in the beam as the end of the beamoscillates in x direction. With virtual instrumentation, the computer operator can control themotion of the shaker and simultaneously record data from the strain gages. This data can beexamined with two different approaches, through determination of the damping ratio anddetermination of the deflection of the end of the beam.Two metallic resistance type strain gages are used in conjunction with a half Wheatstonebridge as shown in Figure 2. Briefly, the gages are mounted on a specimen and under pre-loadconditions; the balance potentiometer is adjusted such that eo is zero. Strain in the specimenelongates the strain gage, altering the electrical resistance in the gage. This change in the gageresistance unbalances the bridge, and results in a voltage at eo. This voltage at eo is proportionalto the strain. Using two gages in this configuration results in a doubled bridge output, ascompared to using a single gage, and also compensates for temperature effects, and torsional andaxial components. As the beam vibrates with harmonic motion, the output from the strain gagesis a sine wave with amplitude proportional to the strain and a period inversely proportional to thefrequency of the vibrations. 1Figure 1 Cantilever Beam Setup. Strain gages are mounted on the top and bottom of the beam. The motion of the oscillator is y(t) and the deflection on the end of the beam is x(t). The width of the beam is b, and the thickness is t.Figure 2 Wheatstone Bridge. A half bridge with two strain gages. After the bridge is balanced such that voltage across eo is zero, strain in gages results in voltage across eo.By driving the shaker with a sine wave, the aluminum beam vibrates with simple harmonicmotion. Left to vibrate freely, without applied external forces, the beam will vibrate at its naturalfrequency, n, and its amplitude of the response will decrease with time, as energy in the systemis lost. The rate at which this amplitude decreases is known as the logarithmic decrement. Thelogarithmic decrement,, can be obtained by measuring two displacements separated by nnumber of complete cycles and applying: noxxnln1 with ox and nx the amplitudesseparated by n cycles. From , the damping ratio, , can be found from: 2. For oursystem with the displacement of the shaker, y, it can be shown that the deflection, x isapproximated by:    22222121rrryx with nr. After impacting the beam, and allowing2it to vibrate freely, the measurement will determine the natural frequency of the beam,logarithmic decrement and damping ratio of the beam. Along with the driving displacement, y,this data can be used to determine the deflection, x, at the end of the beam for a given frequency. The deflection in the beam can also be determined by measurements of strain, , in the beam.The maximum deflection of a cantilever beam with a point load on the end of the beam is:EIPLx33 with P, the load on the beam, L, the length of the beam, E, the modulus of elasticity forthe material, and I, the second moment of area of the beam. The maximum strain in a cantileverbeam is: EIMc, (2tc  with t the thickness of the beam). From these two equations, and withPLM  we can obtain the deflection, x, with:cLx32. The computer station drives theoscillator at a series of determined frequencies and records the maximum strain at eachfrequency. With the user input of data from the Beam Data VI and measurements of the beam,along with the acquired driving displacement, the Frequency Data VI calculates the deflection atthe end of the beam and the magnitude ratios, (x/y),