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Natural Frequencies of a Cantilever Beam OBJECTIVE: The objective of this experiment is to determine the natural frequencies of the first three vibrational modes of a cantilever beam using the swept-sine technique. The swept-sine technique applies a sinusoidal excitation to the system under test and measures the system's response as the frequency of the stimulus is swept across a specified frequency range. A mechanical variable (e.g., displacement, acceleration) that varies sinusoidally is the stimulus in mechanical systems; a sinusoidally-varying voltage or current is the excitation in an electrical system. In this case, a cantilever beam (a beam clamped at one end and free at the other) constitutes the mechanical system that will be subjected to a sinusoidal displacement. A homogeneous beam with constant cross section is a continuum that can vibrate in a theoreticallyinfinite number of modes (amplitude distributions along the vibrating beam). Figure 1 below illustrates the first three modes. Notice that modes beyond the first exhibit nodes which are locations along the beam where the vibrational amplitude is zero. Fig. 1. Illustration of vibratory modes 1–3 for a cantilever beam excited with sinusoidal displacement APPARATUS: Beam specimens (steel and aluminum), vibration exciter, power amplifier; signal generator, and frequency meter. Fig. 1. Illustration of vibratory modes 1–3 for a cantilever beam excited with sinusoidal displacement. The locations marked for modes 2 and 3 represent the nodes. The number of nodes for any vibrational mode is one less than the mode number (e.g., zero nodes for mode 1, two nodes for mode 3). Each mode of vibration has its own natural frequency. The natural radian frequencies of the first three modes of a cantilever beam are as follows [1]: The scaling factors factors in Eqs. (1a)–(1c) come from solution of the partial differential equation that describes the transverse (i.e., at right angles to the main axis) of the beam. The area moment of inertia of the cross-section of a rectangular beam is given by Eq. (2): Experimental procedure The experimental apparatus is depicted in Fig. 3. An arbitrary waveform generator (Agilent 3220A ) produces a sinusoidal output voltage that is the input voltage of a power amplifier. The output of the power amplifier (LDS PO300), in turn, drives an electromechanical shaker (which is much like an audio speaker without the cone). The amplitude of the generator is set to an indicated output voltage of 500mVpp. The frequency of the generator is set to a frequency below the expected natural frequency of the first mode. The frequency is then slowly increased until the beam's first mode is excited. This will be made apparent by the appearance of resonance (whose theoretical explanation is beyond the scope of this laboratory procedure and of the report) at the first-mode natural frequency of the beam. The peak-to-peak deflection of the beam will reach its maximal value at resonance which occurs when the beam is excited at the first-mode natural frequency. The second and third modes' natural frequencies are found in the same manner, although greater generator output voltages will be necessary to elicit apparent second- and third-mode responses from the cantilever. Fig. 3. Experimental apparatus. The midpoint of the cantilever beam is clamped at the center of the shaker core. The effective length L of the cantilever is measured from the edge of the clamp to the free end of the beam. Turn the power amplifier off and adjust the aluminum specimen to a different length (make sure it is not too short) and tighten. Make sure to observe the specimen to make sure no cracks are initiating. Record the new length and repeat the measurement process for the first three bending modes. Finally, switch the specimen to the steel specimen, record its length, and measure the first three bending modes. Calculate the theoretical natural frequencies, discuss the results and comment on the accuracy of the results. Example: This example was created using ANSYS 7.0 and theory in the above equation. The purpose of this example is to compare numerical analysis versus the theory of a simple modal analysis of the cantilever beam shown below. The material used here is steel. Note: mass = density * volume Pre-lab Assignment 1) Given the two beams (Aluminum and steel) (0.125”X 1”X 20”), with the following material properties: For Steel E = 206.8 (10^9) Pa and ? = 0.7830 kg/cm. E= 30X10^6 psi, ? =0.00074 lb/ft For Aluminum: E = 73 (10^9) Pa and ? = 0.2750 kg/cm E=10.6(10^6) psi and ? = 0.00026 lb/ft The resonant frequencies for a simple cantilever beam is given by w= natural frequency in cycles per second E = modulus of elasticity of the material in psi I = moment of inertia in inches4 L =length in inches ? = mass per unit length , for steel =0.284/(12*32) =0.00074 lb/ft or 0.7830 kg/cm For aluminum ? = 0.1/(12*32) =0.00026 lb/ft or .275kg/cm Cn = coefficient for the different resonant modes ( C1 = 3.52, C 2 = 22.4, C 3 = 61.7 ) Calculate the first three natural frequencies using the above equation. 2) Are the number of degrees of freedom of a system and the num ber of its norm al modes related? If so, how? 3) How can a normal mode be recognized physically? 4) What data will you record during the lab? 5) What observations will you make? 6) What do you expect to happen when you dr ive a system at one of its natural frequencies? 4 2 L C EI n n p ? ? = Example 2: The material used here is steel in SI Units Use excel and theory in the above equation to compare numerical analysis versus the theory of a simple modal analysis of the cantilever beam shown above with the following dimensions: Example 3:- The material used here is steel (US Units) Prof. Orabi ME 315 Mechanics Lab – spring 06 Page5 of 6 For AL (US Units) Prof. Orabi ME 315 Mechanics Lab – spring 06 Page6 of 6 For AL (US Units)

Natural Frequencies of a Cantilever Beam April 23 2013 Measurement of natural frequency is necessary to check the failure that may occur due to vibrations. It can also be useful to evaluate the modulus of elasticity of thin films when it is attached to a cantilever beams changing its natural frequency. This project deals with the evaluation of natural frequencies of a cantilever beam with the help of swept sine technique. Lab Report TABLE OF CONTENTS 1. Introduction 3 2. Theory 3 3...

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Natural Frequencies of a Cantilever Beam April 23 2013 Measurement of natural frequency is necessary to check the failure that may occur due to vibrations. It can also be useful to evaluate the modulus of elasticity of thin films when it is attached to a cantilever beams changing its natural frequency. This project deals with the evaluation of natural frequencies of a cantilever beam with the help of swept sine technique. Lab Report TABLE OF CONTENTS 1. Introduction 3 2. Theory 3 3...