ME301BeamVibrationLaboratoryManual

ME301BeamVibrationLaboratoryManual - 1. Vibrating Beam...

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1. Vibrating Beam Natural Frequency and Damping 1.1. Purposes Study free and forced vibration of a simple beam. Different beam materials and different end-masses are used to investigate natural frequency and linear damping theory. Perform experiments in support of AE/ME455 Mechanical Vibrations, ME324 Dynamics of Mechanical Systems, MA232 Elementary Differential Equations and MA330 Advanced Engineering Mathematics. 1.2. Background The primary goal of vibrations testing is to determine the response of a flexible structure to changing forces or displacements. The structure may fail if its natural frequency is reached. The effect of damping will change the way the structure responds. 1.2.1. Vibration Effects Two vibrations concepts: 1. Natural Frequency and 2. Damping. can be investigated prior to construction of a device. The theory contains simplifications such as mass lumping, linear elastic behavior and one-dimensional motion. These assumptions along with a momentum change-force balance (or energy equation) yield a second order differential equation of the form (see the lab textbook page 88) where y(t) is the unknown 1D displacement versus time, m is the lumped mass, c is the
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damping coefficient, k is the elastic spring constant and F is the applied forcing function that might be a sinusoid (shaker table) or impulse (hammer strike, step function). This differential equation can be solved by numerical means but the parameters m , c and k must be known and are assumed constant over time to ease solution. One way to solve the equation is to examine the special case of zero forcing function. This is called free vibration and gives insight into the interaction of the m, c and k values. An example is when a guitar string is picked. Just after the pick is released from the string, the initial string deflection turns into a free vibration. The general solution to the resulting (with F=0) ‘homogeneous differential equation’ is and C 1 and C 2 are the two constants of integration. With two B parameters, this solution has the ability to describe both natural frequency and damping. Usually, for oscillations, B 2 is complex so the exponentials inside the bracket are sine and cosine functions (see Euler’s formula and Argand diagram). For damped oscillations, the amplitude decreases over time and this is when B 1 >0. The first effect, natural frequency, is the oscillation frequency when there is no damping . Thus if damping is zero (or very small), c 0 , then B 1 =0 and B 2 becomes imaginary. Then the undamped solution becomes (with a natural frequency, n , defined) Hertz units are the number of oscillation cycles per second. We see directly that natural frequency is dependent on the spring stiffness and the mass. We also see that for the damped oscillation, there is a relation among the trigonometric function parameters By defining a phase shift, the cosine and sine solutions could be reduced to one sine or cosine function using trigonometric relations for angle addition. For example, the forward and reverse transformation (C 1 , C 2 ) (C, ) are
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ME301BeamVibrationLaboratoryManual - 1. Vibrating Beam...

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