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harmonic_excitation

# harmonic_excitation - Chapter 4 Single Degree of Freedom...

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Chapter 4 Single Degree of Freedom Systems: Forced Response W e consider the same single degree of freedom system we had earlier but subject it now to a force, F ( t ). Frequently in practice, the force is a complicated function of time, and we will look at some possibilities and the consequent response of the system. 4.1 Harmonic Excitation The most common kind of excitation (in vibration, one often uses the word excitation to describe a force) for vibrating systems is harmonic. This is a consequence of periodic motions that are very common (by design) in most engineering systems. In addition, it is important to study harmonic excitation since the solution obtained here forms the mathematical basis for getting the solution to situations when the forces are not harmonic. 4.1.1 Equation of Motion Consider the SDOF model again with a harmonic force as shown in Fig. 4.1. Then, the equation of motion is m ¨ x + c ˙ x + kx = F 0 sin ωt (4.1) where, F 0 is the amplitude of the force, and ω is the frequency of excitation. Note that a more general harmonic force would be F 0 sin( + φ ). We will consider that case later. Next, we write the equation in standard form: ¨ x +2 ζω n ˙ x + ω 2 n x = ω 2 n F 0 k sin (4.2) 53

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54 CHAPTER 4. SINGLE DEGREE OF FREEDOM SYSTEMS: FORCED RESPONSE F(t) Figure 4.1: A Forced mass-spring-damper system 4.1.2 Solution From the elementary theory of diﬀerential equations, the solution can be sought in the form, x ( t )= x c cos ωt + x s sin (4.3) Then, ˙ x ( t - ωx c sin + s cos (4.4) ¨ x ( t - ω 2 x c cos - ω 2 x s sin (4.5) Substitution into the equation of motion and separation of the sine and cosine functions yields the following set of linear algebraic equations in the unknown coeﬃcients. - ω 2 + ω 2 n 2 ζωω n - 2 n - ω 2 + ω 2 n ‚• x c x s = 0 ω 2 n F 0 /k (4.6) Solving the set of algebraic equations, we get the following. x c x s = 1 Δ 0 - ω 2 + ω 2 n - 2 n +2 n - ω 2 + ω 2 n 0 ω 2 n F 0 /k (4.7) where, Δ 0 = ( - ω 2 + ω 2 n ) 2 +(2 n ) 2 (4.8) Next, we deﬁne a convenient nondimensional parameter called the frequency ratio, r , that is a ratio of the forcing frequency to the undamped natural frequency of the system.
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harmonic_excitation - Chapter 4 Single Degree of Freedom...

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