Section 3.7: Mechanical and Electrical Vibrations

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1 Section 3.7: Mechanical and Electrical Vibrations Second order linear equations with constant coefficients serve as mathematical models for mechanical and electrical oscillations. For example, the motion of a mass on a vibrating spring, or the flow of electric current in a simple series circuit. Consider a mass m attached to a spring fixed at one end as shown below. Let y(t) denote the displacement of the mass from its equilibrium position at time t 0. When the mass is displaced, the spring is stretched or compressed and it exerts a force that resists the displacement. By Hooke s Law, the spring force F s is proportional to the displacement. That is, F s = ky, (1) where k > 0 is the stiffness of the spring. Almost all mechanical mechanical systems experience friction (damping). The damping force F d always acts in the direction opposite to the direction of motion of the mass. This force is usually modeled as proportional the velocity. That is, F d = by, (2) where b 0 is the damping coefficient. The other forces acting on the mass-spring oscillator are considered as external to the system. These forces may be due to the motion of the mount to which the spring is attached, or could be applied directly to the mass. These external foces are grouped into a single known function F (t). By Newton s Second Law of motion, my = F s + F d + F (t) my = ky by + F (t) It follows that the equation of motion of the mass is my + by + ky = F (t), (3) where m, b, and k are constants. To complete the formulation of the mass-spring problem, we require two initial conditions. In particular, the initial position y 0 and initial velocity v 0 y(0) = y 0 y (0) = v 0 (4) 1

2 Undamped Free Vibrations Suppose that there is no external force nor damping, so that F (t) = 0 and b = 0. The equation of motion of the mass is reduced to The characteristic equation is my + ky = 0 (5) mr 2 + k = 0, and its roots are r = ±ωi, where ω = k/m. Thus, the general solution is where c 1 and c 2 are arbitrary constants. We can express y(t) in the more convenient form where R 0 by letting c 1 = R cos δ and c 2 = R sin δ. Indeed, y(t) = c 1 cos(ωt) + c 2 sin(ωt), (6) y(t) = R cos(ωt δ), (7) R cos(ωt δ) = R cos δ cos(ωt) + R sin δ(sin ωt) = c 1 cos(ωt) + c 2 sin(ωt) Solving for R and δ in terms of c 1 and c 2, we obtain R = c c 2 2 tan δ = c 2 (8) c 1 where the quadrant in which δ lies is determined by the signs of c 1 and c 2. As expected, the graph is a displaced cosine wave that describes a simple harmonic motion y t The motion is periodic with frequency ω and period T = 2π/ω. The constant R is the amplitude of the motion and δ is the phase, which measures the displacement of the wave from its normal position. 2

3 Example: A mass of 100 g stretches a spring 5 cm. If the mass is set in motion from its equilibrium position with a downward velocity of 10 cm/s, and if there is no damping, determine the position y of the mass at any time t. Plot y versus t. Find the frequency, period, amplitude, and phase of the motion. When does the mass first return to its equilibrium position? 3

4 Damped Free Vibrations In most applications of vibrational analysis, there is some type of frictional or damping force affecting the vibrations. This force may be due to a component in the system or to the medium that surrounds the system, such as air or some liquid. If damping is present, the equation of motion of the mass is my + by + ky = 0 (9) The characteristic equation is mr 2 + br + k = 0, and its roots are r = b ± b 2 4mk = b 2m 2m ± b2 4mk 2m The form of the general solution depends on the nature of these roots. (10) Case 1: Underdamped Motion 4

5 Case 2: Overdamped Motion Case 3: Critically Damped Motion 5

6 Example: A mass weighing 16 lb stretches a spring 3 in. The mass is attached to a viscous damper with a damping coefficient of 2 lb s/ft. If the mass is set in motion from its equilibrium position with a downward velocity of 3 in/s, find its position y at any time t. Plot y versus t. Determine the quasifrequency and quasiperiod. Also find the time τ such that y(t) < 0.01 in for all t > τ. 6

7 Example: A 2 kg mass is attached to a spring with stiffness 40 N/m. The damping constant for the system is 8 5 N s/m. If the mass is pulled 10 cm downward and given a downward velocity of 2 m/s, what is the maximum displacement from equilibrium that it will obtain? 7

4.9 Free Mechanical Vibrations

4.9 Free Mechanical Vibrations Spring-Mass Oscillator When the spring is not stretched and the mass m is at rest, the system is at equilibrium. Forces Acting in the System When the mass m is displaced