The student will experimentally determine the parameters to represent the behavior of a damped oscillatory system of one degree of freedom.
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1 Practice 3 NAME STUDENT ID LAB GROUP PROFESSOR INSTRUCTOR Vibrations of systems of one degree of freedom with damping QUIZ 10% PARTICIPATION & PRESENTATION 5% INVESTIGATION 10% DESIGN PROBLEM 15% CALCULATIONS 25% RESULTS 15% DISCUSSION OF RESULTS 10% CONCLUSIONS 10% TOTAL 100% OBJECTIVE The student will experimentally determine the parameters to represent the behavior of a damped oscillatory system of one degree of freedom. FUNDAMENTALS In real life, it is possible to appreciate that free vibrations are not infinite. This phenomenon can be easily verified by observing that the free vibrations decrease their amplitude of oscillation with respect to time. The differential equation used to represent a mass-spring system such as that shown in Figure 1. F = ma mx + kx = 0 (1) (2) Fig. 1. (a) System mass-spring of one degree of freedom. (b) Free body diagram of said system. The solution of Eq. (1), as discussed in the previous practice, has the form. x(t) = C 1 e iω nt + C 2 e iω nt (3) Where C1 and C2 are constants that are defined from the initial conditions of the oscillatory system and through the use of the identity. e ±iαt = cos αt + sin αt (4) 1
2 It can be seen from Eq. (4) that the displacement is periodic (given the nature of the sine and cosine functions) in time and infinity. However, this does not agree with most of the vibration systems within our reach, which oscillate freely and stop as time goes on. This suggests then the existence of forms of energy dissipation (also known as mechanisms of damping) in the vibratory systems which produce the end of the oscillatory movements of these systems. During damping the energy of the vibratory system is dissipated as friction, heat or sound. Damping mechanisms exist in several ways, for example: Coulomb damping or dry friction. - In this case the damping force is constant. Solid damping or hysteresis. - This is caused by the internal friction of a solid when opposing to vibration. Turbulent damping. - In this case the damping force is proportional to the square of the average velocity. Damping in viscous fluid. - In this case the damping force is proportional to the speed. The most commonly used damping mechanism is viscous damping, in which the damping force is proportional to the speed of movement. These types of damping phenomena occur strictly when a laminar flow of a viscous fluid through a groove is present, for example: an impact absorber, the damping occurring around a piston in a cylinder, the damping mechanism of Automatic closing doors, automobile dampers, etc. Cushioning A basic vibratory system such as that shown in Figure 2 essentially counts with: Mass (m), which is directly related to the kinetic energy of the system. Spring (k), whose function is to store potential energy. Damper (c). - whose function is to dissipate the vibrational energy of the system. From an analysis of the free-body diagram shown in figure 2 it is possible to obtain the equation of motion. mx + cx + kx = f(t) Eq. (5) is a non-homogeneous second-order differential equation, so its solution contains two parts. The first one, which refers to the case of free vibration, is when f(t) = 0 is obtained to solve a second-order homogeneous differential equation, which corresponds physically to the case of a vibration damped system (Figure 2) Which can better represent the oscillations we traditionally appreciate in real life. (5) 2
3 Fig. 2.- (a) System mass spring damping of one degree of freedom. (b) Free body diagram of said system. The homogeneous differential equation that we must solve for this practice is given by: mx + cx + kx = 0 (6) A traditional approach to solve the differential equation (6) is to assume that the solution has the form. x = e rt (7) With, x = re rt x = r 2 e rt (8) (9) Gets Where r is a constant. When replacing Eqs. (7) and (8) in the differential equation (6) e rt (mr 2 + cr + k) = 0 (10) Eq. (9) is satisfied for all values of t when the characteristic equation is zero, that is r 2 + c m r + k m = 0 (11) Eq. (10) has the roots r 1,2 = c 2m ± ( c 2m ) 2 k m (12) In such a way that it is possible to find the following general solution for the displacement. c ( X = e 2m )t (C 1 e t ( c 2m )2 k m + C 2 e t ( c 2m )2 m) k (13) Where C 1 and C 2 are constants that depend on the initial conditions of the movement, while the terms within the radical of Eq. (12) will indicate three different damping cases with their respective characteristic curves, which will be discussed below. 3
4 OVERCURRENT CASE: ( c 2m )2 > k m In this case the radical is real and the roots r 1,2 expressed in Eq. (11) will be real and distinct. The movement of the system is dominated by damping. This means that the system will approach its equilibrium position exponentially without any oscillation occurring and will never return to its original position (from which the movement occurred). Examples of this type of damping can be observed in the mechanisms that serve for the automatic closing of doors. This motion is expressed by Eq. (12) and can be seen graphically in Figure 3 (a). CRITICALLY DAMAGED CASE: ( c 2m )2 = k m In this case the radical is equal to zero and the roots r 1,2 will be real and equal. In this case it is said that the system is critically damped. This value of the damping constant is known as the critical damping constant c cr and its value depends exclusively on k and m. c cr 2 = 4mk c cr 2 = 4mk The relation between the damping of the system and its constant damping criticism is known as damping ratio and is called by the Greek letter ζ ζ = c c cr Which is a dimensionless parameter. In a critically damped oscillation, the damped system is brought to its equilibrium position exponentially in a minimum time without oscillation occurring. In this case, the displacement will also be represented by Eq. (12), which can be written as Eq. (15) by remembering that the radical is equal to zero. (14) (15) (16) c ( X = (C 1 + C 2 )e 2m )2 (17) The movement of a critically damped oscillatory system can be seen in Figure 3 (b). SUB DAMPING CASE: ( c 2m )2 < k m In this case a harmonic oscillatory motion will be shown around an equilibrium position in which the amplitude will decrease over time in each oscillation. In this case, since the radical is negative, the roots r 1,2 are complex and conjugate having the form. r 1,2 = α ± iβ (18) 4
5 Where, α = c 2m β = k m r2 4m 2 (19) (20) And the solution in this case will have the form. X = e αt [C 1 e iβt + C 2 e iβt ] (21) Which can also be written - using Eq. (3) as c ( X = e 2m )t [A cos(ωt) + B sin(ωt)] (22) Where, ω d = β = k m c2 4m 2 = ω n 1 ζ2 (23) Torsional Pendulum If a rigid body oscillates around a specific reference axis, the resulting motion is said to be a torsional vibration. In this case, the displacement of the body is measured in terms of an angular coordinate denoted by </s>. In this type of vibration, the restorative moment can be due to the twisting of an elastic body or the imbalance of a force. Fig. 3.- (a) Torsional pendulum not damped. (b) Free body diagram of said system [1]. In figure 3 is shown a disk with mass moment of inertia J0 mounted at the end of a solid circular arrow, which is recessed at the other end. If the movement of the disk is angular and is described by the coordinate θ, from the theory of torsion of circular arrows it is possible to obtain the relation. M t = GJ bar l (24) 5
6 Where M t is a torque producing the rotation θ, G is the shear modulus, l is the length of the arrow and J bar is the polar moment of inertia of the cross section of the bar, which is given by J bar = πd4 32 (25) d is the diameter of the arrow. If the disk is displaced by an angle from its equilibrium position, the arrow has a restoring torque (resembling a spring restoring force) of magnitude M t. Consequently, the stiffness constant for a torsional spring such as Bar of figure 3 is. k t = M t θ = GJ bar = πgd4 l 32 (26) The equation of motion of a torsional pendulum is obtained through Newton's second law or through some energetic method. Considering the free-body diagram shown in Figure 3(b) for a torsional pendulum with damping you get. J 0 θ + c t θ + k t θ = 0 (27) And writing Eq. (24) as, θ + c t J 0 θ + k t J 0 θ = 0 (28) Where the moment of inertia of the disk J 0 is J 0 = ρhπd4 32 = WD2 32 (29) From Eq. (25) the natural oscillation frequency. ω n = k t J 0 (30) And finally, ζ = c t = c t = c t c ct 2J 0 ω n 2 k t J 0 (31) 6
7 MATERIAL AND EQUIPMENT TO BE USED Mass-Spring system Torsional pendulum Vibration analyzer Accelerometer Vernier caliper Oil REPORT Mass-Spring system Determine the natural frequency using the cycles method, then determine the value of the constant spring stiffness using the two methods seen in Practice 1. Using the vibration analyzer, determine the value of the natural oscillation frequency (ω n ). Using the vibration analyzer, determine the value of the damped oscillation frequency (ω d ). For this use an oil vessel in which the fins of the mass are fully in contact with the oil. Determine the value of the damping ratio (ξ). Determine the value of the critical damping constant (c cr ). Determine the value of the damping constant provided by the oil (c). Discuss the obtain results, under which conditions the percentage of error decrease and which conditions affects the experimental analysis. Conclude according to the objectives of the practice and add a personal conclusion of why is important what we learn? Torsional Pendulum Determine the value of the torsional stiffness constant (k t ) using the equations (22) and (23). Using the vibration analyzer, determine the value of the natural oscillation frequency (ω n ). Using the vibration analyzer, determine the value of the damped oscillation frequency (ω d ). To do this use a container with oil in which the disc is immersed in oil. Determine the value of the damping ratio (ξ). Determine the value of the critical torsional damping constant (c ct ). Determine the value of the torsional damping constant provided by the oil (c t ). Discuss the obtain results, under which conditions the percentage of error decrease and which conditions affects the experimental analysis. Conclude according to the objectives of the practice and add a personal conclusion of why is important what we learn? Note: Extension of the discussion of results and conclusions of the two systems should be 1 page. 7
8 RESULTS Complete the results table below and evaluate the reliability of the techniques applied in the spaces designated for this propose. Table 1. Cycles Method Mass-Spring Time (s) Cycles Average Mass (kg) Period (s) Natural Frequency (rad/s) Spring Constant (N/m) Table 2. Deformation Method Mass-Spring m 1 (kg) δ 1 (m) m 2 (kg) δ 2 (m) m δ Spring constant (N/m) Table 3. Vibration Analyzer Mass-Spring Frequency Amplitude (Hz) (G-S) No damped Damped ω n (rad/s) ω d (rad/s) Table 4. Vibration Analyzer Torsional Pendulum Frequency Amplitude (Hz) (G-S) No damped Damped ω n (rad/s) ω d (rad/s) INVESTIGATION Explain how accelerometers work (Extension 1 page) Search for types of accelerometers and their definition (Extension Half page) Note: All should be referenced following the IEEE format. 8
9 DESIGN PROBLEM Although we find a large number of examples of damping daily, one of the most illustrative is the automobile. In our case, the car we are going to study has a mass of 1520 kg and is supported by its four springs and four shock absorbers. If the static elongation of the springs due to the car's own weight is 0.20 m, how would you determine the damping constant required in each damper to obtain the critical damping case? Assume the car has only one degree of freedom. Before determining the critical damping constant (c cr ), mention what the assumptions you took into account to arrive at your result. REFERENCES [1] Rao, Singiresu S. Mechanical Vibrations, Fourth Edition, Pearson. USA (2003). [2] Steidel, Robert F. An introduction to mechanical vibrations. Third Edition, John Wiley, USA (1989). [3] Thomson, William T. Theory of vibrations: applications. Second Edition, Prentice Hall, USA (1982). [4] Kelly, Graham S. Fundamentals of mechanical vibrations. Second Edition. McGraw Hill. USA (2000). [5] Stile, Hidgon. Ingeniería Mecánica, tomo II: Dinámica Vectorial. Prentice Hall, (1982). 9
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