7. Vibrations DE2-EA 2.1: M4DE. Dr Connor Myant

Transcription

1 DE2-EA 2.1: M4DE Dr Connor Myant 7. Vibrations Comments and corrections to Lecture resources may be found on Blackboard and at

2 Contents Introduction... 2 Types of Vibrations... 2 Springs... 4 Motion of a spring... 5 Series and Parallel... 6 Simple Harmonic Motion... 7 Un-damped Free Vibrations... 9 Motion of a pendulum for large and small oscillations: Energy Method: Damped Systems d < ω d ω

3 Introduction In this chapter we are going to explore the maths involved in vibrations. Vibration is a mechanical phenomenon whereby periodic motion occurs about a reference point. Vibrations are a major concern for design engineers; as almost all mechanical objects, which involve interacting components, experience some form of vibration. The oscillations may be periodic, such as the motion of a pendulum or random, such as the movement of a tire on a gravel road. Vibration can be desirable: for example, the motion of a tuning fork. In many cases, however, vibration is undesirable, wasting energy and creating unwanted sound. The study of vibrations is concerned with the oscillating motion of elastic bodies and the force associated with them. All bodies possessing mass and elasticity are capable of vibrations. As design engineers most of the time you will encounter vibrations you will also encounter springs and dampens. Springs are used to store elastic energy and when required release it. There is a wide range of types of spring that are readily available from specialist suppliers or that can be designed and manufactured fit-for-purpose. Design Engineering Example: Tennis players often use small rubber inserts in theirs rackets to dampen the vibrations in the strings. The main purpose is to reduce the vibration of a racquet s strings and subsequently reduce or eliminate the ping sound you hear when you strike a ball. There is little evidence that the use of tennis racket vibration dampeners reduce the risk of shock related injuries (such as tennis elbow) or help improve your game. You can also be sure vibration dampeners won t help you play like Novak Djokovic, but it s a nice thought right? Types of Vibrations Free vibration takes place when a system oscillates under the action of forces inherent in the system itself due to initial disturbance, and when the externally applied forces are absent. The frequencies of these vibrations are known as a system s natural frequency. Examples of this type of vibration are pulling a child back on a swing and letting go, or hitting a tuning fork and letting it ring. The mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness. One type of mechanism that would undergo free vibration and oscillate at its natural frequency is a spring, fixed at one end and having a mass attached to the other; this would be a single degree of freedom (SDoF) oscillator. For a single degree of freedom oscillator, a system in which the motion can be described by a single coordinate, the natural frequency depends on two system properties: mass and stiffness; (providing the system is undamped). 2

4 However, nearly all vibrating systems under go damped vibration due to the dissipation of energy from friction and other resistances. Damping has very little effect on natural frequency of the system, and hence the calculations for natural frequencies are generally made on the basis of no damping. Forced vibration takes place under the excitation of external forces. If the frequency of excitation coincide with one of the natural frequencies of the system, a condition of resonance is encountered and dangerously large oscillations may result, which results in failure of major structures, i.e., bridges, buildings, or airplane wings etc. Thus, the calculation of natural frequencies in the study of vibrations; and the usage of dampening in limiting the vibration amplitude at resonance are of great importance. Design Engineering Example: The Taccoma Narrows Bridge collapsed, in 1940, due to the damaging effects of vibrations caused by wind blowing across it. The wind caused vibrations to occur at a specific frequency that meant the bridge began to oscillate with catastrophic consequences. This can happen to almost any material, structure or object; and is known as natural resonant frequencies. Through careful design considerations, we can minimise the deleterious effects of vibrations, either through geometrical considerations or the addition of mechanical aids which dampen the vibrations. 3

5 Springs Springs are elastic devices used to store mechanical energy, or exert a force. The force produced by a spring can be compressive or tensile and linear or radial. Alternatively springs can be configured to produce a torque. When a spring is compressed or stretched slightly from rest, the force it exerts is approximately proportional to its change in length (this approximation breaks down for larger deflections). The rate or spring constant of a spring is the change in the force it exerts, divided by the change in deflection of the spring. That is, it is the gradient of the force versus deflection curve. An extension or compression spring has units of force divided by distance, for example lbf/in or N/m. k = F/x (7.1) Where k = spring rate (N/m), F = applied load (N), x = deflection (m). Torsion springs have units of torque divided by angle, such as N m/rad or ft lbf/degree. k = T/θ (7.2) The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/n. C = 1/k (7.3) Virtually any material can be used to make springs. However the ideal material would have a high ultimate strength and yield point and a low modulus of elasticity in order to provide maximum energy storage. Important aspects in spring design include determination of the spring material and dimensions in order to ensure that it will not fail due to either static or fluctuating loads for the lifetime required for the application, that it will not buckle or deform beyond allowable limits, that the natural frequencies of vibration are sufficiently in excess of the frequency of motion that they control and that cost and aesthetic aspirations are met. 4

6 Design Engineering Example: Products such as Door Closers often employ springs in them. The example shown in this image has a spring configured to provide a torque; a torsional spring is employed. The amount of force exerted by the spring with respect to the door s position (i.e. the rate) can be selected by the designer. Springs can be configured to provide almost any force required; this could be linear, dual rate or even progressive. Motion of a spring Lets imagine a hanging spring with the top end fixed, a mass is hung at the free end that causes the spring to change length by a distance x, Figure 7.1. Using Newton s 2 nd law we can see that the restoring force from the spring from its equilibrium point is given by Hooke's law. Figure 7.1. Mass hanging spring F = ma = m d2 x dt2 = kx (7.4) where m is the mass and k is the spring constant. This law holds (approximately) as long as x remains below the spring's elastic limit. Even when the elastic limit is not exceeded, pulling a coil spring far enough to uncoil it results in a much larger spring constant than the spring's "coiled" value. It follows that the differential equation describing the spring's motion is; 5

7 d 2 x dt 2 + k m x = 0 (7.5) This equation describes a particular type of motion; known as a simple harmonic oscillator. Series and Parallel Lets consider figure 7.1, but now we have springs connected in series or parallel. We can find the effective spring constant of the system. The force exerted by two springs attached in parallel exert a force; F = m d2 x dt 2 = (k 1x + k 2 x) Thus, the effecting spring constant is given by; k eq = k 1 + k 2 (7.8) Figure 7.2. Springs in parallel. Now consider two springs placed in series with the mass on the bottom of the second. The force is the same on each of the two springs. Therefore; F = m d2 x dt 2 = k 1x 1 = k 2 x 2 solving in terms of x 1, and rearranging for F = k eq (x 1 + x 2 ), where x = x 1 + x 2, you will find that; 6

8 k eq = ( 1 k k 2 ) 1 (7.9) Figure 7.3. Springs in series. Design Engineering Example: A sprung loaded mattress is a good example of springs employed in parallel. Designing a mattress that offers good support and comfort requires good knowledge of the spring loading capacity and behaviour. The springs will experience numerous loading cycles and it is important that their yield strength or fatigue strength is not reached. Simple Harmonic Motion The simplest form of periodic motion is harmonic motion, called simple harmonic motion (SHM). It can be expressed as (figure 7.4); x = Esin(ωt + φ) (7.10) Where E is the amplitude of the vibration and φ is a constant. 7

9 Figure 7.4. Graph of x as a function of ωt. An alternative form of this equation is often given as; x = Asin(ωt) + BCos(ωt) (7.11) where A and B are arbitrary constants, related to E and φ as; A = Ecos(φ), B = Esin(φ) We can then see that; E = A 2 + B 2 (7.12) Simple harmonic motion is often represented by projection on line of a point that is moving on a circle (Figure 7.5). We draw a circle whose radius equals the amplitude and assume the line from O to P rotates in the counter-clockwise direction with constant angular velocity ω. The time taken for one cycle is called the period of the vibrations and given as; T = 2π/ω (7.13) If we take the inverse of the period we get the frequency; the number of cycles per unit time, given as; f = 1/T = ω/2π (7.14) 8

10 The frequency is usually expressed in cycles per second, or Hertz (Hz). We can see that the angular velocity can also be given as ω = 2πf; consequently ω is also a measure of frequency. From Equation (7.11) we can see that; x = ωacos(ωt) ωbsin(ωt) (7.15) x = ω 2 Asin(ωt) ω 2 BCos(ωt) (7.16) Then by comparing Eqs. (7.11) and (7.16) we find the condition for SHM; x = ω 2 x (7.17) Figure 7.5. Correspondence of SHM with circular motion of a point. In this case the SHM repeats itself every 2π radians. Un-damped Free Vibrations The most basic vibration model consist of a mass suspended from a spring that undergoes vertical motion, Figure 7.6. In the static equilibrium position the weight of the mass, F m = mg, is balanced by the spring force, F S = kx 0. An external input is supplied, either displacement or velocity. Now the mass will experience forces from inertial effects (F i = mẍ), due to the motion; and a new spring force due to the change in displacement F s = kx. 9

11 Figure 7.6. Take the direction of x to be positive in the downward direction; and therefore, also velocity, x, acceleration, x, and force, F. Using Newton s 2 nd Law of motion we can sum the forces; F = mx = mg kx 0 kx We know from the point of static equilibrium that mg = kx 0. Therefore, mx = kx rearranging we get; x = kx m (7.18) Which is the same form as equation (7.5). Comparing Equations (7.17) and (7.18), we see that ω 2 = k/m. This corresponds to the natural frequency of the system, and satisfies the SHM condition. So for SHM conditions; T = 2π m k and f = 1 2π k m (7.19)(7.20) Using the earlier consideration that mg = kx 0 ; we can see that k/m = g/x 0. 10

12 The above analysis is valid for all kinds of single degree of freedom systems including beams or torsional members. For torsional vibrations the mass may be replaced by the mass moment of inertia and stiffness by stiffness of torsional spring. For stepped shaft an equivalent stiffness can be taken or for distributed mass an equivalent lumped mass can be taken. Motion of a pendulum for large and small oscillations: Figure 7.7 shows another type of single degree of freedom (SDoF) oscillator; a simple pendulum and the subsequent free body diagram of the bob. Figure 7.7. (a) Simple Pendulum (b) Free body diagram Tension in spring is given by; T = mgcosθ Newton s 2 nd law tells us that the moment balance about the centre of rotation O is given as; Iθ = M 0 = ( mgsinθ)l Where I is the polar mass moment of inertia, of the mass m about O, and l is the length from the pivot to the centre of the mass m, so that; ml 2 θ = mglsinθ, or θ + g sinθ = 0 (7.21) l which is the equation of motion of the pendulum for large oscillation. For small amplitudes of oscillation, Equation (3.8) reduces to, as sinθ θ; 11

13 θ + g l θ = 0 So, θ = g l θ Comparing this to Eq. (3.5) we obtain equations for the natural frequency of a pendulum as; ω = g l And subsequently for T and f. Energy Method: In a conservative system (i.e. with no damping) the total energy is constant, and a differential equation of motion can also be established by the principle of conservation of energy. This means the sum of the kinetic, T, and potential energies, V, is constant; T + V = constant (7.22) Hence; d (T + V) = 0 dt Kinetic energy T is stored in mass by virtue of its velocity. Potential energy V is stored in the form of strain energy in elastic deformation or work done in a force field such as gravity, magnetic field etc. Our interest is to the find natural frequency of the system, writing Eq. (3.9) for two positions. i.e; T 1 + V 1 = T 1 + V 2 = constant (7.23) where, 1 & 2 represents two instants of time. Let 1 represents a static equilibrium position (choosing this as the reference point of potential energy, here V 1 = 0) and 2 represents the position corresponding to maximum displacement of mass and at this position velocity of mass will be zero and hence T 2 = 0. Equation (2.10) reduces to; 12

14 T = 0 + V 2 If mass is undergoing harmonic motion then T 1 & V 2 are maximum values. Hence; T max = V max. Damped Systems In reality all systems will not vibrate indefinitely. They will slow down and eventually stop as a result of the frictional forces, or damping mechanisms, acting on the system. In some cases, engineers intentionally include damping mechanisms in vibrating systems to control output; for example the shock absorbers in a car are designed to damp the suspension too much damping and you feel a stiff ride, too little and the ride is bouncy. Generally mathematical modelling of damping is quite complicated and not suitable for vibration analysis. Simplified mathematical model (such as viscous damping or dash-pot) have been developed which leads to simplified formulation. Figure 7.8 shows the mass suspended from a spring system from Figure 7.6, with the addition of a damping element (dash pot). The force required to lengthen of shorten a damping element is defined to be the product of a constant, c, and the rate if change of the length of the element, x. We can write the equation of motion of the mass as; F = mx + cx + kx = 0 We want this in terms of ω, so using ω = (k/m) and d = c/2m, we get; x + 2dx + ω 2 x = 0 (7.24) This equation describes the vibration of many damped, one degree of freedom systems. The form of its solution, and consequently the character of the predicted behaviour of the system the equation describes, depends on whether the constant d is less than, equal to, or greater than ω. 13

15 Figure 7.8. Damped spring-mass oscillator, and the subsequent free-body diagram of the mass. d < ω Subcritical damped systems occur when d < ω. Assuming the solution of the form; x = Ce λt (7.25) Into Equation (7.24) we obtain; λ 2 + 2d + ω 2 = 0 This quadratic equation yields two roots for the constant λ that we can write as; λ = d ± iω d Where the damped angular velocity is; ω d = ω 2 d 2 (7.26) Because d < ω, the constant ω d is a real number. The two roots for λ give us two solutions of Equation (7.25). The resulting general solution of Equation (7.25) is x = e dt (Ce iω dt + De iω dt ) 14

16 Where C and D are constants. By using the Euler identity e iθ = cos θ + i sin θ, we can express this solution in the form; x = e dt (Asinω d t + Bcosω d t) (7.27) Where A and B are constants. Equation (7.27) is the product of an exponentially decaying function of time and an expression identical in form to the solution we obtain for an undamped system. The exponential function describes the expected effect of damping: The amplitude of the vibration attenuates (decreases) with time. Damping has an important effect in addition to causing attenuation. Because the oscillatory part of the solution is identical in form to equation (7.11) except that the term ω is replaced by ω d, it follows that the period and frequency of the damped system are; T d = 2π ω d f d = ω d 2π d ω When d > ω the system is said to be supercritically damped. We can again use Equation (7.25) to find a solution to Equation (7.24); λ 2 + 2d + ω 2 = 0 (7.28) However, now we write the roots of this equation as; λ = d ± h Where, h = d 2 ω 2 the resulting general solution is; x = Ce (d h)t + De (d+h)t (7.29) 15

17 Where C and D are constants. When d = ω, a system is said to be critically damped. Then the constant h = 0, so Equation (7.28) has a repeated root λ = d, and we obtain only one solution of the form Equation (7.25). Now, it can be shown that the general solution of Equation (7.24) is; x = Ce dt + Dte dt (7.30) Equations (7.29) and (7.30) indicate that the motion of a critically (d = ω) and supercritically (d ω) damped system is not oscillatory. These equations are expressed in terms of exponential functions and do not contain sines or cosines. Figure 7.9 shows the effect of increasing amounts of damping on the behaviour of a vibrating system. In summary, the motions of many damped one-degree of freedom systems can be modelled by Equation (7.24). Once the equation of motion has been expressed in that form, the values of d and ω are known in terms of the physical parameters of the system. The values determine the state of damping, which indicates the form of the solution of Equation (7.24). Figure 7.9. Amplitude history of a vibrating system that is (a) undamped; (b) subcritically damped; (c) critically damped; (d) supercritically damped. 16

18 Design Engineering Example: A car s suspension is a classic example of a spring-mass-damper system often employed in mechanics textbooks. The behaviour of the suspension system can be tuned (not too bouncy, not too hard) through the characteristics of the spring and the shock absorber. 17