ME 38 Machine Design Vibration handout (vibrations is not covered in text) The ollowing are two good textbooks or vibrations (any edition). There are numerous other texts o equal quality. M. L. James, et al., Vibration o Mechanical and Structural Systems, Harper-Collins College Publishers. R. Vierck, Vibration Analysis, Harper and Row. The ollowing is an outline o vibration subjects covered in ME38. Only single degree o reedom will be investigated: Basic concepts o vibrations and waves o What are the basic eatures o harmonic motion? Analysis o simple harmonic motion o undamped ree vibration o What is the natural requency? Basic behavior o viscous damped ree vibration. o What is the qualitative behavior o a damped system? Analysis o orced vibration without dampening. o What is the vibration amplitude i an undamped system is excited with orce input? Analysis o orced vibration with viscous damping. o What is the vibration amplitude i a damped system is excited with orce input? Isolation and transmission o vibration orces o orced vibration with viscous damping. o How much orce is transmitted rom the vibrating system to the surrounding support structure? ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6
VIBRATION BASICS In this section we will study basic eatures o harmonic motion. Why is vibration worth studying? What eects does it have (good and bad)? What is meant by degree o reedom (DOF)? Sketch a DOF system: How many degrees o reedom does a rigid body have, and what are they? Sketch a DOF system: How many degrees o reedom does a real structure have, say or example a simple cantilever beam or a guitar string? What is meant by mode? How many modes were apparent in the Tacoma Narrows Bridge? Describe them. What is meant by undamental requency? ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6
What are overtones? Waves: Vibration is a oscillating motion, thereore it is essential to deine basic concepts o sinusoidal waves. x(t) = A sin(t + ) x(t) = displacement or amplitude as a unction o time (meters) t = time (seconds) = angular requency (radians per second) = phase angle A = peak amplitude (meters) x(t) time = period, how much time per cycle = requency, cycles per second ( Hz = cycle/sec) = / = ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 3
SIMPLE HARMONIC MOTION OF UNDAMPED FREE VIBRATION In this section, the main objective is to determine natural requency o a system. Simple mass-spring system at equilibrium (not moving): +x -x equilibrium I a system is displaced rom equilibrium it will move to seek out equilibrium. I there is no damping, the system will continually convert energy back and orth between kinetic energy o the moving mass and elastic energy o the spring. +x -x +x -x x(t) time Assumptions we will make or analysis: Degree o reedom, which requires (why, how will system behave dierently i assumptions are not valid?): Rigid support Massless, linear elastic spring Displacement o mass is in one direction only Mass is rigid No losses (no damping, etc.) ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 4
k = spring stiness (lb/in, N/m, etc.) n = natural angular requency = / n (radians/second) n = natural requency = / n (cycles/second, Hz) What is natural requency? How will the mass aect natural requency? Will increased mass increase the natural requency or decrease it? How will spring stiness aect natural requency? Will increased stiness increase the natural requency or decrease it? Sketch FBD & kinetic diagram: Solutions to equations o motion: x = A sin( n t) x = A n cos( n t) x = -An sin( n t) displacement as a unction o time velocity as a unction o time acceleration as a unction o time Equilibrium equation: inertial orces = - elastic orces (momentum opposes the elastic restoring orce) m x = - kx = m(-a n sin( n t)) = - k (A sin( n t)) Solving the equation gives: n = k/m; n = k / m Eq. Does this make sense? Did you expect an increase in stiness to increase the natural requency? Did you expect an increase in mass to decrease the natural requency? ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 5
I the mass-spring system was rotated and placed on rictionless rollers such that gravity is perpendicular to the direction o motion, a) How would natural requency change? equilibrium b) How would amplitude (A) o oscillation change? c) How would the equilibrium position change? Example Given: Find: Assume: An elevator car with mass o 4000kg Suspended by a steel cable, 0m long, 6cm cross-section Natural requency o the system Sketch: Solution: ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 6
Example Given: Find: Assumptions: An aluminum cantilever beam with cross section o 0cm by 0cm A 00kg reciprocating single piston air compressor is placed at the end o the beam The compressor spins at 3600RPM Beam length such that n = Sketch: Solution: ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 7
ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 8
VISCOUS DAMPED FREE VIBRATION (TRANSIENT CONDITION) The main objective o this section is to understanding damping. We will ocus on qualitative behavior, not quantitative. This is the only condition we will look at where the motion is transient (non-steady state). What typically creates viscous damping? What is the damping orce proportional to or viscous damping? Are there other types o damping? What is the orce proportional to or riction damping? Will we study other orms o damping in ME38? No. Tidbits o shock absorber history: 90 irst patent or hydraulic shock absorber 95 hydraulic shocks used on automobiles Oil Shat Piston Holes in piston allow oil to low rom one side to the other (viscous orces are produced). Various relationships between velocity and orce can be created by varying design: Force progressive linear digressive Velocity Progressive and digressive shock absorbers can be created by using shims (laps) to partially cover the passage holes in the piston giving a non-linear response. This can also be used to create a dierent response depending on the direction o travel (up verses down or example). We will only consider linear response (orce is directly proportional to velocity). ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 9
Gas shock absorbers use gas to pressurize the oil (typically nitrogen). What two eects does this have? Oil gas Piston Shat The system we will study consists o a mass, spring, and dashpot (shock absorber). The dashpot is viscous (orce is proportional to velocity). Sketch FBD & kinetic diagram k c +x -x equilibrium k = spring constant (as beore) c = damping constant Equilibrium equation: acceleration orce = - elastic orce - damping orce m x = - kx - c x dividing by mass gives: x + c/m x + k/m x = 0 The critical damping constant, c c is: c c = m n Eq. The damping actor,, is: = c/c c Eq. 3 What is critical damping? ( critical is not the same as very important ) What are the units o the damping constant, c? ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 0
I a mass, spring, dashpot system is displaced rom equilibrium and released (zero initial velocity), on the same graph sketch the response or: a) < (under-damped) (Zeta) b) = (critically damped) c) > (over-damped) x(t) time I a critically damped system is displaced rom equilibrium and released with the ollowing initial velocities, sketch the response a) b) c) x > 0 (velocity away rom equilibrium position) x = 0 (released without initial velocity) x < 0 (velocity towards equilibrium position) x(t) time ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6
ANALYSIS OF FORCED VIBRATION WITHOUT DAMPENING (STEADY STATE CONDITION) In this section we will study the eect o harmonic orced input (an external orce applied sinusoidally). We are primarily interested in determining the maximum displacement o the system. What are some potential sources o harmonic orces? Sketch FBD & kinetic diagram +x -x equilibrium P(t) = P 0 sin( t) P(t) is a harmonic orce input. Question: what is? How is it dierent than n? P is the harmonic orce. It produces a push and a pull in an oscillatory manner with an angular requency o. is sometimes reerred to as the orcing requency. Equilibrium equation: m x = - kx + P(t) = -kx + P 0 sin( t) Solving: P0 P0 / k P0 / k x(t) = sin( t) = sin( t) = k m k / m n sin( t) Eq. 4 Let 0 = P 0 /k 0 is the static displacement due to the static orce P 0. Eq. 5 Let r = / n r is known as the requency ratio. Eq. 6 In this class, we are NOT concerned with x(t), but we are concerned with amplitudes. ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6
Substituting 0 and r into the previous equation or x(t) gives: 0 x(t) = sin( t) Eq. 7 r Now let = 0 r Eq. 8 is the peak amplitude o displacement it is not a unction o time. Finally, x(t) = sin( t) Eq. 9 Three conditions may exist; r < (orcing requency is less than the natural requency; < n ) r > (orcing requency is greater than the natural requency; > n ) r = (orcing requency is equal to the natural requency; = n ) I r < P(t) time sketch x(t) x(t) time I r > sketch x(t) How is the phase dierent than or r <? P(t) time x(t) time ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 3
I r = Then: o t x( t) cos t P(t) x(t) time time An important question to answer is: What is the amplitude o the actual vibration? The magniication actor, MF, is the ratio o the actual vibration amplitude () normalized by the displacement ( 0 ). In all cases MF is a unction o the requency ratio, r. Remember r =. From Equation 7, the vibration without damping is: n o x( t) sin ( t) ; where 0 = P 0 /k r By deinition o magniication actor (ratio o to 0 ): MF ; where = o 0 r (per Equation 8) Then: MF r (MF is always positive so absolute values are used) Eq. 0 NOTE: MF is NOT a unction o time. = dynamic peak displacement (magnitude o vibration displacement). o = static displacement created by static orce o magnitude P 0. Both and 0 are magnitudes, they do not vary with time. What do you expect the magniication actor to be? Should it always be greater than, less than, or will it sometime be greater than and sometimes less than? ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 4
Notice 0 is not a unction o r ( or n ) but that is a unction o r. Thereore, MF is a unction o the requency ratio, r. What happens (what is the vibration amplitude) at: Frequency ratio, r = / n Static displacement, 0 Actual displacement amplitude, Magniication Factor, MF r = 0 (static load); ( = 0) 0 r = ( = n ) 0 r < ( < n ) 0 r = / 0 r >> ( >> n ) 0 Sketch MF vs. r 3 MF 0 / r ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 5
ANALYSIS OF FORCED VIBRATION WITH VISCOUS DAMPING (STEADY STATE CONDITION) The prior section determined the magniication actor or undamped systems. The objective o this section is to determine the magniication actor when viscous damping is present. We will also determine the maximum magniication actor o a given system. Sketch FBD & kinetic diagram: k c +x -x equilibrium P(t) = P 0 sin( t) Equilibrium equation: m x kx cx P(t) The magniication actor or this condition is: MF Eq. o ( r ) (r) NOTE: magniication actor is NOT a unction o time, it is a constant assuming steady state conditions. As previously deined: c c c r n And: c c m n 3 Sketch MF or: a) no damping b) = 0. c) = 0.707 MF 0 r / ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 6
Show that the magniication actor deined in Equation, becomes the equation deined in Equation 0 i there is no damping: Equation : MF o ( r ) (r) Equation 0: MF r By using the magniication actor (MF) we can determine the amplitude o vibration or a orced vibration system with or without viscous damping. In real systems, mass, spring constant and damping constant typically do not vary but the orcing requency ( ) will change. For example, your car has a certain mass, springs, and shock absorbers, but the engine runs at dierent speeds. Thereore, the next question we may want to answer is or a given system (ixed mass, spring, and dashpot), what is the maximum magniication actor (MF max ) or any orcing requency ( )? For a damping actor greater than 0.707 ( ) the magniication actor will remain less than or all values o r. Eq. However, or the maximum magniication actor (MF max ) is: max MF max Eq. 3 o The last question we will answer is at what requency ratio (r) will the magniication actor be maximized? The answer depends upon the damping actor: or The magniication actor is maximum at r = 0 (MF max = and occurs at r MFmax = 0). Thereore, with any non-zero value o r, the dynamic response (vibration) will be less than the static displacement due to a static orce o P 0 and will decrease with increasing. ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 7
or The magniication actor will be maximized at the requency ratio o r MFmax : r MF max Eq. 4 To summarize, the magniication actor (MF) is the ratio o the displacement amplitude o the vibrating system to the displacement o the system due to a static load o magnitude P 0. It is a unction o the system characteristics (mass, spring, dashpot) and the orcing requency ( ). For a given system (with constant mass, spring, dashpot), we can determine the maximum magniication actor or all orcing requencies (MF max ) and we can determine the requency that maximizes the magniication actor (r MFmax ). ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 8
ISOLATION AND TRANSMISSION OF VIBRATION FORCES OF FORCED VIBRATION WITH VISCOUS DAMPING (STEADY STATE CONDITION) So ar we have analyzed the natural requency o a mass-spring system and determined the vibration amplitude as a unction o mass (m), spring (k), dashpot (c), and orcing requency ( ) or a orced vibration system with or without damping. The last question we will answer is what is what is the maximum orce transmitted rom the vibrating system to the structure holding it in place? We will consider orced vibration with viscous damping. Although there is damping, the system will continue to oscillate indeinitely due to the external orce, P. We will investigate the orce transmitted by this steady state oscillation. k c +x -x equilibrium P = P 0 sin( t) In addition to the inertial orce, there are three orces acting on the mass: the spring orce (proportional to displacement), the orce rom the dashpot (proportional to velocity) and the external orce, P. All o these orces are a unction o time. The ree body diagram (FBD) is: kx c x Plus: The equilibrium equation is: P = P 0 sin( t) m x kx cx P o sin ( w t) The inertial orce (m x ) and the external orce (P) are reacted by the spring and dashpot. Only the spring and dashpot are attached to the surrounding structure, and thereore the sum o these two orces is the transmitted orce: ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 9
F transmitted kx cx Eq. 5 Although the transmitted orce will vary with time (harmonically), we are interested only in the maximum value. Let F T be the amplitude o the transmitted orce (it does NOT vary with time, it is a constant). Similarly to the magniication actor (MF), we will normalize the transmitted orce amplitude by the static orce, P 0. This ratio is the so called transmissibility ratio (TR): Transmissibility ratio: F P T TR NOTE: this is NOT a unction o time. o We can express the transmissibility ratio as a unction o the system s characteristics, m, k, c, and : F k ( c ) T TR Eq. 6 P o ( k m ) ( c ) We can also express it in terms o the normalized characteristics, and r: F (r) T TR Eq. 7 P o ( r ) ( r) Sketch the transmissibility ratio as a unction o requency ratio or: a) = 0 b) = 0.5 3 TR 0 / r Note that or all values o damping (), the transmissibility ratio equals one (TR=) at both r = 0 and at r. The maximum transmitted orce will always occur at < n regardless o the system characteristics (m, k, c). The requency ratio or maximizing the transmitted orce, is r TRmax and is given by: ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 0
TR max occurs at: / ( 8 ) r TR max (this is always less than ). Eq. 8 Example: Given: An air compressor is placed on an isolation table with spring and dashpots on all 4 corners The total mass o table and compressor is 800 kg. Springs: k = 60 KN/m Dashpot (shock absorbers): c = 4000 kg/sec Forcing unction is given as P(t) = P o sin( t) where = 370 rad/sec (3600 rpm) and P o = 500N Find: a) Damping ratio, b) What orcing requency ( ) results in maximizing the displacement? c) Determine the vibration displacement amplitude at the orcing requency in part b. d) What orcing requency ( ) results in maximizing the transmitted orce? e) Determine the transmitted orce at the orcing requency in part d. Assume: Sketch: Solution: ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6
ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6
CONCLUSION & SUMMARY We have studied the ollowing systems or one degree o reedom: Analysis o simple harmonic motion o undamped ree vibration o The natural requency i a unction o mass and spring constant: o n = k / m Basic behavior o o viscous damped ree vibration system. o When a damped system is displaced rom equilibrium, it will seek out equilibrium and eventual will come to rest at equilibrium (unless acted upon by a varying external orce). o I it is under-damped, it will oscillate beore coming to rest. o I it is critically damped, it will not oscillate (i initial velocity is zero) but rather it will approach equilibrium asymptotically. o I it is over-damped, it will not oscillate (i initial velocity is zero) and will take a longer time to reach equilibrium than a critically damped system. Analysis o orced vibration without dampening. o The vibration amplitude () o orced vibration without dampening is a unction o the system characteristics (m and k) as well as the orcing requency ( ). The ratio o the amplitude normalized by the displacement created by a static orce, P 0 is called the magniication actor, MF. Although the system oscillates (x(t)), the magniication actor is based on the amplitudes and, thereore, is constant or the system as long as the orcing requency ( ) remains constant. o MF 0 r Analysis o orced vibration with viscous damping. o The magniication actor or a system with viscous damping is deined in the same way as or no damping, but the equation is more complex. o MF o ( r ) (r) o For < 0.707, the magniication actor will remain less than or all values o r. o For < 0.707, the maximum magniication actor or a given system (m, k, c) or all requency ratios (r) is: max MF max o And occurs at: r MF max ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 3
Isolation and transmission o vibration orces o orced vibration with viscous damping. o The amplitude o the orce transmitted rom the vibrating system to the surrounding support structure is F T. By normalizing F T by the static orce, P 0, the transmissibility ratio, TR, is described as: FT TR P o k ( k m ( c ) ) ( c ) ( r (r) ) (r) o Transmissibility ratio does NOT vary with time; it is a constant or a given system and orcing requency. Even though there is viscous damping, the system is assumed to have steady state oscillation due to harmonic external orce, P. o For a given system (m, k, c) the transmissibility ratio is a unction o the orcing requency, and it becomes maximum at the requency ratio o r TRmax : r TR max ( 8 ) / o r TRmax is always less than regardless o system characteristics (m, k, c). In other words, the maximum transmitted orces or a system will always occur when the orcing requency is less than the natural requency. ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 4
Magniication Factor (MF) 5 4 3 0 z=0 z= Magniication Factors or various damping actors z=0 z=0. z=0.5 z= 0 0.5.5.5 Frequency ratio (r) MF o ( r ) (r) Transmissibility Ratio (TR) 5 4 3 Transmissibility ratio or various damping actors 0 0 0.5.5.5 Frequency ratio (r) TR ( r (r) ) r) ME 38 Machine Design, Vibration Handout, Dr. K. Lulay, Rev. D Spring 6 5