RESPONSE OF A CRACKED CANTILEVER BEAM TO FREE AND FORCED VIBRATIONS

1. ABSTRACT


Cracks present in machine parts affect their vibrational behaviour like the fundamental frequency and the resonance. This case is usual in machine parts subjected to fatigue type of loading. In this paper, the resonance response of a cracked cantilever beam has been studied based on fracture mechanics quantities like strain energy release rate, stress intensity factor and compliance. The vibration behaviour of a cantilever beam with cracks at fixed end and at the mid span has been analyzed. The analysis is done for both free and forced vibrations. The spring stiffness and the fundamental frequency decreases with increase in crack length. The amplitude of vibration increases and the occurrence of resonance gets shifted with increase in crack length.





2. INTRODUCTION


Fatigue type of loading of engineering structures and machine parts is likely to introduce cracks at the highly stressed regions. Many times manufacturing methods like welding may also introduce cracks like defects. In engineering machinery and civil structures, members in the form of cantilever type of beams are widely used. Common examples are Propeller shaft, Turbine blades, Cantilever bridges, Tall building structures etc.

Failure is caused by means of a progressive crack formation which is usually fine and microscopic in size. The failure may occur even without prior indication. The fatigue of material is affected by the size of the component, relative magnitude of static and fluctuating loads and the number of load reversals.


Whenever a crack is present in a machine part, the simple stress distribution no longer holds good and the neighborhood of the discontinuity is different. This irregularity in the stress distribution caused by abrupt changes of form, called stress concentration, causes the failure of the structure even before the theoretical fracture stress is reached.


So the effect of the fatigue type of loading is the formation of a crack that will in the course of time cause the destruction of the structure or the machine part.Vibation is one thing that can never be completely eliminated. Every machine is accompanied by vibration. So the study of response to vibration of such parts with cracks becomes indispensable. The timely intervention to repair the faulty machine part or structure prevents the loss of life as well as money.









3. BASIC CONCEPTS


1.FRACTURE: Fracture is the separation or fragmentation of a solid body into two or more parts under the action of stress. The process of fracture can be considered to be made up of two components, crack initiation and crack propagation. Fractures can be classified into two general categories, ductile fracture and brittle fracture. A ductile fracture is characterized by appreciable plastic deformation prior to and during the propagation of the crack. An appreciable amount of gross deformation is usually present at the fracture surfaces. Brittle fracture in metals is characterized by a rapid rate of crack propagation, with no gross deformation and very little microdeformation.Brittle fracture is to be avoided at all cost, because it occurs without warning and usually produces disastrous consequences.


2.CRACK: Crack is a surface defect. Fatigue type of loading of engineering structures and machine parts is likely to introduce cracks at highly stressed regions. Many times, manufacturing methods like welding may introduce a crack. Crack reduces the strength of the structure or the machine part.


3.STRAIN ENERGY RELEASE RATE: When a body is subjected to tensile stress, within the elastic limit, the work done on the body is stored as elastic strain energy. But when a crack is there and due to the action of the stress the crack propagates, the work done or the elastic strain energy stored is dissipated to form two new surfaces. The process of crack extension is inelastic.Strian energy release rate is defined as” the rate of transfer of energy from the elastic stress field of the cracked structure to the inelastic process of crack extension.” Unit is Jm-2.


4. CRACK EXTENSION FORCE: Crack extension force is the strain energy release rate per unit thickness of the material. It is usually represented by (G).


5. STRESS INTENSITY FACTOR: The stress intensity factor is a convenient way of describing the stress distribution around a flaw. If two flaws of different geometry have the same value of stress intensity factor, then the stress field around each of the flaws is identical. Stress intensity factor is usually represented by (K).






Generally K=ασ(пa)1/2


α - a parameter which depend on specimen, crack geometry.
σ - Stress
a- crack length


6.COMPLIANCE: Compliance is the inverse of spring stiffness. It is usually represented by(C).


7. VIBRATION: Vibration is a means of dissipating energy. Any motion which repeats itself after an interval of time called vibration or oscillation. In vibration, there is a continuous change of kinetic energy to potential energy and vice versa.


8. FUNDAMENTAL FREQUENCY: For a continuous system there are infinite numbers of frequencies. The lowest frequency of such a system is called Fundamental frequency.


9. RESONANCE: When the natural frequency of a system and the frequency of excitation become equal, the condition is called resonance. During resonance the amplitude of vibration is very large.



3.1 TERMS USED


a-crack length Mb- Bending moment
A-(1-a/W) P-Force
C-Compliance t- Thickness of beam
E-Young’s Modulus W- Width of beam
G- Crack extension force Y- Deflection/amplitude
I- Moment of inertia Y0- Static deflection without crack
In- Integral Y0 (a)-Deflection /amplitude with crack
k-Spring stiffness α-(A-3-A3)
K-Stress intensity factor β-L/W
L- Length of the beam ω0-Fundamental frequency
M- Mass of the beam ω0 (a) - Fundamental frequency with crack
3.2 RELATION BETWEEN STRESS INTENSITY FACTOR AND COMPLIANCE




Consider an elastic body containing a crack of length as shown in figure.1.The load and the corresponding load point deflection are P and Y, respectively. Now if the crack extends by a small distance da, the load required to cause the same deflection gets reduced and the difference in the elastic strain energy,dU, goes to make the crack propagate through the distance da .Thus the elastic strain energy release rate dU/da is given by

dU/da= (P2/2)dC/da (3.1)

The crack extension force, G, is related to the stress intensity factor K through the Young’s modulus of the material. Thus one has

G=1/t * dU/da =K2/E (3.2)

The above equation leads to

dC/da =2t/E * K2/P2 (3.3)

Thus the change in the compliance is related to the stress intensity factor. The stress intensity factor of a cracked body depends on the shape and size of the body, the type of loading and the length of the crack, and its ratio with the width (a/W). Thus for a cantilever beam with a crack at the top surface, the stress intensity factor is given by



4. ANALYSIS OF VIBRATION OF THE BEAM


From figure .1 it can be seen that when the length of the crack increases, the force required to cause the same deflection is reduced. Thus the main consequence of this crack is to alter the stiffness of the beam leading to changes in fundamental frequency. In the absence of the crack the fundamental frequency of the cantilever beam is given by

ω0 = 2.04 (K/M) 1/2 (4.1)



The spring stiffness decreases with an increase in crack length ‘a’. As a result of this, the fundamental frequency also decreases. Thus vibration response of the cantilever beam gets altered due to the formation and growth of crack.

With the bending moment given as Mb=PL, one gets



Where



Thus one gets the Compliance as
The compliance C is in m/N.C0 is the value (1/k0) at the end of the cantilever beam when a/W=0.

Thus the Compliance C is given by


Or, with the moment of inertia, I=tW3/12 one gets

The Compliance can be given in the non dimensional form as



Where β=L/W and for a beam without a crack



And so, one has the ratio Compliance Ca in the presence of a crack to the Compliance C0 without crack as


Taking the spring stiffness k of the cracked beam as the inverse of the Compliance, one gets

and the ratio k (a)/k(0) as






The relationship between the Compliance and the crack length and between the spring stiffness and the crack length are shown in figure.2








4.1 FUNDAMENTAL FREQUENCY


The fundamental frequency ω0 given by eqn 4.1 depends on the spring stiffness, k, which is a function of a/W. The variation of ω (a)/ω0 with a/W is shown in figure 4. The fundamental frequency decreases with increasing of crack length.








The increase in static deflection Y0(a) with crack length a/W, as a ratio of static deflection at a/W=0 is shown in figure 5


4.2 FREE VIBRATIONS


4.2.1 CRACK AT FIXED END



In the case of a cantilever beam of elastic material with no damping, free vibrations will continue with a period of oscillation τ which is inversely proportional to the frequency. Thus the ratio of the period of oscillation,τ (a) of the beam with end crack to the period of oscillation τ (0), of the beam without a crack, can be given as


τ (a)/ τ(0)= ω0 / ω(a)={k0/k(a)}1/2 (4.11)


Figure 6 shows schematically the period of oscillations for a/W=0 and a/W=0.4



4.2.2 CRACK AT MID SPAN


Consider the vibration of an elastic cantilever beam with a crack at mid span, as shown in figure 7 .The problem will be similar to two spring mass system as shown .The portion of the beam nearer to the fixed end will have the spring stiffness k1 equal to k0 where as the portion on the other side at the free end will have the spring stiffness k2 =k (a).



Thus taking the beam with the mid span crack as two spring mass system, the ratio of the amplitudes Y1 at the centre and Y2 at the free end is obtained as


(4.12)


Consider the cantilever beam with L/W=β=8. For the mid span crack with span length on either side being L/2, the ratio of the spring stiffness values k1/ k2 is given as


k1/ k2= k0/ k (a).

(4.12)


Where β1=4. The maximum amplitudes in the two modes are also shown in figure .7.


4.3 FORCED VIBRATION


4.3.1 CRACK AT FIXED END


Since the introduction of a crack reduces the spring stiffness ,’k’ and the natural frequency of the beam , beams with cracks will resonate at a lesser exciting frequency , ‘ω’, as the natural frequency ‘ω(a)’ , of the cracked beam decreases with increasing crack length . The occurrence of resonance will get shifted to a lower value of ω/ω0(less than 1), where ‘ω’ is the natural frequency and ‘ω0’ is the natural frequency of the uncracked beam.


The amplitude variation Y, in the normal case, can be written as


(4.13)


In a similar way the amplitude variation Y(a), in the presence of a crack at the fixed end can be given in terms of the static deflection Y0 (a) as

(4.14)


The variation of the amplitude Y (a) with the exciting frequency ω/ω0is shown in figure 8 for a/W=0.4 and β=8.






4.3.2 CRACK AT MID SPAN


With the crack at the mid span, the vibrations of the cantilever beam can be analysed as two spring mass system. With the force ‘F0 sin ωt’ , acting at the free end of the beam , the amplitudes ‘Y1’ and ‘Y2’at the mid span and at the free end, respectively can be written as

(4.15)

and
(4.16)


Where Yst= F0/k1.As a particular case with a/W=0.2, and β=8, one gets k1/k2=1.276= (ω1/ω2)2, and the two natural frequencies are ω=1.49 ω1, and ω=0.6 ω1.



Figure 9 shows the variation of the amplitudes Y1/Yst and Y2/Yst with the exciting frequency ω/ω1











CONCLUSION


The vibration behaviour of a cantilever beam with cracks at the fixed end and at the mid span has been analysed based on the fracture mechanics concepts of crack extension force and stress intensity factor. The analysis shows that


1.With increasing crack length, the spring stiffness and the natural frequency of the beam decreases.

2. In free vibrations, the period of oscillation of the beam with a crack is higher than that of the beam without a crack.

3. In forced vibration of a beam with crack at the fixed end, the occurrence of resonance gets shifted to a lower value of ω/ω0, where ω0 is the natural frequency of the beam without the crack.
.
4.In the case of a beam with a crack at the mid span, the vibration behaviour can be analysed treating beam as two spring mass system. The analysis yields the amplitude in free vibration with two modes and in forced vibration, the occurrence of resonance and vibration pattern with the exciting frequency.


5. REFERENCES


JOURNAL- “Defense Science Journal”, Vol 54, No.1, January 2004
TEXTS - a) George.E.Dieter, “Mechanical Metallurgy” Mc Graw Hill Pub.
b)V.P .Singh,”Mechanical vibrations”, Dhanpatrai & Co.
c) R.S. Khurmi, J.K.Gupta,”A Text Book of Machine Design”
S.Chand Publcations.
d) John N. Macduff, John R.Curreri, “Vibration Control”
Mc Graw Hill Publications