Prof. Dr. Md. Mokhlesur Rahman Advanced Foundation Engineering - CE - 4303DUETCE-4303ADVANCED FOUNDATION ENGINEERING

NOTE NO 05

VIBRATION THEORY

PREPARED BY: Dr. Md. Mokhlesur Rahman Professor Civil Engineering Department Dhaka University of Engineering & Technology (DUET), Gazipur.

Soil Dynamics:Soil Dynamics is the branch of soil mechanics which deals with the engineering properties and behavior of soil under dynamic stress, including the analysis of the stability of earth supported and earth retaining structures. The study of Soil dynamics include the machine foundations, impact loadings, dynamic soil properties, slope stability, bearing capacity, settlement, vibratory compaction, pile driving analysis and field testing, ground anchor systems, seismic design parameters, liquefaction, sheet pile walls and laboratory testing.Nature/sources of types of dynamic loading:Dynamic loads on foundation and soil structure may act due to Earthquake Bomb blast Operation of reciprocating and rotary machines and hammers Construction operation such as pile driving Quarrying Fast moving traffic including landing aircraft Wind Loading due to wave action of water etc. The nature of each of these loads is quite different from the nature of the loads in the other cases. Earthquakes constitute the single most important source of dynamic loads on structures and foundation. Every earth quake is associated with a certain amount of energy released at its source and can be assigned a magnitude (m) which is just a number. Table gives an idea of the energy associated with a particular magnitude M (Richter)5.06.06.336.57.07.58.08.6

E (1020ergs)0.082.58.0014.180446250020000

Earthquake:The vibration of earth that accompanies an earthquake is one of the most terrifying natural phenomena known. From geological point of view, earthquakes provide the evidences of the instability of the earths crust and a logical starting point for any examination of the dynamics of the earth.Most earthquakes take place along faults in the upper 25 miles of the earth's surface when one side rapidly moves relative to the other side of the fault.

Due to ground motion during an earthquake Footing may settle Building may tilt Soils may liquefy Soils lose ability to support structures light structures may floatProblems of dynamic loading of soils and soil structures: Earthquake, ground vibration, wave propagation through soil Dynamic stress, deformation and strength characteristics Earth pressure problems and retaining walls Dynamic bearing capacity and design of shallow footing Embankments under earthquake loading Piles foundation under dynamic loads Liquefaction of soils Machine foundationBasic concepts:For a proper understanding and appreciation of the different aspects of design of foundation and soil structures subjected to dynamic loads, it is necessary to be familiar with the simple theoretical concepts of harmonic vibration.Basic Definitions:Vibration or Oscillation: It is the time dependent, repeated motion of translational or rotational type.Periodic motion: It is the motion which repeats itself periodically in equal time intervals.Period (T): The time elapsed in which the motion repeats itself is called the period of motion or simply period.Cycle: The motion completed in the period is called the cycle of motion or simply cycle.Frequency (f): The number of cycles of motion in a unit of time is known as the frequency of vibration. It is usually expressed in hertz (i.e. cycles per second).The period and the frequency are interrelated as, Free vibration: Free vibration occur under the influence of forces inherent in the system itself, without any external force. However, to start free vibrations, some external force or natural disturbance is required. Once started, the vibrations continue without an external force.Forced vibration: Forced vibrations occur under the influence of a continuous external force.Natural frequency: If an elastic system vibrates under the action of forces inherent in the system and in the absence of any externally applied force, the frequency with which it vibrates is its natural frequency.Resonance: When the frequency of the exciting force is equal to one of the natural frequencies of the system, the amplitudes of motion become excessively large. This condition is known as resonance.Damping: The resistance to motion which develops due to friction and other causes is known as damping. Viscous damping is a type of damping in which the damping force is proportional to the velocity. It is expressed as, Where, Degree of freedom (n): The number of independent co-ordinates required to describe the motion of a system is called a degree of freedom. A system may in general have several degrees of freedom; such a system is called a multi degree freedom system.

Figure - 1: System with different degree of freedom (a) One degree of freedom n=1;(b) Two degree of freedom n=2; (c) Three degree of freedom n=3;(d) Infinite degree of freedom n=.Principal modes of vibrations: A system with more than one degree of freedom vibrates in complex modes. However, if each point in the system follows a definite pattern of common natural frequency, the mode is systematic and orderly and is known as the principal mode of vibration. A system with n degrees of freedom has a principal modes and n natural frequencies.Normal mode of vibrations: When the amplitude of some point of the system vibrating in one of the principal modes is mode equal to unity, the motion is then called the normal mode of vibration.Resonance: When the frequency of the exciting force is equal to one of the natural frequencies of the system, the amplitudes of motion become excessively large. This condition is known as resonance. It is important to avoid or control or minimize this situation. In this condition a large magnitude of force and amplitude of motion can be generated which is destructive to the structure.

Minimization or Control of resonance:It is important; therefore, to avoid or minimize these situations by either avoiding the equalization of the forcing and natural frequencies by use of appropriate damping mechanism that will reduce the size of the otherwise increased effects at resonance. Various damping mechanism are available, either inherent in the vibrating system or especially designed into the system. Reference will be later to specific mechanisms, but for the moment it is sufficient that damping loads to the dissipation of energy per cycle of motion and usually leads to a reduction or decay in amplitude of the motion.

Methods of avoid resonance: Isolate resonant component Change exciting frequency using springs, pads, pneumatic, suspending components Use Vibration absorption Increase system damping Reduce forcing amplitude Avoid forcing a system at a natural frequencyUn-damped free vibration of a spring mass system:

Figure - 2: Spring Mass Systema) Un-stretched Springb) Equilibrium Positionc) Mass in Oscillating Positiond) Mass in maximum Downward Positione) Mass in upward positionf) Free body diagram of mass corresponding to (c)

Un-damped Free vibration of a spring mass system:

Figure - 3: Free Vibration of a mass-spring system

Figure - 3 shows a foundation resisting on a spring. Let the spring represent the elastic properties of soil. If the area of the foundation is equal to A, the intensity of load transmitted to the sub grade The static deflection zstat of the spring is, Where, ( is defined as force per unit deflection)

If the foundation is disturbed from its static equilibrium position, the system will vibrate. According to Newtons second law of motion,

... (1)Where, g = acceleration due to gravityz = Displacement t = timem = mass = In order to solve the equation (1), we get ... (2)Where, A and B are arbitrary constant and is the circular natural frequency of system.Now, = = ...(3)From equation (1) and (3) we can write,

Figure - 4: Plot of Displacement, amplitude and cycle for the free vibration of mass spring system

From figure - 4 the magnitude of maximum displacement is equal to z. This usually referred to as the single amplitude. The peak to peak displacement amplitude is equal to 2z. The time required for the motion to repeat itself is called the period of the vibration. When the time required to complete one cycle of motion is, the time period T of this motion can be written as

Frequency,

Example: - 1A mass supported by a spring has a static deflection of 0.5 mm, Determine its natural frequency of vibration.Solution:Given,g = 9810 mm/sec2zstat = 0.5 mm (Ans)Example: - 2Find the fundamental frequency of vibration of a vertical cantilever as shown in figure (5) that supports a mass m which is large relative to total mass of the cantilever. Data - mass, m = 1000 kg, length, L= 20 m, Flexural rigidity, EI = 125102 N-m2.

W = 1000 Kg

EI = 125102 N-m2L = 20 mFigure - 5: Vertical Cantilever supporting massSolution:Distribution mass of cantilever can be ignored. The lateral stiffness, k of the cantilever at the level of m is, Now, frequency,(Ans)Example: - 3Determine the spring constant for the system of springs shown in figure

Figure - 6: Equivalent Spring ConstantSolution:Let us consider that a unit load is applied at c.It is shared at a and b in the ratio of and of respectively.The deflection of points a and b are of and respectively.Therefore, deflection of point c is. = = Hence, the resulting equivalent spring constant at c is, If x1 = x2 = x and k1 = k2 = k, then On the application of a unit load in figure b the total deflection is, Hence, equivalent spring constant, If k1 = k2 = k, then

Example: - 4Write the equation of motion for the systems shown in figure (7) and determine the natural frequencies.

Figure - 7: Equivalent Spring ConstantSolution:On application of unit load at c for spring k1 and k2 the deflection = Hence, equivalent spring constant for k1 and k2, Total deflection is =Hence, the equivalent spring constant, We have, Where, A & B = Arbitrary constant

Example: - 5We will consider the motion of the piston of a reciprocating machine on a soil foundation. The soil displacement of the position from the extreme position is

Also,

Figure - 8: Motion of a Piston of a reciprocating machine

Find the equation of velocity and acceleration.

Solution:

If we expand, the right hand side, by the binomial theorem we get, or,Substituting in the expression for S, we get, The series in brackets contains and even powers of .Now, Now,If is very small, and higher powers can be neglected.Putting that is a function of and where n = 1, 2, 3 .Velocity, Acceleration,

Damped Free Vibration or Free Vibration with Viscous DampingField frictional resistance to motion of a body produces a viscous damping force that is directly proportional to relative velocity, when it is low. At higher relative velocities, the damping force may be proportional to the square of the velocity. Internal frictional resistance of materials, associated with the internal molecular structure, will also lead to decay of vibration.

If the force of damping is proportional to velocity, it is termed viscous damping,

Thus ....(1)Where c is a constant of proportionality, referred to as the viscous damping co-efficient.Figure (9) shows a single degree of freedom oscillator to which is added a dashpot that induces the damping forces. From the force body diagram the equation of motion is .....(2)

Figure - 9: Spring mass Dashpot system

If we define a critical viscous damping co-efficient, is, .....(3)And a damping ratio, .....(4) where,

So equation (2) becomes, .....(5)The general solution of this equation is, .....(6)Where A & B are constants to be determined from the initial conditions at and. are the roots of the auxiliary equation,The three cases,D = 1.0 for Critically Damped ConditionD>=1.0, Over Damped Condition D1 are case of over Damped condition.For damping, the motion or velocity comes rapidly to zero.If D=1, the system is under damped, i.e. gives oscillatory motion.Then Where, are complex and conjugate roots.For the initial conditions and 0, the solution of equation (7) gives, or, Where, and is the phage angle.The motion is oscillatory with exponentially decaying amplitude, the general nature of which shown in the following figure (11)

Figure - 11: Damped Free Oscillation

Damping ratio (D) or Damping factor (D) and its significance:Damping factor may be defined as The ratio of the actual damping coefficient (C) to the critical damping coefficient (Cc).Mathematically,

Force

DeformationFigure - 12: Energy absorption of Materials As usual case, D1, the system is over damped and non-oscillatory and motion is a periodic motion.If D=1, the system is critically damped and non oscillatory and motion is non periodic.If D1, the system is over-damped, the motion of the system is called non-oscillatory or a periodic motion. When D=1, the system is critically damped, the motion of the system is called critically damped motion. When D