NON-LINEAR VIBRATION. DR. Rabinarayan Sethi,

Transcription

1 DEPT. OF MECHANICAL ENGG., IGIT Sarang, Odisha:2012 Course Material: NON-LINEAR VIBRATION PREPARED BY DR. Rabinarayan Sethi, Assistance PROFESSOR, DEPT. OF MECHANICAL ENGG., IGIT SARANG M.Tech, B.Tech in Mechanical Engg Dr. Rabinar ayan Sethi, Assistance Pr ofessor

2 DEPT. OF MECHANICAL ENGG., IGIT Sarang, Odisha:2012 NON-LINEAR VIBRATION (3-0-0) Introduction to concept of trajectories, phase space, singular points limit cycle; Linear stability analysis introduction to bifurcations; Analytical methods including perturbation techniques, heuristic approaches like harmonic balance equivalent linearization; Stability of periodic solutions: Floquet s theory, Hill s Mathieu s equations; Nonlinear free forced responses of the Duffing s van der Pol equation; Introduction to chaos Lyapunov exponents. Dr. Rabinar ayan Sethi, Assistance Pr ofessor

3 Review on Linear Vibrating Systems In this lecture review of the linear Vibrating system has been carried out. Here Problem related single, two multi degree of freedom system have been discussed for both free forced vibrant response. Also, analyzing of continuous system has been carried out. Review of SDOF system with Harmonic forcing Example 1.2.1: Find the response of single degree of freedom systems with harmonic forcing. Solution Figure (a) Spring-mass damper system (b) Force polygon Figure 1.2.1(a) show a spring-mass-damper system subjected to harmonic forcing. Let represent mass, stiffness damping factor of the system. The equation of motion of the system can be given as...(1.2.1) Taking OX as the reference line, the force polygon shows all the forces viz., spring force(kx), damping force(cωx), inertia force(mω2x) external force(f). From the figure, it is clear that the angle between the external force the displacement vector is Φ. The steady state response of the system can be given by...(1.2.2) Here,...(1.2.3)... (1.2.4) Recalling,

4 One can write... (1.2.5) The total response of the system is the summation of transient steady state response which is given below....(1.2.6) As the ratio F/k is the static deflection (Xo) of the spring, is known as the magnification factor or amplitude ratio of the system. Figure shows the magnification factor ~ frequency ratio phase angle ( Φ )~ frequency ratio plot. Following observations can be made from these plots. For undamped system (i.e., ζ=0) the magnification factor tends to infinity when the frequency of external excitation equals natural frequency of the system ( ). But for underdamped systems the maximum amplitude of excitation has a definite value it occurs at a frequency For frequency of external excitation very less than the natural frequency of the system, with increase in

5 frequency ratio, the dynamic deflection ( X ) dominates the static deflection (X0 ), the magnification factor increases till it reaches a maximum value at resonant frequency ( ). For ω >, the magnification factor decreases for very high value of frequency ratio (say the vibration is very much attenuated. ), One may observe that with increase in damping ratio, the resonant response amplitude decreases. Irrespective For For, of, phase angle phase angle of ζ, value at, the phase angle.. approaches 180 for very low value of ζ. From phase angle ~frequency ratio plot it is clear that, for very low value of frequency ratio, phase angle tends to zero at resonant frequency it is 90 0 for very high value of frequency ratio it is 180. The resonant frequency The resonant amplitude of vibration

6 Figure 1.2.2: (a) Magnification factor ~ frequency ratio (b) phase angle ~frequency ratio for different values of damping ratio. Here, it may be noted that one can convert this linear spring-mass-damper system into a nonlinear system by introducing nonlinearity in mass, stiffness damping terms. Support Motion: Figure 1.2.3: A system subjected to support motion Figure 1.2.4: Freebody diagram Consider a system as shown in Fig where the support is moving with. It is required to find the motion of mass m which is supported by spring damper with spring constant k damping factor c. The equation of motion of the system is given by...(1.2.7) Substituting z=x-y... (1.2.8) In Eq. (1.2.8) one obtains...(1.2.9)

7 The solution of this equation can be given by...(1.2.10) where,... (1.2.11) Taking... (1.2.12) One can obtain (1.2.13) From which one obtains (1.2.14) where Figure 1.2.5: Amplitude ratio ~ frequency ratio plot for system with support motion From figure 1.2.5, it is clear that when the frequency of support motion nearly equals to the natural frequency of the system, resonance occurs in the system. This resonant amplitude decreases with increase in damping ratio for. At, irrespective of damping ratio, the mass vibrate with an amplitude equal to that of the support for, amplitude ratio becomes less than 1, indicating that

8 the mass will vibrate with an amplitude less than the support motion. But with increase in damping, in this case, the amplitude of vibration of the mass will increase. So in order to reduce the vibration of the mass, one should operate the system at a frequency very much greater than the system. This is the principle of vibration isolation. times the natural frequency of One may consider a number of problems where the system can be reduced to that of a single-degree of freedom system. Next we will review about two degree of freedom system continuous systems. WO DEGREE OF FREEDOM SYSTEMS Tuned Vibration Absorber: Figure (a) shows a spring mass system which can be thought of the model of a harmonically excited system. To absorb the vibration, generally another spring-mass is added to the primary system as shown in Fig (b). This system is a two degree of freedom system the equation of motion of this system can be given by the following equation. Fig (a) Single spring-mass system subjected to harmonic forcing, (b) secondary spring-mass added to the system shown in (a)....(1.2.15) In the absence of damping the steady state response of the primary system can be given by Or, where,... (1.2.16)... (1.2.17)

9 Hence, when r=1 or, the response of the system tends to infinity. Now one may find the steady state response of the system by substituting in Eq.(1.2.15) which yield...(1.2.18) Or,...(1.2.19) If we want to make the primary system stationary i.e.,, from Eq. (1.2.19) one can write. This is the condition for tuned vibration absorber. But in general the amplitude of response of the primary secondary system can be written as...(1.2.20) Figure (a) Response of the primary system (b) response of the primary system in the presence of secondary mass damper system. Figure (a) shows the steady state response of the primary system (b) shows the response in the presence of secondary spring-mass system. It is clearly observed that when =1, there is no vibration of the primary system. Hence, at this frequency the secondary system absorbs the vibration of the primary system so the system is known as tuned vibration absorber. For multi degree of freedom system one may revise the following points Normal mode of vibration: In this mode of vibration all the masses of the multi-degree of freedom system vibrating with same frequencies passes the equilibrium position at the same time. For example, in case of a double pendulum the two modes of vibration are shown in fig

10 Fig (a) First mode (b) second mode of vibration of a double pendulum If the first second links make an angle of with respect to the vertical axis, then the frequencies first second normal modes can be found as given below.,... (1.2.21)... (1.2.22) (1.2.23) The resulting free vibration of the system is a combination of normal modes having different modal participation. Hence... (1.2.24) Or, where...(1.2.25) is the participation factor of the nth mode is the nth normal mode obtained by finding the eigenvalues corresponding to the nth eigenvector of the dynamic matrix the eigenvector is equal to the square of the modal frequency of the system.. It may be noted that Orthogonality principle of the normal modes: Normal modes are orthogonal. Hence, one may obtain diagonal / uncoupled mass stiffness matrices by using this principle. Modal matrix: (P )The nth column vector of this matrix is the eigenvector corresponding to the n th eigenvalue of the dynamic matrix. So,... (1.2.26) Generalized mass matrix: It is a diagonal matrix which is given by

11 Generalized stiffness matrix : It is a diagonal matrix which is given by Weighted modal matrix ( ): It is obtained when each column of the modal matrix is divided by the square root of the corresponding generalized mass (i.e., n th column of Pmatrix is divided by square root of the n th generalized mass). It may be noted that. Modal analysis: It is used to uncouple the equation of motion of the coupled multi-degree of freedom system. For example consider the coupled equation of motion Where the mass matrix M, stiffness matrix K damping matrix Care coupled matrices (i.e., they have off-diagonal terms). Now one may use any of the following procedure. Procedure 1: Find the modal matrix P Assuming Rayleigh damping one my write Substitute x=py in Eq. equation....(1.2.28) premultiply So one obtains in both sides of the resulting...(1.2.28) Here, all the matrices are diagonal matrices one may solve the resulting equations as that of single degree of freedom system. Procedure 2 Find the weighted modal matrix Assuming Rayleigh damping one my write Substitute in Eq. () resulting equation. premultiply So one obtains Or, in both sides of the (1.2.29)... (1.2.30) Unlike the previous procedure here one has to calculate only the equations as that of single degree of freedom system. vector then solve the resulting Normal mode summation method: Most of the time it is not required to consider all the modes of the system as only the first few modes play dominant role in the resulting vibration. Hence, instead of taking an n x n P or matrix, one may consider n x m matrix corresponding to the first m modes only. Now one may follow the above mentioned procedure where the number of resulting equations will be m only. Hence, this reduces the computational time memory.

12 Continuous or Distributed Mass System : Figure shows few examples of continuous system. The first column shows the beams with fixedfixed, simply supported, fixed-free (cantilevered) free-free end conditions. Fig.: : Example of continuous system The second column shows a base excited beam with a tip mass last column shows a SCARA robot. In all these cases the system can be modelled as a continuous or distributed mass system. Unlike the case of multi-degree of freedom system or discrete mass system where the governing equations are written as ordinary differential equation, here, partial differential equations are used represent the motion of the system. Few typical cases are discussed below. For the following systems the governing equations are represented by wave equations. Lateral vibration of taut string Longitudinal vibration of rod Torsional Vibration of Shaft The wave equation is given by... (1.2.31) Here w is the displacement which is a function of both space variable x time t. The general solution of the system is the following equation. where the mode shape. These constants can be obtained by applying the boundary conditions. is given by..(1.2.32)

13 For transverse vibration y of the beam with youngs modules E, Moment of inertia I mass per unit length ρ, length L due to pure bending one may use Euler Bernoulli Beam which is given by the following equation....(1.2.33) The general solution of the system is the following equation. where the mode shape is given by... (1.2.34) The frequency of the system can be obtained from the following equation....(1.2.35) Table gives the values of mode. Table : 1.2.1: Values of Beam Configuration Simply supported Cantilever Free-free Clamped-clamped Clamped-hinged Hinged-free for different end conditions of the beams for the first three for different end conditions of the beams First mode Second mode Third mode Mode shapes Fig : First four mode shapes for a simply supported beam.

14 Exercise problems 1. Considering the springs damper to be nonlinear write the equation of motion of a single, two multi-degree of freedom system. 2.Write the equation of motion of the tuned vibration absorber. Replace the secondary spring by a spring with cubic nonlinearity write the resulting equation of motion. 3. Consider a pendulum vibration absorber. Considering the link to be flexible, derive the equation of motion on the system. 4. Plots the mode shapes of a (a) cantilever beam (b) cantilever beam with tip mass (c) beam with fixed roller supported end conditions. In this lecture the qualitative analysis of nonlinear conservative system is introduced commonly observed nonlinear phenomena are briefly described. Qualitative analysis of nonlinear conservative systems Consider a nonlinear conservative system which is given by the equation. Multiplying (1.3.1) the resulting equation Upon integrating one obtains (1.3.2)... (1.3.3) This represents that the sum of the kinetic energy potential energy of the system is constant. Hence, for a particular energy level h, the system will be under oscillation, if the potential energy is less than the total energy h. From the above equation, one may plot the phase portrait or the trajectories for different energy level study qualitatively about the response of the system. Example 1.3.1: Perform system qualitative analysis to study the response of the dynamic...(1.3.4) Solution: For this system. (1.3.5) Figure shows the variation of potential energy F(x) with x. It has optimum values corresponding to While x equal to zero represents the system with minimum potential energy, the other two points represent the points with maximum potential energy

15 Fig Potential well phase portrait showing saddle point center corresponding to maximum minimum potential energy. Now by taking different energy level h, one may find the relation between the velocity v displacement x as...(1.3.6) Now by plotting the phase portrait one may find the trajectory which clearly depicts that the motion corresponding to maximum potential energy is unstable the bifurcation point is of saddle-node type (marked by point S) the motion corresponding to the minimum potential energy is stable center type (marked by point C). There are several approximate solution method based on the perturbation techniques to solve the nonlinear equation of motions of the system. Types of Nonlinear response Fixed point response Periodic response Quasiperiodic response Chaotic response

16 Fig : (a) fixed-point trivial response (b) fixed-point non-trivial response, (c) periodic response Fig : (a) periodic response with multi-frequency (b) quasi-periodic response (c) chaotic response. Figures 1.3.2(a, b) show the time response where the steady state response is a fixed point response. While in the first case the steady state response leads to a trivial state, in the second case it is a non-trivial state. In Fig. (1.3.2 c) a periodic response with single frequency is shown. A periodic response with multifrequency is shown in Fig. (1.3.3(a)). Figure 1.3.3(b) shows the time response when the ratio of the considered two frequencies is an irrational number. Such responses are known as aperiodic or quasi-periodic response. The plot shown in Fig (c) is a chaotic response which is a deterministic bounded response but without following any specific pattern.

17 A Matlab code is given below to plot the time responses of fixed-point, periodic, quasi-periodic chaotic responses. One may change the parameters to obtain a wide range of responses. To characterize these responses one may use Time response Phase portrait Poincare' section Lyapunov exponent A detailed discussion to numerically obtain time response, phase portrait, Poincare' section Lyapunov exponent have been carried out in module 5. Classification of fixed point response For a dynamical system substituting one can obtain the steady state solution or solving F(x;m)=0. This solution is stable or unstable can be equilibrium Point xo by substituting studied by performing the solution by substituting x = xo+ Δx in the equation x=f(x,m) which yields the equation Hyperbolic fixed point: when all of the eigenvalues of A have nonzero real parts it is known as.. hyperbolic fixed point. Sink: If all of the eigenvalues of A have negative real part. The sink may be of stable focus if it..has nonzero imaginary parts it is of stable node if it contains only real eigenvalues which.. are negative. Source: If one or more eigenvalues of A have positive real part. Here, the system is unstable.. it may be of unstable focus or unstable node. Saddle point: when some of the eigenvalues have positive real parts while the rest of the...eigenvalues have negative. Marginally stable: If some of the eigenvalues have negative real parts while the rest of the.....eigenvalues have zero real parts Typical Frequency response curves Fig : A typical frequency response curves showing the stable unstable branch (solid line stable, dotted line unstable branch) In nonlinear systems, while plotting the frequency response curves of the system by changing the control parameters, one may encounter the change of stability or change in the number of equilibrium points. These points corresponding to which the number or nature of the equilibrium point changes, are known as bifurcation points. For fixed point response, they may be divided into static or dynamic bifurcation points depending on the nature of the eigenvalues of the system. If the eigen values are plotted in a complex plane with their real imaginary parts along X Ydirections, a static bifurcation occurs, if with change in the

18 control parameter, an eigenvalue of the Jacobian matrix crosses the origin of the complex plane. In case of dynamic bifurcation, a pair of complex conjugate eigenvalues crosses the imaginary axis with change in control parameter of the system. Hence, in this case the resulting solution is stable or unstable periodic type. A detailed discussion on the stability bifurcation of fixed point periodic responses are given in module 4. Fig : Basin of attraction Commonly used nonlinear equation of motion Duffing equation (Free vibration with quadratic cubic nonlinear term)... (1.3.7) Duffing equation with damping weak forcing terms. (1.3.8) Duffing equation with damping strong forcing terms (1.3.9) Duffing equation with multi-frequency excitation... (1.3.10) Rayleigh's equation

19 ...(1.3.11) van der Pol's equation...(1.3.12) Hill's equation...(1.3.13) Mathieu's equation...(1.3.14) Mathieu's equation with cubic nonlinearies forcing terms... (1.3.15) Development of Equation of Motion for Nonlinear vibrating systems In this module following points will be discussed for deriving the governing equation of motion of a system Force moment based approach Newton's 2nd Law Generalized d'alembert's Principle Energy based Approach Lagrange Principle Extended Hamilton's Principle Temporal equation using Galerkin's method for continuous system Ordering techniques, scaling parameters, book-keeping parameter Examples of Commonly used nonlinear equations: Duffing equation, Van der Pol's...oscillator, Mathieu's Hill's equations Force Momentum based Approach In this approach one uses Newton's second law of motion or d'alembert's principle to derive the equation of motion. This is a vector based approach in which first one has to draw the free

20 body diagrams of the system then write the force moment equilibrium equations by considering the inertia force inertia moment of the system. According to Newton's second law when a particle is acted upon by a force it moves so that the force vector is equal to the time rate of change of the linear momentum vector. Consider a body of mass m positioned at a distance r from the origin of the coordinate system XYZas shown in Figure is acted upon by a force F. According to Newton's 2 nd Law, if the body has a linear velocity v, linear momentum vector p=mv, the external force is given by the following equation. Figure 2.1.1: A body moving in XYZ plane under the action of a force F Considering to be the absolute position vector of the particle in an inertial frame, the absolute velocity vector can be given by... (2.1.2) the absolute acceleration vector is given by...(2.1.3) Assuming mass to be time invariant, Hence...(2.1.4) Equation (2.1.4) can also be written as, where is the inertia force. This is d' Alembert's principle which states that a moving body can be brought to equilibrium by

21 adding inertia force to the system. In magnitude this inertia force is equal to the product of mass acceleration takes place in a direction opposite to that of acceleration. Now two examples are given below to show the application of Newton's 2nd law or d' Alembert's principle to derive the non linear equation of motion of some systems. Example : Use Newton's 2nd law to derive equation of motion of a simple pendulum Figure : (a) simple pendulum (b) Free body diagram Solution: Figure (a) shows a simple pendulum of length l mass m Figure 2.1.1(b) shows the free body diagram of the system. The acceleration of the pendulum can be given by free body diagram total external force acting on the mass is given by. From the...(2.1.5) Now using Newton's second law's of motion i.e.,...(2.1.6) Now equating the real imaginary parts one can get the equation of motion the expression for the tension. The equation of motion is given by... (2.1.7) the expression for tension can be given by...(2.1.8) Taking nonlinear term can be given by, the nonlinear equation of motion of the system up to 7th order

22 ...(2.1.9) Or,...(2.1.10) It may be noted that for higher power of θ, the coefficient become very small hence the higher order terms can be neglected. Keeping up to 5th order, the equation can be written as... (2.1.11) which is a form of Duffing equation with cubic quintic nonlinearities. One may derive the same equation using the fact that the moment of a force about a fixed point equal to the time rate of change of the angular momentum about poin written as is. In mathematical form it can be. Refereeing to Figure 2.1.2(b)...(2.1.12) Or, Now,...(2.1.13)...(2.1.14) Or,...(2.1.15) Or,... (2.1.16) Or,... (2.1.17) Keeping up to cubic nonlinearity Eq. (2.1.17) can be written as... (2.1.18) Taking the length of the pendulum 1 m acceleration due to gravity as 10 m/s2 the equation of motion can be written as

23 ...(2.1.19) It may be noted that the coefficient for the cubic order term is very less than that of the linear term. A MatLab code is given below to obtain the variation of restoring force with θ which will give an idea regarding the approximation one has to take while writing the equation of motion. The equation is similar to a Duffing equations with soft type cubic nonlinearity the nonlinear governing equation of motions of multi-degree of freedom nonlinear systems will be derived by using Newton's 2nd Law or d'alembert's principle. The approach is similar to that of the single degree of freedom system. One can derive the equation of motion by drawing the free body diagrams then writing the force or moment equilibrium equations by including the inertia force. Let us consider following simple examples to derive the equation of motions. Example 2.2.1: Derive the equation motion of system shown in Fig Consider the last spring to be nonlinear where the spring force is given by to be linear.. Consider other spring damper behaviour Figure A multi degree of freedom system Solution Considering the equilibrium of the mass, Figure 2.2.3: Free body diagram of part with mass Equating the forces acting on mass as shown in Fig one obtains =0....(2.2.1)

24 Similarly considering the free body diagram for the 2nd mass the equation of motion can be written as... (2.2.2) From the free body diagram shown in Fig , the equation of motion for the 3rd mass can be given by =0...(2.2.3) Figure 2.2.4: Free body diagram of part with mass It may be noted that as the last spring is connected to both second third masses, the obtained second third equations are nonlinear. So the equation of motions of the system can be written as... (2.2.4)... (2.2.5)... (2.2.6) Derivation of the equation of motion of continuous system using d'alembert's principle. In this lecture, with help of example we will derive the governing equation of motion of a continuous or distributed mass system using d'alembert's principle. It may be noted that in previous two lectures we considered discrete system in which the governing equation of motions are in the form of ordinary differential equations. But in continuous system the governing equations are in the form of partial differential equation as the state vector (e.g., displacement) depends not only on time but also on the space co-ordinates. For example in case of axial vibration of a bar the axial displacement of the bar depends on the time location of the point on the bar at which the displacement has to be measured. Also, it may be noted that, unlike discrete system where the natural frequencies of the system has a definite value, in case of continuous system the system has infinite number of natural frequencies. Depending on particular applications, one may convert the analysis of continuous system to that of a multi-degree of freedom system by considering finite participating modes in the analysis. Derivation of equation of motion using Extended Hamilton's Principle The purpose of this lecture is to use extended Hamilton's principle to derive the equation of motion of different system. According to this method, for a system with kinetic energy T, potential energy U virtual work done by the non-conservative force using the following equation., the governing equation motion can be obtained by

25 ...(2.4.1) Here, are the time at which it is assumed that the virtual displacements represented by n physical co-ordinates ( co-ordinates ) vanishes. Using Lagrangian ( for a system ) mgeneralized of the system, the above equation can be written as... (2.4.2) Equation (2.4.1) (2.4.2) are the equation for Extended Hamilton's principle. For a conservative system as, Eq. (2.4.2) reduces to...(2.4.3) which is known as the Hamilton's Principle. This method is particularly useful for continuous systems where one can obtain both governing equation of motion boundary conditions of the system. Example 2.4.1: Derive the equation of motion of a simple pendulum using extended Hamilton's principle. Solution: In this case the kinetic energy T potential energy U of the system can be given by...(2.4.4)... (2.4.5) Now, applying Hamilton's principle one can write...(2.4.6)

26 ... (2.4.7) The first term (marked in red colour) tends to zero as displacement is arbitrary, hence the coefficient of. As the virtual term should vanish. Therefore one obtains... (2.4.8) as the equation of motion of the simple pendulum. Taking up to 5 th order terms this equation can be written as...(2.4.9) Example 2.4.2: Derive the equation of motion for the transverse vibration of an Euler-Bernoulli beam with fixed-free boundary condition subjected to axial periodic load as shown in Fig Solution: Let us first derive the equation of motion of the system considering small displacement of the system. The kinetic energy T of the beam can be given as follows:...(2.4.10) where, m is the mass of the beam per unit length time. represents the differentiation with respect to

27 Fig: 2.4.1: Schematic diagram of a cantilever beam under transverse vibration due to application of a periodic axial load The potential energy of the system is due to the strain energy of the system is given by...(2.4.11)... (2.4.12) Hence the Lagrangian of the system Assuming inextensible beam condition, there will be no elongation in the axial direction along the neutral axis of the beam. The longitudinal deformation u of the beam due to transverse deformation w can be expressed as...(2.4.13) Rearranging Eq. (2.4.13) using a first order Taylor series expansion, the following relationship can be obtained....(2.4.14) The work done due to the nonconservative axial force can be given by...(2.4.15) Using Hamilton's principle...(2.4.16) Using Eq (2.4.12) Eq. (2.4.15) in Eq. (2.4.16) one can write

28 ...(2.4.17)...(2.4.18)...(2.4.19) Using integration by parts Eq.(2.4.19) can be written as,... (2.4.20) Or,...(2.4.21) As the virtual displacement is arbitrary, hence the right h side of the equation will be zero only if...(2.4.22) which is the equation of motion of the system. The boundary conditions can be obtained from the term marked in blue colour in Eq. (2.4.21). Now taking the periodic axial load as, Eq. (2.4.22) can be written as

29 ..... (2.4.23) Due to the presence of a periodic term (marked in pink colour) as the coefficient of the term containing the response (marked in blue colour), the system is a parametrically excited system. Exercise Problems: Problem : Derive the equation of motion of a cantilever beam subjected to magnetic field using extended Hamilton's principle. Fig : Schematic diagram of a cantilever beam subjected to magnetic field. Hints: The expression for kinetic strain energy of the system can be taken similar to that taken in example (2.4.24)...(2.4.25) Considering conductive material, the magnetoelastic load applied to the beam is equivalent to the horizontal force n the distributed moment m which are expressed in terms of the longitudinal displacement ( u ) transverse displacement ( w ) as (Zhou Wang [1] ), Moon Pao [2]...(2.4.26)...(2.4.27)

30 Here, are respectively the permeability of the free space the beam materials. The non-conservative work done due to the applied axial periodic load the above mentioned magnetoelastic loads moments can be given by...(2.4.28) Derivation of Equation of motion using Lagrange Principle Both Hamilton's principle Lagrange principle are based on energy principle for deriving the equation of motion of a system. As energy is a scalar quantity, the derivation of equation of motion is more straight forward unlike the derivation based on Newton's 2nd Law or d' Alembert's principle which are vector based approach. In the Newton's or d'alembert's approach, with increase in degrees of freedom of the system it is very difficult time consuming to draw the free body diagrams to find the equation of motion using force or moment equilibrium. Hence it is advantageous to go for energy based approach. While in Hamilton's principle one uses a integral based approach, in Lagrange principle a differential approach is followed. Hence, use of Lagrange principle is easier than the Hamilton's principle. Though all these methods in principle can be applied to any system, however it is better to use Newton's 2 nd Law or d'alembert's principle for single or two degree of freedom systems, Lagrange principle for multi degree of freedom extended Hamilton's principle for continuous systems. In Lagrange principle, generally the equations of motion are derived using generalized coordinates. Let us consider a system with N physical coordinates n generalized coordinates. The kinetic energy T for a system of particles can be given by... (2.5.1) Where are the position mass mi ( i =1,2,.., N ).Considering coordinates, one may write, velocity vector of a typical particles of as the displacement velocity in terms of kth generalized...(2.5.2) So using generalized coordinate one may write,...(2.5.3) Hence,...(2.5.4)

31 The virtual work ( ) performed by the applied force virtual displacement or can be written in terms of generalized forces... (2.5.5) where,....(2.5.6) The over bar in shows that the work done is a path function. Substituting (2.5.4) (2.5.5) into the extended Hamilton's Principle,,, k =1,2,.. n... (2.5.7) one obtains the following equation.... (2.5.8) Now,...(2.5.9) Substituting (2.5.9) in (2.5.8) we have... (2.5.10) Considering the arbitrariness of the virtual displacement values of, equation ( ) will be satisfied for all provided...(2.5.11)

32 Equation (2.5.11) is known as Lagrange's equation. Considering both conservative force force nonconservative force, the total generalized can be written as...(2.5.12) recalling potential energy depends on coordinates alone, the work done by the conservative force is equal to the negative of the potential energy V. Hence, one may write... (2.5.13) So the conservative generalized forces have the form, k =1, 2,.. n...(2.5.14) Substituting Eq. (2.5.12) Eq. (2.5.13) in Eq. (2.5.10) we have...(2.5.15) As the potential energy does not depend on velocity, using Lagrangian rewritten as, Eq. (2.5.15) can be...(2.5.16) Using dissipation energy D, this equation further can be written as... (2.5.17) Using both external forces moments one may write the generalized force as...(2.5.18) Mi is the vector representation of the externally applied moments, the axis along which the considered moment is applied. is the system angular velocity about

33 -- Lagrange equation can be used for any discrete system whose motion lends itself to a description in terms of generalized coordinates, which include rigid bodies. -- can be extended to distributed parameter system, but such system, they are not as versatile as the extended Hamilton's Principle Let us take some examples to derive the equation of motion using Lagrange principle. Example 2.5.1: Derive the equation of motion of a spring-mass-damper system with spring force given by damping force given by is given by static equilibrium point as x.. The external force acting on the system. Consider mass of the system as m displacement from the Solution: In this single degree of freedom system one can take x as the generalized co-ordinate. From the given expressions for different forces acting on the system, the expressions for kinetic energy T, potential energy V, dissipation energy D can be given by the following expressions.... (2.5.19)...(2.5.20)...(2.5.21)...(2.5.22) Using Lagrange equation (2.5.17)...(2.5.23)...(2.5.24)...(2.5.25)

34 ...(2.5.26) Development of temporal equation of motion using Galerkin's method for continuous system In this lecture one will learn the development of temporal equation of motion using generalized Galerkin's method for continuous system. It may be noted that unlike discrete system where the equations are ordinary differential equations, in case of continuous or distributed mass system the governing equations are partial differential equation as they depend on both time space variables. Hence it is required to reduce the partial differential equation to ordinary differential equation for finding the solution of the system easily. In case of vibrating system these equations are generally reduced to their temporal form by using Galerkin's method. In this method following steps have to be followed. Assume an approximate function for the mode shape of the continuous system. Here one may take single or multi-mode approximation. Substitute the mode shape(s) in the governing partial differential equation of motion to obtain the residue. Minimize the residue by using a weight function equate it to zero to obtain the temporal equation of motion. One may take orthogonal functions for mode shapes weight function to simplify the integration to obtain the coefficients of the temporal equation. In nonlinear systems with many terms, one may use symbolic software like Mathematica Mapple to derive the equation of motion. One may write a Matlab program having inbuilt integration schemes to obtain the coefficients. The method is illustrated with the help of the following example. Example 2.6.1: Consider the transverse vibration of a beam with roller supported at one end attached mass periodically varying load at the other end. The roller supported end is subjected to periodic motion. The governing equation of motion using d'alembert's principle is given in Eq. (2.3.6). We have to derive the temporal equation of motion of the system. Fig : Schematic diagram of a roller supported beam with tip mass transverse follower load. Solution

35 Figure shows the system with a payload of mass m at the tip where a compressive force P= is applied. Also this system is s ubjected to a harmonic base excitation = at the roller supported left end. Here Z are the amplitude frequency of the base excitation,, are the static dynamic force amplitude, is the frequency of the periodic force acting at the free end of the manipulator. The motion is considered to be in the vertical plane. Using d'alembert principle the equation of this system can be given by.....(2.6.1) Here, are the Young modulus, moment of inertia, mass density, area of cross-section, length of the cantilever beam damping factor of the system respectively, are used as integration variables. To determine the temporal equation of motion, one may discretize the governing equation of motion (2.6.1) by using following assumed mode expression.....(2.6.2) Here, r is the scaling factor; is the time modulation of the ith mode of the cantilever beam with tip mass which is given by is the eigenfunction...(2.6.3) One may determine from the following equation.... (2.6.4) The following non-dimensional parameters are used in the further analysis. Substituting the above mentioned nondimensional parameters equation (2.6.2) into equation (2.6.1) one may obtain the residue equation R. Now taking as the weight function using the generalized Galerkin's method, one may write the following equation.

36 ...(2.6.5) This equation can be written in the following form which is the non-dimensional temporal equation of motion of the system.... (2.6.6) [Derivation of only one/two terms are shown below. Taking only the first two terms in Eq. (2.6.1) substituting (2.6.2) one may obtain the residue equation...(2.6.7) Substituting in the above equation...(2.6.8) Taking weight function as obtains, multiplying in the above equation integrating over the domain one... (2.6.9) Where

37 ...(2.6.10)... (2.6.11) It may be noted that while is in dimensional form procedures have to be followed to find all other terms. is in the nondimensional form. Similar Considering single mode discretization i.e. by substituting m =1, the above equation reduces to...(2.6.12) Eq. (2.6.12) is the required temporal equation of motion. The coefficients used in this equation are described below. The natural frequency ( ) of the lateral vibration of an elastic beam... (2.6.13) Damping ratio ( ζ ),...(2.6.14) Coefficient of the nonlinear geometric term =...(2.6.15) Coefficient of the nonlinear inertia term

38 =,...(2.6.16) Coefficient of the nonlinear inertia term =... (2.6.17) Coefficient of the term =... (2.6.18) Coefficient of the direct forced term,,... (2.6.19) Coefficient of the parametric excitation,. (2.6.20) Where,,,,,,,,,,,

39 ,,,,,,,,,. One may find that the non-dimensional temporal equation (2.6.12) has a linear term, a linear parametric term forced a nonlinear parametric excitation term along with cubic geometric inertial nonlinear terms. Here the system is subjected to a two-frequency excitation. One may note that the temporal equation of motion contains many nonlinear terms it is very difficult to find the exact solution. Hence one may go for approximate solution by solving equation (2.6.12) using perturbation method. Ordering scaling technique in nonlinear equations In the previous lectures we learned about the derivation of equation of motion of both discrete distributed mass system. In the later case the equation has been reduced to its temporal form. In these equations the coefficients of different terms used in the differential equations may not be of the same order hence sometimes some terms get neglected in comparison to other terms. But for accurate solution one should take as many terms as possible hence it is required to know the ordering scaling techniques. So in this lecture following points will be discussed with the help of examples. Ordering techniques, scaling parameters, Book-keeping parameter. Commonly used nonlinear equations: Duffing equation, Van der Pol's oscillator, Mathieu's Hill's equations Let us consider the equation we have derived for the simple pendulum. It can be written as

40 ... (2.7.1) Keeping up to quintic nonlinearity Eq. (2.7.1) can be written as...(2.7.2) Taking the length of the pendulum 1 m acceleration due to gravity as 10 m/s2, the equation of motion can be written as...(2.7.3) In Eq. (2.7.3), the coefficient of the linear term θ is 10, the coefficient of cubic nonlinear term is the coefficient of quintic term is As the coefficients of quintic cubic terms are very very less than the linear term, one can neglect these terms to obtain the approximate solution. But to obtain the accurate solution one should consider these terms. One can use scaling parameter book-keeping parameters to make the coefficient of nonlinear linear terms of the same order so that the effect of these nonlinear terms can be taken into account. To use scaling factor, let us take be written as substitute this in Eq. (2.7.3). Now the resulting equation can...(2.7.4)... (2.7.5) Or, Now by taking different values of p, the coefficient of the nonlinear terms can be changed significantly without changing the coefficient of the linear part. For example, taking p=10 the above equation becomes...(2.7.6) Taking Eq. (2.7.5) can be written as... (2.7.7) While in Eq. (2.7.6) the coefficient of linear non-linear terms have large differences, in Eq. (2.7.7), these coefficients are closer to each other. Hence by suitably choosing the value of p, it is possible to bring the coefficient of the linear nonlinear terms to the same order in that case, instead of neglecting the higher order terms, one can consider these terms solve the equation to obtain more accurate response. Considering Eq. (2.7.3), as the coefficients of the cubic quintic order terms are very very small in comparison to the coefficient of the linear term, one can use a book-keeping parameter the coefficients. In this case one may write Eq. (2.7.3) as ε( ) to order

41 ...(2.7.8) Taking ε =0.1, Eq. (2.7.8) can be rewritten as... (2.7.9) In Eq. (2.7.9) now the numerical part of the coefficients ( ) are approximately same orders as that of the linear terms (i.e. 10). So in this way one can use the book-keeping parameter to order the nonlinear terms in a given nonlinear differential equation of motion. Commonly used nonlinear equation of motion Duffing equation (Free vibration with quadratic nonlinear term)...(2.7.10) Duffing equation (Free vibration with cubic nonlinear term)...(2.7.11) Duffing equation (Free vibration with both quadratic cubic nonlinear terms)...(2.7.12) Duffing equation with damping weak forcing terms...(2.7.13) Duffing equation with damping strong forcing terms... (2.7.14) Duffing equation with multi-frequency excitation... (2.7.15) Rayleigh's equation...(2.7.16)

42 Substituting in Eq.(2.7.16) differentiating the resulting equation with respect to time one will obtain the van der Pol's equation as follows...(2.7.17) Hill's equation... (2.7.18) Mathieu's equation...(2.7.19) Mathieu's equation with cubic nonlinearies forcing terms... (2.7.20) Lorentz equation... (2.7.21) Here are parameters Generic equation for one dimensional pitchfork bifurcation Generic equation for saddle-node bifurcation Generic equation for transcritical bifurcation Equation for Hopf bifurcation Approximate methods for solving nonlinear equations

43 In this module different approximate perturbation methods will be used to solve the nonlinear equations of motions derived in the previous module. Initially the straight forward expansion method will be used the following listed methods will be discussed in this module. Straight forward Expansion Lindstedt Poincare' Method Modified Lindstedt-Poincare method Method of Multiple Scales Method of Averaging Harmonic Balance method Intrinsic Harmonic Balance method Generalized Harmonic Balance method Multiple time scale- Harmonic Balance THE LINDSTEDT POINCARE' METHOD : This method was developed by Anders Lindstedt (June 27, 1854 May 16, 1939) Jules Henri Poincaré (29 April July 1912) for uniformly approximating periodic solutions to ordinary differential equations when regular perturbation approaches fail. Here a new independent variable is introduced where initially ω is an unspecified function of ε which is a book- keeping parameter ( ). As the new governing equation contains ω in the coefficient of the second derivative, this permits the frequency the amplitude to interact which is a property observed in nonlinear systems. One can choose the function in such a way as to eliminate the secular terms [Nayfeh Mook, 1979]. This method is explained by taking the following ordinary differential equation of Duffing type. By using Assuming the expansion for ω as...(3.2.1) equation (3.2.1) becomes... (3.2.3) where,, are unknown constants at this point. Moreover, similar to the straight forward expansion, x can be represented by an expansion having the form... (3.2.2)...(3.2.4) where are independent of ε. Then (3.2.2) becomes... (3.2.5) Equating the coefficients of to zero one obtains

44 ... (3.2.6)... (3.2.7)... (3.2.8) The general solution of Eq. (3.2.6) can be written in the form...(3.2.9)...(3.2.10) Due to the presence of the underlined term in equation (3.2.10), the response will be unbounded setting will contain the secular term. Hence, this term must be eliminated which can be done by. The solution of the remaining part of equation (3.2.10) can be written as follows....(3.2.11) Substituting the expression for, one obtain, one must put or... (3.2.13) Hence from (3.2.3), (3.2.9) (3.2.11) one obtains... (3.2.14) where into (3.2.8) recalling that (3.2.12) In equation (3.2.12) the underlined term will yield an unbounded solution to eliminate this secular term from...(3.2.15) Imposing the initial condition one obtains from (3.2.14)

45 ...(3.2.16)...(3.2.17) One should solve these equations (3.2.16) (3.2.17) to obtain a β which will be used further in (3.2.14) to obtain the nonlinear response of the system. Similar to the qualitative description of the motion, it may be noted that the Lindstedt-Poincare method produced (a) a periodic expression describing the motion of the system, (b) a frequencyamplitude relationship (c) higher harmonics in the higher order terms of the expression (d) a drift or steady-streaming term. (Nayfeh Maook 1979) Modified Lindstedt Poincare' technique The Lindstedt-Poincare' (L-P) method described in previous lecture can be applied to weakly nonlinear systems. To apply this method to strongly nonlinear system, the L-P method has been modified by many researchers. Here the method proposed by Cheung et al. (1991) is discussed. In this modified Lindstedt-Poincare' method the coefficient of the nonlinear term α can be written as a function of the book keeping parameter ε component of the expansion of the nonlinear frequency or the forcing frequency ( ). Similar to L-P method here also nondimensional time is used in the governing equation (3.4.1) to obtain the following equation.... (3.3.1) or in general the equation can be written as... (3.3.2) Unlike in L-P method, here ε may not be small. Following four steps have been proposed in this method. 1. In contrast to the stard L-P where expansion of ω is carried out, here it is proposed to exp.... (3.3.3) 2. A new parameter α is introduced.... (3.3.4) It may be noted that α is the ratio of the 2nd term to the first two terms in the expansion given in Eq. (3.3.3). From Eq. (3.3.4) one can write...(3.3.5)... (3.3.6) So, 3.3.7) (

46 Here, ( ) are unknown which will be obtained in the subsequent steps. Substituting Eq. (3.3.6) Eq. (3.3.7) in Eq. (3.3.2), one can write...(3.3. 8) Or, From Eq. (3.3.5) it can be observed that as Hence irrespective of the value of, α value is small. Hence by introducing this parameter α, one can reduce the strongly nonlinear system to a weakly nonlinear system on which the regular L-P or other perturbation method can be used. 3. Exp x into a power series using α...(3.3.9)...(3.3.12).(3.3.13) The usual steps in L-P method may be applied to solve these equations to obtain the solution of Eq. (3.3.2) to any desired order of α. 4. In the fourth last step, the initial value (i.e., into two parts as follows. ) are separated... (3.3.14)...(3.3.15) Where a is the initial value of the sum of all odd harmonic terms of x the sum of all even harmonic terms of.... (3.3.11). Also as...( ) Now substituting (3.3.10) in (3.3.9) equating the coefficients of like power of α, one can obtain the following set of linear differential equations., is the initial value of.... (3.3.16) For detailed application of this method one may refer the work by Cheung et al. (1991), Chen Cheung (1996). Franciosi Tomasiello (1998) used Mathematica to analyze strongly nonlinear two degree of freedom system using modified L-P method. Latif (2004) Yang et al. (2004) also used this method. Amore Ara (2005) used an improved L-P method in which they applied linear delta expansion (LDE) to L-P method it is shown that this method can be applied to a wider range of nonlinear equations it converges to the exact solution more rapidly than the conventional L-P method. Chen et al. (2007) used multi-dimensional L-P method. Xu (2007), Öziş Yıldırım (2007) used He's modified L-P method for strongly nonlinear system. Pušenjak (2008) extended L-P method for nonstationary response of strongly nonlinear system.

47 THE METHOD OF MULTIPLE SCALES In method of multiple scales, the original time is written in terms of different time scales which are considered to be multiple independent variables, or scales, instead of a single variable. Here, the new independent variables ( parameter εas ) of time are written using the book-keeping...(3.4.1) Hence, the derivatives with respect to t can be written in terms of the partial derivatives with respect to the as follows....(3.4.2)... (3.4.3) Let us apply this method to the Duffing equation with quadratic cubic nonlinearities. Example 3.4.1:... (3.4.4) Similar to previous method here, one may assume that the solution of (3.4.4) can be represented by an expansion having the form...(3. 4.5) We note that the number of independent time scales needed depends on the order to which the expansion is carried out. For example for, one may consider (3.4.3) (3.4.5) into (3.4.4) equating the coefficients of the following sets of equations. Order of... (3.4.7) Order of to zero, one obtains... (3.4.6). Substituting Order of...(3.4. 8) The solution of (3.4.6) can be written as....(3.4.9) Here A is an unknown complex function (3.4.9) into (3.4.7) leads to...( ) Here cc denotes the complex conjugate of the preceding terms. The particular solution of (3.4.10) has a secular term containing the factor to be eliminated. Hence one can obtain is the complex conjugate of A. Substituting. To have a bounded solution this term has... (3.4.11)

48 Therefore A must be independent of written as. With...(3.4.12) Substituting the expression for recalling that the particular solution of (3.4.10) can be from equation (3.4.9) (3.4.12) into (3.4.8) we obtain...(3.4.13) To eliminate the secular terms from...(3.4.14) To solve Eq. (3.4.14), it is convenient to write A in the polar form as...(3.4.15) where a β are real function of. Substituting (3.4.15) into (3.4.14) separating the result into real imaginary parts, we obtain, we must put... (3.4.16) where the prime denotes the derivative with respect to. As, a is a constant or... (3.4.17) Here is a constant. Now using from (3.4.15) we find that...(3.4.18) Substituting Eq. (3.4.18) in the expressions for obtains in Eqs. (3.4.9), (3.4.12) (3.4.5), one... (3.4.19) Here...(3.4.20) This solution is in good agreement with the solution obtained using the Lindstedt-Poincare' procedure. The method of multiple scales though a little more involved, has advantage over the

49 Lindstedt-Poincar method, for example it can treat damped systems conveniently. (Noyfeh Marc 1979) Example 3.4.2: Find the expression for the frequency-response curve for a nonconservative system using method of multiple scales. Solution: Consider the governing equation of motion of a nonconservative system which can be given by.....(3.4.21) Following stard procedure of method of multiple scales one may write...( ) Substituting (3.4.3) (3.4.22) into (3.4.21) equating the coefficients of zero, one obtains the following sets of equations. to... (3.4.23)...(3.4.24) The solution of Eq. (3.4.23) can be given by (3.4.27) Substituting Eq. (3.4.27) in Eq. (3.4.24) following equation is obtained. (3.4.28) One may use Fourier series to write the forcing function as follows....(3.4.29) where,...(3.4.30) Hence to eliminate secular term from Eq. (3.4.28) one may write...(3.4.31) For a first order approximation, one may consider A to be a function of its polar form as only can write A in

50 ... (3.4.32) Substituting ( ) in (3.4.31) one can write... (3.4.33) Or,... (3.4.34)... (3.4.35) Separating the real imaginary parts one may write... (3.4.36)...(3.4.37) The first order approximation solution can be written as...(3.4.38) METHOD OF MULTIPLE SCALES APPLIED TO FORCED VIBRATION In this lecture the method of multiple scales is applied to a forced vibration system. One may follow similar procedure as in the previous lecture. But in this case additional secular terms will arise which will give different resonance conditions. In the following example the primary resonance condition for the

51 forced Duffing equation is illustrated. It may be noted that unlike linear system in case of nonlinear system multiple equilibrium solution will arise. Example : Find the frequency-amplitude relation for primary resonance condition for the forced Duffing equation.... (3.5.1) Solution For primary resonance condition, the frequency of external excitation Ω should be nearly equal to that of natural frequency of the system. Hence, to show the nearness of Ω to parameter σ, by using book-keeping parameter it can be written that, one may use a detuning...(3.5.2) Now exping u using the book-keeping parameter different time scales one may write... (3.5.3) Now substituting Eqs.(3.4.3) (3.5.3) in Eq. (3.5.1) separating the like power of ε, following equations are obtained.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,(3.5.4)...(3.5.5) The solution of Eq. (3.5.4) can be given by... (3.5.6) Substituting (3.5.6) in Eq. (3.5.5) one obtains...(3.5.7) Or,...(3.5.8)

52 In Eq. containing (3.5.8), the term containing is a secular term term is a nearly secular term as it will approach to a secular term when. To have a bounded solution these two terms should be eliminated by imposing the following condition....(3.5.9) Substituting in Eq. (3.5.9) separating the real imaginary parts, the following first order differential equations are obtained....(3.5.10)...(3.5.11) One may write these two equations in their autonomous form by substituting equations are. The resulting...(3.5.12)...(3.5.13) Equations (3.5.12) (3.5.13) are known as the reduced equations can be used for finding the response stability of the system. By analytically or numerically solving these equations one may obtain the amplitude phase of the response of the system. The first order response of the system can be given by...(3.5.14) It may be noted that for steady state, the amplitude phase of the system do not depend on the time hence the time derivative terms i.e., write should be equal to zero. Hence, for steady state one can... (3.5.15)

53 ...(3.5.16) Now squaring adding Eqs. (3.5.15) Eq. (3.5.16), the following closed form equation is obtained....(3.5.17) It may be noted that this equation is a 6th order polynomial in terms of a is quadratic in terms of detuning parameter σ. Hence, solving the quadratic equation, one can obtain the following relation for the frequency response curve....(3.5.18) Harmonic balance method is the most commonly used method to study the nonlinear vibration problems. Here, the response of the system is assumed in terms of a Fourier series using this expression in the governing differential equation separating the coefficients of the harmonic terms one can obtain the unknown coefficients frequency amplitude relation of the nonlinear system. One may assume the response in the following form....(3.6.1) Then substituting (3.6.1) in the governing equation equating the coefficient of each of the lowest M + 1 harmonics to zero, one obtains a system of M + 1 algebraic equations relating ω the these equations are solved for ω in terms of. Usually. The accuracy of the resulting periodic solution depends on the value of the number of harmonics in the assumed solution. The method is illustrated using the following examples. Example 3.6.1: Find the expression for frequency amplitude relation for the single degree of freedom system represented by Duffing's equation with both quadratic cubic nonlinearities using harmonic balance method by taking one, two three terms in the expansion of the Fourier series.... (3.6.2) Solution : Taking only one term expansion, from equation (3.6.1) one has...(3.6.3)

54 Substituting equation (3.6.3) into equation (3.6.2) yields,... (3.6.4) Equating the co-efficient of to zero, one obtains...(3.6.5) which for small becomes...(3.6.6) Comparing (3.6.6) with (3.2.14) we conclude that only part of the nonlinear correction to the frequency has been obtained. Now taking two terms following Nayfeh Mook (1979) by putting... (3.6.7 ) in (3.6.2) one obtains...(3.6.8) Equating the constant term (terms with magenta colour) the coefficient of colour) to zero, one obtains the following equations. (terms with blue...(3.6.9)...(3.6.10) When is small, neglecting terms containing, from Eqs. (3.6.9) (3.6.10) one can write...(3.6.11)

55 ...(3.6.12) Hence,... (3.6.13) It may be noted that this expression for frequency is not same as that we obtained by using method of multiple scales or L-P method. Hence to obtain a consistent solution by using the method of harmonic balance, one need either to know about the solution a priori or one has to take many terms in the Fourier series make a convergence analysis. Otherwise one might obtain an inaccurate approximation. Using two harmonic terms...(3.6.14) where. Substituting Eq. (3.6.14) in Eq. (3.6.2) equating the coefficient of the constant part, coefficient of equations. equal to zero, one obtains the following Constant terms... (3.6.15) Coefficient of... (3.6.16) Coefficient of...(3.6.17) Assuming to be small, one can observe from Eqs. ( ) that. So neglecting the terms of (3.6.15) as follows. higher order terms one can write are of the order of in terms of form Eq.... (3.6.18)

56 Now multiplying equation. in Eq.(3.6.16) subtracting it from Eq.(3.6.17) one obtains the following (3.6.19) Substituting the expressions for in Eq. (3.6.16) one obtains...(3.6.20) By substituting, this expression is same as that obtained by applying method of multiple scales Lindstedt Poincare' technique. Now substituting the expression of in Eq. (3.6.14) one obtains...(3.6.21) Though the harmonic balance method is the most commonly used method for analyzing the nonlinear structural vibration, it has several disadvantages. First the formulation is very tedious not only for a multi degree of freedom nonlinear system but also with higher harmonic terms taken into account. Second, to obtain a consistent solution, one needs to know a priori which harmonic terms to be included in the analysis. Third a separate analysis is required to study the stability of the system.

57 his is one of the techniques for variation of parameters there are many techniques such as van der Pol's technique, Krylov-Bogoliubov, the generalized method of averaging, the Krylov-Bogoliubov-Mitropolsky technique, etc.(nayfeh 1973). A detailed study of this method can be found in the book of Nayfeh (1973). Few of these techniques are described here with examples. Ver Pol's Technique Consider the equation... (3.7.1) Assuming ε to be small the frequency of external excitation nearly equal to the natural frequency which can be written by using a detuning parameter σ as follows... (3.7.2) Initially the solution of Eq. (3.7.1) can be assumed to that of the equation considering ε equal to zero but with variable coefficient as given below....(3.7.3) Here Hence, are assumed to be slowly varying function of time.. Differentiating Eq.( 3.7.3) twice one obtains...(3.7.4)...(3.7.5) Substituting Eq. ( ) in Eq. (3.7.1) one obtains (3.7.6 ) or,...(3.7.8) Now using,

58 ,, in Eq. (3.7.8) keeping in mind that of...(3.7.9) are then equating the coefficient to zero one obtains the following two equations....(3.7.10)...(3.7.11) For steady state,. Using Eq. (3.7.2) in the above equations, one may obtain...(3.7.12) Taking the equilibrium solution to be reduced to the following equations. writing, Eqs. (3.7.10) (3.7.11)...(3.7.13)... (3.7.14) squaring adding Eqs. (3.7.13) (3.7.14) one obtains...(3.7.15)...(3.7.16) This is the frequency response equation of the system governed by van der Pol's equation. For forcing function, Eq. (3.7.16) reduces to...(3.7.17) Krylov Bogoliubov Technique Let us consider a general equation

59 ...(3.7.18) According to this method, one may assume the solution of this equation same as the solution of the linear equation by substituting, but in this case the constant terms are assumed as function of time. So the solution of this equation can be written as... (3.7.19) Also it is assumed that...(3.7.20) Differentiating (3.7.19) one may write... (3.7.21) Differentiating (3.7.20) one may write...(3.7.22) Substituting, compairing Eq. (3.7.20) Eq. (3.7.21)... (3.7.23) Also, from Eq. (3.7.18) Eq. (3.7.22) (3.7.24) Or, From...(3.7.25) Eq. (3.7.23) Eq. (3.7.23) - Eq. (3.7.25) Carrying out the operation Eq. (3.7.25) yields...(3.7.26) Carrying out the operation Eq. (3.7.23) + Eq. (3.7.25) yields

60 ... (3.7.27) For small ε, are small; hence a β vary much more slowly with time t than In other words, a β hardly change during the period of oscillation of.. Hence, one may average the equations (3.7.26) (3.7.27) over the period T. Considering a, β, to be constant during this averaging one obtains the following equations.... (3.7.28) Similarly... (3.7.29) Hence, from equation (3.7.28) (3.7.29) one can write the averaged equations as follows....(3.7.30)... (3.7.31) It may be noted that the above two equations are obtained by multiplying to the forcing function ( f ) integrating it from 0 to substitute. But in the forcing function one should. Example 3.7.1: Let us apply Krylov Bogoliubov Technique to Duffing equation with cubic nonlinearity. Solution:

61 Here the equation is given by...(3.7.32) Hence,...(3.7.33) Using equation (3.7.30) (3.7.31) one can write... (3.7.34)... (3.7.35) One may use the following Matlab code to find the integration syms p int(cos(p)*(3*cos(p)+cos(3*p)),0,2*pi) (ans = 3*pi) Or instead of writing in terms of symbolically using Matlab as follows., one may directly integrate * From Eq. (3.7.34) (3.7.35) one may obtain Hence, using equation (3.7.19), the solution of this equation can be given by...(3.7.36) So the frequency of oscillation of the system is. But it may be noted that this frequency expression is not correct. Hence one has to use better approximation to obtain the accurate solution. In the next lecture generalized averaging the KBM method will be described which give better results than the KB method.

62 STABILITY ANALYSIS OF FIXED POINT RESPONSE In the previous lecture, we have learned about different perturbation methods to obtain the solution of the nonlinear differential equations of motion. Unlike linear system, where only one solution exists, in nonlinear case one may observe multiple solutions. Also, the solution may be a fixed-point response; it may be periodic, quasi-periodic or chaotic in nature. Out of these multiple solutions, some solution may be stable, other may be unstable. A stable solution is one, which remain bounded when the response is slightly perturbed. If the response grows with slight perturbation, the response is unstable. In this module the stability bifurcation analysis of different types of responses will be discussed. In this lecture the stability of fixed point response will be discussed. Let us start with a simple linear system by considering a simple spring-mass system. The governing equation in this case is...(4.1.1) If no external force is acting on the system, the system response is periodic with a frequency taking the amplitude depends on the initial displacement, if or velocity. Now, the system will have a bounded solution

63 when, the of response will grow. Taking numerical example. Figure shows the time response phase portrait. Fig : Time response Phase portrait using Eq. (4.1.1) with Bifurcation analysis of fixed point response In nonlinear systems, while plotting the frequency response curves of the system by changing the control parameters, one may encounter the change of stability or change in the number of equilibrium points. These points corresponding to which the number or nature of the equilibrium point changes, are known as bifurcation points. For fixed point response, they may be divided into static or dynamic bifurcation points depending on the nature of the eigenvalues of the system. If the eigenvalues are plotted in a complex plane with their real imaginary parts (as shown in fig 4.2.1) along X Y directions, a static bifurcation occurs, if with change in the control parameter, an eigenvalue of the Jacobian matrix crosses the origin of the complex plane. In case of dynamic bifurcation, a pair of complex conjugate eigenvalues crosses the imaginary axis with change in control parameter of the system. Hence, in this case the resulting solution is stable or unstable periodic type.

64 Fig : Variation of eigenvalues in a complex plane Static Bifurcation For a system, following two conditions to be satisfied at the static bifurcation point.... (4.2.1) The Jacobian matrix zero real parts at has a zero eigenvalue while all of its other eigenvalues have non. Hence the point is nonhyperbolic. There are mainly three different types of static bifurcation points viz., saddle-node, pitch-fork transcritical bifurcation. One can distinguish saddle-node bifurcation from other bifurcation point by finding which is a vector of first partial derivatives of the components of F with respect to the control parameter µ then by constructing a matrix. Stability analysis of Periodic response In this lecture stability analysis of periodic response will be discussed. As pointed out in the previous lecture, after Hopf bifurcation, a periodic response occurs in the system. Also, in many resonance conditions, the system yields a periodic response. Hence, one should study the stability bifurcation of the periodic response. Let us consider the forced Duffing equation as given below....(4.3.1) Using Method of multiple scales, the solution of this equation for the super harmonic resonance condition ( ) can be written (Nayfeh Mook 1979) as

65 ...(4.3.2) Here a γ can be obtained from the following reduced equations....(4.3.3) For steady state as are zero, Eq. (4.3.3) can be written as...(4.3.4) Here are the steady state solutions. Now eliminating obtain the following equation for the frequency response curve from the above equations, one may... (4.3.5) It may be noted at this stage that though the actual solution u is periodic (refer Eq ), one may study the stability of the fixed point response given below. as discussed in the previous lecture by perturbing the solution as... (4.3.6) Substituting Eq. (4.3.6) in the reduced equations (4.3.3) one can obtained the following sets of equation. Limit cycles Bifurcation of Periodic Response In this lecture a special case of periodic response i.e., the limit cycles is briefly discussed. Also, the bifurcation analysis of the periodic response has been described. As discussed in the previous lecture, it may be noted that a solution is said to be periodic if it repeats with certain time period T. Hence, for a periodic solution x=x(t ),...(4.4.1) So, for a periodic solution given by Eq. (4.4.1) it should have the following properties. Minimum period T Form a closed orbit in phase portrait. Could be treated as a fixed point in Poincare' section

66 Figure (a) shows a periodic response The phase portrate is shown in Fig (b) where. Fig : (a)time response (b) Phase portrait Quasi-periodic Chaotic response In the previous lectures in this module we have discussed about fixed point periodic responses in this lecture we will discuss about the quasi-periodic chaotic responses. Quasiperiodic Response Consider the response of a system which contains two frequency terms as given below. This response will be periodic if the ratio between the two frequencies is a rational number if the frequencies are incommensurable i.e., if this ratio is an irrational number, then the response will be quasiperiodic. For example consider irrational number. The time response is shown in Figure In this case the ratio is which an

67 Fig : Time response of a typical quasi-periodic response Fig : (a) Phase portrait Poincare' section of the response Free Vibration of Nonlinear Conservative system In this lecture we will learn about the free vibration response of nonlinear conservative systems. Initially the qualitative analysis will be demonstrated later by using one of the perturbation methods the free vibration response of the single degree of freedom system will be illustrated using different examples. Qualitative Analysis of Nonlinear Systems Consider the nonlinear conservative system given by the equation... (6.1.1) Multiplying in Eq. (6.1.1) integrating the resulting equation one can write as

68 ...(6.1.2) Or.,...(6.1.3) This represents that the sum of the kinetic energy potential energy of the system is constant. Hence, for particular energy level h, the system will be under oscillation, if the potential energy is less than the total energy h. From the above equation, one may plot the phase portrait or the trajectories for different energy level study qualitatively about the response of the system using the following equation.... (6.1.4) It may be noted that velocity exists, or the body will move only when. One will obtain equilibrium points corresponding to or when. For minimum potential energy a center will be obtained for maximum potential energy a saddle point will be obtained. The trajectory joining the two saddle points is known as homoclinic orbit. The response is periodic near the center. \ FREE VIBRATION OF NONLINEAR SINGLE DEGREE OF FREEDOM CONSERVATIVE SYSTEMS WITH QUADRATIC AND CUBIC NONLINEARITIES. In this lecture the free vibration response of a nonlinear single degree of freedom system with quadratic cubic nonlinearities will be discussed with numerical examples. As studied in module 3, the equation of motion of a nonlinear single degree of freedom system with quadratic cubic nonlinearities can be given by...(6.2.1) Here is the natural frequency of the system are the coefficient of the quadratic cubic nonlinear terms. Also ε is the book-keeping parameter which is less than 1. Using method of multiple scales the solution of this equation can be written as...(6.2.2)

69 Using different time scales where Eq. (6.2.2) in Eq. (6.2.1) separating the terms with different order of ε one can write the following equations. Order of... (6.2.3) Order of... (6.2.4) Order of... (6.2.5) The solution of (6.2.3) can be written as....(6.2.6) Here A is an unknown complex function (6.2.4) leads to is the complex conjugate of A. Substituting (6.2.3) into... (6.2.7) Here cc denotes the complex conjugate of the preceding terms. To have a bounded solution one should eliminate the secular term hence... (6.2.8) Therefore A must be independent of. With the particular solution of (6.2.7) can be written as...(6.2.9) Substituting the expression for that from equation (6.2.6) (6.2.9) into (6.2.5) recalling we obtain

70 ...(6.2.10) To eliminate the secular terms from, we must put...(6.2.11) Using where a β are real function of real imaginary parts, one obtains... (6.2.12) where the prime denotes Here in Eq. (6.2.11) separating the result into the derivative with respect or is a constant. Now using to. As, a is a constant...(6.2.13) one may write...(6.2.14) Substituting Eq. (6.2.14) in the expressions for in Eqs. (6.2.6), (6.2.9) (6.2.2), one obtains...(6.2.15) Here... (6.2.16) This solution is in good agreement with the solution obtained using the Lindstedt-poincare' procedure. [ Nayfeh Mook, 1979] Example : Taking the,,in equation find the response of system.

71 Using Eq. (6.2.16) the variation of frequency with amplitude is shown in figure (6.2.1). Taking two values of a (viz., a =0.009 a =.029) the time response has been plotted in figure (6.2.2). It may be noted Figure 6.2.1: Variation of amplitude with frequency for Figure (a): Time response (b) Phase portrait corresponding to initial amplitude a=0.009 a=.029 By changing the quadratic nonlinear terms from.5 to 2.5 keeping all other parameter same figure (6.2.3) shows the variation of the frequency with amplitude of oscillation.