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Structural Dynamics Introduction Newton’s first law, or more particularly its corollary, is the cornerstone of static analysis: if a body is in equilibrium then the some of the forces acting on the body must be zero. Similarly, Newton’s second law is the cornerstone of dynamic analysis. Newton’s second law states that the rate of change of momentum of a body, in a particular direction, is equal to the applied force in that direction. d (mv ) = Force dt Equation (1) If the mass of the body is constant then this equation simplifies to m dv = Force dt or simply mass × acceleration = Force D’Alembert introduced the concept of inertial force, inertial _ force = − mass × acceleration so that the dynamic equation given above could be formulated as an equilibrium equation i.e. Force + inertial _ force = 0 Force − mass × acceleration = 0 which is Clearly these formulations are identical. Simple Harmonic Motion Consider the behaviour of the simple dynamic system shown in Figure 1. m k Figure 1. If the mass is given an initial displacement from its static position and then released. Then the mass will oscillate about its equilibrium position. x m k Figure 2. If the spring is linear then the force applied to the mass by the spring is; − kx = Force Equation (2) and hence, from Newton’s second law, the acceleration of the mass is &x& = − k x, m which has the form; &x& = −ω 2 x ω= where Equation (3) k m Equation (4) Thus the acceleration is a constant times the displacement. If the vertical displacement of the mass is plotted against time then the result is as shown in Figure 3. 2 pi A wt Figure 3. The displacement of the mass can is described by the expression x = A sin ωt where ω is the circular frequency (units of radians/sec). Equation (5) Note: A is the amplitude (maximum displacement) The natural period of the oscillation is τ= 2π = 2π m k Equation (6) 1 2π k m Equation (7) &x& + ω n 2 x = 0 Equation (8) ω and the natural frequency of the oscillation fn = 1 τ = The general solution to the differential equation is given by x= x& (0) ωn sin ω n t + x(0 ) cos ω n t Equation (9) Energy Formulation The fundamental differential for the dynamic system in the previous example was obtained directly. Other methods of formulating the differential equations of dynamic equations exist. One potential approach is the energy approach. A conservative system is one in which energy is not lost. Un-damped elastic systems can often be considered as conservative systems. In such a system the total energy does not change. Thus, if T is the total kinetic energy of the system and U is the potential energy of the system then d (T + U ) = 0 Equation (10) dt In particular, if the reference position for the potential energy is chosen such that U=0 at the static position then the equation can be formulated as T + U = Constant and hence, Tmax = U max Equation (11) Multi Degree of Freedom Systems The previous section explored a single degree of freedom system. However, most dynamic systems cannot be described in terms of a single degree of freedom. However, by using D’Alembert’s principle we can formulate the dynamic equation by developing equivalent static equilibrium equations that incorporate inertial forces. Thus, we obtain the general equilibrium equation for a multi-degree-of-freedom system, which is similar in form to the standard static equation; [K ]{x} = {F } Equation (12) Where the forces acting on the body include the inertial forces, thus [K ]{x} = { f }− [M ]{&x&} Equation (13) A general system will include damping, which is typically taken to be dependent on {x&}. Thus, the general dynamic equation for a linear multi-degree-of-freedom system is described by the equation; [M ]{&x&}+ [C ]{x&}+ [K ]{x} = { f } Equation (14) Where both the forcing function { f (t )} and the displacements {x(t )} are functions of time. Before considering how this dynamic equation can be solved let us consider the various terms, or at least some of the components of the terms. The matrix [K ] is the stiffness matrix of the structure, hence, it is completely defined by the structure and should be relatively easy to calculate, either by hand or by extracting the stiffness matrix from a structural analysis package having constructed an appropriate structural model. The mass matrix [M ] is also defined by the mass of the components of the structure and again can be calculated either by hand, which is relatively easy if a lumped mass model is used, or taken from a structural analysis package. The damping matrix [C ] is more difficult to quantify and for this reason, and because damping will generally tend to limit the response of the structure, damping is often omitted and the un-damped response sought. Finally, it is assumed that the forcing function { f (t )} is known. However, many useful analyses can be carried out for the particular case where there is no external forcing function. [M ]{&x&}+ [C ]{x&}+ [K ]{x} = 0 Equation (15) In this case the free vibration of the structure is considered. A number of approaches to tackling the fundamental equation for a multi-degree-offreedom linear system will now be considered. These approaches are not necessarily independent. Time Stepping The fundamental equation, Equation (14), [M ]{&x&}+ [C ]{x&}+ [K ]{x} = { f } gives a snapshot in time. At a particular instant in time the forces, real and inertial must balance. However, what is generally sought are expressions for {x(t )}. There a many approaches to getting such results from the fundamental equation but perhaps the most versatile, robust and conceptually simple approach is that of time stepping. Put simply, if we know the displacements, velocities and accelerations of the system at a particular instant then we can calculate the displacements, velocities and accelerations at a short time interval later. This is most easily seen by referring to a single degree of freedom system. If the velocity, acceleration and displacement of the mass a simple system such as that shown below are known at time t. At a time t + ∆t later, the new displacement of the system can be described by a Taylor series x(t + ∆t ) = x(t ) + x& (t ) ⋅ ∆t + &x&(t ) ⋅ ∆t 2 ∆t 3 + &x&&(t ) +K 2! 3! Equation (16) Expressed simply, if we know our position, velocity and acceleration at an instant then we can make a very good guess at where we will be an instant later. In the general multi-degree-of-freedom case the fundamental equation can be rearranged into the following form. {&x&} = [M ]−1 [{ f }− [C ]{x&}− [K ]{x}] Equation (17) Thus, if we know the velocity and displacements at an instant we can calculate the accelerations. Knowing the accelerations at time t we can then predict the velocity and displacement of the system at an instant ∆t later. The crudest approach would be to assume that the force doesn’t change during the time-step, a better approximation assumes the force to vary linearly during the time step, with more sophisticated algorithms modelling the variation of the acceleration more accurately. Runge-Kutta One popular algorithm for undertaking this time stepping is the Runge-Kutta method. By introducing a dummy variable y = x& , Equation (17) can be expressed at two equations, namely x& = y y& = f ( x, y, t ) Equation (18a) Equation (18b) In the neighbourhood of xi and yi, x and y can be expressed using Taylor expansions. Thus, letting h = ∆t d 2 x h2 dx x = xi + h + 2 +K dt 2 dt i i Equation (19a) d 2 y h2 dy y = yi + h + 2 +K dt i dt i 2 Equation (19b) Truncating the series after the first derivative but using an average value for the first derivative gives, dx x = xi + h dt Average _ over (i −i +h ) Equation (20a) dy y = yi + h dt Average _ over ( i −i + h ) Equation (20b) Clearly truncating the series this early is not a great idea in general unless the values for the first derivatives are good averages for the average value during the time step, i.e. not just the value at the start of the time step. Remember too that y is in fact x& and dy thus the average value of is in essence an average value for &x& . dt dx dy Good values for the averages of and can be obtained by using Simpson’s dt dt rule. Thus, 1 dy dy dy dy = + + 4 dt Average _ over ( i −i + h ) 6 dt ti dt ti + h 2 dt ti + h Equation (21) dy is evaluated The Runge-Kutta procedure is very close to this except that dt ti + h 2 twice, the second time using an updated estimate for the force. The procedure is best explained using the following table taken from Theory of Vibration with applications by W.T. Thomson, published by Unwin Hyman. Table 1. Runge-Kutta – From Theory of Vibration with application by W.T. Thomson Using the Runge-Kutta scheme x and y are evaluated four times during the time-step. T1 at the start of the time-step, T2 and T3 half way through the time-step, and T4 at the end of the time-step. At T1 the forces f (T1 , X 1 , Y1 ) , which govern the accelerations F1, dy i.e. the values, are evaluated on the basis of the initial values of xi and yi. At dy time T2 the xi + h and yi + h values half-way through the time-step are calculated based 2 2 on the accelerations at the start of the time-step. At time T3 the values of xi + h and 2 yi + h 2 are re-evaluated using the accelerations based on the displacements and velocities calculated at T2. Finally, at T4 the value of xi + h and yi + h are calculated using accelerations based on the T3 accelerations. Finally, the values of xi + h and yi + h are taken to be h [Y1 + 2Y2 + 2Y3 + Y4 ] 6 h = yi + [F1 + 2 F2 + 2 F3 + F4 ] 6 xi + h = xi + Equation (22a) yi + h Equation (22b) Note: The Runge-Kutta is not unconditionally stable and the output from this routine has to be vetted. Nevertheless, it tends to give good results and can be used for non-linear analysis. Furthermore, this method is self-starting. In general, the duration of the timestep should be limited to less than one tenth of the period of oscillation. Natural Response – Free Vibration The homogenous equation, Equation (15) [M ]{&x&}+ [C ]{x&}+ [K ]{x} = 0 describes the response of a linear system in the absence of external forces. One trivial solution to this system of equations is {x}, {x&} and {&x&} equal to zero. However, there are other solutions. The natural modes of the system are the patterns of oscillation that the structure will tend to oscillate in if disturbed. The natural frequencies of a system are very important. If an external loading function has a periodic component matching one or more of the natural frequencies then resonance may occur with the effect that the amplitude of the oscillation will increase until it is either restricted by damping, or the structure fails. See the video clip on the Tacoma Narrows collapse in the Advanced Structures directory) The natural modes of vibration and their corresponding frequencies can be found by solving the equation [[K ] − ω [M ]]{x} = 0 2 (or equivalently [A] − λ [I ] = 0 ). [M ]−1 [K ] − ω 2 [I ] = 0 , which Equation (23) is sometimes written as For the non-trivial case where {x} ≠ 0 this equates to finding the values for ω 2 for which the determinate of [K ] − ω 2 [M ] = 0 Equation (24) Once the values of the natural frequencies, ω s, have been found the corresponding mode shapes (eigenvectors) can be obtained by substituting the known value of ω back into Equation (23). For example, for a three degree of freedom system and the fundamental natural frequency (smallest value of ω 2 ) Equation (23) becomes, k11 − ω12 m11 2 k 21 − ω1 m21 k31 − ω12 m31 2 k13 − ω1 m13 x1 0 2 k 23 − ω1 m23 x2 = 0 2 k33 − ω1 m33 x3 0 k12 − ω1 m12 2 k 22 − ω1 m22 2 k32 − ω1 m32 2 This equation has no unique solution because the right hand side equals zero. The mode shape gives the relative magnitudes of the displacement at the different degrees of freedom. Therefore it is necessary to seed the problem by setting the magnitude of one of the displacements to 1. Thereafter it is possible to solve for the other magnitudes uniquely. Thus for example, 1 k − ω 2 m 1 21 21 2 k31 − ω1 m31 0 x1 1 k 23 − ω1 m23 x2 = 0 2 k33 − ω1 m33 x3 0 0 k 22 − ω1 m22 2 k32 − ω1 m32 2 2 Equation (25) Orthogonal Properties of Eigenvectors The eigenvectors of a system are orthogonal with respect to both the mass and stiffness matrices. If the eigenvectors of a system are φ1 , φ 2 , φ3 , K , φ n then, {φi }T [M ]{φ j } = 0 for all i ≠ j Equation (26a) {φi }T [K ]{φ j } = 0 for all i ≠ j Equation (26a) and Modal Analysis The displaced shape of a structural system can be described in term of its natural mode shapes (eigenvectors). Thus, if the mode shapes are {φ1 }, {φ 2 }, {φ3 }, K , {φ n } then the displaced shape {x} can be written in the form {x} = Y1{φ1} + Y2 {φ2 } + Y3 {φ3 } + K + Yn {φn } Equation (27a) where Y1 , Y2 , Y3 , K , Yn are scalar values, or written in matrix notation {x} = [φ ]{Y } Equation (27b) where [φ ] = [{φ1} {φ2 } {φ3 } φ1 ( x1 ) L{φ n }] = M φ1 ( xn ) φ 2 ( x1 ) M φ ( x ) 2 n φ n ( x1 ) φ3 ( x1 ) M L M φ ( x ) φ ( x ) n n 3 n Since, the displacement of the system can be expressed in terms of the mode shapes the response of the system in time can also be expressed in term of the mode shapes. Thus, x(t ) = [φ ]Y (t ) Equation (28) Of course versions of Equations (26) and (27) could be constructed using any set of independent vectors. However, using expressing the response of a system in terms of its eigenvectors has two main advantages. Advantage 1 – Problem Reduction The eigenvectors of a structural system define the characteristics of the system. Furthermore, the eigenvector associated with the 1st natural frequency (i.e. the lowest frequency) is the most indicative of the systems characteristics, followed by the eigenvector associated with the 2nd natural frequency (2nd lowest frequency) and so on down to the last eigenvector. In many cases, truncating Equations (26) or (27) so that the response of the structure is described in terms of a limited number of eigenvectors does not have a large effect on the overall accuracy of the representation. The advantage is that a problem with a large number of degrees of freedom can be reduced to a problem with a very small number of degrees of freedom. This can be particularly beneficial where techniques such as time-stepping, which may be computationally expensive, are involved. Advantage 2 – Separation of the modal responses Consider Equation (14) for the particular case where the damping is zero [M ]{&x&} + [K ]{x} = { f } Equation (29) This equation can be rewritten in terms of the eigenvectors. [M ][φ ]{Y&&}+ [K ][φ ]{Y } = { f } Equation (30) If all the terms of Equation (30) are now pre-multiplied by [φ ] particularly interesting. −1 [φ ]T [M ][φ ]{Y&&}+ [φ ]T [K ][φ ]{Y } = [φ ]T { f } the result is Equation (31) Taking Equation (31) term by term. The first term contains the product mG1 0 [φ ]T [M ][φ ] = [M G ] = 0 0 mG 2 0 0 0 O M L mGn Equation (32) which is referred to as the generalised mass matrix. This matrix is diagonal (i.e. it has the property that it has non-zero entries on the leading diagonal only). The generalised stiffness matrix, k G1 0 T [φ ] [K ][φ ] = [K G ] = 0 0 kG 2 0 0 0 O M L kGn Equation (33) Has the same property. This property arises from the orthogonal properties of the eigenvectors. In a similar manner the pre-multiplying the forcing function by [φ ] yields a modal force vector. f G1 f [φ ]T { f } = {FG } = G 2 Equation (34) M f Gn T Thus Equation (31), which has the same solution as Equation (30), is in a form where the response of each mode is independent of all the other modes. For example, the response of the first mode of the system will be governed by the equation, mM 1Y&&1 + k M 1Y1 = f M 1 Equation (35) Equation (35) describes the response of a single degree of freedom system. Therefore identifying the response of the structure is equivalent to summing the response of n (or less than n) single-degree-of-freedom systems. The overall response in the natural coordinates {x} can be obtained from the modal response {Y } via Equation (28).