Questions and Answers of Fundamentals Of Vehicle Dynamics And Modelling

=+2. to apply the boundary conditions to the general solution in order to get an equation governing the frequency parameter ???? = ω2m2 EA as a function of the mass ratio ???? = M m.
=+1. to write down the equation for vibration motion of the system and the associated boundary conditions.
=+Figure 4.56 Graphical determination of the roots ????s of a clamped– free beam with tip mass( M m = 0.25).
=+Problem 4.2 Let us consider a uniform bar of length with stiffness and mass characteristics(EA, m) clamped at one end and with a mass M attached at the tip (similar to what was depicted for a
=+in which case the characteristic values ????s are located at their intersection. Figure 4.56 shows the determination of the characteristic values ????s for ???? = M m = 0.25. The intersections of
=+4. The roots of Problem (P4.1.a) can be obtained for example by plotting the functions:f1(????) = ???????? and f2(????) = 1 + cos ???? cosh ????−cosh ???? sin ???? + cos ???? sinh ????(P4.1.j)
=+beam since the inertia of the tip mass becomes infinite and thus blocks the displacement. It can be verified that the numerator of Equation (P4.1.i) is nothing else than the eigenvalue equation
=+3. The numerator of Equation (P4.1.i) corresponds to the eigenvalue Equation (E4.5.c)obtained for the clamped– free beam. Therefore when ???? = 0 the solution obtained corresponds to the
=+The characteristic equation associated to (P4.1.g) can be put in the form:???????? = s1(????)s2(????) − c2 1(????)s1(????)c2(????) − c1(????)s2(????) (P4.1.h)or, in terms of trigonometric and
=+to which corresponds the homogeneous system:[ s1(????) c1(????)c1(????) + ????????s2(????) s2(????) + ????????c2(????)] [C D]= 0 (P4.1.g)
=+and applying next the boundary conditions (P4.1.c) provides the set of equations:????2 (Cs1(????) + Dc1(????))= 0 (P4.1.e)????3 (Cc1(????) + Ds2(????))= −????????4 (Cs2(????) + Dc2(????)) (P4.1.f)
=+2. Expressing the solution in terms of Duncan functions (4.248) and applying the boundary conditions (P4.1.b) yields:????(????????) = Cs2(????????) + Dc2(????????) 0
=+4. to show how the roots (????s, s = 1, ∞) can be obtained graphically as functions of the ????parameter.
=+3. to verify the correctness of the result obtained when ???? → 0 and when M → ∞.
=+2. to apply the boundary conditions to the general solution in order to get an equation governing the frequency parameter ???? such that ????4 = ω2m4 EI as a function of the mass ratio ???? = M
=+1. to write down the equation for vibration motion of the system and the associated boundary conditions in nondimensional form.
=+Let us consider a uniform beam of length with stiffness and mass characteristics (EI, m) clamped at one end and with a mass M attached at the tip (Figure 4.55). You are asked:
=+– Compute (using a numerical software) the complex eigenvalues and the complex eigenmodes of the damped system. Compare the results with the approximation one would obtain using the assumption
=+– Write the system matrices of the system and determine the undamped modes and frequencies. Compute the associate modal damping ratios assuming that c = 0.06√km.
=+Consider the one-dimensional problem with two degrees of freedom depicted in Figure 3.20.a.
=+– For the damped system, find the damped modes and associated eigenvalues in the case k1 = k2 = k.
=+– For the undamped system, compute the eigenfrequencies and eigenmodes when k1 ≠ k2 and when k1 = k2 = k.
=+Let us consider now the mass suspended in the horizontal and vertical directions by springs of stiffness k1 and k2 respectively (Figure 3.20.b). A damper is attached to the mass at an angle of 45
=+– To an ordinary product in the Laplace domain corresponds a convolution product in the time domain:−1 [F1(s)F2(s)]= ∫t 0f1(????)f2(t − ????)d
=+To that purpose, it is advised to make use of the following identities:– Laplace transform of an exponential with negative real part:[e????t] = 1 s − ????ℜe(????) < 0
=+Obtain through Laplace transformation the response to the normal equation:{????̈ + 2????ω0????̇ + ω2 0???? = ????(t)????(0) = ????0, ̇ ????(0) = ????̇0(P3.3.a)in the form:????(t) = ????0d(t)
=+Provide the explicit expression of the coefficients ckr. Demonstrate that their knowledge allows determining the system eigenmodes.
=+Problem 3.4 Develop the spectral expression of response to initial conditions (q0, q̇ 0) of a damped system recast in first-order form Ar + Bṙ = 0. Show that the response at a single DOF can be
=+where the dynamic admittance matrix is now written as an expansion in terms of modal damping ratios ????s, and modes and frequencies of the associated conservative system
=+with ????̃s defined as in (3.80). This expansion results from a decomposition in the basis of the complex conjugate modes z(s) of the damped system, with the associated eigenvalues ????s.Show
=+Problem 3.5 Consider the dynamic influence matrix (or FRF) H(ω) as given in (3.82):H(ω) = ∑n s=1{1(iω − ????s)z(s)zT(s)????̃s+1(iω − ????s)z(s)z T(s)????̃s}(P3.5.a)
=+Show that if such a damping model is used even for large modal damping ratios (which then might not be an accurate representation of reality), the modes of the undamped system are still equal to
=+Problem 3.6 Let us assume that the viscous damping matrix of a model is given by the modal expansion (3.21). As explained in Section 3.1.3 such a damping model is a good approximation of damping in
=+Problem 3.12 Apply the time-marching scheme developed in Exercise 3.11 to the system of Figure 3.1 (with c = 0.1) released from an initial position corresponding to the application of a static load
=+You are asked to apply (P3.11.a) to Equation (3.3) recast in the first-order form:M????̇ + C???? + Kq = p(t)ẋ = ???? (P3.11.b)and show that it leads to a time-marching scheme described by the
=+Problem 3.11 It will be shown in Chapter 7 that the simplest integration scheme with adequate stability and accuracy properties to time integrate a first-order differential equation ẋ = f(x, t)
=+Problem 3.10 Using the results of Exercises 3.8 and 3.9, develop the expression of the response under arbitrary excitation as described by Equation (P3.3.a) in Exercise 3.3.
=+208 Mechanical Vibrations: Theory and Application to Structural Dynamics Deduce from it the impulse response h of the system, defined as the solution of:ḧ + 2????ω0ḣ + ω2 0h = ????(t)where
=+Problem 3.9 Repeat Exercise 3.8 to get the response to a nonzero initial velocity:{????̈ + 2????ω0????̇ + ω2 0???? = 0????̇(0) = ????̇0
=+– Develop the final expression (in real form) of the response in terms of ???? and ω.
=+– Compute the coefficients Ai in order to match the initial condition.
=+– Solve the associate eigenvalue equation to compute the ????i. Observe that they are of the form ????1,2 = −???? ± iω.
=+– Assume a general solution of the form ∑i Ai e????it.
=+Problem 3.8 Consider the normal Equation (3.18), which is also the equilibrium equation of a single DOF system. Develop its solution expressing the response to a nonzero initial displacement{????̈
=+Figure 3.21 A schematic representation of a wing-fuselage assembly (3-dof free-free system).
=+– Draw a Nyquist plot of the transfer functions q1 f2 and q2 f2 over the frequency range[0.5–10] Hz.q1 km M = 8 m mk q2 q3
=+– Assuming that the elastic eigenfrequencies match quite well the experimentally measured ones and that their damping coefficients have been measured to be ????1 = 2%, ????2 =5%, build a damping
=+advantage of the system symmetry, making use of orthogonality to get the mode shapes and of the Rayleigh quotient to compute the eigenfrequencies).
=+m = 2 Hz, determine the eigenfrequencies and mode shapes of the system (hint: this can be done most easily by simple reasoning if taking
=+mass (a simplified representation of the bending stiffness of the wing). The system possesses a rigid body mode for which the damping is assumed to be zero. You are asked to:– For a wing
=+Let us consider the free-free system of Figure 3.7, which could be representative of the fundamental behaviour of bending vibration for an airplane in flight. Take M = 8m as the fuselage mass, m
=+3. Normalize the eigenmodes in the form:P = Wdiag⎛⎜⎜⎜⎝1√|????T(r)A????(s)|⎞⎟⎟⎟⎠to compute the normalized matrices J = PTAP and ???? = PTBP
=+2. Concentrate next your analysis to the stability zone ???? > ω2 by assuming a sufficiently large value of ????. Compute the projection of the system matrices on the eigenmodes by forming the
=+???? < ω1 and ???? > ω2, while a negative eigenpair appears for ω1
=+1. Observe that all eigenvalues ????r have multiplicity 2 and are all positive for rotating speeds
=+148 Mechanical Vibrations: Theory and Application to Structural Dynamics using Matlab® or equivalent numerical software to get the matrices of eigenvalues and eigenvectors ???? and W.
=+Problem 2.20 Generalize the formula (2.115) to integrate a normal equation (2.105) when it corresponds to a rigid body mode (hint: apply l’Hôpital’s rule and take the limit for ???? →
=+Find the equilibrium position and determine their stability. Compute the eigenfrequencies and eigenmodes around the stable equilibrium position(s).
=+Problem 2.19 Consider the mass sliding on an elastically supported rod depicted in Figure 1.15.e as discussed in Problem 1.10. Take:k1 = k2 = k mg = ka = 2a
=+Problem 1.9. Find the equilibrium position and determine their stability. Then, taking ???? = ????∕2 and mg = k, compute the eigenfrequencies and eigenmodes around the stable equilibrium
=+Problem 2.18 The pendulum on a nonlinear spring depicted in Figure 1.15.d was analyzed in
=+Problem 2.17 For the rotational pendulum of Figure 1.15.b described in Problem 1.7, compute the equilibrium positions and determine their stability. Then compute the eigenfrequencies and
=+Problem 2.16 In Problem 1.6 of a sliding mass on a massless rod is considered in a gravity field (Figure 1.15.a). The generalized coordinates are taken as ???? and . Find the equilibrium positions
=+3. Compute the eigenfrequencies and eigenmodes around the stable configuration assuming m1 = m2 = m and 1 = 2 = .
=+2. Write the linearized free vibration equations and show that, for the stable equilibrium position, they are equivalent to Equations (P2.3.d) obtained for the relative angles.
=+1. Find the equilibrium positions and discuss their stability.
=+Problem 2.15 Consider the double pendulum when taking the absolute angles as degrees of freedom (see Problem 1.8 and Figure 1.15.c.).
=+Problem 2.14 For the model approximating a beam as constructed in the solved Problem 2.5, consider that N = 2, namely that the model is consisting of only two segments. Calculate by hand the
=+6. Using the results at intermediate steps, determine the eigenmodes for the case f0 = f4 =0.
=+5. Determine the eigenvalues of the system for all possible boundary conditions. Validate your results.
=+Figure 2.18.f.iv.Undamped Vibrations of n-Degree-of-Freedom Systems 147
=+4. Use the transfer matrix obtained at step 3 to express the transfer matrix of the system of
=+3. Making use of the results from steps 1 and 2, construct the transfer matrix of a spring-mass system (Figure 2.18.f.iii).
=+2. Express the transfer matrix of a vibrating mass (Figure 2.18.f.ii).
=+1. Express the transfer matrix of a spring (Figure 2.18.f.i).
=+2.7. As an application, you are asked to:
=+Problem 2.13 The eigenvalue equation of simply connected systems made of springs and masses can easily be obtained using the concept of transfer matrix as developed in Exercise
=+10. Assuming that the building has 5 storeys, to compute numerically the effective modal masses in order to determine which modes contribute to 85% to the effective mass.
=+9. To express the reaction force at the foundation.
=+8. To verify that the eigenvalues are:ωk = 2√ k m sin ( 2k − 1 2N + 1.????2)and to provide the explicit expression of the eigenmodes.
=+7. To express the boundary conditions, and show that this provides the analytical solution to the internal eigenvalue problem.
=+6. To verify that it obeys to a general solution of the form:qj = A sin(j???? + ????) j = 0, … N
=+5. To write in explicit form the equilibrium equation for mass j in the middle of the structure.
=+4. To formulate the internal eigenvalue problem.
=+3. To compute the stiffness and mass matrices condensed on the boundary.
=+2. To determine the static condensation matrix S at foundation level (hint: matrix S can be obtained by simple inspection in this case).
=+1. To express the equations of motion of the system in matrix form.
=+Problem 2.12 A N-storey building is modelled as a series of masses interconnected by springs as displayed on Figure 2.18.e. All masses and springs are equal except for the foundation which has a
=+For the stable equilibrium position(s) of the system, compute the eigenfrequencies and eigenmodes.
=+4. Assuming from now on that:m = I 42 e = 2
=+3. Find the equilibrium positions of the system and determine their stability
=+2. Find the linearized equations of motion of the system.
=+1. Write the kinetic and potential energies of the system 146 Mechanical Vibrations: Theory and Application to Structural Dynamics
=+Problem 2.11 In the system shown in Figure 2.18.d a rotation speed ???? is imposed on the left disk which is connected through a shaft to a second disk of rotational inertia I around its centre.
=+3. compute, for those values, the eigenfrequencies and eigenmodes around the stable equilibrium.
=+2. show that if:a0 = R e = R 2 = R m1 = m2 = m mg = kR 16 there exist two unstable and one stable equilibrium positions,
=+1. write the kinetic and potential energies of the system,
=+Problem 2.10 Consider the double pendulum where the first mass is connected to an eccentric spring as depicted in Figure 2.18.c. The spring has a stiffness k and is attached at a distance e above
=+5. Find the eigenfrequency of the system around the stable equilibrium positions.
=+4. Analyze the linear static stability of the system around the equilibrium positions (when a2kx > b2ky and when a2kx < b2ky).

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