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 ???? 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).
=+3. Find the static equilibrium positions of the mass.
=+2. Find the kinetic and potential energies of the system and obtain the equations of motion using the Lagrange equations.
=+1. Consider the free body diagram of the mass and write the Newton equations describing the dynamics in the x and y directions. Apply then the virtual work principle, namely project the dynamic
=+Problem 2.9 A mass m is constrained to move on an ellipse (see Figure 2.18.b). The ellipse is described by:x2 a2 + y2 b2 = 1 and is in a horizontal plane (no gravity is acting on the mass). The
=+Figure 2.18 Selected exercises.Undamped Vibrations of n-Degree-of-Freedom Systems 145
=+e. N-storey building model under seismic excitationf. Transfer relationships for spring-mass systems ky kx ky ba ym mx mxθR Ik, a0 ax e em1 m m2θ1θθ2k????????qN q1 q0 km m0− f0, q0 f1,
=+a. A two spring-mass systemb. Mass on an ellipsec. Double pendulum with eccentric springd. Centrifugal pendulum(i) (ii)(iii)(iv)
=+Consider the one-dimensional system made of two masses and two linear springs depicted in Figure 2.18.a. As degree of freedom we choose the displacement of the upper mass and the relative
=+where is a given length. A linear spring with stiffness k is attached to the mass and fixed to the point (0, 0). The spring has a zero natural length. The gravity acts in the direction −y.Prove
=+In the system described in Figure 1.14.a a mass moves on a parabolic curve described by the equation:y(x) = x2 − 4
=+– To develop the system equations of motion in Lagrange form.
=+– To express the potential and kinetic energies of the system.
=+– To express the Cartesian velocities of both masses.
=+– To express the position coordinates ????ik of both masses in terms of the generalized coordinates ????1 and ????2 as displayed on Figure 1.14.b.

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