Questions and Answers of Fundamentals Of Vehicle Dynamics And Modelling

=+– to develop a small computer program to compute the eigenvalues ω̃ 2 i of Equation (P5.8.a)versus the rotation speed parameter ????̃ 2 for a specified value of eccentricity ratio ???? over a
==+– to write a routine to numerically compute (using Gauss quadrature) the matrix K̃ g assuming that the beam element is hinged at its left-end with eccentricity e from the rotation axis (note:
=+where ????̃ and ω̃ are respectively the nondimensional rotating velocity and frequency defined by:????̃ 2 = ????2m4 EI and ω̃ 2 = ω2m4 EI
=+– to demonstrate that the eigenvalue problem admits the nondimensional form:(K̃ + ????̃ 2K̃ g − ω̃ 2M̃ )q̃ = 0 (P5.8.a)
=+Approximation of Continuous Systems by Displacement Methods 409
=+where K̃ g is a nondimensional matrix function only of the eccentricity ratio:???? = e
=+– to demonstrate that the geometric stiffness matrix of the system can be put likewise in the non-dimensional form:Kg = m????22K̃ g
=+For sake of simplicity, the system is modelled using a single beam finite element. Its stiffness and mass matrices being directly available, only the geometric stiffness matrix resulting from the
=+Problem 5.8 Let us consider again the system made of a rotating hinged blade with eccentricity e from the rotation axis flapping in the vertical direction (Figure 4.30), taking now into account
=+– to compute the amplitude of the forced response when the system is excited by a unit moment at the right end of the beam (assuming again A = I2 and L = 4).
=+and to depict the mode shapes;
=+– to compute the eigenfrequencies and eigenmodes of the system when A = I2 and L = 4
=+– to build the mass and stiffness matrices considering two elements of length for the beam and one element for the bar;
=+Problem 5.7 Let us consider the assembly of a beam and a bar as depicted in Figure 5.30.d.The beam is clamped to its left and hinged in its middle and at its right end. The bar is hinged at its
=+– Plot the eigenmodes of the system and compare them to the eigenmodes of the unsupported clamped-clamped beam.
=+– Compute the eigenfrequencies and eigenmodes of the model when k = EI∕44.
=+408 Mechanical Vibrations: Theory and Application to Structural Dynamics
=+– Build the mass and stiffness matrix of a two-finite element model for the transverse motion.
=+Perform the following computations:
=+Problem 5.6 Consider the model of a clamped-clamped beam as described in Figure 5.30.c.The beam is supported on the ground with a soil stiffness represented by a distributed translational spring
=+Figure 5.30 Unsolved finite element problems.
=+– to compute the response amplitude of the system when a harmonic torque is generated with a forcing frequency equal to ω1∕2 by applying unit loads parallel to the beam on masses M.b.2EI, m EI
=+– to compute the response amplitude of the system when a unit transverse harmonic force is applied on mass M with a forcing frequency equal to ω1∕2;
=+– to compute the eigenmodes and eigenfrequencies of the model when M = (16∕105) mand to draw the mode shapes;
=+– to write the stiffness and mass matrix of the system;
=+Using two beam finite elements of length , you are asked
=+Problem 5.5 Let us consider the system of Figure 5.30.b made of two beams of length , clamped on both sides. The left beam has a bending stiffness of 2EI and a distributed mass m per unit length.
=+– To compare the forced vibration deformation of the system obtained previously with the static response to a unit transverse force on M.
=+– To compute the response amplitude of the system when a unit transverse harmonic force is applied on mass M with a forcing frequency equal to ω1∕2.
=+7 The same reasoning could be followed to get the stiffness matrix (5.194) of the beam with shear deformation.Approximation of Continuous Systems by Displacement Methods 407
=+– To compute the eigenmodes and eigenfrequencies of the model when M = (32∕105) mand draw the mode shapes.
=+v– To write the stiffness and mass matrix of the system. This requires in particular to construct the mass matrix of the right beam for the point mass located in its middle.
=+Using two beam finite elements of length , you are asked:
=+Let us consider the system made of two beams of length , clamped at both sides (Figure 5.30.a). The left beam has a bending stiffness of 2EI and a distributed mass m per unit length. The right
vv4. to deduce from it the eigenvalue equation for the bar clamped at one end and with a mass M at the other end.
=+4. to deduce from it the eigenvalue equation for the bar clamped at one end and with a mass M at the other end.
=+3. to express the transfer matrix of a bar with a mass M = ????m attached at one end.
=+2. To deduce from its individual elements the eigenvalue equation of the bar under different boundary conditions.
=+1. To develop the explicit expression of the transfer matrix T(????).
=+where T is a 2 × 2 matrix that can be written along the same principles as presented for discrete systems in Exercise 2.7 of Chapter 2. The difference lies in the fact that here, the elements of
=+Figure 4.58 Transfer matrix representation of a bar.transfer relationship of the form:[Nk+1 uk+1]= T(????)[Nk uk,](P4.9.a)
=+Figure 4.57 A beam on anelastic foundation (a.) and on elastic supports (b.).Continuous Systems 333−N0, u0 −Nk, uk Nk+1, uk+1 −Nn–1, un–1 Nn N1 , un , u1. . . . . .
=+Problem 4.9 Let us suppose that a bar is decomposed in slices as displayed on Figure 4.58, and consider a piece of length with uniform characteristics (EA, m). In harmonic regime, a???? ????x x k
=+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 k → ∞.
=+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 stiffness ratio ????.
=+1. to write down the equation for vibration motion of the system and the associated boundary conditions in nondimensional form.
=+Problem 4.8 Let us consider a uniform beam of length with stiffness and mass characteristics (EI, m) resting on elastic supports of stiffness k (Figure 4.57.b). The non-dimensional stiffness
=+5. to determine the eigensolutions and observe that the stiffness of the support modifies the system eigenspectrum, but not its eigenshapes.
=+4. to express the general solution and apply the boundary conditions.
=+3. to define an appropriate nondimensional frequency parameter taking into account the foundation eigenfrequency.
=+2. to introduce the concept of foundation eigenfrequency ωf =√ k m and give its physical meaning.
=+1. to write down the equation for vibration motion of the system and the associated boundary conditions.
=+Problem 4.7 Let us consider a uniform beam, free-free, of length with stiffness and mass characteristics (EI, m) supported by a continuous elastic foundation of stiffness k per unit length
=+6. to represent graphically the first two eigenmodes.
=+5. starting from the asymptotic approximation obtained before, to compute the exact values(to 5 significant digits) of ????1 and ????2. Determine the error characterizing the asymptotic solution.
=+4. to setup an iteration procedure to compute a given root ????s and determine their asymptotic behaviour of the roots as s increases.
=+3. to demonstrate the existence of eigensolutions corresponding to rigid body motion and show how the nonzero roots (????s, s = 1, ∞) can be obtained graphically.332 Mechanical Vibrations: Theory
=+2. to apply the boundary conditions to the general solution in order to get the transcendental equation governing the frequency parameter ???? such that ????4 = ω2m4 EI .
=+1. to write down the equation for vibration motion of the system and the associated boundary conditions.
=+Problem 4.6 Let us consider a free-free uniform beam of length with stiffness and mass characteristics (EI, m). You are asked:
=+2 and at x = 0.95 obtained successively with the mode displacement and the mode acceleration methods, with increasing number of modes in the expansion (e.g. 10, 20 and 50 modes). Compare the
=+5. Plot in a similar way its derivative ????x(Su) characterizing the convergence of the axial stress N(x). Observe the behaviour of both functions at x = . Explain it in relationship with their
=+4. Assuming truncation of the modal expansion, plot the function ????(Su) successively for 10, 20 and 50 modes.
=+3. Deduce the response of the system from the general form given by Equation (4.102).
=+2. Determine the quasi-static response of the system.
=+1. Determine the eigensolutions of the free vibration problem with homogeneous boundary conditions and put them in orthonormal form.
=+Figure 4.13, in the sense that the static response has a similar shape and the wave propagation of the initial disturbance will also be observed. However, due to the difference in boundary
=+Problem 4.5 Let us consider the case of a bar in extension clamped at one end and submitted in a stepwise manner to an imposed displacement u0 at the other end.EA????2u????x2 − mü = 0 0 ≤ x
=+4. Deduce the expression of the mass matrix M from the evaluation of the term in ω2 in the general expression (4.122).Continuous Systems 331
=+3. Deduce from Equation (P4.4.c) the expressions of the bending moments (M1, M2) and shear forces (T1, T2) (taking into account the orientation of the outward normal to the cross-section!). Deduce
=+2. Derive the expression obtained to compute the bending moment and shear force:M(x) = EI d2????dx2 T(x)=−dM dx (P4.4.c)
=+Express the solution of (P4.4.a) in the form:????(????????) = P1(x)????????1 + P2(x)????????1 + P3(x)????????2 + P4(x)????????1 (P4.4.b)where Pi(x), i = 1 … 4 are interpolation polynomials.
=+1. The development supposes obtaining the solution of the associated quasi-static problem:EI d4????dx4 = 0 0 < x < ????(0) = ????????1 ????() = ????????2????(0) = ????????1 ????() =
=+– observe that the stiffness and mass matrices are those given in (4.154) and (4.161).Problem 4.4 The impedance matrix for the Euler-Bernoulli beam of length with uniform characteristics (EI,
=+– show that it gives rise to a linearized impedance relationship of the form(K − ω2M)q = p
=+– assuming ???? to be small and make a series expansion of Z(????) up to second-order;
=+3. To show the link to the expansion (4.167) by:
=+2. To provide the physical interpretation of its individual elements.
=+– express the applied axial forces (N1, N2) at both ends of the bar (respecting the sign conventions of Figure 4.11);– write then the expression of (N1, N2) in terms of (u1, u2) in the matrix
=+1. To obtain the exact impedance matrix of the bar by considering the general solution to the differential equation (P4.3.a) and apply the following steps:– specify the general solution by
=+EA is the nondimensional frequency parameter.You are asked:
=+describing the equilibrium in the bar for a harmonic motion, when no volume forces are present. This equation is identical to the free vibration equation (4.129), but now we consider
=+5. to observe the asymptotic behaviour of the solutions when ???? → 0 and when M → ∞.Problem 4.3 The impedance matrix of a continuous bar was found in (4.167) as an expansion in terms of its
=+4. to determine analytically the eigenvalues ????s in the particular case ???? = 1.
=+3. to show how the eigenvalues (????s, s = 1, ∞) can be obtained graphically as functions of the ???? parameter.
=+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 → ∞.

Showing 300 - 400 of 535