# Lecture 32. TORSION EXAMPLES HAVING MORE THAN ONE DEGREE OF FREEDOM

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1 Lecture 32. TORSION EXAMPLES HAVING MORE THAN ONE DEGREE OF FREEDOM Figure 5.40 (a) Two-disk, torsional vibration example, (b) coordinates and free-body diagram for Torsional Vibration Examples We worked through a one-degree-of-freedom, torsionalvibration example in section 5.4, starting with the model of figure Figure 5.35 illustrates a two-degree-of-freedom extension to this example. The upper disk has mass m 1, radius R 1, and is connected to ground by a circular shaft of radius length L 1, and shear modulus G 1. The lower disk has mass m 2, radius R 2, and is connected to disk 1 by a circular shaft of radius, length L 2, and shear modulus G 2. The rotation angles and define the orientations of the two disks. The shafts have zero elastic deflections and moments when these angles are zero. 148

2 As with the example of figure 5.10, twisting the upper disk through the angle will develop the reaction moment, (5.144) acting on top of the upper disk. The negative sign in this equation implies that the moment is acting in a direction opposite to a positive rotation direction. The reaction moment acting on the bottom of disk 1 and the top of disk 2 is proportional to the difference between the rotation angles and. Assuming that is greater than, the reaction moment acting on disk 1 from the lower shaft is (5.145) The positive sign for the moment implies that it is acting in the + direction, i.e., acting to rotate disk 1 in a positive + direction. The negative of this moment acts on the top of disk 2. In addition, assume that the applied moments M 1 ( t ) and M 2 ( t ) are acting, respectively, on disks 1 and 2. Individually summing moments about the axis of symmetry for the two bodies including these external moments yields: 149

3 (5.146) The matrix statement of these equations is (5.147) The inertia matrix is diagonal, and the stiffness matrices is symmetric. 150

4 Figure 5.36 Unrestrained, two-disk torsional-vibration example. The governing differential equations for the model of figure 5.36 are obtained from Eqs.(5.147) by setting the upper stiffness equal to zero. Eq.(5.147) becomes (5.151) Substituting the assumed solution, into the homogeneous version of Eq.(5.151) nets (5.152) The characteristic equation is obtained by setting the determinant equal to zero and is: 151

5 For the numbers in Eq.(5.150), (5.150) the eigenvalues are, and the natural frequencies are. Substituting into Eq.(5.152), gives The determinant of the coefficient matrix is clearly zero, and the first eigenvector can be defined from either scalar equation by setting, obtaining, and the first eigenvector is The second mode is obtained by substituting the second 152

6 eigenvalue into Eq.(5.152), (plus substituting for from Eq.(5.150)) obtaining and Again, the determinant of the coefficient matrix is zero, and the second eigenvector is The matrix of eigenvectors is The first step in obtaining the modal differential equations is taken by introducing the modal coordinates, via the coordinate transformation,, Substituting into Eq.(5.151) and then premultiplying by the transpose matrix gives the uncoupled modal differential equations: 153

7 Observe that the first modal coordinate (with the zero eigenvalue and natural frequency) defines rigid-body rotation of the rotor with zero relative rotation between and. The second modal coordinate defines relative motion with the two disks moving in opposite directions. The complete model is 154

8 Figure 5.37 (a). Two, two-disk rotors in frictionless bearings with coupled motion. (b). Kinematic constraint between disks 2 and. (c). Reaction-force components between disks 2 and. Figure 5.37 shows two rotors, each with two disks connected by a shaft. The rotations of disks 1 and 2 on rotor 1 are defined by the counter clockwise rotations. The rotations of disks and 3 of rotor 2 are defined by the clockwise rotations. The radii of disks 2 and are, respectively,. The two disks are geared together with no relative slipping between their edges; hence, their rotation variables are related by the kinematic constraint equations: (5.153) Figure 5.37C illustrates the reaction force acting between disks 2 and. Stating the equations of motion for the disks of the two rotors gives: Rotor 1 ( + counter clockwise rotations and moments) 155

9 (5.154a) Rotor 2 (+ clockwise rotations and moments) (5.154b) Eqs.(5.153) and (5.154) comprise five equations in the five unknowns. Equating in the second of Eqs.(5.154a) and the first of Eqs.(5.154b) eliminates this variable. Also, Eq.(5.153) can be used to eliminate yielding the following three coupled differential equations: These equations define a three-degree-of-freedom problem with the matrix statement 156

10 (5.155) Because neither of the component rotors are connected to ground, one of the eigenvalues is zero, and the remaining two roots (and their associated eigenvectors) can be determined analytically. 157

11 Lecture 33. BEAMS AS SPRINGS VIBRATION EXAMPLES Figure 5.38 Lumped-parameter rotor model including two disks. Unlike linear extension/compression springs that deliver a reaction force opposing the direction of displacement, or torsional springs that produces a reaction torque to oppose twisting, deflecting or rotating the end of a beam normally produce both a reaction moment and a reaction force. The coupling of displacements and rotations and reaction forces and moments can be confusing, and we will start with a simple example of a cantilevered beam with an attached weight. Displacement of the beam s end creates a reaction force but no reaction moment. 158

12 Example A. Cantilever Beam supporting a disk that does not rotate. Zero moments about the right end. The bar has mass m. The beam has length l and section modulus EI. If it is pulled down and then released (the cable remains in tension), what is the natural frequency of free motion for the disk? Figure XP 5.2b Cantilever beam with the disk hanging from its end. From strength of materials, a beam with a zero moment at its free end and a lateral load f will deflect a distance δ. δ and f are related by In this equation, defines the displacement flexibility coefficient for the beam s end. We want the displacement stiffness coefficient (for zero moment at the beam s 159

13 end) For motion about the equilibrium position, the free-body diagram for the disk (massless beam) is shown below. ( δ remains small enough that the cord remains in tension.) Free-body diagram. From the free-body diagram, the equation of motion is, and the natural frequency is 160

14 Suppose the beam has length and a circular cross section with diameter. It is made from steel with modulus of elasticity. The disk has radius, thickness and is also made from steel (density ). As a first step, the bending section modulus is The mass is Substituting, the natural frequency is 161

15 where the dimensions of a kg are. Figure 5.39a represents the more general situation with a disk rigidly attached to the end of a cantilevered beam. If the disk is displaced from its equilibrium position, the disk will rotate in the Y-Z plane through the angle Because of the disk s moment of inertia, a moment will now result at the beam s end due to disk rotation. Figure 5.39b provides the free-body diagram for the displaced and rotated disk, including the applied force and moment pair and reaction force and moment pair. Figure 5.39 Cantilevered beam supporting a thin circular disk at its right end. Free-body diagram for the disk. The equations of motion are easily stated from figure 5.39b. Applying and, the disk s equations of motion are: 162

16 (5.156) Defining the reaction force and moment in terms of the displacement and rotation coordinates is the principal difficulty in completing these equations. Figure 5.40 (a). Cantilevered beam with an applied end force. (b). Applied moment Figure 5.40a illustrates the beam with a concentrated load applied at its end, yielding (from strength of materials) the displacement and rotation. (5.157a) We used the displacement result in the first example. Similarly, figure 5.40b illustrates a moment M applied to the beam s end, yielding 163

17 Combining these results as gives and In matrix format, these equations are This coefficient matrix is a flexibility matrix. An flexibility-matrix entry is the displacement (or rotation) at point i due to a unit load (or moment) at point j. Multiplying through by gives 164

18 (5.158) This coefficient matrix is the stiffness matrix. The form of this equation tends to be confusing if we think of it as defining the applied loads as the output due to input displacements and rotations. However, the following statement makes sense when defining the reaction force and moment of figure 5.39b due to displacements and rotations (5.159) Substituting this result into Eqs.(5.156) gives the matrix equation of motion, (5.160) or 165

19 (5.160) An entry for the stiffness matrix is the negative reaction force (or moment) at point i due to a displacement (or rotation) at station j, with all other displacements and rotations equal to zero. Example Problem 5.1 The cantilevered beam of figure 5.39 has length, a circular cross section with diameter. It is made from steel with modulus of elasticity. The disk has radius, thickness and is also made from steel (density = ). The following engineering-analysis tasks apply: a. Determine the inertia and stiffness matrices and state the matrix equation of motion. 166

20 b. Determine the eigenvalues, natural frequencies, and eigenvectors. Solution. As a first step, the bending section modulus is Continuing, the stiffness coefficients are: The inertia-matrix entries are: 167

21 Substituting these results into Eq.(5.160) defines the model as (i) Substituting the solution homogeneous version of Eq.(i) nets into the (ii) The characteristic equation is The eigenvalues and natural frequencies defined by the roots of this equation are: 168

22 (iii) The first natural frequency the initial example result is slightly lower than Substituting and into Eq.(ii), the corresponding eigenvectors are: (iv) As illustrated in figure XP 5.1a, the displacement and rotation 169

23 are in phase for the first mode and out of phase for the second. Figure XP 5.2a Calculated mode shapes of Eq.(iv); not to scale. Figure XP 5.2c Cantilever beam with the end disk forced to move up and down but prevented from rotating.. The disk in Figure XP 5.2c is constrained by rollers that prevent rotation. For, the example has only one degree of freedom, and Eq.(5.160) (5.160) gives: 170

24 The first equation is the equation of motion for. The second equation defines the reaction moment that the constraint rollers must provide to keep the disk from rotating. The natural frequency is defined by The moment restraint on the disk has doubled the lowest natural frequency. Note that we have determined the displacement stiffness for a cantilever beam whose end is deflected with zero rotation to be 171

25 Example Problem 5.5. The framed structure has two square floors. The first floor has mass and is supported to the foundation by four solid columns with square cross sections. These columns are cantilevered from the foundation and are welded to the bottom of the first floor. The second floor has mass and is supported from the first floor by four solid columns with square cross sections. These columns are welded to the top of the first floor and are hinged to the second floor. The bottom and top columns have length and. The top and bottom beams cross-sectional dimensions are and. They are made from steel with a modulus of elasticity. A model is required to account for motion of the foundation due to earthquake excitation defined by. Tasks: a. Select coordinates, draw a free-body diagram, derive the equations of motion. b. State the equations of motion in matrix format and solve for the eigenvalues and eigenvectors. Draw the eigenvectors. 172

26 Figure XP5.5 (a) Front view of a two-story framed structure excited by base excitation, (b) Coordinates, (c) Free-body diagram for, (d) Eigenvectors 173

27 Solution. Figure XP5.5 b illustrates the coordinates selected to locate the first and second floors with respect to ground. All beams connecting the foundation and the first floor are cantilevered at both ends, similar to the beam in figure 5.47 with a stiffness. The free-body diagram of figure XP5.5c was drawn assuming that the first floor has moved further than the ground and defines the reaction force due to all four cantilevered beams acting at the bottom of floor 1. Each beam connecting the floors has a cantilevered end attached to floor 1 and a pinned end attached to floor 2, similar to the pinned-end beam of figure 5.44, with a stiffness coefficient. The free-body diagram in figure XP5.5c was developed assuming that the second floor has moved further than the first floor and provides The negative of this force is acting at the top of floor 1. Summing forces for the two floors gives: 174

28 Putting these equations in matrix form gives (i) This outcome is similar to Eq.(3.126) for two masses connected by springs. Filling in the numbers gives: 175

29 Continuing, the stiffness coefficients are: Plugging these results into Eq.(i) gives (ii) Substituting the assumed solution the homogeneous version of this equation gives into (iii) Since,, and and are also not zero, a nontrivial solution, for Eq.(ii) requires that the determinant of the coefficient matrix must equal zero, producing 176

30 This characteristic equation defines the two eigenvalues and natural frequencies: Alternately substituting and into Eq.(iii) gives the eigenvectors Figure XP5.5d illustrates these two eigenvectors, showing the relative motion of the two floors somewhat better that the twomass eigenvectors of figure

31 Lecture 34. 2DOF EXAMPLES Example 1 k 1 k 2 A k B 3 m, R m, R The cylinders roll without slipping. Select coordinates, draw free-body diagrams, and derive the equations of motion. Coordinates and free-body diagram for 178

32 Equations of motion, cylinder 1 (2) Equations of motion cylinder 2 (3) Four equations, 6 unknowns Rolling-without slipping kinematic constraints: (3) Eliminate in (1), and in (2) 179

33 Use (3) to eliminate, and rearrange For and the matrix form is 180

34 Alternative development: Take moments about C the point of contact General moment equation for clockwise moments However, Hence, 181

35 Similarly for cylinder 2 Matrix Format This is the same equation 182

36 Example 2 D 3 d 1 k A L 1 L 2 Y D 1 D 2 k B d 2 D 3 P Assume small rotations, select coordinates, draw free-body diagrams, and derive the equations of motion. Gravity is vertically down, and the body is in equilibrium. 183

37 Coordinates: is the vertical mass-center displacement from equilibrium. θ is the rotation angle for the body from the horizontal. Free-body diagram for motion about equilibrium Equations of motion: 184

38 Small-angle kinematics: Substituting, Gathering terms, Matrix Format A good deal of effort is required on the homework problem to get and. 185

39 Lagrange s Equations Example 1 k 1 k 2 A k B 3 m, R m, R Coordinates Kinetic Energy Potential Energy 186

40 Lagrangian EOM Result from Example 2 187

41 Kinematics: Proceeding 188

42 EOMs from 189

43 Lecture 35. More 2DOF EXAMPLES A Translating Mass with an Attached Compound Pendulum Figure 5.53 Translating cart of mass M supported by frictionless rollers and supporting a compound pendulum of length l and mass m. (a) Equilibrium position, (b) General position (c) Cart free-body diagram, (d) Pendulum freebody diagram. 190

44 Cart EOM (5.201) Pendulum Moment Equation Moment about g: will draw the unknown and unwanted reaction components into the moment equation. Moment about o using is quicker. An inspection of the pendulum in figure 5.53c gives: Hence, stating the moment equation about the pivot point o gives (5.202) 191

45 where We now have two equations (the first of Eq.(5.201) and Eq.(5.202)) for the three unknowns:. The X component of the gives the last required EOM as equation of the pendulum (5.203) However, this equation introduces the new unknown, which can be eliminated, starting from the geometric relationship Differentiating this equation twice with respect to time gives Substituting for into Eq.(5.203) gives Now, substituting this gives definition into the first of Eq.(5.201) 192

46 (5.204) Eqs.(5.202) and (5.204) comprise the two governing equations in. Their matrix statement is: (5.205) We have now completed Task a. Assuming small motion for this system means that second order terms in X and θ are dropped. Introducing the small angle approximations ;, and dropping second order and higher terms in θ and yields: 193

47 (5.206) which concludes Task b. 194

48 APPLYING LAGRANGE S EQUATION OF MOTION TO EXAMPLES WITH GENERALIZED COORDINATES (NO KINEMATIC CONSTRAINTS). Coupled Cart/Pendulum Figure 6.3 Translating cart with an attached pendulum (no external force) This system has the two coordinates X,θ and two degrees of freedom. Hence, the two coordinates X,θ are the generalized coordinates, and their derivatives are the generalized velocities of Lagrange s equations. The following engineering task applies: Use Lagrange s equations to derive the equations of motion. 195

49 The kinetic energy of the cart is easily calculated as. The kinetic energy of the pendulum follows from the general kinetic energy for planar motion of a rigid body (5.183) where is the velocity of the body s mass center with respect to an inertial coordinate system. The pendulum s mass center is located by Hence and Hence, the system kinetic energy is 196

50 Using a plane through the pivot point as datum for the gravity potential energy function gives. The potential energy of the spring is ; hence, the system potential energy is and Proceeding with the Lagrange equations developments, the partial derivatives with respect to generalized velocities are: and the derivatives of these terms with respect to time are: 197

51 Once again, note the last terms in these derivatives. The partial derivatives of L with respect to the generalized coordinates are By substitution, the governing equations of motion are: (6.29) The right-hand terms are zero, because there are no nonconservative forces. Eqs.(6.29) are stated in matrix notation as 198

52 (6.30) which coincides with Eq.(5.160) ( without the external force of figure 5.38) that we derived earlier from a free-body diagram/newtonian approach. Again, the results are obtained without recourse to free-body diagrams, and only velocities are required for the kinematics. 199

53 A Swinging Bar Supported at its End by a Cord Figure 5.54a shows a swinging bar AB, supported by a cord connecting end A to the support point O The cord has length l 1 ; the bar has length l 2 and mass m. This system has the two degrees of freedom φ and θ. The engineering tasks for this system is: Derive the governing differential equations of motion. Figure 5.54 Swinging bar supported at its end by a cord. (a) Equilibrium, (b) Coordinate choices, (c) Free-body diagram From, the force component equations are: 200

54 The acceleration components can be obtained by stating the components of R g as: Differentiating these equations twice with respect to time gives (5.163) Substitution gives 201

55 Eliminate T c from these equations by: (i) multiply the first by, (ii) multiply the second by, and (iii) add the results to obtain (5.164) This is the first of our required differential equations. One could reasonably state a moment equation about either g, the mass center, or the end A. Stating the moment about A has the advantage of eliminating the reaction force, and we will use the following version of the moment Eq.(5.24) (5.165) 202

56 The moment due to the weight is negative because it acts opposite to the positive counterclockwise direction of θ. The vector goes from A to g and is defined by (5.166a) To complete Eq.(5.165), we need to define. The cord length is constant; hence, the radial acceleration component consists of the centrifugal-acceleration term. Similarly, the circumferential acceleration term reduces to. Resolving these terms into their components along the X and Y axes gives (5.166b) Figure 5.55 Polar kinematics for the cord to determine. 203

57 Eqs.(5.166) give Substituting this result into Eq.(5.165) gives (5.167) This is the desired moment equation for the bar and is the second and last equation of motion with. Eqs.(5.164) and (5.167) can be combined into the following matrix equation. 204

58 The inertia-coupling matrix can be made symmetric by multiplying the top row by l 1. Eliminating second-order terms in θ and φ in this equation gives the linear vibration equations 205

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