1. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -1- Chapter 4 Vibration Under General Forcing Condition 4

2. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -2- Learning Objectives Find the response of single-DOF systems subjected to general periodic forces using the Fourier series Use the method of convolution or Duhamel ( 杜哈明 ) integral to solve vibration problems of systems subjected to arbitrary forces Find the response of systems subjected to earthquakes using response spectra Solve undamped and damped systems subjected to arbitrary forces, including impulse, step, and ramp forces, using Laplace transform Understand the characteristics of transient response, such as peak time, overshoot, settling time, rise time, and decay time, and procedures for their estimation Apply numerical methods to solve vibration problems of systems subjected to forces that are described numerically Solve forced-vibration problems using MATLAB

3. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -3- Chapter Outline 4.1IntroductionIntroduction 4.2Response Under a General Periodic ForceResponse Under a General Periodic Force 4.3Response Under a Periodic Force of Irregular FormResponse Under a Periodic Force of Irregular Form 4.4Response Under a Nonperiodic ForceResponse Under a Nonperiodic Force 4.5Convolution IntegralConvolution Integral 4.6Response SpectrumResponse Spectrum 4.7Laplace TransformsLaplace Transforms 4.8Numerical MethodsNumerical Methods 4.9Response to Irregular Forcing Conditions Using Numerical MethodsResponse to Irregular Forcing Conditions Using Numerical Methods

4. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -4- 4.1 Introduction 4.1

5. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -5- 4.1 Introduction Many practical systems are subjected to several types of forcing functions that are not harmonic. The general forcing functions may be periodic (nonharmonic) or nonperiodic. The nonperiodic forces include forces such as : A step force: a suddenly applied constant force A ramp force: a linearly increasing force An exponentially varying force A nonperiodic forcing function may be acting for short, long, or infinite duration. Shock: a forcing function or excitation of short duration compared to the natural ( 固有的 ) time period of the system.

6. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -6- 4.1 Introduction Some examples of general forcing functions include the motion imparted ( 傳遞 ) by a cam to the follower; the vibration felt by an instrument when its package is dropped from a height; the force applied to the foundation of a forging press; the motion of an automobile when it hits a pothole; and the ground vibration of a building frame during an earthquake, etc. If the forcing function is periodic but not harmonic, it can be replace by a sum of harmonic functions using the harmonic analysis procedures (Section 1.11). Using the principle of superposition, the response of the system can then be determined by superposing the responses due to the individual harmonic forcing functions.

7. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -7- The vibration in a cam to the follower The typical force fluctuation in a valve mechanism during the cam operating cycle. The dynamic force affecting valvecam follower of a large diesel engine during the cam operating cycle at engine running speed 500 r/min is shown.

8. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -8- The response of a system subjected to any type of nonperiodic force is commonly found using the following methods: (1) Convolution integral; (2) Laplace transform; (3) Numerical methods The first two methods are analytical ones, in which the response is expressed in a way that helps in studying the behavior of the system under the applied force with respect to various parameters and in designing the system. The third method can be used to find the response of a system under any arbitrary force for which an analytical solution is difficult or impossible to find. However, the solution found is applicable only for particular set of parameter values, which makes it difficult to study the behavior of the system when the parameters are varied.

9. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -9- 4.2 Response Under a General Periodic Force 4.2

10. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -10- Fourier Series Expansion: When the external force F(t) is periodic with period =2/, it can be expanded in a Fourier Series (Section 1.11) The response of systems under general periodic forces is considered herein for both first- and second-order system 4.2 Response Under a General Periodic Force

11. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -11- Fig. 4.1 Examples of first-order systems Displacement excitation

12. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -12- Fig. 4.2 Examples of second-order systems

13. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -13- 4.2 Response Under a General Periodic Force Consider a spring-damper system subjected to a periodic excitation as shown in Fig. 4.1(a). The equation of motion of the system is given by where y(t) is the periodic motion (or excitation), which is expressed as Fourier series ( 與 Eq. (4.1) 做比較 )

14. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -14- Example 4.1 Response of a First-Order System under Periodic Force Find the response of the spring-damper system, shown in Fig. 4.1(a) subjected to a periodic force with the equation of motion given by Eq. (4.5) Solution Using the principle of superposition, the steady-state solution of Eq. (4.5) can be found by summing the steady-state solutions corresponding to the individual forcing terms on the right-hand of Eq. (4.5)

15. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -15-

16. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -16-

17. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -17-

18. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -18-

19. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -19-

20. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -20- j j 用以求出 C

21. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -21- 4.2 Response Under a General Periodic Force Example 4.3 Response of a Second-Order system Under Periodic Force The equation of motion of Fig. 4.2(a) can be expressed as To determine the response of Eq. (4.8) using the principle of superposition. Where the forcing function F(t) is periodic.

22. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -22- Using the results of Section 3.4, we can express the solutions of Eqs.(E.2) and (E.3), respectively, as

23. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -23- Thus the complete steady-state solution of Eq.(4.8) is given by Operation frequency

24. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -24- 4.2 Response Under a General Periodic Force Example 4.5 Total Response Under Harmonic Base Excitation Find the total response of a viscously damped single-DOF system subjected to a harmonic base excitation (3.6 Section) for the following data: Solution The equation of motion of the system is given by (see Eq. (3.65)): The steady-state response of the system can be expressed as (using Eq.(E.9) of example 4.3) Eq. (E.1) is similar to Eq.(4.8) with Initial conditions

25. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -25- 4.2 Response Under a General Periodic Force Example 4.5 Total Response Under Harmonic Base Excitation Solution For the given data, We find

26. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -26- 4.2 Response Under a General Periodic Force Example 4.5 Total Response Under Harmonic Base Excitation Solution The solution of the homogeneous equation is given by (see Eq. (2.70)): The total solution can be expressed as the superposition of x h (t) and x p (t) as: where X 0 and Φ 0 are unknown constants

27. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -27- 4.2 Response Under a General Periodic Force Example 4.5 Total Response Under Harmonic Base Excitation Solution Using Eqs.(E.4) and (E.5), we find The velocity of the mass can be expressed from Eq.(E.4) ( 使用初始條件，求出 X 0 與  0 )

28. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -28- 4.2 Response Under a General Periodic Force Example 4.5 Total Response Under Harmonic Base Excitation Solution The solution of (E.6) and (E.7) yields X 0 =0.488695 and Φ 0 =1.529683 rad. Thus the total response of the mass under base excitation, in meters, is given by

29. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -29- 4.3 Response Under a Periodic Force of Irregular Form 4.3

30. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -30- 4.3 Response Under a Periodic Force of Irregular Form In some cases, the force acting on a system may be quite irregular and may be determined only experimentally. Examples of such forces include wind and earthquake-induced forces. In such cases, the forces will be available in graphical form and no analytical expression can be found to describe F(t). Sometimes, the value of F(t) may be available only at a number of discrete points t 1, t 2, t 3 …t N. In all these cases, it is possible to find the Fourier coefficients by using a numerical integration procedure, as described in Section 1.11.

31. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -31- 4.3 Response Under a Periodic Force of Irregular Form If F 1, F 2, F 3 …F N denote the values of F(t) at t 1, t 2, t 3 …t N, respectively, where N denotes an even number of equidistant ( 等距離 ) points in one time period  (=N △ t), as shown in Fig. 4.5. The application of trapezoidal ( 梯形的 ) rule gives:

32. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -32- 4.3 Response Under a Periodic Force of Irregular Form An irregular forcing function: Once the Fourier coefficients a 0, a j, and b j are known, the steady- state response of the system can be found using Eq. (E.9) of Example 4.3 with

33. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -33- 4.3 Response Under a Periodic Force of Irregular Form Example 4.6 Steady-State Vibration of a Hydraulic Valve Find the steady-state response of the valve in the figure (see example 4.4) if the pressure fluctuations in the chamber are found to be periodic. The valves of pressure measured at 0.01 second intervals in one cycle are given below. period pressure

34. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -34- 4.3 Response Under a Periodic Force of Irregular Form Example 4.6 Steady-State Vibration of a Hydraulic Valve Solution (see Example 1.20) Since the pressure fluctuations on the valve are periodic, the Fourier analysis of the given data of pressures in a cycle gives: Other quantities needed for the computation are 乘上 A 變成力

35. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -35- 4.3 Response Under a Periodic Force of Irregular Form Example 4.6 Steady-State Vibration of a Hydraulic Valve Solution We have also

36. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -36- 4.3 Response Under a Periodic Force of Irregular Form Example 4.6 Steady-State Vibration of a Hydraulic Valve Solution The steady-state response of the valve can be expressed as (Using Eq. (E.9) of Example 4.3) (see example 4.4)

37. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -37- 4.4 Response Under a Nonperiodic Force 4.4

38. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -38- 4.4 Response Under a Nonperiodic Force We have seen that periodic forces of any general waveform can be represented by a Fourier series as a superposition of harmonic components of various frequencies. The response of a linear system is then found by superposing the harmonic response to each of the exciting forces. When the exciting force F(t) is nonperiodic, such as that due to the blast ( 爆破 ) from an explosion, a different method of calculating the response is required. Various methods can be used to find the response of the system to an arbitrary excitation. Some of these methods are as follows: 1. Representing the excitation by a Fourier integral 2. Using the method of convolution integral 3. Using the method of Laplace transforms 4. Numerically integrating the equation of motion

39. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -39- 4.5 Convolution Integral 4.5

40. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -40- 4.5 Convolution Integral A nonperiodic exciting force usually has a magnitude that varies with time; it acts for a specified period and then stops. The simplest form is the impulsive force – a force that has a large magnitude F and acts for a very short time  t. From dynamics, we know that impulse can be measured by finding the change it causes in momentum ( 動量 = 質量 × 速度 ) of the system. If denote the velocities of the mass m before and after the application of the impulse, we have

41. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -41- 4.5 Convolution Integral By designating ( 指定 ) the magnitude of the impulse by F, we can write, in general, A unit impulse acting at t=0 (f) is defined as It can be seen that in order for Fdt to have a infinite value, F tends to infinity (Since dt tends to zero)

42. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -42- The unit impulse, f=1, acting at t=0, is also denoted by the Dirac delta function as

43. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -43- 4.5 Convolution Integral Response to an impulse Consider a viscously damped spring-mass system subjected a unit impulse at t=0, as shown in Figs. 4.6(a) and (b). For an underdamped system, the solution of the equation of motion is given by Eq. (2.72)(free vibration) as

44. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -44- 4.5 Convolution Integral Fig. 4.6 A single-degree-of-freedom system subjected to an impulse

45. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -45- If the mass is at rest before the unit impulse is applied (x= =0 for t<0 or at t=0 - ), we obtain, from the impulse-momentum relation, Thus the initial conditions are given by In view of Eqs. (4.23) and (4.24), Eq.(4.18) reduces to Eq. (4.25) gives the response of a single-DOF system to a unit impulse, which is also known as the impulse response function, denoted by g(t), as shown in Fig. 4.6(c).

46. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -46- 4.5 Convolution Integral Response to an impulse If the magnitude of the impulse is F instead of unity, the initial velocity is and the response of the system becomes If the impulse F is applied at an arbitrary time t =, it will change the velocity at t = by an amount F/m ( 下頁 ). Assuming that x=0 until the impulse is applied, the displacement x at any subsequent time t, cause by a change in the velocity at time , is given by Eq. (4.26) with t replaced by the time elapsed after the application of the impulse  that is, t-. Thus we obtain The displacement due to a change in the velocity at time 0 The displacement due to a change in the velocity at time 

47. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -47- Impulse Response

48. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -48- 4.5 Convolution Integral Example 4.7 Response of a Structure Under Impact In the vibration testing of a structure, an impact hammer with a load cell to measure the impact force is used to cause excitation, as shown in Fig.4.8(a). Assuming m = 5kg, k = 2000 N/m, c = 10 N-s/m and F = 20 N-s, find the response of the system. Example 4.8

49. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -49- 4.5 Convolution Integral Example 4.7 Response of a Structure Under Impact Solution From the known data, Assuming that the impact is given at t = 0, the response of the system (From Eq. (4.26)) (Continued to Example 4.8)

50. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -50- 4.5 Convolution Integral Response to General Forcing Condition Consider the response of the system under an arbitrary external force F(t), shown in Fig. 4.9, the impulse response, for the impulse force F() at time , is given by An arbitrary (nonperiodic) forcing function This force may be assumed to be made up of a series of impulses of varying magnitude.

51. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -51- 4.5 Convolution Integral Response to General Forcing Condition The total response at time t can be found by summing all the responses due to the elementary impulses acting at all times : Letting and replacing the summation by integration, we obtain which represents the response of an underdamped single-DOF system to the arbitrary excitation F(t) and is also called the convolution or Duhamel integral By substituting Eq. (4.25), Eq. (4.27) into Eq. (4.30), we obtain

52. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -52- 4.5 Convolution Integral Response to Base Excitation If a spring-mass-damper system is subjected to an arbitrary base excitation described by its displacement, velocity, or acceleration, the equation of motion can be expressed in terms of the relative displacement of the mass z=x-y as follows (see Section 3.6.2) This is similar to the equation For an undamped system subjected to base excitation, the relative displacement can be found from Eq. (4.31) where the variable z replacing x All of the results derived for the force-excited system are applicable to the base-excited system only for x replaced by z and the term F is replaced by

53. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -53- 4.5 Convolution Integral Example 4.12 Compacting Machine Under Linear Force Determine the response of the compacting machine shown in Figure (a) when a linearly varying force (shown in Figure (b)) is applied due to the motion of the cam. X(t)

54. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -54- 4.5 Convolution Integral Example 4.12 Compacting Machine Under Linear Force Solution Figure (b) is known as the ramp function. Where F denotes the rate of increase of the force F per unit time. By substituting this into Eq. (4.31), we obtain (See problem 4.28)

55. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -55- Problem 4.28

56. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -56-

57. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -57- 4.5 Convolution Integral Example 4.12 Compacting Machine Under Linear Force Solution These integrals can be evaluated and the response expressed as follows: For an undamped system ( d = n ), Eq.(E.1) reduces to Fig. 4.13(c) shows the response given by Eq.(E.2)

58. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -58- 4.6 Response Spectrum 4.6

59. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -59- 4.6 Response Spectrum The graph showing the variation of the maximum response (maximum displacement, velocity, acceleration, or any other quantity) with the natural frequency (or natural period) of a single degree of freedom system to a specified forcing function is known as the response spectrum. Since the max. response is plotted against the natural frequency (or natural period), the response spectrum gives the max. response of all possible single-DOF systems. Once the response spectrum corresponding to a specified forcing function is available, we need to know just the natural frequency of the system to find its max. response. Example 4.14 illustrates the construction of a response spectrum.

60. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -60- 4.6 Response Spectrum Example 4.11 Response Spectrum of a Sinusoidal Pulse Find the undamped response spectrum for the sinusoidal pulse force shown in the figure using the initial conditions

61. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -61- 4.6 Response Spectrum Example 4.14 Response Spectrum of a Sinusoidal Pulse Solution The equation of motion of an undamped system can be expressed as Free vibration

62. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -62- 4.6 Response Spectrum Example 4.14 Response Spectrum of a Sinusoidal Pulse Solution Superimposing the homogeneous solution x c (t) and the particular solution x p (t), as

63. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -63- 4.6 Response Spectrum Example 4.14 Response Spectrum of a Sinusoidal Pulse Solution Using the initial conditions, the constants can be found: Thus,

64. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -64- 4.6 Response Spectrum Example 4.14 Response Spectrum of a Sinusoidal Pulse Solution Since there is no force applied for t>t 0, the solution can be expressed as a free vibration solution where the constants A’ and B’ can be found by using the values of and, given by Eq. (E.8), as initial conditions for t>t 0 :

65. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -65- 4.6 Response Spectrum Example 4.14 Response Spectrum of a Sinusoidal Pulse Solution Where Hence, Therefore, ( 代回 (E.10) 整理 )

66. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -66- 4.6 Response Spectrum Response Spectrum for Base Excitation For a harmonic oscillator (an undamped system under free vibration), we notice that Thus, the acceleration and displacement spectra can be obtained:

67. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -67- 4.6 Response Spectrum Response Spectrum for Base Excitation The velocity response spectrum can be obtained: Thus the pseudo response spectra are given by:

68. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -68- 4.6 Response Spectrum Earthquake Response Spectra The most direct description of an earthquake motion in time domain is provided by accelerograms that are recorded by instruments called strong motion accelerographs. A typical accelerogram is shown in the figure below.

69. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -69- 4.6 Response Spectrum Earthquake Response Spectra A response spectrum is used to provide the most descriptive representation of the influence of a given earthquake on a structure of machine. It is possible to plot the maximum response of a single degree freedom system using logarithmic scales.

70. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -70- 4.6 Response Spectrum Example 4.17 Derailment of Trolley of a Crane During Earthquake The trolley of an electric overhead traveling (EOT) crane travels horizontally on the girder as indicated in the figure. Assuming the trolley as a point mass, the crane can be modeled as a single degree of freedom system with a period 2 s and a damping ratio 2%. Determine whether the trolley derails under a vertical earthquake excitation.

71. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -71- 4.6 Response Spectrum Example 4.17 Derailment of Trolley of a Crane During Earthquake

72. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -72- 4.6 Response Spectrum Example 4.17 Derailment of Trolley of a Crane During Earthquake Solution For = 2 s and ζ = 0.02, Fig.4.16 gives the spectral acceleration as Sa = 0.25 g and hence the trolley will not derail.

73. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -73- 4.6 Response Spectrum Design Under a Shock Environment When a force is applied for short duration, usually for a period of less than one natural time period, it is called a shock load. A shock may be described by a pulse shock, velocity shock, or a shock response spectrum. The pulse shocks are introduced by applied forces or displacements in different forms. A velocity shock is caused by sudden changes in the velocity. The shock response spectrum describes the way in which a machine or structure responds to a specific shock.

74. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -74- 4.6 Response Spectrum Design Under a Shock Environment Typical shock pulses

75. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -75- 4.6 Response Spectrum Example 4.18 Design of a Bracket for Shock Loads A printed circuit board (PCB) is mounted on a cantilevered aluminum bracket, as shown in Figure (a). The bracket is placed in a container that is expected to be dropped from a low-flying helicopter. The resulting shock can be approximated as a half-sine wave pulse, as shown in figure (b). Design the bracket to withstand an acceleration level of 100g under the half-sine wave pulse shown in figure (b). Assume a specific weight of 30 kN/m3, a Young’s modulus of 70 GPa, and a permissible stress of 180 MPa for aluminum.

76. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -76- 4.6 Response Spectrum Example 4.18 Design of a Bracket for Shock Loads

77. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -77- 4.6 Response Spectrum Example 4.18 Design of a Bracket for Shock Loads Solution The self-weight of the beam is given by The total weight is The area moment of inertia of the cross section of the beam is

78. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -78- 4.6 Response Spectrum Example 4.18 Design of a Bracket for Shock Loads Solution The static deflection of the beam can be obtained We adopt a trial and error procedure to determine the values of unknown.

79. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -79- 4.6 Response Spectrum Example 4.18 Design of a Bracket for Shock Loads Solution Assuming d = 10 mm, We have Hence,

80. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -80- 4.6 Response Spectrum Example 4.18 Design of a Bracket for Shock Loads Solution The dynamic load acting on the cantilever is given by: The maximum bending stress at the root of the cantilever bracket can be computed as:

81. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -81- 4.6 Response Spectrum Example 4.18 Design of a Bracket for Shock Loads Solution Since this stress exceeds the permissible value, we assume the next trial value of d as 20 mm. This yields:

82. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -82- 4.6 Response Spectrum Example 4.18 Design of a Bracket for Shock Loads Solution The dynamic load can be determined: The maximum bending stress at the root of the bracket will be: Since this stress is within the permissible limit, the thickness of the bracket can be taken as 20 mm.

83. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -83- 4.7 Laplace Transforms 4.7

84. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -84- 4.7 Laplace Transforms The Laplace transform of a function x(t) is defined as: The general solution can be expressed as

85. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -85- 4.8 Numerical Methods 4.8

86. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -86- 4.8 Numerical Methods The determination of the response of a system subjected to arbitrary forcing functions using numerical methods is called numerical simulation. Numerical simulations can be used to check the accuracy of analytical solutions, especially if the system is complex. Several methods are available for numerically integrating ordinary differential equations. The Runge-Kutta methods are quite popular for the numerical solution of differential equations.

87. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -87- 4.9 Response to Irregular Forcing Conditions Using Numerical Methods 4.9

88. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -88- 4.9 Response to Irregular Forcing Conditions Using Numerical Methods Let the function vary with time in an arbitrary manner. The response of the system can be found:

89. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -89- 4.9 Response to Irregular Forcing Conditions Using Numerical Methods Thus the response of the system at t = t j becomes

90. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -90- 4.9 Response to Irregular Forcing Conditions Using Numerical Methods Example 4.31 Damped Response Using Numerical Methods Find the response of a spring-mass-damper system subjected to the forcing function in the interval, using a numerical procedure. Assume F 0 =1, k=1, m=1, ζ=0.1, and t0=τ n /2, where τ n denotes the natural period of vibration given by The values of x and at t=0 are zero

91. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -91- 4.9 Response to Irregular Forcing Conditions Using Numerical Methods Example 4.31 Damped Response Using Numerical Methods Solution For the numerical computations, the time interval 0 to t 0 is divided into 10 equal steps with 4 different methods are used to approximate the forcing function F(t).

92. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -92- 4.9 Response to Irregular Forcing Conditions Using Numerical Methods Example 4.31 Damped Response Using Numerical Methods Solution In the figure, F(t) is approximated by a series of rectangular impulses, each starting at the beginning of the corresponding time step.

93. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -93- 4.9 Response to Irregular Forcing Conditions Using Numerical Methods Example 4.31 Damped Response Using Numerical Methods Solution In Fig. 4.36, piecewise linear (trapezoidal) impulses are used to approximate the forcing function F(t). The numerical results are given in Table 4.2. The results can be improved by using a higher-order polynomial for interpolation instead of the linear function.

94. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -94- 4.9 Response to Irregular Forcing Conditions Using Numerical Methods Example 4.31 Damped Response Using Numerical Methods Solution

95. 台灣師範大學機電科技學系 C. R. Yang, NTNU MT -95- 4.10 Examples using MATLAB To practice by yourself from Ex. 4.32 to Ex.4.35 The source codes of all MATLAB programs are given at the companion website