Mechanical Vibrations: Response to Nonperiodic Loadings PDF

University of Kufa College of Engineering Mechanical Engineering Department Mechanical Vibrations Lecture 13: Response to Nonperiodic Loadings Lecturers: Dr. Muhannad Lafta Al-Waily Mohammed Fakhir Al-Edhari  University of...

University of Kufa College of Engineering Mechanical Engineering Department Mechanical Vibrations Lecture 13: Response to Nonperiodic Loadings Lecturers: Dr. Muhannad Lafta Al-Waily Mohammed Fakhir Al-Edhari  University of Kufa Mechanical Vibrations Lecture 13: Response to Nonperiodic Loadings Lecturer: Dr. Muhannad Lafta Al-Waily Teaching Assistant: Mohammed Fakhir Al-Edhari 1 Impulse Impulse is a force applied for a very short of time which it is large enough to cause change in momentum. It is determined as the force integrated over the time it is acting. As shown, ∫ 𝑡2 𝐹˜ = 𝑡1 𝐹 (𝑡)𝑑𝑡 𝐹 (𝑡) 𝑡 𝑡1 𝑡2 The impulse is defined as, ∫ 𝐹˜ = 𝐹 (𝑡)𝑑𝑡 We will first consider an impulse that takes place over a very short period of time so that it appears as ∫ 𝑡2 𝐹˜ = 𝑡1 𝐹𝑑𝑡 𝐹 (𝑡) 𝑡 𝜏 From dynamics we know that the impulse is equal to the change in momentum. 𝐹˜ = 𝑚 𝑥¤𝑎 𝑓 𝑡𝑒𝑟 − 𝑥¤ 𝑏𝑒 𝑓 𝑜𝑟𝑒
1 Lecture 13: Response to Nonperiodic Loadings 2 Consider a SDOF spring-mass-damper system that is at rest at time 𝑡=0, so that 𝑥¤ 𝑏𝑒 𝑓 𝑜𝑟𝑒 = 0. The velocity directly after impact can be found to be, 𝐹˜ 𝑥¤𝑎 𝑓 𝑡𝑒𝑟 = 𝑚 Concept 1 𝐹˜ Impulse causes initial velocity change of 𝑥¤ = while maintaining the value of displacement 𝑚 unchanged. Concept 2 𝐹˜ Impulse response is just free vibration of a system with initial velocity (𝑥¤ 𝑜 = ) 𝑚 1.1 Transient Response of Under-Damped System to an Impulse We can consider this to be the initial condition for the mass initial with 𝑥 𝑜 =0. Depending on whether the mass is a) un-damped, b) under-damped, c) critically-damped, d) or overdamped we can use the free-vibration expression to find the response to the impulse. The initial velocity is, 𝐹˜ 𝑥¤𝑎 𝑓 𝑡𝑒𝑟 = 𝑥¤ 𝑜 = 𝑚 For example, if the mass is underdamped, 𝜁 < 1, then we would use the free-response expression for an under-damped mass,   −𝜁𝜔 𝑛 𝑡 𝑥¤ 𝑜 + 𝜁𝜔𝑛 𝑥 𝑜 𝑥(𝑡) = 𝑒 𝑥 𝑜 𝑐𝑜𝑠𝜔 𝑑 𝑡 + 𝑠𝑖𝑛𝜔 𝑑 𝑡 𝜔𝑑 𝐹˜ After substituting in 𝑥 𝑜 =0 and 𝑥¤ 𝑜 = then the transient response would be, 𝑚 ˜ −𝜁𝜔𝑛 𝑡 𝑠𝑖𝑛𝜔 𝑑 𝑡 𝑥(𝑡) = 𝐹𝑒 𝑚𝜔 𝑑 Lecture 13: Response to Nonperiodic Loadings 3 This is the solution when the impulse occurs at time 𝑡 = 0. Suppose the impulse occurs when 𝑡 ≠ 0. We choose a time when 𝑡 > 0 and call the moment when the impulse occurs, 𝜏. The response is static for times less than 𝜏. We can re-write the transient response as, 0 0 𝑡𝑜 Lecture 13: Response to Nonperiodic Loadings 10 For undamped system: 1 𝑔(𝑡 − 𝜏) = 𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 𝑚𝜔𝑛 ∫ 𝑡 𝑥(𝑡) = 𝐹 (𝜏)𝑔(𝑡 − 𝜏)𝑑𝜏 0 ∫ 𝑡 1 = 𝐹𝑜 𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 𝑚𝜔𝑛 0 From 0 ≤ 𝑡 ≤ 𝑡 𝑜 ∫ 𝑡  𝑡 𝐹𝑜 𝐹𝑜 𝑥(𝑡) = 𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 = 𝑐𝑜𝑠𝜔𝑛 (𝑡 − 𝜏) 𝑚𝜔𝑛 0 𝑚𝜔2𝑛 0   𝐹𝑜 𝐹𝑜
𝑥(𝑡) = 𝑐𝑜𝑠𝜔𝑛 (𝑡 − 𝑡) − 𝑐𝑜𝑠𝜔𝑛 𝑡 = 1 − 𝑐𝑜𝑠𝜔𝑛 𝑡 𝑘 𝑘 From t > 𝑡 𝑜 ∫ 𝑡𝑜 ∫ 𝑡   𝑡𝑜 𝐹𝑜 𝐹𝑜 𝑥(𝑡) = 𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 + 0𝑑𝜏 = 𝑐𝑜𝑠𝜔𝑛 (𝑡 − 𝜏) 𝑚𝜔𝑛 0 𝑡𝑜 𝑚𝜔2𝑛 0   𝐹𝑜 𝐹𝑜
𝑥(𝑡) = 𝑐𝑜𝑠𝜔𝑛 (𝑡 − 𝑡 𝑜 ) − 𝑐𝑜𝑠𝜔𝑛 𝑡 = 𝑐𝑜𝑠𝜔𝑛 (𝑡 − 𝑡 𝑜 ) − 𝑐𝑜𝑠𝜔𝑛 𝑡 𝑘 𝑘 Lecture 13: Response to Nonperiodic Loadings 11 Example 4 A compacting machine, modeled as a single-degree-of-freedom system, is shown in Fig.(a). The force acting on the mass 𝑚 (𝑚 includes the masses of the piston, the platform, and the material being compacted) due to a sudden application of the pressure can be idealized as a step force, as shown in Fig.(b). Determine the response of the system assuming the system is underdamped. Solution The applied force is 𝐹 (𝑡) = 𝐹𝑜 for 𝑡 > 0, ∫ 𝑡 𝐹𝑜 𝑥(𝑡) = 𝑒 −𝜁𝜔𝑛 (𝑡−𝜏) 𝑠𝑖𝑛𝜔 𝑑 (𝑡 − 𝜏)𝑑𝜏 𝑚𝜔 𝑑 0    𝑡 𝐹𝑜 −𝜁𝜔 𝑛 (𝑡−𝜏) 𝜁𝜔 𝑛 𝑠𝑖𝑛𝜔 𝑑 (𝑡 − 𝜏) + 𝜔 𝑑 𝑐𝑜𝑠𝜔 𝑑 (𝑡 − 𝜏) = 𝑒 𝑚𝜔 𝑑 (𝜁𝜔𝑛 ) 2 + (𝜔 𝑑 ) 2 0 √︃ 𝐹𝑜  1 − 𝑒 −𝜁𝜔𝑛 𝑡 𝜁 𝑠𝑖𝑛𝜔 𝑑 𝑡 + 1 − 𝜁 2 𝑐𝑜𝑠𝜔 𝑑 𝑡 𝑚
 = 𝑘 Lecture 13: Response to Nonperiodic Loadings 12 Example 5 A building frame is modeled as an undamped single-degree-of-freedom system (Fig.(a)). Find the response of the frame if it is subjected to a blast loading represented by the triangular pulse shown in Fig.(b). Solution 𝜏
 𝐹𝑜 1 − 0 ≤ 𝜏 ≤ 𝑡𝑜    𝐹 (𝜏) = 𝑡𝑜 0  𝜏 > 𝑡𝑜  For an undamped system, ∫ 𝑡 1 𝑥(𝑡) = 𝐹 (𝜏)𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 𝑚𝜔𝑛 0 Response during 0 ≤ t ≤ 𝑡 𝑜 gives ∫ 𝑡   𝐹𝑜 𝜏 𝑥(𝑡) = 1− 𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 𝑚𝜔𝑛 0 𝑡𝑜 Integration by part gives ∫ 𝜏𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝜏)) 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏)) 𝜏𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏))𝑑𝜏 = + 𝜔𝑛 𝜔2𝑛 The response can be written as    𝑡 𝐹𝑜 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝜏)) 1 𝜏𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝜏)) 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏)) 𝑥(𝑡) = − + 𝑚𝜔𝑛 𝜔𝑛 𝑡𝑜 𝜔𝑛 𝜔2𝑛 0 𝐹𝑜  𝑡 1  = 1 − − 𝑐𝑜𝑠𝜔𝑛 𝑡 + 𝑠𝑖𝑛𝜔𝑛 𝑡 𝑚 𝑘 𝑡𝑜 𝜔𝑛 𝑡 𝑜 Lecture 13: Response to Nonperiodic Loadings 13 Response during 𝑡 > 𝑡 𝑜 gives    𝑡 𝐹𝑜 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝜏)) 1 𝜏𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝜏)) 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏)) 𝑜 𝑥(𝑡) = − + 𝑚𝜔𝑛 𝜔𝑛 𝑡𝑜 𝜔𝑛 𝜔2𝑛 0   𝐹𝑜 = 𝑠𝑖𝑛(𝜔𝑛 𝑡) − 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝑡 𝑜 )) − 𝑡 𝑜 𝜔𝑛 𝑐𝑜𝑠(𝜔𝑛 𝑡) 𝑚 𝑘𝜔𝑛 𝑡 𝑜 Example 6 Find the response of the mass-spring system when it is subjected to the force shown in Fig. 𝐹 (𝑡) 𝑘 𝐹𝑜 𝑚 𝑡 𝑡1 𝑡2 Solution  𝜏    𝐹𝑜 0 ≤ 𝜏 ≤ 𝑡1  𝑡1   𝐹 (𝜏) = 𝐹𝑜 𝑡1 ≤ 𝜏 ≤ 𝑡2    0  𝜏 > 𝑡2  For an undamped system, ∫ 𝑡 1 𝑥(𝑡) = 𝐹 (𝜏)𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 𝑚𝜔𝑛 0 Response during 0 ≤ t ≤ 𝑡1 gives ∫ 𝑡 𝐹𝑜 𝜏 𝑥(𝑡) = 𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 𝑚𝜔𝑛 0 𝑡1 Integration by part gives ∫ 𝜏𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝜏)) 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏)) 𝜏𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏))𝑑𝜏 = + 𝜔𝑛 𝜔2𝑛 The response can be written as, Lecture 13: Response to Nonperiodic Loadings 14  𝑡 𝐹𝑜 𝜏𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡𝑎𝑢)) 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏)) 𝑥(𝑡) = + 𝑚𝜔𝑛 𝑡1 𝜔𝑛 𝜔2𝑛 0 𝐹𝑜   = 𝜔𝑛 𝑡 − 𝑠𝑖𝑛𝜔𝑛 𝑡 𝑚 𝑘𝜔𝑛 𝑡1 Response during 𝑡1 ≤ t ≤ 𝑡2 gives ∫ 𝑡1 ∫ 𝑡 1 𝜏 1 𝑥(𝑡) = 𝐹𝑜 𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 + 𝐹𝑜 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏))𝑑𝜏 𝑚𝜔𝑛 0 𝑡1 𝑚𝜔𝑛 𝑡1  𝑡  𝑡 𝐹𝑜 𝜏𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡𝑎𝑢)) 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏)) 1 𝐹𝑜 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡 1 )) = + + 𝑚𝜔𝑛 𝑡1 𝜔𝑛 𝜔2𝑛 0 𝑚𝜔𝑛 𝜔𝑛 𝑡1     𝐹𝑜 𝐹𝑜 = 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝑡1 )) + 𝑡1 𝜔𝑛 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡1 )) − 𝑠𝑖𝑛𝜔𝑛 𝑡 + 1 − 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡1 )) 𝑚 𝑘𝜔𝑛 𝑡1 𝑘 Response during 𝑡 > 𝑡2 gives ∫ 𝑡1 ∫ 𝑡2 1 𝜏 1 𝑥(𝑡) = 𝐹𝑜 𝑠𝑖𝑛𝜔𝑛 (𝑡 − 𝜏)𝑑𝜏 + 𝐹𝑜 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏))𝑑𝜏 𝑚𝜔𝑛 0 𝑡1 𝑚𝜔𝑛 𝑡1  𝑡  𝑡 𝐹𝑜 𝜏𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡𝑎𝑢)) 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝜏)) 1 𝐹𝑜 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡1 )) 2 = + + 𝑚𝜔𝑛 𝑡1 𝜔𝑛 𝜔2𝑛 0 𝑚𝜔𝑛 𝜔𝑛 𝑡1   𝐹𝑜 = 𝑠𝑖𝑛(𝜔𝑛 (𝑡 − 𝑡1 )) + 𝑡1 𝜔𝑛 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡 1 )) − 𝑠𝑖𝑛𝜔𝑛 𝑡 𝑘𝜔𝑛 𝑡1   𝐹𝑜 + 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡2 )) − 𝑐𝑜𝑠(𝜔𝑛 (𝑡 − 𝑡1 )) 𝑚 𝑘 Lecture 13: Response to Nonperiodic Loadings 15 Homework Conceptual Questions 1. Any nonperiodic function can be represented by (A) harmonic function (B) sum of periodic function (C) convolution integral (D) Fourier series 2. The transient part of the solution arises from (A) forcing function (B) initial conditions (C) bounding conditions (D) both A and B 3. An impulse 2 N-s impacts an undamped mass at t=𝜋 s. The mass is initially at rest. What is the initial velocity of the mass just after impact? (A) 𝑥¤ 𝑜 = 0.25 m/s (B) 𝑥¤ 𝑜 = 0.5 m/s (C) 𝑥¤ 𝑜 = 4 m/s (D) 𝑥¤ 𝑜 = 2 m/s 4. What is the position of the mass at time t=5𝜋/4 s if the impact occurs at t=𝜋s? (A) 0.25 m (B) 0 m (C) 0.639 m (D) 1.14 m 5. The change in momentum is called impulse. (A) True (B) False Lecture 13: Response to Nonperiodic Loadings 16 6. The response of a system under arbitrary force can be found by summing the responses due to several elementary impulses. (A) True (B) False Problems 1. A SDOF system with 𝑚 = 10 kg, 𝑘 = 40,000 N/m, and 𝑐 = 300 N.s/m is at rest in equilibrium when a 80 N s impulse is applied. This is followed by a 40 N s impulse 0.02 s later. Determine the response of the system. 2. Sandblasting is a process in which an abrasive material, entrained in a jet, is directed onto the surface of a casting to clean its surface. In a particular setup for sandblasting, the casting of mass 𝑚 is placed on a flexible support of stiffness k as shown in Fig.(a). If the force exerted on the casting due to the sandblasting operation varies as shown in Fig.(b), find the response of the casting. 3. A wave consisting of the wake from a passing boat impacts a seawall. It is desired to calculate the resulting vibration. Fig. illustrates the situation and suggests a model. Calculate the response of the seawall–dike system to such a load. Lecture 13: Response to Nonperiodic Loadings 17 4. A compressed air cylinder is connected to the spring-mass system shown in Fig.(a). Due to a small leak in the valve, the pressure on the piston, 𝑝(𝑡), builds up as indicated in Fig. (b). Find the response of the piston for the following data: 𝑚 = 10 kg, 𝑘 = 1000 N/m, and 𝑑 = 0.1 m. Lecture 13: Response to Nonperiodic Loadings 18 5. Find the transient response of an undamped spring-mass system for 𝑡 > 𝜋/𝜔 when the mass is subjected to a force 𝐹𝑜 𝜋  (1 − 𝑐𝑜𝑠𝜔𝑡) for 0 ≤ 𝑡 ≤    𝐹 (𝑡) = 2 𝜋 𝜔   𝐹𝑜 for 𝑡 >  𝜔 6. Use the Dahamel integral method to derive expressions for the response of an undamped system subjected to the forcing functions shown in Figs. (a) to (c). 7. Fig. shows a one degree of freedom model of a motor vehicle traveling in the horizontal direction. Find the relative displacement of the vehicle as it travels over a road bump of the form y(s) = Y sin𝜋𝑠/𝛿. Lecture 13: Response to Nonperiodic Loadings 19 8. A vehicle traveling at a constant speed 𝑣 in the horizontal direction encounters a trian- gular road bump, as shown in Fig. Treating the vehicle as an undamped spring-mass system, determine the response of the ve- hicle in the vertical direction. 9. An automobile, having a mass of 1000 kg, runs over a road bump of the shape shown in Fig. The speed of the automobile is 50 km/hr. If the undamped natural period of vibration in the vertical direction is 1.0 sec, find the response of the car by assuming it as a single-degree-of-freedom undamped system vibrating in the vertical direction. 10. A camcorder of mass 𝑚 is packed in a container using a flexible packing material. The stiffness and damping constant of the packing material are given by 𝑘 and 𝑐, respectively, and the mass of the container is negligible. If the container is dropped accidentally from a height of ℎ onto a rigid floor (see Fig.), find the motion of the camcorder. Lecture 13: Response to Nonperiodic Loadings 20 11. An airplane, taxiing on a runway, encounters a bump. As a result, the root of the wing is subjected to a displacement that can be expressed as ( 𝑦(𝑡 2 /𝑡 𝑜2 ) for 0 ≤ 𝑡 ≤ 𝑡𝑜 𝑦(𝑡) = 0 for 𝑡 > 𝑡 𝑜 Find the response of the mass located at the tip of the wing if the stiffness of the wing is 𝑘 (see Fig.). 12. Find the response of the rigid bar shown in Fig. using convolution integral for the following data : 𝑘 1 = 𝑘 2 = 5000 N/m, 𝑎 = 0.25 m, 𝑏 = 0.5 m, 𝑙 = 1 m, 𝑀 = 50 kg, 𝑚 = 10 kg, and 𝐹𝑜 = 500 N. Lecture 13: Response to Nonperiodic Loadings 21 13. Find the response of the mass shown in Fig. under the force 𝐹 (𝑡) = 𝐹𝑜 𝑒 −𝑡 using convolution integral. Data: 𝑘 1 = 1000 N/m, 𝑘 2 = 500 N/m, 𝑟 = 5 cm, 𝑚 = 10 kg, 𝐽𝑜 = 1 kg-m2 , 𝐹𝑜 = 50 N. 14. The frame, anvil, and base of the forging hammer shown in Fig.(a) have a total mass of 𝑚. The support elastic pad has a stiffness of 𝑘. If the force applied by the hammer is given by Fig.(b), find the response of the anvil. Lecture 13: Response to Nonperiodic Loadings 22 15. A bird strike on the engine of an airplane can be considered as an impulse (Fig.(a)). If the stiffness and damping coefficient of the engine mount are given by 𝑘 = 50000 N/m and 𝑐 = 1000 N-s/m, and the engine mass is 𝑚 = 500 kg, find the response of the engine. Assume the mass of the bird as 4 kg and the velocity of the airplane as 250 km/hr. 16. A bronze ball of mass 𝑚 𝑜 is dropped on the mass of a single-degree-of-freedom system from a height h as shown in Fig. If the ball is caught after its first bounce, determine the resulting displacement response of the mass 𝑚. Assume that the collision is perfectly elastic and the system is at rest initially. Data: 𝑚 = 2 kg, 𝑚 𝑜 = 0.1 kg, 𝑘 = 100 N/m, 𝑐 = 5 N-s/m, and ℎ = 2 m. Lecture 13: Response to Nonperiodic Loadings 23 17. A machine tool with a mass of 50 kg is mounted on an undamped foundation of stiffness 3000 N/m. During operation, it is subject to one of the machining force shown in Fig. Use the principle of superposition and the convolution integral to determine the response of the system to each force. 18. A power line pole with a transformer is modeled by 𝑚 𝑥¥ + 𝑘𝑥 = − 𝑦¥ where 𝑥 and 𝑦 are as indicated in Fig. Assuming the initial conditions are zero, calculate the response of the relative dis- placement (𝑥 − 𝑦) if the pole is subject to an earthquake-based excitation of     𝑡 𝐴 1 −   for 0 ≤ 𝑡 ≤ 2𝑡 𝑜 𝑦¥ (𝑡) = 𝑡𝑜  0  for 𝑡 > 2𝑡 𝑜  Lecture 13: Response to Nonperiodic Loadings 24 19. A machine tool with a mass of 30 kg is mounted on an undamped foundation of stiffness 1500 N/m. During operation, it is subject to one of the machining force shown in Figs (A) through (F). Use the principle of superposition and the convolution integral to determine the response of the system to each force. Reference Singiresu. S. Rao , Mechanical Vibrations, 5th Edition, Prentice Hall.