WeightPleFigure 4: Schematic of a pile driverf(t) ↓mdum² + bv=f(1)bypileFigure 5: First-order model of the pile insoil5. Figure 4 shows a pile driver that is used to drive piles into soil to provide foundation supportfor buildings. Two major components of a pile driver are a pile and a weight. The pile is inthe form of a long rod going into the soil. The weight is to impact the pile at the top withan impulsive force, thus driving the pile into the soil. Figure 5 shows a first-order model ofthe pile in the soil. The impulsive force produced by the weight is f(t). The resistance of thesoil is modeled via a viscous damping coefficient. b. The weight of the pile is balanced by thenormal forces in the soil, so it does not need to be considered. In addition, the elasticity ofthe soil is ignored. Let z(t) and v(t) be the downward displacement and velocity of the pile,respectively. The equation of motion governing the pile velocity v(1) iswhere m is the mass of the pile. Answer the following questions.(a) What is the time constant 7 of the system?(b) Immediately after the impact from the weight at t=0, the load f(t) disappears andthe pile picks up an initial velocity . What is the velocity (1) afterwards? Derive thevelocity v(1) using the initial velocity 0.(c) When the velocity v(1) is less than 5% of to, the pile is considered as stopped and it is safeto apply another impact force. With this safety guideline, what is the minimal durationneeded between two consecutive impulsive loads? Represent the minimal duration interms of the time constant 7.(d) The weight has a mass mo, and the weight is raised for a height ho and released. Whenthe weight is dropped, the impulse produced by the weight is I = m mo√2gho, where gis the gravitational acceleration. Assume that the pile is stationary before the weightis dropped. What is the velocity to of the pile immediately after the impact? Pleasejustify your answer.(e) Based on the velocity v(1) you obtain in part (b), how long does it take for the pile tostop entirely (i.c., to reach zero velocity) under the first-order model in equation (5)?What distance does the pile travel before it stops? Represent the distance in terms ofthe initial velocity and the time constant. 7. [Hint: fedt = c/a + constant.]T.(f) The manager on-site wants to increase the distance travelled per impact. Engineer Asuggests that a heavier pile be used. Engineer B suggests the opposite. Who is correctand why? Use the time constant obtained in part (a) and the velocity to obtained inpart (d) to justify your answer.

Figure 4 shows a pile driver that is used to drive piles into soil to provide foundation support for buildings. Two major components of a pile driver are a pile and a weight. The pile is in the form of a long rod going into the soil. The weight is to impact the pile at the top with an impulsive force, thus driving the pile into the soil. Figure 5 shows a first-order model of the pile in the soil. The impulsive force produced by the weight is f(t). The resistance of the soil is modeled via a viscous damping coefficient. b. The weight of the pile is balanced by the normal forces in the soil, so it does not need to be considered. In addition, the elasticity of the soil is ignored. Let a(t) and v(1) be the downward displacement and velocity of the pile, respectively. The equation of motion governing the pile velocity v(t) is m+bv=f(1) where m is the mass of the pile. Answer the following questions. (a) What is the time constant 7 of the system? (b) Immediately after the impact from the weight at t=0, the load f(t) disappears and the pile picks up an initial velocity to. What is the velocity v(t) afterwards? Derive the velocity v() using the initial velocity vo (e) When the velocity v() is less than 5% of to, the pile is considered as stopped and it is safe to apply another impact force. With this safety guideline, what is the minimal duration needed between two consecutive impulsive loads? Represent the minimal duration in terms of the time constant. 7. (d) The weight has a mass mo, and the weight is raised for a height ho and released. When the weight is dropped, the impulse produced by the weight is I = mo√2gho, where g is the gravitational acceleration. Assume that the pile is stationary before the weight is dropped. What is the velocity to of the pile immediately after the impact? Please justify your answer. (e) Based on the velocity v(1) you obtain in part (b), how long does it take for the pile to stop entirely (i.c., to reach zero velocity) under the first-order model in equation (5)? What distance does the pile travel before it stops? Represent the distance in terms of the initial velocity p and the time constant r. [Hint: fe" dt = c/a+constant.] (f) The manager on-site wants to increase the distance travelled per impact. Engineer A suggests that a heavier pile be used. Engineer B suggests the opposite. Who is correct and why? Use the time constant obtained in part (a) and the velocity to obtained in part (d) to justify your answer.