Exercise of mechanical vibration Nov. 1st and 8th Q2: A stick with a mass m is supported by a frictionless pin, around which the stick rotates. An end of the stick is connected Q1: Consider a balancer depicted below. Answer its equation of to the wall by a spring. When 0, its motion and natural angular frequency. The mass of the stick can k be ignored and a small angle assumption is available: sin ≅ . length is natural. Gravity force acts on the mass. Answer the equation of motion of the system about θ and the period of free vibration.
m
Q3: A pulley, which rotates around the center of mass, with a string Q4: Consider a balancer that has two weights at each side and hangs mass m. The mass moment of inertia of the pulley is rotates around O. The weight of the stick is ignorable. Find the denoted by I. There is no slippage between the pulley and string. equation of motion of a balancer (やじろべえ) and its natural The string is supported by a spring of coefficient k. Answer the angular frequency p. period of free vibration of the system.
k O
R l m m a a
m g Given that θ ≪ 1, sinθ ≂ θ and cosθ ≂ 1 – θ2/2 Q5: When the arm of the balancer is circular, answer l such that the Q6: Consider the following system which consists of a disk of mass m and natural frequency becomes 1 Hz. spring. The spring is fixed to the disk at its center of gravity, around which the disk rotates. The motion of the disk is constrained along x‐ axis. The slippage between the disk and ground does not occur. Mass moment of the disk is I. (1) Answer the equation of motion about x. x (2) Answer the kinetic energy of translation (Tt) and rotation (Tr) using l and , respectively. k m r (3) Answer the potential energy of the spring (U). m m l l (4) Find the maximum T and U values. (5) When = cos , find the natural frequency of the system (p) based on the relationship of T ＝ .
Q7: 半径Rの円筒内で転がりながら微小振動をする質量m，半径r の円 Q8: Consider a homogeneous stick of which volume can be negligible. 柱を考える．円筒は，滑らないものとする．(1)‐(4)の問いに答えよ． Its lengths and mass are l and m, respectively. Answer its mass moment of inertia about the center O. (1) 並進の運動エネルギー を求めよ． Q9: Consider a rectangle plate made of a homogeneous material. Its (2) 円筒の慣性モーメントを I とするとき， R thickness is small enough to be neglected. The mass and r 円筒の回転の運動エネルギー を 求めよ． dimensions are m and 2a by 2b. X and Y axes are on the surface and along with its edges. Answer the mass moments of inertia 位置エネルギーU を求めよ． (3) about each of the three axes. (4) = cos として，円筒の最大角度 であり，最大角速度 Y Z となる．このとき， o X を について解け．な お， はこの系の固有角周波数であ る． b a 出題: テキストの演習問題 2.8a Q10： Figure (1) shows a simplified seesaw. The main plate is of length Q11: Translate the following paragraph on pulley dynamics into 2l and mass M. Its width and thickness are small enough to be Japanese. neglected. The supporting pivot O is at d from the plate. The The figure provides an example of pulley dynamics. The pulley’s gravity acceleration is g. Using the small inclination of the seesaw center is fixed, and forces f1 and f2 are the tensions in the cord on , establish the equation of motion. Also, answer the natural either side of the pulley. From law of rotational motion, frequency. . Furthermore, as is shown in Fig. (2), when a mass point is at f3 each end, answer the equation of motion and the natural The tension forces are equal if is negligible. frequency. This condition is satisfied if either the pulley rotates at a constant speed or the pulley (1) (2) θ inertia is negligible compared to the other R inertias in the system. The pulley inertia will d d be negligible if either its radius or its mass is M M m small. The force on the support at the pulley center is f = f + f .If is negligible, then the f1 f2 3 1 2 l m l support force is 2f1. l l
Q12: The inverted pendulum is connected by two springs as Q13: A slender rod of mass m and length l, moves around a circular shown in the figure. Assume small angles of vibration and track under the influence of gravity. The small rollers at the neglect the rod mass. ends of the rod are confined to the circular track and roll without friction. Neglect the mass of the rollers. The radius of a) Derive the equation of motion of the track is r and 3 . Find an equation of motion of the the system about . rod as a function of the coordinate, θ, as shown in the figure. b) What is the relation among , , , and for the characteristic roots to include at least one 3 positive real value? c) Describe the behavior of the 2 system when the characteristic roots include positive real values. Q14: A T‐shaped plate is made from two solid plates M1 and M2 of Q15: A cuckoo clock pendulum consists of two pieces glued uniform density. Their dimensions are shown in the figure. M1 together: and M2 have mass 2m and 2/3m, respectively. G1 and G2 are the centers of mass of each plate. ・ a slender rod of mass m1 and length l,and a) Find the location of the center of mass of the combined ・ a circular disk of mass m2 and radius r,centeredatthe slender rod’s midpoint. plate, G12, which lies somewhere on the vertical line. Express it as a distance s measured from the top of the The pendulum is attached at one end to a fixed pivot (center of object. rotation), O. , as shown below. The gravity acts. b) Find the mass moment of inertia of the plate about its a) Find the mass moment of inertia about O. gravity center, G . 12 a b) Find the pendulum’s angular momentum about O. M1 c) Find the equation of motion of the pendulum as a function s b G1 of θ. G12
b G2 M2
b
Q16: Consider a pendulum made by a homogeneous plate. Its dimensions are, a, b,andh as shown in the figure. The plate rotates around O under the gravity force, and small‐angle‐ O assumption applies. The material density is ρ = m /(abh) as the mass and thickness of the plate are m and h, respectively. θ is 0 at the equilibrium position of the system. θ (1) Obtain the plate’s mass moment of inertia about O. l r m2 (2) Obtain the equation of motion in terms of θ. (3) Find the natural angular frequency of the pendulum. (4) Solve the rotational behavior of the object θ(t) with the initial m1 conditions 0 and 0 0. (5) Solve the rotational behavior of the object with the initial conditions 0 0and 0 . Q17: Consider a homogenous plate of skewed T‐shape as is shown in the figure. It is composed of two parts M1 and M2, each of which has the same dimensions and mass m. (1) Obtain the mass moment of inertia about an axis passing O through O. (2) Obtain the centroid (center of gravity) of the plate using x and y. a θ O X
a Y M1 b/2 b/2 b/2 b M2
Q18: Consider a cylinder that rolls without slipping. Let x =0denote (1) Obtain the equation of motion in terms of x. the rest position of the cylinder. The mass of the cylinder is m. Neglect the mass of the spring. The mass moment of inertia of (2) Determine the natural angular frequency of the cylinder. the cylinder is I. (3) Let δ is the static deflection of the spring. Answer the kinetic and potential energy of the system.
(4) Suppose cos , answer the maximum kinetic and x potential energy of the cylinder as well as the minimum k potential energy. R (5) Solve for p to obtain the natural angular frequency of the cylinder. θ φ Q19: As shown in the figure, one end of the spring of constant k is 3) Answer the equation of motion about x. Use m, M, k, and a, fixed to the wall. The other end hangs the mass of m through and not T and T’. the a pulley. The radius and mass of the pulley are a and M, 4) Answer kinetic energy of K in this system. respectively. The pulley rotates smoothly around its fixed 5) Answer potential energy of U in this system. central axis and the mass vibrates along the vertical direction. 6) By Lagrange’s method, answer the motion equation. The string does not slip on the pulley. x and θ are the T displacement of the mass and the rotation angle of the pulley M from their statistically equivalent position. θ 1) What is the moment of inertia of the pulley about its T’ rotation axis? 2) When the displacement of the mass is x, answer tension T using x and k. Furthermore, answer the relation between x Displacement from the T’ and θ. statically equivalent position x
Q20: Consider a disk with a handle. The disk rotates around its Q21: Consider a thin circular disk of radius a and mass m. Answer the center, which is fixed to the ground. Two ends of the disk are mass moment of inertia around axis Y. linked to the wall by springs of coefficient k. Answer the following problems. Q22: Consider a cylinder of radius a and height l. Answer the 1) Mass of the disk and handle is m, for each. Answer the mass moment of inertia around axis Y. mass moment of inertia of the disk with a handle around the rotation axis. Y 2) Establish the equation of motion about θ. d Y 3) Answer the natural frequency of the disk. rdrd dr Length l a X a Z θ
Radius r Q23: A grandpa accidentally dropped a rice ball (weight m, radius r), Q24: A rod of mass M and length 2a revolves around its frictionless and it rolled on a curved surface of which radius was R. The rice center O. Thickness of the rod is small enough to be neglected. A ball did not slip on the surface. The angle of the rice ball is weight m is attached at b from the center. The gravity force acts on denoted by θ. the mass and rod, and they behave as a pendulum. Answer the (1) Assume that the rice ball is following problems. a homogeneous sphere. (1) Answer the mass moment of Answer the moment of inertia of the pendulum around O inertia of the rice ball . M a around its centroid. R (2) Answer the centroid of the (2) Answer the equation of θ pendulum from O. O (3) Using a small angle assumption motion of the rice ball Rice about θ. (sinθ ~ θ), determine the b a ball m (3) Using a small angle φ natural frequency of the assumption (sinθ ~ θ), pendulum. g r determine the natural (4) Determine b, such that the frequency of the ball. mg pendulum rotates most quickly.
Schedule Answer Q1: Q2: • 7th, Nov. (Lecture) 13:00‐ 0 0 • 8th, Nov. (Exercise) 1st mini quiz. 9:30‐
• 14th, Nov. (Lecture) 13:00‐ • 15th, Nov. (Exercise) 9:30‐ • 21th, Nov. (Lecture) 13:00‐ • 22th, Nov. (Exercise) 9:30‐ • 28th, Nov. (Lecture) 13:00‐ 減災館での体験授業, 減災館に集合 Q3: Q4: • 29th, Nov. (Exercise) 9:30‐ 0 0 • … • 26th, Dec. (Lecture and exercise) 9:30‐12:00 • The whole schedule is listed in the course web site. • Search with “名古屋大学 講義 振動学 岡本” Q5: Q8: Q9: 解答省略 8 12
Q6: Q10: Q11: 解答省略 (1) (3) (1) 1 1 2 2 ． Q12: 0 3 (4) a) 2 0
(2) (2) b) 2 0 1 , 2 2 c) The pendulum falls down. 1 1 2 2 2 3 Q7: テキストに回答が掲載されています．
Q16: Q17: Q13: (1) (1) sin 0 m m I 4a2 b2 I 20a2 5b2 ab O 12 O 12 Q14: 3 (2) (2) a) 4 mga 回転中心Oから下記の位置S 0 d1 and d2 are the distances 2I b) O OX from G12 to G1 and G2, 1 3 respectively. (3) r1 a 4 4 mga COG1 (3a+b)/4 p r2 Y 2 19 2I O 12 9 6 6 S a/4 (4) Q15: b 1 cos COG2 a) 3 2 4 (5) 1 sin Q18: b) c) sin 0 3 2 4 2 Q6と同じ運動方程式になります． Q19: Q20: 1 T kx 1 2 2 1) I Ma 2) 2 2 x a 1 0 2 1 1 3) M mx kx 0 2 3
3 4) 2 1 5) U kx2 2 Q21: Q22: 6) 1 1 1 4 4 12
Q23: Q24: (1) (2) (1) (2) 1 2 2 I mr 3 5 (3) (4) (3) 1 mR mg sin F 3 5g 5g 3 I rF 0 p 7R 7R R r