Dynamics of Rigid Bodies PDF

26/09/2024 INTRODUCTION TO DYNAMICS OF RIGID BODIES DYNAMICS ENGR. PAUL GERALD M. OCHOA...

26/09/2024 INTRODUCTION TO DYNAMICS OF RIGID BODIES DYNAMICS ENGR. PAUL GERALD M. OCHOA It is divided into two branches: WHAT IS DYNAMICS? 1. Kinematics is the geometry of motion and used Dynamics is the branch of mechanics which deals to define the motion of a particle or body without with the study of bodies in motion. The experiments consideration of the forces causing the motion. which form the foundation of dynamics require the (e.g. displacement, velocity, and acceleration). use of three kinds of units: force, length, and time. (Singer) (Singer) It is divided into two branches: The motion of the particle may either be rectilinear motion or curvilinear motion. 2. Kinetics is the analysis of the forces causing the Rectilinear motion refers to the particle as it moves motion. (Hibbeler, 2015). It relates the force acting along a straight line while particle in a curvilinear on a body to its mass and acceleration. (Singer) motion moves along a curved line in two or three dimensions. (Beer, 2019) 1  26/09/2024 Application of Dynamics in Engineering 1.Structural design of any vehicle such as DEFINITION OF TERMS automobile or airplane. (Hibbeler,2015) Body. It denotes a system of particles which form 2.Design of mechanical devices such as motors, an object of appreciable size. (Singer) pumps, movable tools, industrial manipulators, and machinery. (Hibbeler,2015) Particle. It usually denotes an object of point size. (Singer). It has a mass but negligible size and shape. 3.Predictions of the motions of artificial satellites, projectiles, and spacecraft. (Hibbeler,2015) Position. It is used to specify the location of a particle at any given instant DISPLACEMENT Displacement is defined to be the change in position Mathematically, displacement can be defined as: of an object. (Hibbeler, 2015). Say, a particle moves ∆𝒔 = 𝒔𝒇 − 𝒔0 from the initial position 𝑠0 to a final position 𝑠𝑓, where: then the displacement is the difference in distance Δs – displacement sf – final position between the two positions. so – initial position VELOCITY If the final position (sf) is to the right of the initial Velocity is defined as a vector measurement of the rate and position (s0), the displacement is positive. Likewise, if direction of motion; the speed at which something moves in one the final position (sf) is to the left of the initial position direction. The speed of a car traveling north on a major freeway (s0), the displacement is negative. (Hibbeler,2015) and the speed a rocket launching into space can both be measured using velocity. The displacement is a vector quantity. (Hibbeler,2015). This means it has a direction as well as a magnitude and is represented visually as an arrow that points from the initial position to the final position. (Khanacademy.org) 2  26/09/2024 In calculus terms, velocity is the first derivative of The average velocity of the particle over the time position with respect to time. You can calculate interval Δt is defined as the quotient of the velocity by using a simple formula that includes displacement Δs and the time interval Δt as rate, distance, and time. The average velocity is expressed in meters per second (m/s) or in feet per second (ft/s) The most common way to calculate the constant velocity of an object moving in a straight line is with The instantaneous velocity v of a particle at the this formula: instant t is determined by allowing the time interval Δt to become infinite similarly small. Thus, where: v - velocity or rate of speed s - distance travelled t - time it takes to complete the movement The instantaneous velocity is also expressed in m/s or ft/s. Difference Between Speed and Velocity Speed is a scalar quantity that indicates the rate of The SI (international) units for velocity are m/s motion distance per time. Its units are length and (meters per second), but velocity may also be time. Speed is often described simply as the expressed in any units of distance per time [e.g. distance traveled per unit of time. It is how fast an miles per hour (mph), kilometers per hour (kph), and object is moving. kilometers per second (km/s)]. Velocity is a vector quantity that gives the rate of motion of a particle in a certain direction. 3  26/09/2024 Consider the velocity v of the particle at time t and ACCELERATION also its velocity v + Δv at a later time t + Δt. Acceleration is defined as a vector quantity that indicates the rate of change of velocity. It has dimensions of length and time over time. Acceleration is often referred to as “speeding up”, but it really measures changes in velocity. An object is accelerating if it is changing its velocity. If the velocity of the particle is known at any two points, the average acceleration during the time interval ∆t is Acceleration can be experienced every day in a vehicle. defined as: The average acceleration You step on the accelerator and the car speeds up, increasing its velocity. is expressed in m/s2 or in ft/s2. Refresher ∆𝒔 = 𝒔𝒇 − 𝒔0 1.A student has aSample Problems displacement of 304 m north in 180 s. What was the student's average velocity? 2.Layla jogs with an average velocity of 2.4 m/s east. What is her displacement after 46 seconds? 3.Phillip walks along a straight path from his house to his school. How long will it take him to get to school if he walks 428 m west with an average velocity of 1.7 m/s west? 4.A trucker drives along a straight highway for 0.25 h with a KINEMATICS OF PARTICLES displacement of 16 km south. What is the trucker’s average velocity? 5.A bird flies with an average velocity of 7.5 m/s east from one branch to another in 2.4 s. It then pauses before flying with an average velocity of 6.8 m/s east for 3.5 s to another branch. What is the bird’s total displacement from its starting point? 4  26/09/2024 KINEMATICS OF PARTICLES RECTILINEAR MOTION OF PARTICLES Rectilinear motion is another name for straight-line Depending on the path of the particles, the motion motion. This type of motion describes the movement of may either be: a particle or a body. 1.Rectilinear Motion A body is said to experience rectilinear motion if any two particles of the body travel the same distance along 2.2. Curvilinear Motion two parallel straight lines. Following are the rectilinear motion examples: Types of Rectilinear Motion There are three types of rectilinear motion and they are: Use of elevators in public places is an example of rectilinear motion. 1. Uniform rectilinear motion Gravitational forces acting on objects resulting in free fall is an example of rectilinear motion. 2.Uniformly accelerated rectilinear motion Kids sliding down from a slide is a rectilinear motion. 3. Rectilinear movement with non-uniform acceleration Motion of planes in the sky is a rectilinear motion The three-basic equation of rectilinear motion are as follows: 𝑣𝑓 = 𝑣 + 𝑎𝑡 𝑠=𝑣 𝑡+ 𝑎𝑡 𝑣𝑓 = 𝑣 + 2𝑎𝑠 Where, t - the time of motion in seconds s - the distance(m) covered during the time t. vo - the initial velocity(m/s), at t=0. vf - the final velocity(m/s) after time t. a - the rate of acceleration(m/s2 ) 5  26/09/2024 x = x + v t + ½at² The position equation x = x₀ + v₀t + ½at² consists of three terms, each representing different aspects of an object's motion. The first term, x₀, represents the initial position of the object; it's where the object starts from. The second term, v₀t, accounts for the displacement due to the object's initial velocity (v₀) acting over a period of time (t). The third term, ½at², represents the additional displacement caused by the object's constant acceleration (a) over the square of the elapsed time (t²). Car B is travelling a distance d ahead of A. Both cars A freight train travels at v = 60(1 − 𝑒 ) ft/s, where are travelling a at 60 𝑓𝑡 𝑠 when the driver of B t is the elapsed time in seconds. Determine the suddenly applies the brakes, causing his car to distance traveled in three seconds, and the decelerate at 12 𝑓𝑡 𝑠 2. It takes the driver of car A acceleration at this time. 0.75s to react (this is the normal reaction time for drivers). When he applies his brakes, he decelerates at 15 𝑓𝑡 𝑠 2. Determine the minimum distance d between the cars to avoid collision. Rectilinear Movement with Non-Uniform Case 1: The displacement is given in terms of Acceleration time; s = f(t) to find velocity and acceleration. When an object travels at an irregular speed and acceleration it is known as rectilinear movement with non-uniform acceleration. When bodies are acted upon by variable forces, they move with variable acceleration.). Since the acceleration may vary in many ways, there is no general equation as in constant acceleration. (Singer) 6  26/09/2024 Case 3: The velocity is given in terms of time; v= f(t); to find acceleration and distance. This case is a Case 2: The acceleration is expressed in terms combination of Case I and Case II. Differentiating of time; a = f(t) to find velocity and distance. the given velocity-time relation determines the acceleration while integrating it determines the displacement Case 4: One of the principal variables is expressed in terms of adjacent variable; a =f(v) or v=f(s). The procedure here is to use either Case 5: The given variables are not adjacent, a =f(s). In this case, we substitute the given relation in 𝑣𝑑𝑣 = 𝑎𝑑𝑠 , separate the variables and integrate to to relate to the given variable in terms of the time. obtain one variable in terms of its adjacent variable. Comparison Between Uniform Acceleration CURVILINEAR MOTION and Non-uniform acceleration: Curvilinear motion is defined as motion that occurs when a particle travels along a curved path. The curved path can be in two dimensions (in a plane), or in three dimensions. 7  26/09/2024 Following are the curvilinear motion examples: cyclist racing on curved motion of cycle wheels tracks of velodrome motion of an earth moving around the earthworm sun motion of a spring a car taking a turn on a Curved jet motion, road a ball thrown upwards at Large sea waves, an angle Running race in curved throwing of a javelin track motion of a snake 8