1. History of Dynamics; Motion in Moving Reference Frames

Introduction

The professor introduces the course and explains how mechanical engineering courses work.

• The subjects 2001 through 2005 are foundational to mechanical engineering.

• Mechanical engineering uses observations of the world to pose problems and produce models that explain them.

• Models are tested against observations and measurements, and improved over time.

Problem Solving Methodology

The professor explains the three-step methodology for solving dynamic problems in mechanical engineering.

• To solve a problem, you need to describe the motion, choose the physical laws to apply (such as f equals ma or conservation of energy), and apply the correct math.

• Most dynamic problems can be broken down this way.

• This methodology fits into our models box, which is tested against observations and measurements.

History of Dynamics

The professor gives a brief history of dynamics by discussing key figures in chronological order.

• Copernicus was Polish and proposed that the sun is at the center of the solar system instead of Earth.

• Brahe was an imperial mathematician who wrote about astronomy around 1600.

• Kepler was a German astronomer who discovered three laws of planetary motion.

• Galileo was an Italian physicist who made important contributions to mechanics, including discovering that objects fall at the same rate regardless of their mass.

• Descartes was a French philosopher who made contributions to geometry and mechanics.

• Newton was an English physicist who developed calculus and formulated three laws of motion that form the basis for classical mechanics.

• Euler was a Swiss mathematician who made important contributions to many areas of mathematics, including calculus, number theory, graph theory, and mechanics.

-Lagrange was an Italian-French mathematician who made important contributions to many areas of mathematics, including calculus, number theory, graph theory, and mechanics.

The History of Dynamics

This section provides an overview of the history of dynamics and the contributions made by various scientists.

Key Figures in Dynamics

• Johannes Kepler used observations to come up with the three laws of planetary motion, including a statement on conservation of angular momentum.

• Galileo Galilei turned his telescope towards Jupiter in 1609 and discovered its four moons, which helped disprove the Ptolemaic view of the solar system.

• René Descartes developed analytic geometry and introduced Cartesian coordinates, which combined algebra, coordinates, and geometry.

• Sir Isaac Newton formulated his three laws of motion in 1666 and taught us about linear momentum. Euler put Newton's three laws into mathematics and taught us about angular momentum.

• Joseph-Louis Lagrange used energy methods to derive equations of motion without using Newton's laws or direct methods.

Course Outline

• The course will start with kinematics using analytic geometry before moving onto Newton's three laws and direct methods for finding equations of motion.

• Conservation of momentum will be discussed as well as torque being dh/dt in most cases.

• Angular momentum will be covered extensively since it is important for understanding rigid body rotations.

• In the last third of the course, Lagrange's energy method will be introduced as another way to derive equations of motion.

• Vibration examples will be studied throughout the course to apply these different methods to modeling and solving interesting vibration problems.

Introduction to Modeling and Motion

In this section, the professor introduces the course by discussing an example of modeling motion using physical laws and math. He also explains the importance of choosing a coordinate system to describe motion.

Describing Motion

• To describe motion, a coordinate system must be assigned so that the object's movement can be tracked. The professor chooses an xyz Cartesian coordinate system.

• The origin of the coordinate system marks both the origin and names the frame. This is important because reference frames are used throughout the course.

• An inertial reference frame is fixed to Earth and not moving. It will be used in this course to describe motion.

• The x-coordinate will be used to describe the motion of a mass from its zero spring force position.

Applying Physical Laws

• Newton's second law (sum of external forces = mass times acceleration) will be applied to solve problems in this course.

• A free body diagram (FBD) is needed to apply physics laws. FBDs show all external forces acting on an object.

Vibration Problem Example

In this section, the professor uses a vibration problem as an example for applying physical laws and solving equations of motion.

Problem Description

• The problem involves a spring, mass, and natural frequency.

• To begin solving this problem, we need to follow a modeling method that arrives at an equation of motion for it.

Equation of Motion

• To find an equation of motion for this problem, we need to apply physical laws such as Newton's second law.

• We also need to use math skills to solve these equations.

Free Body Diagrams

In this section, the professor discusses how to create free body diagrams (FBDs) and the importance of sign conventions.

Creating FBDs

• FBDs show all external forces acting on an object.

• To create an FBD, start with known forces such as gravity and draw them in the direction they act.

• For less obvious forces, assume positive values for deflections and velocities. Then, determine which way the resulting force acts based on the positive deflection.

Sign Conventions

• The professor emphasizes that sign conventions are important when creating FBDs.

• Confusion about signs is a common mistake people make when creating FBDs.

• The professor has simple rules for creating FBDs that help avoid confusion about signs.

Deriving the Equation of Motion

In this section, the professor explains how to derive the equation of motion using Newton's laws and energy considerations.

Deriving the Equation of Motion Using Newton's Laws

• The constitutive relationship for spring force is fs=kx and fd=bx dot.

• The sum of forces in the x direction is fs+fd-mg.

• The equation of motion is mx double dot + bx dot + kx = mg.

Deriving the Equation of Motion Using Energy Considerations

• The total energy of the system is a sum of kinetic and potential energies.

• If there are no damping forces, then the total energy must be constant.

• By taking time derivative, we can solve for equation of motion without looking at conservation momentum or Newton's laws.

Conclusion

• We have described two methods for deriving equations of motion: using Newton's laws and using energy considerations.

Introduction to Reference Frames and Vectors

In this section, the speaker introduces the topic of reference frames and vectors in kinematics. The Cartesian coordinate system is discussed as a means of describing motion.

Describing Positions with Vectors

• A fixed frame called O-xyz or O for short is used as an inertial frame of reference.

• The positions of points A and B are described by vectors R with respect to point O.

• Velocity and acceleration can be obtained by taking time derivatives of RBO dt.

Derivatives of Vectors in Moving Frames

• Taking derivatives of vectors in moving frames becomes more complex when dealing with rigid bodies that are moving and rotating.

• Derivatives of vectors in moving frames require accounting for the rotation and movement of the body.

Calculus Basics

• Basic calculus concepts such as adding vectors, dot products, and derivatives are important to remember.

• The derivative of a product of two things involves taking the derivative of one function times another plus the derivative of the other function times one.

Using Cartesian Coordinates

• Cartesian coordinates provide a simple way to calculate velocities when everything is described using them.

Introduction to Velocity and Acceleration in Cartesian Coordinates

In this section, the speaker introduces the concept of velocity and acceleration in Cartesian coordinates. The speaker explains how to calculate velocity and acceleration using fixed Cartesian coordinates.

Velocity in Cartesian Coordinates

• The velocity of an object is calculated as the time derivative of its position.

• The formula for calculating velocity in Cartesian coordinates involves taking the product of two things: R dot Bx times I plus R dot By times J plus R dot Bz times K.

• When taking derivatives of unit vectors like I, J, and K, their derivatives are zero because they are constant.

Acceleration in Cartesian Coordinates

• To calculate acceleration in Cartesian coordinates, take another derivative of the position vector.

• The formula for calculating acceleration in Cartesian coordinates is R double dot x term in the plus R double dot By in the J plus r double dot Bz in the K.

Velocity and Polar Coordinates

• In polar coordinates, unit vectors change direction over time which makes it more complicated to calculate velocity and acceleration.

• Velocity is always tangent to an object's path at any given instant.

Key Takeaways

• Velocity is calculated as a time derivative of position.

• Acceleration is calculated as a second derivative of position.

• In polar coordinates, unit vectors change direction over time which makes it more complicated to calculate velocity and acceleration.

Velocity of B with respect to A

In this section, the professor explains how to calculate the velocity of B with respect to A using vector expressions.

Vector Expressions

• The velocity of B with respect to O is the velocity of A with respect to O plus the velocity of B with respect to A.

• To find the velocity of B with respect to A, subtract the velocity of A from the velocity of B.

• The resulting relative velocity is 6 feet per second in the J direction.

Perception and Reference Frames

• The perception of speed depends on one's position in a fixed reference frame.

• The same relative velocity can be perceived differently from different positions in a fixed reference frame or even from a moving point at constant velocity.

• Any fixed point in a frame will see the same vector of velocity as any other point in that frame.

Moving Reference Frames

In this section, the professor introduces translating coordinate systems attached to rigid bodies and explains how they can be used to compute velocities.

Translating Coordinate Systems

• Moving reference frames can be attached to rigid bodies and used for computations within that system.

• Velocities computed within a translating coordinate system will yield identical results when converted back into another coordinate system.

• Different coordinate systems can be used interchangeably for computations by converting answers between them.

Understanding Translation and Rotation in Rigid Bodies

In this section, the professor explains how to describe the motion of rigid bodies by combining translation and rotation. He also defines translation and rotation and explains how they can be combined to describe general motion.

Describing Motion of Rigid Bodies

• A rigid body's motion can be described by combining a translation and a rotation.

• If you can describe a rigid body's translation and rotation, you have its complete motion.

Definition of Translation

• When a body translates, any two points on the body move in parallel paths.

• Curvilinear translation occurs when a translating body moves through curved paths.

Definition of Rotation

• Pure rotation occurs when every point on the body experiences the same rotation rate.

• Rotating bodies do not necessarily have fixed axes of rotation; their axis can move as well.

General Motion

• General motion is a combination of both translation and rotation. It is important to understand each piece to describe the complete motion of the system.

Next Steps

• The next step is to take derivatives of vectors that are rotating, which will allow us to do velocities and accelerations under those conditions. Read chapter 16 for more information on this topic.