Mechanical Translational Systems
Mechanical Translational Systems
Currently, there are two systems of units: One is the international system (SI) or metric system, the other is the British engineering system (BES) or the English system. We prefer to use the SI system in this course, buy may use either of them whenever necessary. Also, within each system, we define certain base units. In SI system, they are meter, kilogram, and second. In BES, they are foot, pound, and second. Almost all the other units used in mechanical system can be derived from these base units, and are categorized as derived units. In this course, the following basic and derived units are used more often.
3.2 Mechanical Elements
3.2.1 Inertia Elements
The inertia elements include masses for translation and moments of inertia for rotation. The mass is usually denoted by m with unit as kg or slug. The moment of inertia often represented as J with unit as kg − m.
3.2.2 Spring Elements
A linear spring is a mechanical element that can be deformed by an external force such that the deformation is directly proportional to the force or torque applied to it.
For a translational spring, the relation between the acting force F and the net displacement x is
F = kx = k(x1 − x2) (3.2.1) where k is a proportionality constant called a spring constant, the unit of k is N/m or lb/in. For a rotational motion with a torsional spring, the relation between the acting torque T and the net angular displacement θ is T = kθ = k(θ1 − θ2) (3.2.2) where k is also a proportionality constant called a torsional spring constant.
Note: When a linear spring is overstretched over certain point, it will become nonlinear. In this course, we will assume the springs are always working within its linear limit. Further, although all practical springs have inertia and damping, we assume that the effect of them are negligibly small. Therefore, all the springs in this course will be ideal springs with neither mass nor damping and will obey the
Currently, there are two systems of units: One is the international system (SI) or metric system, the other is the British engineering system (BES) or the English system. We prefer to use the SI system in this course, buy may use either of them whenever necessary. Also, within each system, we define certain base units. In SI system, they are meter, kilogram, and second. In BES, they are foot, pound, and second. Almost all the other units used in mechanical system can be derived from these base units, and are categorized as derived units. In this course, the following basic and derived units are used more often.
3.2 Mechanical Elements
3.2.1 Inertia Elements
The inertia elements include masses for translation and moments of inertia for rotation. The mass is usually denoted by m with unit as kg or slug. The moment of inertia often represented as J with unit as kg − m.
3.2.2 Spring Elements
A linear spring is a mechanical element that can be deformed by an external force such that the deformation is directly proportional to the force or torque applied to it.
For a translational spring, the relation between the acting force F and the net displacement x is
F = kx = k(x1 − x2) (3.2.1) where k is a proportionality constant called a spring constant, the unit of k is N/m or lb/in. For a rotational motion with a torsional spring, the relation between the acting torque T and the net angular displacement θ is T = kθ = k(θ1 − θ2) (3.2.2) where k is also a proportionality constant called a torsional spring constant.
Note: When a linear spring is overstretched over certain point, it will become nonlinear. In this course, we will assume the springs are always working within its linear limit. Further, although all practical springs have inertia and damping, we assume that the effect of them are negligibly small. Therefore, all the springs in this course will be ideal springs with neither mass nor damping and will obey the