Curvilinear Motion

General Curvilinear Motion

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. This type of motion is more complex than rectilinear (straight-line) motion.

Three-dimensional curvilinear motion describes the most general case of motion for a particle.





To find the velocity and acceleration of a particle experiencing curvilinear motion one only needs to know the position of the particle as a function of time.

Let’s say we are given the position of a particle P in three-dimensional Cartesian (x,y,z) coordinates, with respect to time, where

Position of a particle in three dimensions for curvilinear motion


The velocity of the particle P is given by

Velocity of a particle in three dimensions for curvilinear motion


The acceleration of the particle P is given by

Acceleration of a particle in three dimensions for curvilinear motion


As you can see, if we know the position of a particle as a function of time, it is a fairly simple exercise to find the velocity and acceleration. You simply take the first derivative to find the velocity and the second derivative to find the acceleration.


The magnitude of the velocity of particle P is given by

Magnitude of velocity of particle in three dimensions for curvilinear motion


The magnitude of the acceleration of particle P is given by

Magnitude of acceleration of particle in three dimensions for curvilinear motion


Note that the direction of velocity of the particle P is always tangent to the curve (i.e. the path traveled, denoted by the blue curve in the figure above). But the direction of acceleration is generally not tangent to the curve.

However, the acceleration component tangent to the curve is equal to the time derivative of the magnitude of velocity of the particle P (along the curve). In other words, if vt is the magnitude of the particle velocity (tangent to the curve), the acceleration component of the particle tangent to the curve (at) is simply

Acceleration component of particle tangent to curve for curvilinear motion

In addition, the acceleration component normal to the curve (an) is given by

Acceleration component of particle normal to curve for curvilinear motion

where R is the radius of curvature of the curve at a given point on the curve (xp,yp,zp).

The figure below illustrates the acceleration components at and an at a given point on the curve (xp,yp,zp).

Normal and tangential acceleration components of particle for curvilinear motion


For the specific case where the path of the blue curve is given by y = f(x) (two-dimensional motion), the radius of curvature R is given by

Radius of curvature at any point on two dimensional curve for curvilinear motion

where |x| means the “absolute value” of x. For example, |-2.5| = 2.5, and |3.1| = 3.1.

However, it is usually not necessary to know the radius of curvature R along a curve. But nonetheless, it is informative to understand it on the basis of its relationship to the normal acceleration (an).


Curvilinear Motion In Polar Coordinates

It is sometimes convenient to express the planar (two-dimensional) motion of a particle in terms of polar coordinates (R,θ), so that we can explicitly determine the velocity and acceleration of the particle in the radial (R-direction) and circumferential (θ-direction). For this type of motion, a particle is only allowed to move along the radial R-direction for a given angle θ.

For a particle P defined in polar coordinates (as shown below), we can derive a general equation for its radial velocity (vr), radial acceleration (ar), circumferential velocity (vc), and circumferential acceleration (ac).

Note that the circumferential direction is perpendicular to the radial direction.

Motion in polar coordinates for curvilinear motion


The position of the particle P is given with respect to time, where

Position of a particle in polar coordinates for curvilinear motion


To find the velocity, take the first derivative of x(t) and y(t) with respect to time:

Velocity of a particle in polar coordinates for curvilinear motion


To find the acceleration, take the second derivative of x(t) and y(t) with respect to time:

Acceleration of a particle in polar coordinates for curvilinear motion


Without loss of generality we can evaluate the velocities and accelerations at angle θ = 0, knowing that (at this angle) radial velocity and radial acceleration is in the x-direction, and circumferential velocity and circumferential acceleration is in the y-direction.

Setting θ = 0 we have:

Radial velocity and acceleration of a particle in polar coordinates for curvilinear motion

Circumferential velocity and acceleration of a particle in polar coordinates for curvilinear motion


Equations (1), (2), (3), and (4) fully describe the curvilinear motion of a particle P in polar coordinates.

The term dθ/dt is called angular velocity. It has units of rad/s. One rad (radian) = 57.296 degrees.

The term d2θ/dt2 is called angular acceleration. It has units of rad/s2.


Since vr and vc are perpendicular to each other, the magnitude of the velocity of particle P is given by

Magnitude of velocity of a particle in polar coordinates for curvilinear motion


Since ar and ac are perpendicular to each other, the magnitude of the acceleration of particle P is given by

Magnitude of acceleration of a particle in polar coordinates for curvilinear motion



Example Problem For Curvilinear Motion

A slotted link is rotating about fixed pivot O with a counterclockwise angular velocity of 3 rad/s, and a clockwise angular acceleration of 2 rad/s2. The movement of the link is causing a rod to slide along the curved channel, as shown. The radius of the channel as a function of θ is given by, R = 0.7θ (with R in meters and θ in radians). Determine the velocity and acceleration components of the rod at θ = 45°


Example problem for rotating slotted link in polar coordinates for curvilinear motion


Solution

The angle θ = 45° is equal to π/4 radians. In the equations, counterclockwise angular velocity is positive, and clockwise angular acceleration is negative (since it acts to “slow down” the rotational speed of the link).


The radial velocity of the rod is given by equation (1):

Example problem for rotating slotted link in polar coordinates for curvilinear motion 2

(The radial velocity is in the direction of increasing R).


The circumferential velocity of the rod is given by equation (3):

Example problem for rotating slotted link in polar coordinates for curvilinear motion 3

(The circumferential velocity is in the direction of increasing θ).


The radial acceleration of the rod is given by equation (2):

Example problem for rotating slotted link in polar coordinates for curvilinear motion 4

(The radial acceleration is in the direction of decreasing R).


The circumferential acceleration of the rod is given by equation (4):

Example problem for rotating slotted link in polar coordinates for curvilinear motion 5

(The circumferential acceleration is in the direction of increasing θ).



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