Question

Solve the following differential equations.

A spring has a constant of 4 N/m. The spring is hooked a mass of 2 kg. Movement takes place in a viscous medium that opposes resistance equivalent to instantaneous speed. If the system is subjected to an external force of (4 cos(2t) - 2 sin(2t)) N. Determine:

a. The position function relative to time in the transient state or homogeneous solution

b. Position function relative to time in steady state or particular solution

c. The position function relative to time

Please show the whole procedure.

Answer #1

A spring-mass system has a spring constant of 3 N/m. A mass of 2
kg is attached to the spring, and the motion takes place in a
viscous fluid that offers a resistance numerically equal to the
magnitude of the instantaneous velocity.
(a) If the system is driven by an external force of (12 cos 3t −
8 sin 3t) N, determine the steady-state response.
(b) Find the gain function if the external force is f(t) =
cos(ωt).
(c) Verify...

A mass of 4 Kg attached to a spring whose constant is 20 N / m
is in equilibrium position. From t = 0 an external force, f (t) =
et sin t, is applied to the system. Find the equation of motion if
the mass moves in a medium that offers a resistance numerically
equal to 8 times the instantaneous velocity. Draw the graph of the
equation of movement in the interval.

3. A mass of 5 kg stretches a spring 10 cm. The mass is acted on
by an external force of 10 sin(t/2) N (newtons) and moves in a
medium that imparts a viscous force of 2 N when the speed of the
mass is 4 cm/s. If the mass is set in motion from its equilibrium
position with an initial velocity of 3 cm/s, formulate the initial
value problem describing the motion of the mass. Then (a) Find the...

A spring-mass system has a spring constant of 3 Nm. A mass of 2
kg is attached to the spring, and the motion takes place in a
viscous fluid that offers a resistance numerically equal to the
magnitude of the instantaneous velocity. If the system is driven by
an external force of 15cos(3t)−10sin(3t) N,determine the
steady-state response in the form Rcos(ωt−δ).
R=
ω=
δ=

Assume an object with mass m=1 kg is attached to a spring with
stiffness k=2 N/m and lies on a surface with damping constant b= 2
kg/s. The object is subject to the external force F(t) = 4cos(t) +
2sin(t). Suppose the object starts at the equilibrium position
(y(0)=0) with an initial velocity of y_1 (y'(0) = y_1).
In general, when the forcing function F(t) = F*cos(γ*t) +
G*sin(γ*t) where γ>0, the solution is the sum of a periodic
function...

Differential Equations
A spring is stretched 6in by a mass that weighs 8 lb. The mass
is attached to a dashpot mechanism that has a damping constant of
0.25 lb· s/ft and is acted on by an external force of 2cos(2t)
lb.
(a) Find position u(t) of the mass at time t
(b) Determine the steady-state response of this system
Assume that g = 32 ft/s2

A mass of 1 kg stretches a spring 9.8 m. The mass is acted on by
an external force of 4 cos(t) N. If the mass is set in motion from
its equilibrium position with a downward velocity of 2 m/s, find
the position of the mass at any time. Identify the transient (i.e.,
complementary) and steady state (i.e., particular) solutions. Does
the motion exhibit resonance or a beat?

Consider a damped forced mass-spring system with m = 1, γ = 2,
and k = 26, under the influence of an external force F(t) = 82
cos(4t).
a) (8 points) Find the position u(t) of the mass at any time t,
if u(0) = 6 and u 0 (0) = 0.
b) (4 points) Find the transient solution uc(t) and the steady
state solution U(t). How would you characterize these two solutions
in terms of their behavior in time?...

An oscillator of mass 2 Kg has a damping constant of 12 kg/sec
and a spring constant of 10N/m. What is its complementary position
solution? If it is subject to a forcing function of F(t)= 2*sin(2t)
what is the equation for the position with respect to time?
Equation 2(x2) + 12(x1) + 10(x) = 2*sin(2t); x2 is the 2nd
derivative of x; x1 is the 1st derivative of x.

a 3 kg mass is attached to a spring whose constant is 147 N/m,
comes to rest in the equilibrium position. Starting at t
= 0, a force equal to f (t) =
12e−5t cos 2t is applied to
the system. In the absence of damping,
(a)
find the position of the mass when t =
π.
(b)
what is the amplitude of vibrations after a very long
time?

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 3 hours ago

asked 3 hours ago