Simple Harmonic Oscillator with Boundary Conditions

In summary: I don't like seeing the cosine of an angle greater than 1. And the problem as stated has no solution.In summary, we have a problem where we are trying to solve for the amplitude (A) and phase constant (ø) of a spring undergoing simple harmonic motion. We are given two boundary conditions and the frequency of the motion. Using the equations for SHM and identities for cosine and sine, we can set up a system of equations to solve for A and ø. However, when we solve for these values, we get an inconsistent solution which means that the given values may not make physical sense.
  • #1
blizzardof96
22
0
Moved from a technical forum, no template.
How would you solve for the Amplitude(A) and Phase Constant(ø) of a spring undergoing simple harmonic motion given the following boundary conditions:

(x1,t1)=(0.01, 0)
(x2,t2)=(0.04, 5)
f=13Hz

x values are given in relation to the equilibrium point.

Equation of Motion for a spring undergoing SHM
Screen Shot 2018-09-17 at 5.56.47 PM.png
Solution to the 2nd order ODE
Screen Shot 2018-09-17 at 5.56.53 PM.png


Any hints or help would be much appreciated.
 

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  • #2
Our rules say that you must show your attempt at a solution before we are allowed to help you. Show us your work.
 
  • #3
Attempt at a solution
x(t1)=Acos(wt1+ø)

Therefore:
(1) 0.01=Acos(ø)

x(t2)=Acos(wt2+ø)

(2) 0.04=Acos(5w+ø)

w=2pif
w=2pi(13)=81.69

(2) becomes 0.04=Acos(408.4+ø)

This leaves us with the following equations to be solved for A and ø. This is where I am struggling.
(1) 0.01=Acos(ø)
(2) 0.04=Acos(408.4+ø)
 
  • #4
One way to proceed is to divide the second equation by the first to eliminate A. What do you get? Remember the identities ##\cos(a+b) = \cos a*\cos b - \sin a*\sin b## and ##\sin x = \sqrt{1-\cos^2x}.##
 
  • #5
Maybe we shouldn't approach just by using the giving equation. Equation 2 in the OP post but deducing the formule from the start. Finding constants in x(t) equation and then proceed. How this sounds ?
 
  • #6
Arman777 said:
Maybe we shouldn't approach just by using the giving equation. Equation 2 in the OP post but deducing the formule from the start. Finding constants in x(t) equation and then proceed. How this sounds ?
Isn't this what OP is doing by writing down equations (1) and (2)? It's a system of two equations and two unknowns that OP is wondering how to solve.
 
  • #7
kuruman said:
Isn't this what OP is doing by writing down equations (1) and (2)? It's a system of two equations and two unknowns that OP is wondering how to solve.

I am confused about something and it might be helpful to the OP.

Now we have a second order diff eqn and 2 boundry conditions. So we can see that the solution will have a form of
$$ x(t) =c_1sin(\sqrt (\frac {k} {m})t)+c_2cos(\sqrt (\frac {k} {m})t)$$ (Eq.1)

Since its a boundry condition I derived the conditon from $$m(d^2x/dt^2) + kx = 0$$ then I said, $$(d^2x/dt^2) + (k/m)x = 0$$ where λ=k/m and then we get the above equation.

Now we have 2 boundry conditions so I thought we should put those into the Eq.1 and then try to find c1 and c2. when we do that it seems that they are not both zero. So after that we need to do some calculations to turn it into the like the ##cos(a+b)## thing as you said.
$$x(t) = Acos(ωt+Φ)$$
 
  • #8
The cos(a+b) identity was helpful. This is how I solved for ø and A.

x(t)=Acos(ωt+ø)

(1) 0.01=Acos(ø)
(2) 0.04=Acos(ωt+ø) where ωt is some constant k
(3) 0.04=Acos(k+ø)

(
3)/(1)
4=cos(k+ø)/cos(ø)

using the identify cos(a+b)=cosa∗cosb−sina∗sinb we can proceed.

(4) 4cos(ø)=cos(k)*cos(ø)-sin(k)*sin(ø)

dividing through by cos(ø)

(5) 4=cos(k)-sin(k)*tan(ø)

(6) Arctan(cos(k)-4)/sin(k))=ø

ø= -77.36 degrees

Subbing back into original equations we get an ampitude(A) of 0.046.
 
  • #9
Did you check your work? Do you get a displacement of 0.04 when you put your values for A and φ back in the original equation? Your method is correct.

On edit: There is something about this problem that bugs me. If the frequency is 13 Hz, the period is ##T = \frac{1}{13}## s. This means that in time ##t_2 = 5~s## the number of oscillations executed is ##N = t_2/T = 65##. This is an integer which means that the oscillator at ##t_2## must have the same displacement as at time ##t_1 =0##.
 
Last edited:
  • #10
kuruman said:
Did you check your work? Do you get a displacement of 0.04 when you put your values for A and φ back in the original equation? Your method is correct.

On edit: There is something about this problem that bugs me. If the frequency is 13 Hz, the period is ##T = \frac{1}{13}## s. This means that in time ##t_2 = 5~s## the number of oscillations executed is ##N = t_2/T = 65##. This is an integer which means that the oscillator at ##t_2## must have the same displacement as at time ##t_1 =0##.

I checked my work and the solution agreed with initial eqn. The values were arbitrary so its possible that they don't make physical sense.

Thank you for your help.
 
  • #11
blizzardof96 said:
6) Arctan(cos(k)-4)/sin(k))=ø

ø= -77.36 degrees
As @kuruman points out, the problem as stated is inconsistent and has no solution.
What value did you use for k?
 

FAQ: Simple Harmonic Oscillator with Boundary Conditions

What is a simple harmonic oscillator?

A simple harmonic oscillator is a physical system that exhibits periodic motion, meaning it repeats the same pattern over and over again. It is characterized by a restoring force that is directly proportional to the displacement from the equilibrium position.

What are the boundary conditions for a simple harmonic oscillator?

The boundary conditions for a simple harmonic oscillator are the initial position and velocity of the system. These conditions determine the amplitude, frequency, and phase of the oscillation.

How does a simple harmonic oscillator behave near its equilibrium position?

Near its equilibrium position, a simple harmonic oscillator behaves like a linear system and follows Hooke's Law, which states that the restoring force is directly proportional to the displacement from the equilibrium position.

What is the equation of motion for a simple harmonic oscillator?

The equation of motion for a simple harmonic oscillator is given by F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position.

How are the period and frequency of a simple harmonic oscillator related?

The period and frequency of a simple harmonic oscillator are inversely related. The period is the time it takes for one complete oscillation, while the frequency is the number of oscillations per unit time. As the frequency increases, the period decreases, and vice versa.

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